Abstract

Log Algebraic Stacks and Moduli of Log Schemes

by

Martin Christian Olsson

Doctor of Philoshophy in Mathematics

University of California at Berkeley

Professor Arthur Ogus, Chair

In an effort to incorporate the log smooth deformation theory developed by K. Kato and F. Kato into the study of moduli, we develop a theory of log algebraic stacks and an analog of M. Artin’s method for determining representability by an algebraic stack. We also study the moduli of log structures themselves and prove that there exist universal log structures in the sense of algebraic stacks. Using this result we translate results from the theory of stacks into the language of log geometry and thereby lay the foundations for the study of moduli of log schemes. In the second half of the thesis, we apply our foundational work to the study of moduli of stable curves. We construct the Deligne-Mumford compactification of the moduli space of curves of genus g ≥ 2 using log geometry as well as the Deligne-Rapoport compactifications of moduli stacks of elliptic curves with level structure. We use K. Kato’s work on log finite flat group schemes instead of generalized elliptic curves to define level structure on degenerate elliptic curves. In the last chapter we use K. Kato and S. Usui’s work on log Hodge structures to interpret the compactified moduli stack of elliptic curves from the analytic point of view.

Professor Arthur Ogus

Dissertation Committee Chair

Contents

0 Introduction 1 0.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1 Log Algebraic Stacks 8 1.1 Flat descent of log structures . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Review of algebraic stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 The 2-category of log algebraic stacks . . . . . . . . . . . . . . . . . . . . . 20

2 Artin’s method 27 2.1 Review of [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 A relative version of Artin’s theorem. . . . . . . . . . . . . . . . . . . . . . . 32

3 Universal Log Structures 40 3.1 Direct construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2 Deformation theory of log structures . . . . . . . . . . . . . . . . . . . . . . 51 3.3 A second construction of LOG_{S} . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4 Basic properties of LOG_{S} and variants . . . . . . . . . . . . . . . . . . . . . 66 3.5 A log version of Artin’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 74

4 Technicalities on log structures 79 4.1 A theorem about integral morphisms . . . . . . . . . . . . . . . . . . . . . . 79 4.2 Log versions of the Grothendieck existence theorem. . . . . . . . . . . . . . 80 4.3 Log structures on normal crossing schemes. . . . . . . . . . . . . . . . . . . 88

5 Moduli of log curves 107 5.1 Statement of Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.2 Basic properties of log curves. . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.3 Cohomological calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.4 Verification of 3.5.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.5 Comparison with [8], [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6 Modular Curves: First Construction 128 6.1 Full level N structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.2 Solid objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.3 The action of (ℤ/Nℤ)^{2}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.4 Proof that M_{N}^{log} is algebraic. . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.5 Comparison with ([9], III.2.1). . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7 Modular Curves: Second Construction 141 7.1 Valuative Log Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.2 The Kato-Tate Curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.3 Full Level Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.4 M_{0}(N)^{log} and M_{1}(N)^{log}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

8 Analytic theory of log elliptic curves 166 8.1 Classical theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 8.2 Log Hodge structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 8.3 The log Hodge structure associated to a log elliptic curve . . . . . . . . . . 174 8.4 Elliptic log Hodge structures . . . . . . . . . . . . . . . . . . . . . . . . . . 175 8.5 The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Bibliography 184

Acknowledgements

In the course of this work I have had conversations with a number of people and I am grateful to them all. I would especially like to thank my advisor A. Ogus, A. Abbes, J. Borger, and S. Ünver. Throughout my graduate studies I have learned a great deal from frequent conversations with J. Borger and S. Ünver. Some of these conversations have contributed directly to this thesis but many more are hidden in my understanding of foundational material necessary for this work. A. Abbes gave very valuable suggestions after reading an earlier version of this thesis. In particular, he first suggested that the stacks LOG_{S} of chapter 3 might be algebraic. Most of all I want to thank my advisor Professor A. Ogus whose help can be seen throughout what follows. He was the one who first suggested that I look at log algebraic stacks and has continually helped me with my work on this thesis. Most of all, however, I am grateful to him for introducing me to the wondrous subject which is modern algebraic geometry. Indeed my first exposure to the subject came in a course he taught in Fall 1997 and since then he has patiently guided me in my studies and answered innumerable questions.

Chapter 0

Introduction

The subject of this thesis is a theory of log algebraic stacks and its applications to compactification of moduli spaces. The development of such a theory is motivated by the following two phenomena:

1. Compactifying a moduli stack M in a modular fashion essentially amounts to understanding degenerations of the objects that M classifies. In many examples, these degenerate objects which are singular in the category of schemes become smooth objects in the category of log schemes when endowed with suitable log structure.

2. Compactifications of moduli spaces sometimes have a natural log structure. If M ⊂ \overline{M} is a compactification, the substack \overline{M} −M classifying singular objects is often a divisor with normal crossings and hence defines a log structure on \overline{M}.

Examples of these phenomena are the Deligne-Mumford compactification of the moduli space of curves of genus g ≥ 2 ([8]) and the Deligne-Rapoport compactifications of modular curves ([9]). An algebraic stack with a log structure defines in a natural way a stack over the category of log schemes. In this thesis we study two closely related problems about such stacks. The first problem is to give a method for determining when a fibered category over the category of log schemes arises from an algebraic stack with a log structure. In this thesis we develop a version of M. Artin’s theorem ([5], 5.3) for determining representability by a log algebraic stack.

Recall that given a stack F over the category of S-schemes (for some fixed S) M. Artin’s theorem gives necessary and sufficient conditions for F to be an algebraic stack. The key ingredients in his theorem are:

1. Deformation theory of objects of F. In applications the deformations are usually controlled by some type of cohomology groups.

2. Effectivity of formal objects of F. Here the Grothendieck existence theorem ([10], III.5) almost always plays a crucial role.

Our version of M. Artin’s theorem will have the same main ingredients except we work in the logarithmic category. Let S be a fixed log scheme and let F Log_{S} be the category of fine log schemes over S with the strict étale topology (see the text for the definition). Given a stack F/F Log_{S} we prove a theorem giving sufficient conditions for F to be equivalent to the stack defined by a log algebraic stack. The key ingredients are as follows:

1. Log deformation theory. In the applications we use the log smooth deformation theory of K. Kato and F. Kato ([18], [20]).

2. Effectivity of formal objects of F. In the log setting this usually involves an effectivity statement about certain types of formal log schemes, and hence a version of the Grothendieck existence theorem for log schemes is needed. We study versions of such a theorem in chapter 4, but the problem of effectivity of formal log schemes in general seems quite subtle.

3. Identification of special objects in F. If F is a log algebraic stack then there is a distinguished class of objects in F which are those obtained by pulling back the log structure on the stack. We assume that a characterization of these special objects is known. In section 4.3 where we prove a theorem which is useful for identifying the special objects when working with varieties with normal crossings.

For more details about our log version of Artin’s theorem see 3.5.13. The second problem discussed in this thesis is to give modular interpretations of the log algebraic stacks which arise in classical contexts. For example, which morphisms of log schemes f : C → X are obtained by pullback from the universal morphism of log schemes over the Deligne-Mumford compactification of the moduli stack of curves of genus g ≥ 2? In this thesis we give modular interpretations of the log stacks defined by the moduli stacks of stable curves of genus g ≥ 2 and some modular curves arising in the theory of elliptic curves. We now discuss the contents of the individual chapters in more detail. In chapter 1 we introduce the language of log algebraic stacks. The main technical issue in this chapter arises from the fact that we wish to work with Artin stacks and then have to consider log structures on the smooth site. We could just work with the smooth site for the rest of the thesis, but in order to conform with the more standard practice of considering log structures on the étale site we instead prove a descent theorem for log structures. This also allows us to consider stacks over the category of fine log schemes with the strict étale topology as opposed to the smooth topology. Chapter 2 is concerned with a review of M. Artin’s theorem ([5], 5.3) and a slight extension of his results to the case when the base is an algebraic stack instead of a scheme. The third chapter discusses the moduli of log structures themselves. Fix a fine log scheme S with underlying scheme \underline{S} and define a stack LOG_{S} over the category of \underline{S}-schemes as follows. If f : \underline{X} → \underline{S} is an \underline{S}-scheme then the fiber of LOG_{S} over \underline{X} consists of pairs (M_{X}, f^{b}) where M_{X} is a fine log structure on \underline{X} and

f^{b} : f^{∗}M_{S} → M_{X}

is a morphism of log structures. Pullback and morphisms in LOG_{S} are defined in the natural way. The main result of chapter 3 is that the stack LOG_{S} is algebraic. We have included two proofs of this result. The first is based on local constructions using charts, whereas the second is based on deformation theory and M. Artin’s theorem. Combining the results of chapter 2 with the fact that LOG_{S} is algebraic we obtain the version of M. Artin’s theorem for log stacks discussed above. We hope that chapter 3 might have wider applications in log geometry than just to the theory of moduli. In particular, in this chapter we explain how the theory of log smoothness and log flatness in log geometry can be viewed as special cases of the theory of smoothness and flatness for algebraic stacks. In addition, using the results of this chapter one can develop the foundations of the theory of log crystalline cohomology from a theory of crystalline cohomology over algebraic stacks ([31]). This last subject, however, will not be touched upon in this thesis. Chapter 4 gathers together various technical facts about log structures needed in what follows. We prove two theorems which may be viewed as versions of the Grothendieck existence theorem ([10], III.5) for log structures. One of these theorems is due to K. Kato (cited in [28]) though we do not know to what extent our proof agrees with his. In this chapter we also prove a theorem which enables us to determine what the distinguished class of objects alluded to above has to be in the moduli problems we consider. More precisely we prove the following. Suppose f : X → S is a smooth, proper, integral morphism of noetherian log schemes and that M_{X/S} = Coker(f^{∗}M_{S} → M_{X}) is a sheaf of torsion free groups. Assume that for each geometric point s̄ → S the underlying scheme of the fiber Xs̄ is scheme with normal crossings. Then we show that X/S is obtained by base change from a morphism f^{♯} : X^{♯} → S^{♯} of a special form whose underlying morphism of schemes is that of the original f : X → S. In chapter 5 we study the moduli of so-called stable log curves. For each g ≥ 1 we define a stack M_{g}^{log} over the category of fine log schemes which classifies certain smooth log curves. Combining the results of chapter 4 with the log smooth deformation theory of K. Kato and F. Kato we conclude from our version of M. Artin’s theorem that M_{g}^{log} is algebraic. It should be noted that the technical results of chapter 4 are foundational in nature and not specific to M_{g}^{log}. Thus the fact that M_{g}^{log} is algebraic follows more or less formally from our foundational work and the log smooth deformation theory. Once we know that M_{g}^{log} is algebraic we can compare it to the Deligne-Mumford compactification M_{g} ([8]) of the moduli space of curves of genus g (or in the case of g = 1 to the Deligne-Rapoport compactification [9] of the moduli stack of elliptic curves). The universal curve C/M_{g} with its natural log structure coming from the divisor at infinity defines a map M_{g} → M_{g}^{log}. By comparing the deformation theory of objects of M_{g}^{log} with the deformation theory of the underlying schemes we show that this map is an isomorphism, and hence we get a modular interpretation of the log stack defined by M_{g}. In chapter 6 we modify the method of chapter 5 to study the moduli stack of log elliptic curves with full level N structure. We do not use generalized elliptic curves but rather consider a group action to rigidify the moduli problem. Chapter 7 gives an entirely new interpretation of the Deligne-Rapoport compactifications of the moduli stacks of elliptic curves with level structure which only involves irreducible pointed curves. This is made possible by K. Kato’s construction of N-torsion points on the Kato-Tate curve as outlined in ([16]). Using Kato’s work one can attach a finite flat log group scheme E_{N} of rank N^{2} to any irreducible pointed log curve of genus 1. Using this log group scheme of N-torsion points, one can define moduli problems with level structure just as in the smooth case. In chapter 7 this is done for the stacks M_{N}, M_{Γ_{00}(N)}, and M_{Γ_{0}(N)} of ([9]) over ℤ[1/N]. All these stacks turn out to be log étale over M_{1}, and the ramification structure at infinity of the underlying morphisms of algebraic stacks can be explicitly seen in the structure of the log schemes E_{N}. Whereas the construction in chapter 6 does not seem to generalize well to other moduli problems the method of chapter 7 should carry through for any moduli problem on elliptic curves involving torsion subgroups and points. In this chapter we also use K. Kato’s notion of a valuative log space (described in [16]). In chapter 8 we study the analytic theory of log elliptic curves. The main result of the chapter is that if S/ℂ is a log smooth scheme with \underline{S}/ℂ smooth, then the category of log elliptic curves over S is naturally equivalent to the category of log Hodge structures on S^{log} (notation as in [29]) of rank 2 and type {(−1, 0), (0, −1)} satisfying a monodromy condition. This chapter is based on the work of K. Kato and S. Usui ([22]) on log Hodge structures.

0.1 Notation

Throughout this thesis S will denote a base log scheme whose underlying scheme is of finite type over an excellent Dedeking ring. Denote by F Log_{S} the category of fine log scheme over S. Sometimes if the reference to S is clear we write just F Log. For basic definitions and results about log schemes we refer to [20]. In our notation we will however follow that of ([29], [30]). Throughout we denote log spaces (schemes, algebraic spaces, stacks etc.) by unadorned letters (e.g. X) and spaces without log structure by an underlined letter (e.g. \underline{X}). If X is a log scheme with log structure M_{X}, we write the monoid law on M_{X} additively. For this reason, it becomes notationally significant to avoid identifying O_{X}^{∗} with its inverse image in M_{X} under the map α : M_{X} → O_{X}. We denote by λ the inverse to the map

α^{−1}(O_{X}^{∗}) → O_{X}^{∗}.

With these conventions we have for sections a ∈ O_{X}^{∗}, m ∈ M_{X} the formula

α(λ(a) + m) = aα(m).

If f : X → Y is a morphism of log schemes, Ω_{X/Y}^{1} denotes the sheaf of logarithmic differentials and Ω_{\underline{X}/\underline{Y}}^{1} denotes the relative differentials of the underlying morphism of schemes. The symbol M_{X/Y} denotes the relative characteristic. By definition, it is the cokernel in the category of sheaves of integral monoids of the map

f^{∗}M_{Y} → M_{X}.

We also write \overline{M}_{X} = M_{X}/O_{X}^{∗} for M_{X/\underline{X}}. We denote by X^{♯} the log scheme whose underlying scheme is \underline{X} with the log structure f^{∗}M_{Y}. There is a canonical factorization of f

X ⟶ X^{♯} ⟶^{f^{♯}} Y.

Regarding monoids, we write the monoid law additively. If M is a monoid, M^{∗} denotes the group of units in M and M^{+} is the unique maximal ideal M − M^{∗}. We denote the projection map M → M/M^{∗} by q_{M}. A morphism ϕ : M_{1} → M_{2} of monoids is called logarithmic if the map ϕ^{−1}(M_{2}^{∗}) → M_{2}^{∗} is bijective. ϕ is called local if ϕ^{−1}(M_{2}^{∗}) = M_{1}^{∗}.

If P is a sharp monoid, we define a partial ordering on P by p_{1} ≤ p_{2} if there exists p ∈ P such that

p_{1} + p = p_{2}.

We use the notion of algebraic space (locally of finite type) freely. For definitions and results about algebraic spaces see ([3], [25]). Throughout this paper, all algebraic spaces (and hence also schemes) will be assumed locally quasi-separated. Also we use the notion of a log algebraic space which by definition is an algebraic space A with a log structure on A_{et}. If S is a ringed site we denote by LS(S) the category of integral log structures on S. It contains as a full subcategory the category F LS(S) of fine log structures on S. If \underline{X} is a scheme, we denote by \underline{X}_{et} (resp. \underline{X}_{fl}, \underline{X}_{sm}) the small étale (resp. fppf, smooth) site on \underline{X}.

Chapter 1

Log Algebraic Stacks

1.1 Flat descent of log structures

1.1.1. The goal of this section is prove some technical results about log structures comparing the flat, smooth, and étale topologies. Some of the results of this section apply in greater generality to log structures which are not necessarily integral but on which the units O^{∗} act freely. However, in keeping with the restrictions necessary for the results used later we restrict our attention to integral log structures.

Lemma 1.1.2. Let \underline{X} be a scheme, and let ϕ : N → M be a morphism of log structures on \underline{X}. Then ϕ is an isomorphism if and only if \overline{ϕ} : \overline{N} → \overline{M} is an isomorphism.

Proof. The only if direction is immediate. For the other direction, consider the diagram

N ⟶ \overline{N}
ϕ↓  \overline{ϕ}↓
M ⟶ \overline{M}.

Suppose n_{1}, n_{2} ∈ N are section such that ϕ(n_{1}) = ϕ(n_{2}). Then

\overline{n}_{1} = \overline{ϕ}^{−1}(ϕ(n_{1})) = \overline{ϕ}^{−1}(ϕ(n_{2})) = \overline{n}_{2}.

Thus n_{1} = λ(a) + n_{2} for some a ∈ O∗ ; \underline{X}. But then

ϕ(λ(a) + n_{2}) = λ(a) + ϕ(n_{2}) = ϕ(n_{2})

and so a = 1 and ϕ is injective. Surjectivity is immediate.

Lemma 1.1.3. Let \underline{X} be a scheme and π : \underline{X}′ → \underline{X} a faithfully flat cover. Suppose N is a log structure on \underline{X}, and that \underline{U} → \underline{X} is an étale morphism. Then the sequence

N (\underline{U}) → π^{∗}N (\underline{U} ×\underline{X} \underline{X}′) ⇉ p_{1}^{∗}π^{∗}N (\underline{U} ×\underline{X} \underline{X}′ ×\underline{X} \underline{X}′)

is exact (here p_{1}^{∗}π^{∗}N is identified with p_{2}^{∗}π^{∗}N via the canonical isomorphism).

Proof. We can without loss of generality assume that \underline{U} = \underline{X}. Using ([27], 1.5) we reduce to the case when \underline{X} = Spec(A), \underline{X}′ = Spec(B) and A → B is faithfully flat. The sheafification functor P reSh(\underline{X}_{et}) → Sh(\underline{X}_{et}) is exact so it is enough to prove that the sequence is exact in the category of presheaves. Let N_{1} be the presheaf on \underline{X}′ which to every \underline{V} → \underline{X}′ associates the pushout in the category of monoids

α^{−1}(O_{\underline{X}′}^{∗}(\underline{V})) ⟶ (π^{−1}M\underline{X})(\underline{V})
↓
O_{\underline{X}′}^{∗}(\underline{V})

where α : π^{−1}M\underline{X} → O\underline{X}′. By the universal property of pushout and sheafification, π^{∗}M\underline{X} is the sheafification of N_{1}. Similarly, for \underline{V} → \underline{X}′ ×\underline{X} \underline{X}′, define N_{2}(\underline{V}) to be the pushout of

α^{−1}(O_{\underline{X}′×\underline{X}′})(\underline{V}) ⟶ (p_{1}^{−1}π^{−1})M\underline{X}(\underline{V})
↓
O_{\underline{X}′×\underline{X}′}^{∗}(\underline{V})

so that p_{1}^{∗}π^{∗}M\underline{X} is the sheafification of N_{2}. It then suffices to show that

M\underline{X}(\underline{X}) → N_{1}(\underline{X}′) ⇉ N_{2}(\underline{X}′ ×\underline{X} \underline{X}′)

is exact. Since the map A → B is faithfully flat, it is injective and A ∩ B^{∗} = A^{∗} ([27], 2.17). Thus the above sequence can be written as (letting M = M\underline{X}(\underline{X}))

M → M ⊕_{A^{∗}} B^{∗} ⇉ M ⊕_{A^{∗}} (B ⊗_{A} B)^{∗}.

If (m, b) is in the equalizer then there exists h ∈ A^{∗} such that

(m, b ⊗ 1) = (mh, 1 ⊗ bh^{−1}).

This implies that h = 1 and that b ∈ A ∩ B^{∗} = A^{∗}.

1.1.4. If \underline{X} is a scheme, and if M ∈ LS(\underline{X}_{et}), then we can define a presheaf of monoids G(M) on \underline{X}_{fl} by

G(M)(g : \underline{X}′ → \underline{X}) = Γ(\underline{X}′, g^{∗}M).

Lemma 1.1.3 implies that this presheaf is in fact a log structure on \underline{X}_{fl}. In other words, we have functors

LS(\underline{X}_{et}) ⟶^{G} LS(\underline{X}_{fl}) ⟶^{res} LS(\underline{X}_{et})

and the composite functor res ∘ G is the identity functor.

Theorem 1.1.5. 1. Let π : \underline{X}′ → \underline{X} be a faithfully flat locally of finite presentation morphism of schemes. Suppose we are given a log structure M on \underline{X}′ together with an isomorphism

ϕ : p_{1}^{∗}M ≃ p_{2}^{∗}M

on \underline{X}′ ×\underline{X} \underline{X}′ such that on \underline{X}′ ×\underline{X} \underline{X}′ ×\underline{X} X′ we have

p_{13}^{∗}(ϕ) = p_{23}^{∗}(ϕ) ∘ p_{12}^{∗}(ϕ).

Then there exists unique pair (N , s) on \underline{X} where N is a log structure and s is an isomorphism of log structures with descent datum on \underline{X}′

s : π^{∗}N ≃ M.

2. Let \underline{X} be a scheme, and let M_{1} ∈ F LS(\underline{X}_{et}). Then for any M_{2} ∈ LS(\underline{X}_{et}) the

natural map

Hom_{\underline{X}_{et}}(M_{1}, M_{2}) → Hom_{\underline{X}_{fl}}(G(M_{1}), G(M_{2}))

is bijective.

Proof. (1) The previous lemma assures uniqueness of (N, s) if it exists. By ([6], VIII.9.4) the sheaf \overline{M} descends to \underline{X}. Let \underline{U} → \underline{X} be an étale morphism. We identify p_{1}^{∗}M with p_{2}^{∗}M using ϕ and thus obtain a commutative diagram as below and we define N (\underline{U}) to be the equalizer of the first row:

N (\underline{U}) → M(\underline{U} ×\underline{X} \underline{X}′) ⇉ p_{2}^{∗}M(\underline{U}\underline{X} ×\underline{X} \underline{X}′ ×\underline{X} \underline{X}′)
↓  ↓
O\underline{X}(\underline{U}) → O\underline{X}(\underline{U} ×\underline{X} \underline{X}′) ⇉ O\underline{X}(\underline{U}\underline{X} ×\underline{X} \underline{X}′ ×\underline{X} \underline{X}′).

Then we have a natural morphism of monoids α_{N} : N (\underline{U}) → O\underline{X}(\underline{U}). Moreover, both N (\underline{U}) → M(\underline{U} ×\underline{X} \underline{X}′) and O\underline{X}(\underline{U}) → O\underline{X}(\underline{U} ×\underline{X} \underline{X}′) are injective, and O\underline{X}(\underline{U})^{∗} = O\underline{X}(\underline{U} ×\underline{X} \underline{X}′)^{∗} ∩ O\underline{X}(\underline{U}) by faithful flatness. From this it follows that α_{N}^{−1}(O(\underline{U})^{∗}) → O\underline{X}(\underline{U})^{∗} is an isomorphism. Hence N is a log structure and comes with a natural morphism of log structures with descent datum

π^{∗}N → M.

We must check that π^{∗}N ≃ M. Using lemma 1.1.2 it is enough to show that the induced map

(π^{∗}N)/O_{\underline{X}′}^{∗} = π^{−1}(N /O∗ ; \underline{X}) → M/O∗ ; \underline{X}

is an isomorphism. To prove this it suffices to show that N /O∗ ; \underline{X} is the equalizer of the maps

M/O_{\underline{X}′}^{∗} ⇉ p_{2}^{−1}(M/O_{\underline{X}′}^{∗})

since this equalizer is the sheaf obtained by descent from \overline{M}. Let K be this equalizer. Let \underline{U} → \underline{X} be étale. We have to analyze the commutative diagram

K(\underline{U}) → (M/O_{\underline{X}′}^{∗})(\underline{X}′ ×\underline{X} \underline{U}) ⇉ p_{2}^{−1}(M/O_{\underline{X}′}^{∗})((\underline{X}′ ×\underline{X} \underline{X}′ ×\underline{X} \underline{U})
↑  ↑  ↑
N (\underline{U}) →  M((\underline{X}′ ×\underline{X} \underline{U}))  ⇉  p_{2}^{∗}M(\underline{X}′ ×\underline{X} \underline{X}′ ×\underline{X} \underline{U})
↑  ↑  ↑
O∗ ; \underline{X}(\underline{U}) →  O_{\underline{X}′}^{∗}(\underline{X}′ ×\underline{X} \underline{U})  ⇉  p_{2}^{∗}O_{\underline{X}′}^{∗}(\underline{X}′ ×\underline{X} \underline{X}′ ×\underline{X} \underline{U}).

The map

(N /O∗ ; \underline{X})(\underline{U}) → K(\underline{U})

is injective for by faithful flatness O\underline{X}(\underline{U}) ∩ O\underline{X}(\underline{U} ×\underline{X} \underline{X}′)^{∗} = O\underline{X}(\underline{U})^{∗}. The difficulty is in proving surjectivity étale locally on \underline{X}. Let m̄ ∈ K(\underline{U}). Replacing \underline{X} by \underline{U} we may assume that m̄ ∈ K(\underline{X}). Let N^{fl} be the sheaf on X_{fl} defined by

\underline{V} ↦ Eq(M(\underline{V} ×\underline{X} \underline{X}′) ⇉ M(\underline{V} ×\underline{X} \underline{X}′ ×\underline{X} \underline{X}′)).

N^{fl} is a log structure on \underline{X}_{fl}. Define a presheaf S on X_{fl} by

\underline{V} ↦ \{liftings of m̄ to N^{fl}(\underline{V})\}.

Then S is a 𝔾_{m}-pseudo-torsor on X_{fl}. We show that S is in fact a torsor. Let \underline{V} → \underline{X}′ be such that m̄ ∈ \overline{M}(\underline{V}) lifts to M(\underline{V}). Then π_{2}^{∗}m̄ ∈ \overline{M}(\underline{V} ×\underline{X} \underline{X}′) also lifts to M(\underline{V} ×\underline{X} \underline{X}′). For we have a diagram

\underline{X}′ ⇇ \underline{X}′ ×\underline{X} \underline{X}′ ← \underline{V} ×\underline{X} \underline{X}′

from which we deduce that π_{1}^{∗}m̄ = π_{2}^{∗}m̄ in \overline{M}(\underline{V} ×\underline{X} \underline{X}′). Now we have

\overline{M}(\underline{V} ×\underline{X} \underline{X}′) ⟵ M(\underline{V} ×\underline{X} \underline{X}′)
π_{1}^{∗}↑  ↑^{π_{1}^{∗}}
\overline{M}(\underline{V})  ⟵  M(\underline{V})

and hence π_{2}^{∗}m̄ = π_{1}^{∗}m̄ lifts. Choose a lifting m ∈ M(\underline{V} ×\underline{X} \underline{X}′) of π_{2}^{∗}m̄ and consider the diagram

M((\underline{V} ×\underline{X} \underline{X}′)) ⇉ p_{2}^{∗}M(\underline{V} ×\underline{X} \underline{X}′ ×\underline{X} \underline{X}′) \overrightarrow{⟶} s_{3}^{∗}M(\underline{V} ×\underline{X} (\underline{X}′)^{⊗3})
↑ ↑ ↑
O_{\underline{X}′}^{∗}(\underline{V} ×\underline{X} \underline{X}′) ⇉ p_{2}^{∗}O_{\underline{X}′}^{∗}(\underline{V} ×\underline{X} \underline{X}′ ×\underline{X} \underline{X}′) \overrightarrow{⟶} s_{3}^{∗}O∗ ; \underline{X}(\underline{V} ×\underline{X} (\underline{X}′)^{⊗3}).

Let p_{1} and p_{2} denote the double rightarrows in the diagram (so p_{1} = ϕ ∘ p_{1}^{∗} and p_{2} = p_{2}^{∗}). Write

p_{1}(m) = λ(t) + p_{2}(m).

It follows from the standard simplicial identities that for all m ∈ M(\underline{V} ×\underline{X} \underline{X}′)

η_{1}(p_{1}(m)) + η_{2}(p_{2}(m)) + η_{3}(p_{1}(m)) = η_{1}(p_{2}(m)) + η_{2}(p_{1}(m)) + η_{3}(p_{2}(m))

where the η_{i} are the triple arrows. From this we have

η_{1}(λ(t)) + η_{3}(λ(t)) + η_{1}(p_{2}(m)) + η_{2}(p_{2}(m)) + η_{3}(p_{2}(m)) =
η_{2}(λ(t)) + η_{1}(p_{2}(m)) + η_{2}(p_{2}(m)) + η_{3}(p_{2}(m)).

Since our log structures are integral, it then follows that

η_{1}(λ(t)) + η_{3}(λ(t)) = η_{2}(λ(t)).

Thus t defines a class in H^{1}(\underline{V} , 𝔾_{m}). Let \underline{V}′/\underline{V} be an étale cover such that this class is zero in H^{1}(\underline{V}′, 𝔾_{m}) and replace \underline{V} by \underline{V}′. Then we can assume that this class is zero. Therefore, after replacing \underline{V} ×\underline{X} \underline{X}′ by a cover \underline{W} we can find t_{0} ∈ 𝔾_{m}(\underline{W}) such that the image of t in 𝔾_{m}(\underline{W} ×\underline{V} ×\underline{W}) can be written as π_{1}^{∗}(t_{0}^{−1})π_{2}^{∗}(t_{0}). Then

−λ(t_{0}) + m ∈ M(\underline{W})

maps to m̄ ∈ \overline{M}(\underline{W}) and is in the equalizer of

M(\underline{W}) ⇉ M(\underline{W} ×\underline{V} \underline{W}).

We claim that \underline{W} → \underline{X} is the desired cover. That is, there exists m ∈ N^{fl} mapping to m̄ ∈ K(\underline{W}). To see this consider the diagram

M(\underline{W} ×\underline{X} \underline{X}′) ⇉ M((\underline{W} ×\underline{X} \underline{X}′) ×\underline{W} (\underline{W} ×\underline{X} \underline{X}′))
↑  ↑
M(\underline{W})  ⇉  M(\underline{W} ×\underline{V} \underline{W})

where the vertical arrows are induced from

π_{1} : \underline{W} ×\underline{X} \underline{X}′ → \underline{W} , π_{1} × π_{1} : (\underline{W} ×\underline{X} \underline{X}′) ×\underline{W} (\underline{W} ×\underline{X} \underline{X}′) → \underline{W} ×\underline{V} \underline{W} .

By construction there exists m ∈ N^{fl}(\underline{W}) mapping to π_{1}^{∗}m̄ ∈ \overline{M}(\underline{W} ×\underline{X} \underline{X}′). But π_{1}^{∗}m̄ = π_{2}^{∗}m̄ and hence m maps to the image of m̄ in K(\underline{W}). We conclude that S is a torsor. Now any 𝔾_{m}-torsor in the fppf-topology can be trivialized étale locally and hence the map

\overline{N} → K

is surjective.

(2) Let G be the functor defined in 1.1.4. It follows from the formula res ∘ G = id that for any two log structures M_{1}, M_{2} the natural map

Hom_{\underline{X}_{et}}(M_{1}, M_{2}) → Hom_{\underline{X}_{fl}}(G(M_{1}), G(M_{2}))

is injective. If M_{1} is fine, then by replacing \underline{X} by an étale cover, we may assume that there exists a global chart P → Γ(\underline{X}_{et}, M_{1}). It follows that any morphism η : G(M_{1}) → G(M_{2}) is uniquely determined by the induced map

P → Γ(\underline{X}_{fl}, G(M_{2})) = Γ(\underline{X}_{et}, M_{2}).

From this the second part of the theorem follows.

Corollary 1.1.6. For any scheme \underline{X}, the functor G induces equivalences of categories

G : F LS(\underline{X}_{et}) → F LS(\underline{X}_{fl})

and

G : F LS(\underline{X}_{et}) → F LS(\underline{X}_{sm}).

Proof. This will follow from the theorem once we show that any log structure M_{X} on \underline{X}_{et} for which G(M_{X}) is a fine log structure on X_{fl} is in F LS(X_{et}). Observe first that for all geometric points x̄ → \underline{X} the stalk \overline{M}_{X,x̄} is a finitely generated monoid. For if

f : \underline{X}′ → \underline{X}

is an fppf-cover such that f^{∗}M_{X} is fine and if x̄′ → \underline{X}′ is a geometric point lying over x̄ then

\overline{M}_{X,x̄} ≃ \overline{f^{∗}M_{X}}_{}_{x̄′}.

Therefore, in some étale neighborhood of x̄ we can find a finitely generated monoid P and a morphism of sheaves

P → M_{X}

such that if N_{X} denotes the log structure associated to P then the map

N_{X,x̄} → M_{X,x̄}

is an isomorphism. Let f : \underline{X}′ → \underline{X} be an fppf cover such that f^{∗}M_{X} is fine and let x̄′ → \underline{X}′ be a geometric point lying over x̄. Then

f^{∗}N_{}_{x̄′} → f^{∗}M_{X,x̄′}

is an isomorphism, and hence by replacing \underline{X}′ by an étale neighborhood of x̄′ we can assume that f^{∗}N → f^{∗}M_{X} is an isomorphism. It now follows from 1.1.5 that the map N → M_{X} is an isomorphism in some étale neighborhood of x̄.

1.2 Review of algebraic stacks

For the sake of completeness, and because conventions and definitions are not standardized, we review some of the theory of algebraic stacks. References are [8], [11], [26]. For definitions concerning 2-categories we refer to [12]. 1.2.1. Let C and F be a categories and let p : F → C be a functor. For each object X ∈ C, F_{X} denotes the subcategory of F whose objects are x ∈ Ob(F) such that p(x) = X, and whose morphisms are morphisms in F inducing the identity on X. We call p : F → C a groupoid if the following conditions hold:

1. For any morphism f : X → Y in C and y ∈ Ob(F_{Y}) there exists an object x ∈ Ob(F_{X}) together with a morphism f̃: x → y such that p(f̃) = f;

2. Given a diagram in C,

[figure: ϕ′; ϕ; X; Y; Z; α]

and a partial lifting of the diagram to F

[figure: ϕ̃′; ϕ̃; x; y; z]

there exists a unique lifting of α to a morphism α̃ : y → z such that the diagram commutes. In the presence of condition (2), the data f̃: x → y whose existence is assured by (1) is unique up to unique isomorphism. We assume from this point on that when speaking of a groupoid a splitting has been chosen. This means that for each f as in (1) a choice of f̃: x → y has been chosen. We denote this data by f^{∗}y → y. 1.2.2. Suppose now that our category C comes equipped with a Grothendieck topology, and that products and fiber products exist in C. We say that a groupoid p : F → C is a pre-stack if the following additional conditions holds:

3. If X ∈ Ob(C), and x, y ∈ F_{X}, then the functor

C/X → Ens

given by (f : Z → X) ↦ Hom_{F_{Z}}(f^{∗}x, f^{∗}y) is a sheaf for the topology on C. Here C/X denotes the category whose objects are morphisms f : Z → X in C and whose morphisms are commutative diagrams

[figure: f; f′; X; Z; Z′]
in C.

If in addition the following holds

4. If f_{i} : Z_{i} → X is a covering family then any descent datum for objects in F relative to the f_{i} is effective.

then F is a stack. 1.2.3. The collection of pre-stacks over C form the objects of a 2-category [12]. If p_{1} : F_{1} → C and p_{2} : F_{2} → C are two pre-stacks over C, then

Hom(F_{1}, F_{2})

is the category whose objects are functors F_{1} → F_{2} inducing the identity functor on C and whose morphisms are natural transformations of functors. The 2-category of pre-stacks contains as a full 2-sub-category the 2-category of stacks.

Lemma 1.2.4. Let F → C be a pre-stack. Then there exists a universal pair (F^{+}, θ) where F^{+} is a stack over C and θ : F → F^{+} is a 1-morphism inducing for any stack G → C an equivalence of categories

Hom(F, G) → Hom(F^{+}, G).

Proof. F^{+} is constructed as follows. If X ∈ Ob(C), then F_{X}^{+} is the category with objects triples (f_{i} : X_{i} → X, x_{i}, h) where (f_{i} : X_{i} → X) is a covering family in X, x_{i} ∈ F_{X_{i}}, and for each i and j, h_{i,j} is an isomorphism

h_{i,j} : x_{j}|_{X_{i}×_{X}X_{j}} ≃ x_{i}|_{X_{i}×_{X}X_{j}}

satisfying the standard cocycle condition. If (Y_{ij} → X_{i}) is a refinement of the cover X_{i} then by restriction the (x_{i}, h) induce objects with descent datum relative to the cover Y_{i,j} → X. A morphism

(f_{i} : X_{i} → X, x_{i}, h) → (g_{j} : Y_{j} → X, y_{j}, l)

in F^{+} is by definition an isomorphism in F of the objects with descent datum relative to the cover (X_{i} ×_{X} Y_{j} → X) induced by the x_{i} and y_{j}. The map θ is defined by x ∈ F_{X} ↦ (id : X → X, x, id). It is straightforward to check that with these definitions the pair (F^{+}, θ) has the correct universal properties.

1.2.5. If C is a category such that for X ∈ Ob(C) the functor h_{X} associated to X by Yoneda’s lemma is a sheaf, then a useful way to construct stacks is using groupoids in C [26]. We recall this construction. A groupoid in C consists of two objects X^{0}, X^{1} ∈ C together with morphisms

s, t : X^{1} → X^{0}, ϵ : X^{0} → X^{1}, i : X^{1} → X^{1}, μ : X^{1} ×_{X^{0},s,t} X^{1} → X^{1}

satisfying the following axioms:

1. s ∘ ϵ = t ∘ ϵ = id_{X^{0}}, s ∘ i = t, t ∘ i = s, s ∘ μ = s ∘ pr_{2}, t ∘ μ = t ∘ pr_{1}.

2. The two maps X^{1} ×_{X^{0}} X^{1} ×_{X^{0}} X^{1} → X^{1} given by μ ∘ (μ × id) and μ ∘ (id × μ) agree.

3. The two maps

X^{1} ⇉ X^{1} ×_{X^{0}} X^{1} → X^{1}

given by μ ∘ (id × ϵ) and μ ∘ (ϵ × id) agree.

4. The two diagrams

X^{1} ⟶^{i×id} X^{1} ×_{X^{0}} X^{1}
s↓ ↓^{μ}
X^{0} ⟶^{ϵ}  X^{1}
X^{1} ⟶^{id×i} X^{1} ×_{X^{0}} X^{1}
t↓ ↓^{μ}
X^{0} ⟶^{ϵ}  X^{1}

commute.

If (X^{0}, X^{1}, s, t, μ, ϵ, i) is a groupoid, then define a pre-stack {X^{0}/X^{1}} to be the fibered category over C such that for U ∈ Ob(C), {X^{0}/X^{1}}_{U} is the category whose objects are h_{X^{0}}(U) and for which a morphism f → g is an element β ∈ X^{1}(U) such that s(β) = f and t(β) = g. Composition of morphisms is defined using μ. By 1.2.4, there is a canonical stack associated to {X^{0}/X^{1}} which we denote by [X^{0}/X^{1}].

1.2.6. Let C be the category of \underline{S}-schemes with the smooth topology. If \underline{X} is an \underline{S}-scheme, then \underline{X} defines a stack over C. The objects of this stack over \underline{Y} ∈ Ob(C) are \underline{S}-morphisms \underline{Y} → \underline{X}, and the morphisms in the stack are the identity morphisms. If F is another stack over C, then a 1-morphism from the stack associated to \underline{X} to F is equivalent to an object of F\underline{X}. For this reason we do not distinguish between \underline{X} and the associated stack, and will often think of an object of F\underline{X} as a 1-morphism

\underline{X} → F.

Similarly, an algebraic space \underline{X}/\underline{S} defines a stack over C in which the only morphisms are the identity maps.

Definition 1.2.7. We say that a stack F → C is representable by an algebraic space if it is equivalent to the stack associated to an algebraic space \underline{X}/\underline{S}.

Definition 1.2.8. Let C be the category of \underline{S}-schemes with the smooth topology, and let p : F → C be a stack. We say that F is an algebraic stack if the following conditions hold:

1. F is limit preserving. This means that for any filtering direct system of noetherian \underline{S}-rings \{A_{i}\} the natural functor

\underrightarrow{lim} F_{Spec(A_{i})} → F_{Spec(⟶^{lim} A_{i})}

is an equivalence of categories.

2. F is relatively representable. More precisely, if \underline{X} and \underline{Y} are schemes locally of finite type over \underline{S} and if x ∈ F_{X}, y ∈ F_{Y}, then the fibered product stack X ×_{F} Y is representable by an algebraic space over \underline{S} locally of finite type.

3. F admits a smooth cover. This means that there exists an X ∈ C and x ∈ F_{X} such that for every Y ∈ C and y ∈ F_{Y} the induced morphism of algebraic spaces X ×_{F} Y → Y is smooth and surjective.

1.2.9. From here on, we will suppress the reference to the base category C, and simply write algebraic stack with the implicit assumption that the base category C has been given. The collection of algebraic stacks forms a 2-category. 1-morphisms and 2-morphisms between algebraic stacks are just 1-morphisms and 2-morphisms of stacks.

1.2.10. Let P be a property of morphisms of algebraic spaces which is local for the smooth topology on source and target. For example, P could be the property of being smooth, flat, surjective, locally of finite presentation, or locally of finite type. If F_{1}, F_{2} are algebraic stacks, then a 1-morphism f : F_{1} → F_{2} has property P if for some (and hence all) 2commutative diagram of 1-morphisms

\begin{matrix}X & ⟶^{f̃} & Y \\ ↓ & & ↓ \\ F_{1} & ⟶^{f} & F_{2} \end{matrix}

where X → F_{1} and Y → F_{2} are smooth and surjective, the morphism f̃ has property P. 1.2.11. If F is an algebraic stack, we define the smooth site F_{sm} on F as follows. Objects in F_{sm} are smooth 1-morphisms of algebraic stacks f : G → F. A morphism in F_{sm}

(G_{1}, f_{1}) → (G_{2}, f_{2})

is a 1-morphism ϕ : G_{1} → G_{2} of algebraic stacks together with a 2-morphism f_{1} → ϕ ∘ f_{2}. If (G, f) ∈ F_{sm}, then a covering family of (G, f) is a collection of morphisms in F_{sm}, (G_{i}, f_{i}) → (G, f) such that the induced 1-morphism of algebraic stacks

∐ G_{i} → G

is surjective. Note that there is a base for this topology given by f : G → F where G is a scheme. Thus there is a natural pre-sheaf O_{F} defined on this base and by sheafification a sheaf of rings O_{F} on F_{sm} making the category of sheaves Sh(F_{sm}) a ringed topos. Given a morphism of stacks f : F_{1} → F_{2} there is a natural induced morphism of ringed topoi Sh(F_{1,sm}) → Sh(F_{2,sm}). In particular, it makes sense to speak of log structures on an algebraic stack, and the usual pullback and pushforward operations on log structures are valid in this context as well. 1.2.12. If f : F_{1} → F_{2} is a 1-morphism between algebraic stacks, then f induces a functor

f^{−1} : Sh(F_{2,sm}) → Sh(F_{1,sm}).

If f, g : F_{1} → F_{2} are two 1-morphisms, and r : f → g is a 2-morphism between them, then r induces an isomorphism of functors f^{−1} → g^{−1}. By chasing through the definitions, one can show that the diagram

[figure: O_{F_{1}}; f^{−1}O_{F_{2}}; g^{−1}O_{F_{2}}; r]

commutes. In particular, if M is a log structure on F_{2}, then r induces an isomorphism of log structures

r : f^{∗}M ≃ g^{∗}M.

1.2.13. If (X^{0}, X^{1}, s, t, ϵ, μ, i) is a groupoid in algebraic spaces with s and t smooth, then the stack [X^{0}/X^{1}] given by 1.2.5 is an algebraic stack. Moreover, given any algebraic stack F, choose a smooth surjective cover X → F and let X^{1} = X ×_{F} X. Then there are natural maps s, t : X^{1} → X and ϵ, i, μ giving a groupoid in algebraic spaces. Moreover, the natural morphism of pre-stacks {X/X^{1}} → F induces by the universal property a morphism of stacks

[X/X^{1}] → F

and it is in fact an equivalence of stacks. Remark 1.2.14. Our definition of algebraic stack is slightly more general than that found in ([26]). Their definition of algebraic stack is equivalent to an algebraic stack in the sense of 1.2.8 in which the algebraic spaces \underline{X} ×_{F} \underline{Y} occurring in 1.2.8.2 are assume to be separated and of finite type over \underline{X} ×\underline{S} \underline{Y}.

1.3 The 2-category of log algebraic stacks

The most direct definition of log algebraic stack is the following:

Definition 1.3.1 (First Version). A log algebraic stack is a pair F = (\underline{F} , M_{F}) where \underline{F} is an algebraic stack and M_{F} is a log structure on \underline{F}_{sm}. A 1-morphism of log algebraic stacks f : (\underline{F}_{1}, M_{F_{1}}) → (\underline{F}_{2}, M_{F_{2}}) is a pair (f, f^{b}) consisting of a 1-morphism of stacks f : \underline{F}_{1} → \underline{F}_{2} together with a morphism of log structures f^{b} : f^{∗}M_{F_{2}} → M_{F_{1}} on \underline{F}_{1,sm}. A 2-morphism h : (f, f^{b}) → (g, g^{b}) is a natural transformation of functors h : f → g such that the induced diagram of morphisms of sheaves (see 1.2.12)

[figure: f^{b}; g^{b}; M_{F_{1}}; f^{∗}M_{F_{2}}; g^{∗}M_{F_{2}}]

commutes. We denote the 2-category of log algebraic stacks by LAS.

For the study of moduli of log schemes it seems more natural to consider stacks over the category of log schemes. The rest of this section is devoted to describing log algebraic stacks in this manner. 1.3.2. Let F Log be the category of fine S-log schemes. By theorem 1.1.5, there is a natural fully faithful 2-functor from the subcategory of F Log of log schemes locally of finite type over S to LAS. We denote this 2-functor by G. 1.3.3. If F = (\underline{F} , M_{F}) ∈ Ob(LAS), we define a fibered category η(F) → F Log as follows. For any X ∈ F Log, the category η(F)_{X} is defined to be

Hom_{LAS}(G(X), F).

Give F Log the strict, smooth topology in which covering families are collections of morphisms

f_{i} : X_{i} → X

where each f_{i} is strict and smooth, and such that ∐ X_{i} → X is surjective. It then follows tautologically that η(F) is a stack. From this point on we will assume that F Log is given this smooth topology. 1.3.4. If X is a fine log algebraic space, it follows from theorem 1.1.5.2 that X defines a sheaf on F Log which we denote by h_{X}. For any U ∈ F Log, h_{X}(U) is the set of morphisms of log algebraic spaces U → X. By classical descent theory, morphisms of algebraic spaces glue in the smooth topology, and by 1.1.5.2 morphisms of log structures also glue.

Definition 1.3.5. A stack F → F Log is representable by a log algebraic space if there exists a log algebraic space X and an equivalence of stacks h_{X} → F. Note that by our conventions about algebraic spaces, \underline{X} is locally of finite type.

Definition 1.3.6. Let F → F log be a stack, and denote by Aff ⊂ F Log the subcategory of fine log schemes whose underlying scheme is isomorphic to the spectrum of a noetherian ring.

1. F is limit preserving if F is compatible with filtering strict inverse limits in Aff. More precisely, if I is a filtering category and h : I^{op} → Aff is a functor such that for any arrow i → j in I the induced morphism of log schemes h(j) → h(i) is strict, then \underleftarrow{lim}_{I} h exists in Aff, and our requirement is that the induced functor

\underrightarrow{lim}_{I} F_{h(i)} → F_{(⟵^{lim}_{I} h)}

is an equivalence of categories.

2. F is relatively representable if for any pair of 1-morphisms x : X → F, y : Y → F with X and Y log schemes locally of finite type over S, the product functor X ×_{F} Y is representable by a log algebraic space locally of finite type over S.

3. If X ∈ F Log and x : X → F is a 1-morphism, then we say that x is relatively strict if for all Y ∈ F Log locally of finite type over S and y : Y → F the product X ×_{F} Y is representable by a log algebraic space locally of finite type over X, and the induced morphism of log algebraic spaces

X ×_{F} Y → Y

is strict.

4. We say that F admits a smooth, strict cover if F is relatively representable, and if there exists X ∈ F Log and x : X → F with X locally of finite type over S such that for all Y ∈ F Log and y : Y → F with Y locally of finite type over S the induced map of log algebraic spaces

X ×_{F} Y → Y

is smooth, strict and surjective.

5. Denote by LAS′ the full 2-subcategory of the 2-category of stacks over F Log whose

objects are limit preserving, relatively representable, and admit a smooth cover. 1.3.7. Suppose F → F Log is a limit preserving, relatively representable stack which admits a smooth cover. Choose such a cover X^{0} → F and let X^{1} = X^{0} ×_{F} X^{0}. Then as in the case of schemes, we get a groupoid in log algebraic spaces (X^{0}, X^{1}, s, t, ϵ, i, μ) and hence an associated stack [X^{0}/X^{1}] which comes with a natural morphism of stacks

[X^{0}/X^{1}] → F.

It follows from the definition of the stack associated to a pre-stack that this is in fact an equivalence of stacks.

Theorem 1.3.8. 1. If F ∈ LAS then η(F) → F Log is an object of LAS′;

2. η induces an equivalence of 2-categories LAS → LAS′.

Proof. (1) That η(F) is limit preserving follows from the assumption (1.2.8) that \underline{F} is limit preserving and from the following lemma:

Lemma 1.3.9. Let \{A_{i}\} be a filtering direct system of noetherian rings and let A = \underrightarrow{lim} A_{i}. Then the natural functor

β : \underrightarrow{lim} F LS(Spec(A_{i}))_{sm}) → F LS((Spec(A))_{sm})

is an equivalence of categories.

Proof. Let M be a fine log structure on (Spec(A))_{sm}. Choose A → R smooth and surjective such that M|_{R} admits a global chart P → M. Since R is of finite presentation over A there exists for some i an A_{i}-algebra R_{i} such that Spec(R_{i}) → Spec(A_{i}) is smooth and surjective together with an isomorphism

R_{i} ⊗_{A_{i}} A ≃ R.

If i → j is a morphism in our index category we let R_{j} = R_{i} ⊗_{A_{i}} A_{j}. Since P is a finitely presented monoid we can lift the map P → R to R_{i} after perhaps choosing bigger i. This gives a log structure on Spec(R_{i}) inducing M|_{R}. To descend this log structure to some A_{i} it suffices to lift the descent data given for M relative to A → R to some A_{i} → R_{i}. On the other hand, morphisms of fine log structures can locally be approximated by morphisms between finitely presented monoids and by using a similar argument to the above we find that morphisms also can be approximated.

To see that η(F) is relatively representable, let x : X → F and y : Y → F be two 1morphisms with X, Y ∈ F Log locally of finite type over S. If Z ∈ F Log, then giving a morphism Z → X ×_{η(F)} Y is equivalent to giving two morphisms of log schemes f : Z → X, g : Z → Y together with a morphism of algebraic spaces s : \underline{Z} → \underline{X} ×\underline{F} \underline{Y} such that the induced diagram

f^{∗}x^{∗}M_{F} ≃ g^{∗}y^{∗}M_{F} ⟶ f^{∗}M_{X}
↓ ↓
g^{∗}M_{Y}  ⟶ M_{Z}

commutes. Note that by 1.1.9.2 we can abuse notation and not distinguish between log structures on \underline{Z}_{et} and log structures on \underline{Z}_{sm}. It follows that X ×_{η(F)} Y is represented by

(\underline{X} ×\underline{F} \underline{Y} , pr_{1}^{∗}M_{X} ⊕_{pr_{1}^{∗}f^{∗}x^{∗}M_{F}} pr_{2}^{∗}M_{Y}).

From this it also follows that η(F) admits a smooth, strict surjective cover.

(2) Recall ([12]) that η being an equivalence of 2-categories means that

1. Every T ∈ Ob(LAS′) is equivalent to η(F) for some F ∈ Ob(LAS).

2. For any two F_{1}, F_{2} ∈ Ob(LAS) the natural functor

β : Hom_{LAS}(F_{1}, F_{2}) → Hom_{LAS}′(η(F_{1}), η(F_{2}))

is an equivalence of categories.

To verify 1, note that any T ∈ Ob(𝔖𝔏′) is equivalent to the stack obtained from a groupoid (X^{0}, X^{1}, s, t, ϵ, i, μ). Forgetting the log structures defines a groupoid in algebraic spaces and hence a stack, and the log structures on X^{0} and X^{1} induce in a natural way a log structure on [\underline{X}^{0}/\underline{X}^{1}]. Applying η to the resulting object of LAS gives back a stack equivalent to T. That β is an equivalence of categories also follows from looking at representations of stacks as groupoids. If F_{1}, F_{2} ∈ LAS, and if h : \underline{F}_{1} → \underline{F}_{2}, h^{b} : h^{∗}M_{F_{2}} → M_{F_{2}} is a 1-morphism in LAS, then one can construct smooth surjections \underline{X}^{0} → \underline{F}_{1}, and \underline{Y}^{0} → \underline{F}_{2} such that the log structures M_{F_{1}} and M_{F_{2}} restrict to fine log structures on \underline{X}_{et}^{0} and \underline{Y} _{et}^{0} respectively, and morphisms of log schemes giving a model for (h, h^{b})

h_{0} : X^{0} = (\underline{X}^{0}, M_{F_{1}}|_{X^{0}}) → (\underline{Y}^{0}, M_{F_{2}}|_{Y^{0}}) = Y^{0}
h_{1} : X^{1} = (\underline{X}^{0} ×_{F_{1}} \underline{X}^{0}, M_{F_{1}}|_{X^{0}×_{F_{1}}X^{0}}) → (\underline{Y}^{0} ×_{F_{2}} \underline{Y}^{0}, M_{F_{2}}|_{Y^{0}×_{F_{2}}Y^{0}}) = Y^{1}.

These morphisms satisfy h_{0} ∘ s_{X} = s_{y} ∘ h_{1} and h_{0} ∘ t_{X} = t_{y} ∘ h_{1} and in addition satisfy the axioms necessary for a morphism of groupoids in log schemes. The induced functor η(h) is simply the morphism of stacks induced by this morphism of groupoids. From this it follows that β is faithful and essentially surjective. Given a morphism η(F_{1}) → η(F_{2}) we can choose a presentation for it as above. Then forgetting the log structure gives a morphism of stacks h : \underline{F}_{1} → \underline{F}_{2} and on the chosen cover there is a natural morphism of log structures h^{b} with descent datum. In the case when we apply this to η(h, h^{b}) we simply recover h and h^{b}.

If T_{1}, T_{2} ∈ Ob(LAS′) and if f, g : T_{1} ⇉ T_{2} are two functors, then one can find a common presentation for the two functors. That is, groupoids X^{1} ⇉ X^{0}, Y^{1} ⇉ Y^{0} defining stacks equivalent to T_{1} and T_{2} together with morphisms of log schemes f_{0}, g_{0} : X^{0} → Y^{0}, f_{1}, g_{1} : X^{1} → Y^{1} such that (f_{0}, f_{1}) induce the functor f and (g_{0}, g_{1}) induce the functor g. A natural transformation f → g is induced from a morphism of log schemes r : X^{0} → Y^{1} such that certain diagrams commute. Forgetting the log structures defines a 2-morphism in LAS which induces f → g.

Thus we have a second equivalent definition of log algebraic stack in terms of fibered categories over F Log.

Definition 1.3.10 (Second Version). Let F → F Log be a limit preserving stack. Then F is a log algebraic stack if

1. F is relatively representable.

2. There exists a smooth strict cover x : X → F.

1.3.11. If (X^{0}, X^{1}, s, t, ϵ, i, μ) is a groupoid in log algebraic spaces, then it makes sense to take the associated stack with respect to either the strict smooth or étale topology on F Log. However, the existence of a quasi-section of a smooth morphism ([10] IV, 17.16.3) implies that taking the associated stack with respect to either topology gives equivalent stacks. Thus we may work with the strict étale topology on F Log.

Definition 1.3.12 (Third Version). Let F → F Log be a limit preserving fibered category and suppose F is a stack with respect to the strict étale topology on F Log. Then F is a log algebraic stack if

1. F is relatively representable.

2. There exists a smooth strict cover x : X → F.

1.3.13. Suppose F → F Log is a log algebraic stack. Then under the equivalence of 2categories 1.3.8 there is an associated algebraic stack \underline{F} over the category of schemes. This stack can be described as follows. Given F let F̃ be the fibered category over the category of schemes locally of finite type over \underline{S} such that for a scheme \underline{X}, F̃\underline{X} is the category with objects pairs (x, M_{X}) such that M_{X} is a log structure on \underline{X} and x is a 1-morphism

x : (\underline{X}, M_{X}) → F

which is relatively strict. A morphism (x, M_{X}) → (x′, M′_{X}) is a morphism of schemes

s : \underline{X} → \underline{(\underline{X}, M_{X}) ×_{F} (\underline{X}, M′_{X})}

such that s ∘ pr_{1} = s ∘ pr_{2} = id\underline{X}. Note that since we have only defined the notion of relatively strict 1-morphisms for log schemes locally of finite type, F̃ is a fibered category over the category of \underline{S}-schemes locally of finite type. The above arguments show that F̃ is in fact equivalent to \underline{F} restricted to the category of schemes locally of finite type. However, since \underline{F} is limit preserving, its restriction to the category of schemes locally of finite type determines \underline{F} up to equivalence. The log structure on \underline{F} can be recovered as the sheaf

(x, M_{X}) ↦ Γ(\underline{X}, M_{X}).

Chapter 2

Artin’s method

2.1 Review of [5]

2.1.1. Fix a scheme \underline{S} locally of finite type over an excellent Dedekind ring and let F be a limit preserving stack over the category of \underline{S}-schemes with the étale topology. In this section we review the results of ([5]).

Definition 2.1.2. A deformation situation is a diagram in the category of noetherian \underline{S}-rings

B
↓
A′ ⟶ A ⟶ A_{0}

such that

1. A_{0} is reduced.

2. A′ → A and A → A_{0} are surjections with nilpotent kernel.

3. The composite B → A_{0} is surjective.

4. The kernel I = Ker(A′ → A) is an A_{0}-module of finite type.

2.1.3. If A is a ring, a ∈ F_{A}, and A′ → A a morphism of rings, we denote by F_{a,A′} the category whose objects are pairs (a′, σ) where a′ ∈ F_{A′} and σ is an isomorphism in F_{A} between the pullback of a′ to A and a. A morphism (a′, σ) → (b′, τ) is a morphism ϕ : a′ → b′ in F_{A′} such that σ = τ ∘ϕ. We denote by \overline{F}_{a,A′} the isomorphism classes of objects in F_{a,A′}. The following conditions (S1) and (S2) on F will be referred to as Schlessinger’s Conditions: 2.1.4 (S1). Suppose given a deformation situation and a ∈ F_{A}. Then the canonical functor

F_{a,A′×_{A}B} → F_{a,A′} × F_{a,B}

is an equivalence of categories (this condition is called (S1′) in [5]). 2.1.5. If A is a ring, and I an A-module we denote by A[I] the ring with underlying abelian group A ⊕ I and multiplication defined by

(a, i)(b, j) = (ab, aj + bi).

If (S1) holds then \overline{F}_{a_{0},A_{0}[I]} has a natural structure of an A_{0}-module. In addition, if a_{0} ∈ F_{A_{0}} we denote by Aut_{a_{0}}(A_{0}[I]) the set of automorphisms of a_{0} restricted to A_{0}[I] which reduce to the identity on a_{0}. (S1) implies that Aut_{a_{0}}(A_{0}[I]) also has a natural A_{0}-module structure. These facts are fundamental in what follows so we explain them a little more carefully. Let C be the category of noetherian A-algebras R with a surjection R → A_{0} such that the induced map A → A_{0} is the given one. Morphisms in C are defined as morphisms of Aalgebras compatible with the maps to A_{0}. The category C has fiber products defined using fiber products in the category of rings. Now the rings A_{0}[I] with the maps A_{0}[I] → A_{0} are in a natural way A_{0}-module objects in C. This amounts to saying that there is a map (addition)

+ : A_{0}[I] ×_{A_{0}} A_{0}[I] → A_{0}[I], (a + t, a + t′) ↦ a + t + t′

and for each a_{0} ∈ A_{0} a map (multiplication by a_{0})

a_{0} : A_{0}[I] → A_{0}[I], (a, t) ↦ (a, a_{0}t)

such that for all R ∈ C these maps induce a structure of an A_{0}-module on

Hom(R, A_{0}[I]).

Now suppose D is a functor

D : C → Set

such that D(A_{0}) consists of a single element and which commutes with fiber products (in our case D is \overline{F}_{a_{0}} or Aut_{a_{0}}). Then D(A_{0}[I]) has a natural A_{0}-module structure with addition law given by

D(A_{0}[I]) × D(A_{0}[I]) ⟶^{≃} D(A_{0}[I] ×_{A_{0}} A_{0}[I]) ⟶^{D(+)} D(A_{0}[I])

and the action of an element a_{0} ∈ A_{0} given by

D(a_{0}) : D(A_{0}[I]) → D(A_{0}[I]).

Condition (S2) is the following: 2.1.6 (S2). \overline{F}_{a_{0},A_{0}[I]} and Aut_{a_{0}}(A_{0}[I]) are A_{0}-modules of finite type. 2.1.7 (Obstruction Theory). By an obstruction theory for F we mean the following:

A. For each surjection A → A_{0} with nilpotent kernel and element a ∈ F_{A}, a functor

O_{a} : (A_{0}-modules of finite type) → (A_{0}-modules of finite type).

B. For each diagram A′ → A → A_{0} as in 2.1.2, and a ∈ F_{A}, an element o_{a}(A′) ∈ O_{a}(I) such that \overline{F}_{a,A′} is non-empty if and only if o_{a}(A′) = 0.

O_{a} and o_{a}(A′) should be functorial and linear in I. 2.1.8. In fact the existence of an obstruction theory need only hold étale locally. More precisely, let a local obstruction theory for F at a_{00} ∈ F_{A_{00}} (A_{00} reduced) be data as follows:

A. For every diagram

a ⟶ a_{0} ; ↑ ; a_{00}

in F lying over

\begin{aligned}A & ⟶^{α} & A_{0} \\ β & ↑ \\ & & A_{00} \end{aligned}

where β is étale and α is surjective with nilpotent kernel, a functor

O_{a} : (A_{0}-modules of finite type) → (A_{0}-modules of finite type).

B. For each diagram A′ → A → A_{0} and a ∈ F_{A} where A → A_{0} and a are as in (A) and A′ → A is surjective with kernel I (an A_{0}-module of finite type), an element o_{a}(A′) ∈ O_{a}(I) such that \overline{F}_{a,A′} is non-empty if and only if o_{a}(A′) = 0.

This data should be functorial and linear in A_{0} and I.

Remark 2.1.9. The point of distinguishing between local and global obstruction theories is that the functor O_{a} in 2.1.8 may be defined in terms of some additional structure on a_{00}. The main theorem (2.1.11) requires only that local obstruction theories exist for a collection of a_{00}’s which form a base for the étale topology on elements of F. In particular, the local obstruction theories may be defined in terms of some additional structure on elements of F which exists only étale locally. This seems particularly relevant in the context of log geometry (see remark 3.5.14). 2.1.10 (Additional Conditions). If F is to be algebraic, the following conditions must also hold. Let a ∈ F_{A} be an object with A of finite type over S, and let the subscript A_{0}, B_{0}, etc. denote the objects obtained by tensor product with A_{red}.

A. Compatibility with étale localization: If ϕ : A → B is étale, and if b ∈ F_{B} is the

image of a ∈ F_{A}, then

\overline{F}_{b_{0},B_{0}[I⊗B_{0}]} ≃ \overline{F}_{a_{0},A_{0}[I]} ⊗ B_{0},
Aut_{b_{0}}(B_{0}[I ⊗ B_{0}]) ≃ Aut_{a_{0}}(A_{0}[I]) ⊗ B_{0},

and

O_{b}(I ⊗ B_{0}) ≃ O_{a}(I) ⊗ B_{0}.

B. Effectivity of formal objects: If 𝔪 ⊂ A_{0} is a maximal ideal and Â the completion of

A_{0} along 𝔪, then

\overline{F}_{a_{0},A_{0}[I]} ⊗ Â ≃ \underleftarrow{lim} \overline{F}_{a_{0},A_{0}[I/𝔪^{n}I]},

and

Aut_{a_{0}}(A_{0}[I]) ⊗ Â ≃ \underleftarrow{lim} Aut_{a_{0}}(A_{0}[I/𝔪^{n}I]).

C. Constructibility: There exists an open dense set of points p in Spec(A_{0}) with k(p) of finite type over \underline{S} such that

\overline{F}_{a_{0},A_{0}[I]} ⊗ k(p) ≃ \overline{F}_{a_{0},A_{0}[I⊗k(p)]},
Aut_{a_{0}}(A_{0}[I]) ⊗ k(p) ≃ Aut_{a_{0}}(A_{0}[I ⊗ k(p)]),

and

O_{a}(I) ⊗ k(p) ⊂ O_{a}(I ⊗ k(p)).

One form of Artin’s theorem is the following:

Theorem 2.1.11 (Artin, [5]). Let F be a limit preserving stack over the category of S-schemes. Then F is an algebraic stack locally of finite type over S if and only if the following conditions hold:

1. The conditions (S1), (S2), and the conditions on \overline{F}_{a_{0},A_{0}[I]} and Aut_{a_{0}}(A_{0}[I]) in 2.1.10 hold.

2. If Â is a complete local ring with residue field of finite type over S, then the natural

functor

FÂ → \underleftarrow{lim} F_{Â/𝔪^{n}}

is faithful and has dense image. Recall that having dense image means that the functor

FÂ → F_{Â/𝔪^{n}}

is essentially surjective onto the image of

\underleftarrow{lim} F_{Â/𝔪^{n}} → F_{Â/𝔪^{n}}.

3. There exists local obstruction theories O satisfying the conditions in 2.1.10 for a base for the étale topology on elements of F of finite type.

4. If A_{0} is of finite type over S, a_{0} ∈ F_{A_{0}} an object, and ϕ an automorphism of a_{0} which induces the identity in F_{k} for a dense set of points A_{0} → k with k of finite type over S, then ϕ is the identity over some dense open subset of Spec(A_{0}).

Remark 2.1.12. The fact that the conditions of the theorem are also necessary follows from ([5], remark following 5.3). Remark 2.1.13. If conditions 2.1.11.1, 2.1.11.2, and 2.1.11.4 hold then F is relatively representable (i.e. condition 1.2.8.2 holds). In fact to conclude that F is relatively representable condition 2.1.11.2 can be replaced by the condition that the functor is faithful and that given a_{1}, a_{2} ∈ FÂ, ξ_{n} : a_{1,n} ≃ a_{2,n} a compatible family of isomorphisms in F_{Â/𝔪^{n}}, there exists for all n an isomorphism ξ : a_{1} ≃ a_{2} in FÂ inducing ξ_{n}. These statement follow from the proof of the theorem. When F is known to be relatively representable condition 2.1.11.3 can be modified. In order to conclude that F is an algebraic stack it suffices to prove a statement of openness of versality: if R is a ring of finite type over S and if v : Spec(R) → F is a one morphism which is formally smooth at some point p ∈ Spec(R) of finite type then there exists an open set containing p such that v is formally smooth everywhere in that open set. In fact, the above theorem is proved by showing that the assumptions imply relative representability and openness of versality.

2.2 A relative version of Artin’s theorem.

2.2.1. Let \underline{S} be as before, and let B → \underline{S} be an algebraic stack locally of finite type.

Definition 2.2.2. A B-stack is a stack F over the category of \underline{S}-schemes with the étale topology together with a 1-morphism of stacks

m : F → B.

A morphism of B-stacks F_{1} → F_{2} is a 1-morphism of stacks ϕ : F_{1} → F_{2} together with an isomorphism of functors

ψ : m_{2} ∘ ϕ ≃ m_{1}.

Viewing schemes as stacks we get notions of B-schemes and morphisms of B-schemes. 2.2.3. Fix a B-stack (F, m). If X is a B-scheme define F_{X} as follows. Objects are morphisms of B-stacks

X → F.

More precisely this means that objects of F_{X} are pairs (f, f^{b}) where f : X → F is a morphism of stacks and f^{b} is an isomorphism between the structural morphism

x : \underline{X} → B

and m ∘ f. A morphism

(f′, f^{′b}) → (f, f^{b})

in F_{X} is an isomorphism of functors α : f′ → f such that the two isomorphisms

x → f′ ∘ m

given by f^{′b} and α ∘ f^{b} agree. Then F_{X} is a groupoid. If h : X′ → X is a morphism of B-schemes then there is a notion of pullback

h^{∗} : F_{X} → F_{X′}

and so F defines a fibered category over the category of B-schemes. If a ∈ F_{X} we denote by F_{a,X′} the category whose objects are pairs (a′, α) where a′ ∈ F_{X′} and α : h^{∗}a′ ≃ a is an isomorphism in F_{X}. Morphisms in F_{a,X′} are defined to be morphisms in F_{X′} which are compatible with the morphisms to a. We denote by \overline{F}_{a,X′} the isomorphism classes of objects in F_{X′} mapping to the isomorphism class of a in F_{X}.

2.2.4. The category of B-schemes has an étale topology. Covering families {U_{i} → U} are collections of morphisms of B-schemes U_{i} → U such that the underlying family of morphisms of schemes \{\underline{U}_{i} → \underline{U}\} is an étale cover of \underline{U}. The condition that F is a stack with respect to the étale topology on \underline{S}-schemes is equivalent to the statement that the fibered category over the category of B-schemes is a stack with respect to this topology. 2.2.5. By a B-ring A we mean a ring A together with a structure of a B-scheme on Spec(A). It follows from the fact that B is limit preserving that the stack F is limit preserving if and only if for all filtering inductive limits of B-rings {A_{i}} the natural functor

\underrightarrow{lim} F_{A_{i}} → F_{⟶^{lim} A_{i}}

is an equivalence of categories. 2.2.6. Since B is an algebraic stack, for any deformation situation as in 2.1.2 and a ∈ B_{A} the functor

B_{a,A′×_{A}B} → B_{a,A′} × B_{a,B}

is an equivalence of categories. It follows that products exist in the category of B-rings.

Definition 2.2.7. A B-deformation situation is a diagram in the category of B-rings B

↓
A′ ⟶ A ⟶ A_{0}

such that the underlying diagram of rings is a deformation situation in the sense of 2.1.2.

The following condition will be referred to as (S1^{B}): 2.2.8 (S1^{B}). Given a B-deformation situation as above and a ∈ F_{A} the natural functor

F_{a,A′×_{A}B} → F_{a,A′} × F_{a,B}

is an equivalence of categories. 2.2.9. If A is a B-ring as above, and if I is an A_{0}-module of finite type then the natural retraction

Spec(A[I]) → Spec(A)

gives A[I] the structure of a B-ring which we denote by A[I]. With this definition

A[I] ×_{A} A[I] ≃ A[I ⊕ I]

and so just as in 2.1.5 A[I] has the structure of an A_{0}-module object in the category of B-rings. It therefore follows from (S1^{B}) that given a_{0} ∈ F_{A_{0}} the isomorphism classes of objects in F_{a_{0},A_{0}[I]} has a natural structure of an A_{0}-module. We denote this module by E_{a_{0}}(I). Similarly, if Aut_{a_{0}}(A_{0}[I]) denotes the set of automorphisms of a_{0}|_{A_{0}[I]} which induce the identity on a_{0} then Aut_{a_{0}}(A_{0}[I]) also has a natural structure of an A_{0}-module. 2.1.6 in this context is the following: 2.2.10 (S2^{B}). E_{a_{0}}(I) and Aut_{a_{0}}(A_{0}[I]) are A_{0}-modules of finite type.

Conditions (S1^{B}) and (S2^{B}) will be referred to as Schlessinger’s conditions. 2.2.11. By an obstruction theory for F → B we mean the following:

A. For every morphism of B-rings A → A_{0} as in 2.2.7 and a ∈ F_{A} a functor

O_{a} : (A_{0}-modules of finite type) → (A_{0}-modules of finite type).

B. For every diagram A′ → A → A_{0} as in 2.2.7 an element o_{a}(A′) ∈ O_{a}(I) which is zero if and only if there exists a lifting of a to A′.

C. This data is to be functorial and linear in A_{0} and I.

2.2.12 (Additional Conditions). The modules E_{a_{0}}, O_{a}(I), and Aut_{a_{0}}(A_{0}[I]) are required to satisfy the following additional conditions:

A. Compatibility with étale localization: With notation as above and as in 2.1.10, if A → C is an étale morphism of B-rings (i.e. A → C is étale) then for any a ∈ F_{A} with image c ∈ F_{C}

E_{c_{0}}(I ⊗ C_{0}) ≃ E_{a_{0}}(I) ⊗ C_{0},
Aut_{c_{0}}(C_{0}[I ⊗ C_{0}]) ≃ Aut_{a_{0}}(A_{0}[I]) ⊗ C_{0},

and

O_{c}(I ⊗ C_{0}) ≃ O_{a}(I) ⊗ C_{0}.

B. Effectivity of formal objects: If 𝔪 ⊂ A_{0} is a maximal ideal and Â the completion of

A_{0} along 𝔪, then

E_{a_{0}}(I) ⊗ Â ≃ \underleftarrow{lim} E_{a_{0}}(I/𝔪^{n}I),
Aut_{a_{0}}(A_{0}[I]) ⊗ Â ≃ \underleftarrow{lim} Aut_{a_{0}}(A_{0}[I/𝔪^{n}I]),

and

O_{a_{0}}(I) ⊗ Â ↪ \underleftarrow{lim} O_{a_{0}}(I/𝔪^{n}I).

C. Constructibility: There exists an open dense set of points p ∈ Spec(A_{0}) with k(p) of finite type over \underline{S} such that

E_{a_{0}}(I) ⊗ k(p) ≃ E_{a_{0}}(I ⊗ k(p)),
Aut_{a_{0}}(A_{0}[I]) ⊗ k(p) ≃ Aut_{a_{0}}(A_{0}[I ⊗ k(p)]),

and

O_{a}(I) ⊗ k(p) ⊂ O_{a}(I ⊗ k(p)).

Theorem 2.2.13. Let F → B be a limit preserving B-stack, and suppose an obstruction theory O is given as above. Then F is an algebraic stack if the following conditions hold:

1. The conditions (S1^{B}), (S2^{B}), and the conditions on E_{a_{0}}(I) and Aut_{a_{0}}(A_{0}[I]) in 2.2.12 hold.

2. If Â is a complete local B-ring with residue field of finite type over \underline{S}, then the natural

functor

FÂ → \underleftarrow{lim} F_{Â/𝔪^{n}}

is faithful and has dense image.

3. The obstruction theory O satisfies the conditions in 2.2.12. In addition, for every reduced noetherian ring A_{0}, a_{0} ∈ F_{A_{0}} and I an A_{0}-module of finite type the map

B_{m(a_{0}),A_{0}[I]} → O_{a_{0}}(I)

which sends a lifting b of m(a_{0}) to A_{0}[I] to the obstruction o_{a_{0}}((A_{0}[I], b)) ∈ O_{a_{0}}(I) is an A_{0}-module homomorphism. Here (A_{0}[I], b) denotes the B-ring induced by b which comes with a natural morphism of B-rings

(A_{0}[I], b) → A_{0}

where A_{0} denotes the B-ring obtained from A_{0} via m(a_{0}).

4. If A_{0} is of finite type over \underline{S}, a_{0} ∈ F_{A_{0}} an object, and ϕ an automorphism of a_{0} which induces the identity in F_{k} (k viewed as a B-ring) for a dense set of points A_{0} → k with k of finite type over \underline{S}, then ϕ is the identity over some dense open subset of Spec(A_{0}).

Proof. Let \underline{B}′ → B be a smooth cover over B where \underline{B}′ is a scheme locally of finite type over \underline{S}. To prove the theorem, it suffices to show that F is relatively representable and that the stack F′ = F ×_{B} \underline{B}′ is an algebraic stack. For if \underline{T} → F′ is a smooth surjective cover then for any \underline{X} → F all the arrows in

\underline{T}′ ×_{F′} (F′ ×_{F} \underline{X}) → F′ ×_{F} \underline{X} ≃ \underline{B}′ ×_{B} \underline{X} → \underline{X}

are smooth surjective morphisms of schemes. Hence \underline{T}′ also gives a smooth surjective cover of F. To prove that F is relatively representable it suffices by remark 2.1.13 to prove that conditions 2.1.11.1, 2.1.11.4, and the weakened version of 2.1.11.2 discussed in 2.1.13 hold for F. Let us begin by verifying condition (S1). Given a deformation situation (see 2.1.2, except we write C for B so as not to confuse the notation) and a ∈ F_{A} we have to show that

F_{a,A′×_{A}C} → F_{a,A′} × F_{a,C}

is an equivalence of categories. Suppose given a′ ∈ F_{a,A′} and c ∈ F_{a,C}. The map

m ∘ a : Spec(A) → B

gives A the structure of a B-ring A and similarly a′ and c define B-rings A′ and C. Moreover, the isomorphisms a′|_{A} ≃ a and c|_{A} ≃ a define morphisms of B-rings

A′ → A, C → A

and hence we have a B-deformation situation and an object in

F_{a,A′} × F_{a,C}.

By (S1^{B}) there exists an object in F_{a,A′×_{A}C} inducing this object and hence the functor in (S1) is essentially surjective. For the full faithfulness, suppose a_{1}, a_{2} ∈ F_{a,A′×_{A}C}. Given a pair of isomorphisms (f′, f_{C}) ∈ F_{a,A′} × F_{a,B} we get a pair of isomorphisms (m(f′), m(f_{C})) ∈ B_{m(a),A′} × B_{m(a),C} and hence an isomorphism f : m(a_{1}) ≃ m(a_{2}) in B_{a,A′×_{A}C}. The element a_{1} gives in the same manner as in the preceding paragraph A′, A, and C the structure of B-rings A′, A, and C and the morphism f allows us to view a_{2} as an object in

F_{a,A′×_{A}C}

and the morphisms (f′, f_{C}) as morphisms in

F_{a,A′} × F_{a,C}.

From (S1^{B}) it therefore follows that the functor is full. Similarly, given two isomorphisms f_{1}, f_{2} : a_{1} ⇉ a_{2} in F_{a,A′×_{A}C} which define the same morphism in F_{a,A′} × F_{a,C} we must by (S1) for B have m(f_{1}) = m(f_{2}). Therefore we can view f_{1} and f_{2} as morphisms in F_{a,A′×_{A}C} where we give A′, A, C the structure of B-rings via a_{1}. Therefore, (S1^{B}) also implies that the functor is faithfull. This proves that (S1) holds. Now let A_{0} be a reduced noetherian \underline{S}-ring, and let a_{0} ∈ F_{A_{0}}. Denote by Aut_{a_{0}}(A_{0}[I]) the module of infinitesimal automorphisms of the image of a_{0} in A_{0}[I] (the module structure follows from the fact that (S1) holds). m ∘ a_{0} gives A_{0} the structure of a B-ring which we denote by A_{0}. We claim that there is a natural 8-term exact sequence of A_{0}-modules

\begin{aligned}0 & → & Aut_{a_{0}}(A_{0}[I]) & → & Aut_{a_{0}}(A_{0}[I]) & → & Aut_{m(a_{0})}(A_{0}[I]) & → \\ & → & E_{a_{0}}(I) & → & \overline{F}_{a_{0},A_{0}[I]} & → & B_{m(a_{0}),A_{0}[I]} & → & O_{a_{0}}(I). \end{aligned}  (2.1)

Here Aut_{a_{0}}(A_{0}[I]) is the module of 2.2.10 for A_{0}, Aut_{a_{0}}(A_{0}[I]) is the module of infinitesimal automorphisms of the image of a_{0} in A_{0}[I] (ignoring the map to B), Aut_{m(a_{0})}(A_{0}[I]) is the module of 2.1.6 for B, and E_{a_{0}}(I) is as in 2.2.10. The maps in the sequence are the ones induced from functoriality. The only non-trivial ones are the maps Aut_{m(a_{0})}(A_{0}[I]) → E_{a_{0}}(I) and the map B_{m(a_{0}),A_{0}[I]} → O_{a_{0}}(I). The map Aut_{m(a_{0})}(A_{0}[I]) → E_{a_{0}}(I) is defined as follows. Let h : Spec(A_{0}[I]) → B, f : Spec(A_{0}[I]) → F, η : h ≃ m ∘ f = m(a_{0}) be the data defining the element over A_{0}[I] defined by a_{0}. The map sends an automorphism ϕ : m(a_{0}) → m(a_{0}) to the element in E_{a_{0}[I]} defined by the maps h, f together with the isomorphism

ϕ ∘ η.

It follows from unwinding the definition of the module structures that this in fact is an A_{0}- module homomorphism. This amounts to saying that if I′ → I is a morphism of A_{0}-modules then the induced diagram

Aut_{m(a_{0})}(A_{0}[I]) ⟶ E_{a_{0}}(I)
↑ ↑
Aut_{m(a_{0})}(A_{0}[I′]) ⟶ E_{a_{0}}(I′)

commutes which is immediate. The map B_{m(a_{0}),A_{0}[I]} → O_{a_{0}}(I) is described in 2.2.13.3. The exactness of the sequence is tautological.

This sequence implies that (S2) holds since all the modules except Aut_{a_{0}}(A_{0}[I]) and \overline{F}_{a_{0},A_{0}[I]} are known to be of finite type. Similarly, a diagram chase shows that the conditions in 2.1.10 are satisfied. This shows that 2.1.11.1 is satisfied. The faithfulness of the functor in 2.1.11.2 can be shown as follows. If a_{1}, a_{2} ∈ FÂ and f_{1}, f_{2} : a_{1} ⇉ a_{2} are morphisms which define the same morphism in \underleftarrow{lim} F_{Â/𝔪^{n}} we see that m(f_{1}), m(f_{2}) must be equal. View Â as a B-ring via m(f_{1}). Then f_{1} and f_{2} are morphisms in FÂ which induce the same map in \underleftarrow{lim} F_{Â/𝔪^{n}} and hence must be equal. Next suppose a_{1}, a_{2} ∈ FÂ and that a compatible family of of isomorphisms ξ_{n} : a_{1,n} ≃ a_{2,n} are given. Since B is an algebraic stack, m(a_{1}) ×_{B} m(a_{2}) is representable by a scheme and hence there exists a unique isomorphism m(ξ) : m(a_{1}) ≃ m(a_{2}) in BÂ inducing the isomorphisms m(ξ_{n}). It follows that we can view a_{1} and a_{2} as objects in FÂ where Â is the B-ring obtained via m(a_{1}). It now follows from 2.2.13.2 that for all n there exists an isomorphism ξ : a_{1} ≃ a_{2} inducing ξ_{n}. Finally condition 2.1.11.4 can be verified as follows. If A_{0} is reduced and of finite type over \underline{S}, a_{0} ∈ F_{A_{0}}, ϕ an automorphism of a_{0} which induces the identity for a dense set of points, then there exists a dense open set of A_{0} such that m(ϕ) is the identity automorphism of m(a_{0}) over this open set. Replacing A_{0} by the ring of a basic open set inside this open set we can assume that m(ϕ) is the identity map. Thus viewing A_{0} as a B-ring via m(a_{0}) we are reduced to 2.2.13.4. This proves that F is relatively representable. Now fix a smooth surjective cover b : \underline{B}′ → \underline{B} and let F′ = F ×_{B} \underline{B}′. If s : Spec(A) → \underline{B}′ is a morphism then an object in F′_{A} is equivalent to a pair (f, h) where f : Spec(A) → F is a 1-morphism and h : s^{∗}(b) ≃ m(f) is an isomorphism in B. A morphism (f′, h′) → (f, h) in F′_{A} is the same as a morphism η : f′ → f in F_{A} such that m(η) ∘ h′ = h. In other words, if we view A as a B-ring A via the map b, the category F′_{A} is naturally equivalent to the category F_{A}. This implies that all the hypothesis of 2.1.11 hold except possibly 2.1.11.3. To construct an obstruction theory O′ for F′ proceed as follows. An infinitesimal extension A → A_{0} over \underline{B}′ can be viewed via b as an infinitesimal extension of B-rings

A → A_{0}.

Moreover, an element a ∈ F′_{A} defines in a natural way an element in F_{A} which we also denote by a. Define the functor O′_{a} to be the functor O_{a} of 2.2.11 coresponding to this data. Given a deformation situation A′ → A → A_{0} over \underline{B}′ we can view it as a B-deformation situation, and hence get an element o_{a}(I) ∈ O′_{a}(I). It is immediate that (O′_{a}, o_{a}) satisfies the axioms of an obstruction theory. This proves that F′ is an algebraic stack, and hence that F is also algebraic.

Remark 2.2.14. From the proof it follows that it suffices to construct local obstruction theories O_{a} in the sense of 2.1.8 translated into the language of B-schemes.

Chapter 3

Universal Log Structures

3.1 Direct construction

The main result of this section is the following theorem suggested by A. Abbes:

Theorem 3.1.1. Let \underline{S} be a scheme locally of finite type over an excellent Dedekind ring and let M_{S} be a fine log structure on \underline{S}_{fl}. Let LOG_{S} be the fibered category over the category of \underline{S}-schemes for which LOG_{S}(f : \underline{X} → \underline{S}) is the category of fine log structures M_{X} on \underline{X}_{fl} with a morphism

f^{∗}M_{S} → M_{X}.

Then LOG_{S} is an algebraic stack.

We refer to the objects of LOG_{S} as S-log structures. Remark 3.1.2. Note that by 1.1.7 we can view LOG_{S} as classifying log structures on the étale site.

The proof of the theorem will be in two steps (theorems 3.1.3 and 3.1.11).

Theorem 3.1.3. Let \underline{X} be a scheme locally of finite type over an excellent Dedekind ring and let

\begin{matrix}M_{S} & ⟶^{s_{1}} & M_{1} \\ s^{2} & ↓ \\ M_{2} \end{matrix}  (3.1)

be a diagram of fine log structures on \underline{X}. Then the functor I on \underline{X}-schemes defined by

(f : \underline{Z} → \underline{X}) ↦ set of isomorphisms i : f^{∗}M_{1} ≃ f^{∗}M_{2} such that i ∘ s_{1} = s_{2}

is representable by a locally separated algebraic space.

Remark 3.1.4. The functor I of 3.1.3 is not separated in general. For an explicit example where the valuative criterion for separation fails, let A be a discrete valuation ring with uniformizer π, and let M be the log structure associated to the chart

ℕ ⊕ ℕ → A, (i, j) ↦ π^{i+j}.

Then the identity map and the map induced by

(i, j) ↦ (j, i)

are two automorphisms of M both of which induce the identity map on the generic fiber.

We begin the proof of 3.1.3 by considering a special case:

Lemma 3.1.5. If s_{1} and s_{2} are surjective then the functor I is represented by a locally closed subscheme of \underline{X}.

Proof. For i = 1, 2 let

K^{i} = Ker(s_{i}^{gp} : M_{S}^{gp} → M_{i}^{gp}), \overline{K}^{i} = Ker(s̄_{i}^{gp} : \overline{M}_{S}^{gp} → \overline{M}_{i}^{gp}).

From the diagram

0 ⟶ O_{X}^{∗} ⟶ M_{S}^{gp} ⟶ \overline{M}_{S}^{gp} ⟶ 0
=↓ ↓ ↓
0 ⟶ O_{X}^{∗} ⟶ M_{i}^{gp} ⟶ \overline{M}_{i}^{gp} ⟶ 0

and the snake lemma we see that the natural map K^{i} → \overline{K}^{i} is an isomorphism of sheaves. If f : \underline{Z} → \underline{X} is a morphism of schemes then we denote by f^{∗}K^{i} and f^{∗}\overline{K}^{i} the sheaves

Ker(s_{i}^{gp} : f^{∗}M_{S}^{gp} → f^{∗}M_{i}^{gp}) and Ker(s̄_{i}^{gp} : f^{−1}\overline{M}_{S}^{gp} → f^{−1}\overline{M}_{i}^{gp}).

Let F be the functor on \underline{X}-schemes for which F (\underline{Z}) consists of a single element if f^{∗}K^{1} = f^{∗}K^{2} as subsheaves of f^{∗}M_{S}^{gp} and the empty set otherwise. There is a natural morphism of functors

I → F

which we claim is an isomorphism. To see this suppose K^{1} = K^{2}. Then we can define an isomorphism j : M_{1} → M_{2} be sending an element m ∈ M_{1} to s_{2}(m̃) where m̃ ∈ M_{S} is any lifting of m. This does not depend on the choice of m̃. For if ñ is a second such lifting then m̃ − ñ ∈ K^{1} = K^{2} and hence maps to zero on M_{2}^{gp}. Conversely, if we have an isomorphism j : M_{1} ≃ M_{2} then we have

K^{2} = Ker(s_{2}^{gp}) = Ker(j^{gp} ∘ s_{1}^{gp}) = Ker(s_{1}^{gp}) = K^{1}.

It therefore suffices to show that F is representable by a locally closed subscheme of \underline{X}. Let F′ be the functor which to any f : \underline{Z} → \underline{X} associates the unital set if f^{∗}\overline{K}^{1} = f^{∗}\overline{K}^{2} in f^{−1}\overline{M}_{S}^{gp} and the empty set otherwise. We have a diagram of functors

F → F′ → \underline{X}.

We claim that F′ is represented by an open subset of \underline{X} and that F is represented by a closed subscheme of F′. To prove these statements we may pass to a cover of \underline{X}. We claim that locally there exists a chart

P → H^{0}(\underline{X}, M_{S})

such that the image P^{i} of P in

H^{0}(\underline{X}, M_{i})

is a chart for M_{i}. To see this it suffices to construct P and P^{i} at the stalk of some geometric point x̄ → \underline{X} since charts can always be extended to some étale neighborhood. Fix a chart

P^{i} → M_{i,x̄}

which is exact and injective. Note that an exact chart λ : P^{i′} → M_{i,x̄} exists by ([17], II.2.2.10). If P^{i} denotes the image of P^{i′} in M_{i,x̄} then the inclusion

P^{i} → M_{i,x̄}

is again an exact chart. It is exact because

P^{i′} = P^{i′,gp} ×_{M_{i,x̄}^{gp}} M_{i,x̄}

and hence the image of P^{i′} in M_{i,x̄} (which by definition is P^{i}) is equal to

P^{i,gp} ×_{M_{i,x̄}^{gp}} M_{i,x̄} ↪ M_{i,x̄}.

To see that it is a chart, it suffices to show that the map

P^{i}/(P^{i} ∩ O_{X,x̄}^{∗}) → \overline{M}_{i,x̄}

is an isomorphism. Since λ is exact,

P^{i′} = P^{i′,gp} ×_{P^{i,gp}} P^{i}

and hence if G = Ker(P^{i′,gp} → P^{i,gp}) then G ⊂ P^{i′} and

\overline{M}_{i,x̄} ≃ P^{i′}/λ^{−1}(O_{X,x̄}^{∗}) = (P^{i′}/G)/(λ^{−1}(O_{X,x̄}^{∗})/G) ≃ P^{i}/(P^{i} ∩ O_{X,x̄}^{∗}).

It follows that P^{i} → M_{i,x̄} is also a chart. Given our injective and exact chart P^{i} ↪ M_{i}, define P by

P = P^{i,gp} ×_{M_{i,x̄}^{gp}} M_{S,x̄}.

Then by ([20], 2.10) P is a chart for M_{S,x̄} and the image of P in M_{i,x̄} is by the exactness equal to P^{i}. Let K̃^{i} be the kernel of the map

P^{gp} → P^{i,gp}.

We claim that for all geometric points x̄ → \underline{X} the map

K̃^{i} → \overline{K}i ; x̄

is surjective. To see this consider the diagram

0 ⟶ K̃^{i} ⟶ P^{gp} ⟶ P^{i,gp} ⟶ 0
↓ ↓ ↓
0 ⟶ \overline{K}i ; x̄ ⟶ \overline{M}_{S,x̄}^{gp} ⟶ \overline{M}_{i,x̄}^{gp} ⟶ 0

and define

H = Ker(P^{gp} → \overline{M}_{S,x̄}^{gp}), H^{i} = Ker(P^{i,gp} → \overline{M}_{i,x̄}^{gp}).

Since the map P^{gp} → \overline{M}_{S,x̄}^{gp} is surjective it suffices to show that the map H → H^{i} is surjective by the snake lemma. Now H^{i} is generated by differences of elements p̄ ∈ P^{i} for which α(p̄) ∈ O_{X,x̄}^{∗}. But then any p ∈ P lifting p̄ also has this property and consequently is in H. This proves that K̃^{i} → \overline{K}i ; x̄ is surjective for all x̄. Since K̃^{i} is finitely generated this implies that F′ is represented by an open subset of \underline{X}. Replacing \underline{X} by this open subset we can therefore assume that \overline{K}^{1} = \overline{K}^{2}. It follows that if k ∈ K̃^{1} then there exists a unique element k′ ∈ K^{2} and u(k) ∈ O_{X}^{∗} such that

k = k′ + λ(u(k)).

The functor F is therefore represented by the closed subscheme of \underline{X} represented by the conditions

u(k) = 1, k ∈ K̃^{1}.

Note that since K̃^{1} is finitely generated there are only finitely many equations for F.

To prove the general case of the theorem we make various reductions.

Reduction Step 3.1.6. It suffices to consider the case when s_{1} and s_{2} are injective.

Proof. Let M_{S}^{1} (resp. M_{S}^{2}) be the image of s_{1} (resp. s_{2}) in M_{1} (resp. M_{2}). Let \underline{X}′ ⊂ \underline{X} be the locally closed subscheme obtained by applying the lemma to the diagram

M_{S} ⟶ M_{S}^{1}
↓
M_{S}^{2}

Then the morphism of functors I → \underline{X} factors through \underline{X}′ and hence we can replace \underline{X} by \underline{X}′. Over \underline{X}′ there is a unique isomorphism i : M_{S}^{1} → M_{S}^{2} and hence we may replace the diagram in 3.1.3 by the diagram

M_{S}^{1} ⟶ M_{1}
i↓
M_{2}

Reduction Step 3.1.7. It suffices to prove the theorem fppf-locally on \underline{X}.

Proof. I comes with descent datum relative to any fppf-cover of \underline{X}. The result therefore follows from ([5], 6.3) which implies that descent datum for an algebraic space in the fppftopology is effective.

Lemma 3.1.8. Suppose \underline{X} = Spec(A) where A is an artinian local ring. Then I is representable by an affine scheme.

Proof. The property of being an affine scheme over Spec(A) is local in the fppf-topology ([27], 2.23) so it suffices to prove the theorem after replacing \underline{X} by an appropriate cover. If I ≠ ∅ then after replacing A by an étale cover (using 1.1.7) we can assume that \overline{M}_{1} and \overline{M}_{2} are constant sheaves and that we have an isomorphism

ī: \overline{M}_{1} ≃ \overline{M}_{2}

such that s̄_{2} = ī∘ s̄_{1}. Fix a chart Q → M_{S} such that Q → \overline{M}_{S} is bijective (this is possible after a finite flat extension). Then after perhaps replacing A by another finite flat extension we can choose charts

P_{1} ⟶ M_{1}
↑ ↑
Q ⟶ M_{S}

and P_{2} ⟶ M_{2}

↑ ↑
Q ⟶ M_{S}

such that P_{1} → \overline{M}_{1} and P_{2} → \overline{M}_{2} are bijective ([29], 2.13 and 2.14). Identifying P_{1} and P_{2} via i we can therefore assume that the log structures are induced from two morphisms β_{1}, β_{2} : P → A where P is a fine sharp monoid, and that s_{1} and s_{2} are induced from an injective map Q → P. Let Aut(P/Q) be the set of automorphisms of P which leave Q fixed. For ξ ∈ Aut(P/Q), let R_{ξ} denote the ring

A[P^{gp}/Q^{gp}]/{e(p)β_{2}(ξ(p)) = β_{1}(p)e(0)}_{p∈P}

where e(p) denotes the image of an element p ∈ P in A[P^{gp}/Q^{gp}]. Given any map of rings ϕ : R_{ξ} → B we get an isomorphism of log structures over B induced from the map

P → B^{∗} ⊕ P, p ↦ ϕ(e(p)) + ξ(p).

This defines a map

Spec(R_{ξ}) → I

and since P → \overline{M}_{1} and P → \overline{M}_{2} are isomorphisms, every element i ∈ I(B) is induced from a map R_{ξ} → B for a unique ξ (assuming that Spec(B) is connected). It follows that I is represented by the scheme

∐_{ξ∈Aut(P/Q)} Spec(A[P^{gp}/Q^{gp}]/{e(p)β_{2}(ξ(p)) = β_{1}(p)e(0)}_{p∈P}).

Later it will be useful to understand the construction of the above proof using more general charts:

Lemma 3.1.9. Let A be an artinian local ring, and suppose

Q ⟶^{h_{1}×s_{1}} G_{1} ⊕ P_{1}
h_{2}×s_{2}↓  ^{α_{1}}↓
G_{2} ⊕ P_{2} ⟶^{α_{2}} A

is a chart for diagram 3.1 where G_{1}, G_{2} ⊂ A^{∗} are subgroups, and Q, P_{1}, P_{2} are sharp monoids. For any isomorphism

σ : P_{1} → P_{2}

such that s_{2} = σ ∘ s_{1} let C_{σ} be the quotient of

A[P_{1}^{gp}]

by the relations

α_{2}(σ(p))e(p) = α_{1}(p)e(0), p ∈ P_{1}

and

e(q) = α_{2}(h_{2}(q))α_{1}(h_{1}(q))^{−1}, q ∈ Q.

Then the natural map

j : Spec(C_{σ}) → I

defined by the isomorphism over Spec(C_{σ}) induced by

p ↦ e(p) + σ(p)

is an open immersion.

Proof. The statement of the lemma is fppf-local on Spec(A) and hence using the argument of 3.1.8 we can assume that the log structures M_{1} and M_{2} are induced from two maps

β_{1}, β_{2} : P̃ ⇉ A

and that the maps s_{j} : M_{S} → M_{j} are induced from an inclusion Q̃ ↪ P̃. In addition we can assume that the maps

P̃ → \overline{M}_{1}, P̃ → \overline{M}_{2}, Q̃ → \overline{M}_{S}

are bijective. From the preceding lemma we see that I is represented by

∐_{ξ∈Aut(P̃/Q̃)} Spec(A[P̃^{gp}/Q̃^{gp}]/{e(p)β_{2}(ξ(p)) = β_{1}(p)e(0)}_{p∈P̃}).

α_{1} and α_{2} induce isomorphisms

P_{1}/F_{1} ≃ P̃, P_{2}/F_{2} ≃ P̃

where

F_{j} = {p ∈ P_{j}|α_{j}(p) ∈ A^{∗}}.

If Spec(C_{σ}) ≠ ∅ then it follows from the definition that σ(F_{1}) = F_{2}. Assume this is the case. Then σ induces an element ξ ∈ Aut(P̃/Q̃). The map

Spec(C_{σ}) → I

maps into the component corresponding to ξ defined by a map of rings

A[P̃^{gp}/Q̃^{gp}]/{e(p)β_{2}(ξ(p)) = β_{1}(p)e(0)}_{p∈P̃} → C_{σ}.

This map is defined by sending e(p̃) to

u(p̃)u(ξ(p̃))^{−1}e(p)

where p is a lifting of p̃ and u(p̃) and u(ξ(p̃)) are obtained by writing

β_{1}(p̃) = λ(u(p̃)) + α_{1}(p)

in M_{1} and

β_{2}(ξ(p̃)) = λ(ξ(p̃)) + α_{2}(σ(p))

in M_{2}. It now follows from the definition of C_{σ} that this map is an isomorphism.

3.1.10 (Conclusion of proof of 3.1.3). We assume that s_{1} and s_{2} are injective. Replacing \underline{X} by an affine cover we can assume that \underline{X} is quasi-compact. We proceed by induction on

n = sup_{x̄→\underline{X}}#Irr(\overline{M}_{1,x̄})

where Irr(\overline{M}_{1,x̄}) denotes the set of irreducible elements in \overline{M}_{1,x̄}. Observe that if we fix an integer n, then the set of point x ∈ X with

#Irr(\overline{M}_{1,x̄}) < n

is an open set. To see this, suppose x is such a point, and let p̄_{1}, …, p̄_{r} (r < n) be the irreducible elements of \overline{M}_{1,x̄}. Let p_{1}, …, p_{r} ∈ M_{1,x̄} be liftings of the p̄_{i} and let P ⊂ M_{1,x̄} be the sub-monoid generated by the p_{i}. Then P → M_{1,x̄} extends to a chart for M_{1} in some étale neighborhood of x. Since P is generated by r elements it follows from ([17], 2.1.2) that for each point x′ ∈ X in the image of this étale neighborhood we have

#Irr(\overline{M}_{1,x̄′}) < n.

If n = 0 then I is represented by the open subset of \underline{X} where M_{2} is isomorphic to O_{X}^{∗}. Next we prove the result for n assuming that it holds for n − 1. Let \underline{U} ↪ \underline{X} be the open set of points x such that #Irr(\overline{M}_{1,x̄}) < n. Choose a geometric point x̄ → \underline{X} − \underline{U} with I(x̄) ≠ ∅. If no such x̄ exists we are done so we can assume such a point exists. After perhaps replacing \underline{X} by an fppf-neighborhood of x̄ we can assume that M_{1} and M_{2} are induced from two morphisms

α_{1}, α_{2} : P → O_{X}

and that s_{j} : M_{S} → M_{j} are induced from an inclusion Q ↪ P. In addition we can assume that the maps

Q → \overline{M}_{S,x̄}, P → \overline{M}_{1,x̄}, P → \overline{M}_{2,x̄}

are bijective. For each σ ∈ Aut(P/Q) let

T_{σ} = Spec(O_{X}[P^{gp}/Q^{gp}]/{e(p)α_{2}(σ(p)) = α_{1}(p)e(0)}_{p∈P}).

Then there is a morphism

∐_{σ∈Aut(P/Q)} T_{σ} → I

such that for any connected \underline{Z}/\underline{X} whose image in \underline{X} is not contained in \underline{U} the map

∐_{σ∈Aut(P/Q)} T_{σ}(\underline{Z}) → I(\underline{Z})

is bijective. Now for each σ ∈ Aut(P/Q) we claim that the map

T_{σ} ×\underline{X} \underline{U} → I ×_{X} U

is an open immersion (note that by induction I ×_{X} U is representable by an algebraic space). This is equivalent to the map being étale and injective on field valued points. In other words, we have to show that if \underline{Z}/\underline{U} is a scheme isomorphic to the spectrum of an artinian local ring, then the map

T_{σ}(\underline{Z}) = (T_{σ} ×_{X} \underline{Z})(Z) → I ×_{X} Z(Z) = I(Z)

is injective. But this follows from lemma 3.1.9.

We now conclude the proof of representability by observing that I is represented by the algebraic space obtained by gluing each of the T_{σ} to I ×\underline{X} \underline{U} via T_{σ} ×\underline{X} \underline{U}. It remains to verify that I is locally separated. This means that if \underline{U} → I is a morphism, then the map

\underline{U} ×_{I} \underline{U} → \underline{U} ×\underline{X} \underline{U}

is a locally closed immersion. That is, given two isomorphisms of log structures (we can ignore M_{S} for this ) on \underline{U} ×\underline{X} \underline{U}

α, β : M_{1} → M_{2}

the condition that α = β is represented by a locally closed subscheme of \underline{U} ×\underline{X} \underline{U}. But this follows from 3.1.5 applied to the diagram

M_{1} ⟶^{α} M_{2}
β↓
M_{2}  .

Theorem 3.1.11. Let Q be a fine sharp monoid and let

\begin{matrix}Q & ⟶^{s_{1}} & P_{1} \\ s^{2} & ↓ \\ P_{2} \end{matrix}

be a diagram of fine monoids with s_{1} and s_{2} injective. Let

S = Spec(ℤ[Q]), X = Spec(ℤ[P_{1}]), Y = Spec(ℤ[P_{2}])

and let M_{S}, M_{X}, M_{Y} be the natural log structures. Let I/X ×_{S} Y be the functor corresponding to the diagram

M_{S}|_{X×_{S}Y} ⟶ p_{1}^{∗}M_{X}
↓
p_{2}^{∗}M_{Y}

given by theorem 3.1.3. Then the two projections

I → X, I → Y

are flat.

Proof. We proceed by induction on n = #Irr(\overline{P}_{1}). If n = 0, then P_{1} = P_{1}^{∗} and

I ≃ Spec(ℤ[P_{1}^{∗}]) ×_{Spec(ℤ[Q^{gp}])} Spec(ℤ[P_{2}^{gp}]).

Now ℤ[P_{2}^{gp}]/ℤ[Q^{gp}] is flat since Q^{gp} → P_{2}^{gp} is injective. Next we prove the theorem for n assuming the result for n − 1. We can assume that P_{1} ≃ P_{1}^{∗} ⊕ \overline{P}_{1} by replacing P_{1} by G ⊕_{P_{1}^{∗}} P_{1} for a suitable group G. Fix such a splitting P_{1} ≃ P_{1}^{∗} ⊕ \overline{P}_{1}. We can also assume that #Irr(\overline{P}_{2}) = n, for I → X ×_{S} Y maps into the union of the open sets

X ×_{S} Spec(ℤ[P_{2,F}])

where F ⊂ P_{2} is a face such that #Irr(P_{2}/F) ≤ n. Hence we may replace P_{2} by P_{2,F}. In addition we may assume that P_{2} ≃ P_{2}^{∗} ⊕ \overline{P}_{2} by replacing Y by

Y′ = Spec(ℤ[G ⊕_{P_{2}^{∗}} P_{2}])

for a suitable group G. Then if I′ denotes the algebraic space obtained from X and Y′ we have a flat surjection

I′ → I

and since the property of being flat is fppf-local on domain it suffices to consider I′. Fix such a splitting P_{2} ≃ P_{2}^{∗} ⊕ \overline{P}_{2}. For each isomorphism σ : \overline{P}_{1} → \overline{P}_{2} define C_{σ} as in 3.1.9 and let

T_{σ} = Spec(C_{σ}).

Then by 3.1.9 the map T_{σ} → I is an open immersion and the union of the T_{σ} cover the set of points of I lying over points x for which #Irr(\overline{M}_{X}) = n. Therefore by induction it suffices to show that C_{σ} is flat over ℤ[P_{1}]. This is straightforward from the definitions. Writing out the definition of C_{σ} we find that

C_{σ} ≃ ℤ[\overline{P}_{1}][P_{1}^{∗} ⊕_{Q^{gp}} P_{2}^{gp}]

and hence it suffices to show that

ℤ[P_{1}^{∗}] → ℤ[P_{1}^{∗} ⊕_{Q^{gp}} P_{2}^{gp}]

is flat. This follows from the fact that Q^{gp} → P_{2}^{gp} is injective.

3.1.12 (Conclusion of proof of 3.1.1). Since LOG_{S} classifies sheaves in the fppf-topology LOG_{S} is a stack for this topology and 3.1.3 implies that LOG_{S} is relatively representable. By ([5], 6.3) fppf descent datum for an algebraic stack is effective. Therefore it suffices to prove the theorem in the case when there is a chart Q → M_{S} where Q is a fine sharp monoid. Fix a set {P_{j}}_{j∈J} of fine monoids with injective maps i_{j} : Q → P_{j} such that for any inclusion i : Q → P there exists a j ∈ J and isomorphism ϕ : P_{j} ≃ P such that i = ϕ ∘ i_{j}. Let

S̃ = ∐_{j∈J} X ×_{Spec(ℤ[Q])} Spec(ℤ[P_{j}])

with the natural log structure. By 3.1.11 the two projections

S̃ ×_{LOG_{S}} S̃ ⇉ S̃

are flat, and by ([29], 2.13) LOG_{S} is the stack associated to this flat groupoid. By ([5], 6.3) it now follows that LOG_{S} is an algebraic stack.

3.2 Deformation theory of log structures

3.2.1. It follows from Artin’s theorem 2.1.11 and the fact that the stacks LOG_{S} of the preceding section are algebraic that a nice deformation theory of log structures exists. In this section we give a more direct approach to this deformation theory independent of the preceding sections. Throughout this section all log structures will be sheaves in the étale topology.

The following proposition records some standard facts about flatness and deformation situations.

Proposition 3.2.2. Suppose given a deformation situation and flat modules E′, E, E′′ over A′, A, and B respectively together with isomorphisms E′ ⊗_{A′} A ≃ E ≃ E′′ ⊗_{B} A.

1. E′ ×_{E} E′′ is a flat A′ ×_{A} B-module.

2. If L is an A′ ×_{A} B-module together with morphisms such that the following diagram commutes

\begin{matrix}L & ⟶ & E′′ \\ ↓ & & ↓ \\ E′ & ⟶ & E \end{matrix}

and L ⊗ B ≃ E′′ then L ≃ E′ ×_{E} E′′.

3. Suppose A′ → C′, A → C, and B → C′′ are étale algebras of finite presentation such that C ≃ C′′ ⊗_{B} A ≃ C′ ⊗_{A′} A. Then A′ ×_{A} B → C′ ×_{C} C′′ is étale and of finite presentation. Moreover, the set of étale A′ ×_{A} B algebras of this form are a base for the étale topology on A′ ×_{A} B.

Proof. (1) If R is any ring and I ⊂ R a nilpotent ideal, then an R-module E is flat if and only if E/IE is flat over R/I and the natural map

γ : gr_{I}^{0}(E) ⊗_{R/I} gr_{I}^{∗}(R) → gr_{I}^{∗}(E)

is an isomorphism ([1], V theorem 3.2). Here gr_{I}^{∗}(E) denotes

⊕_{n}I^{n}E/I^{n+1}E.

Applying this criteria to E′ we deduce that

I^{i}E′/I^{i+1}E′ ≃ E ⊗_{A} (I^{i}/I^{i+1}).

The kernel of A′ ×_{A} B → B is the ideal I ×_{A} {0} which is isomorphic to I. We apply the above criteria to E′ ×_{E} E′′. By direct calculation we have

gr_{I×\{0\}}^{i}(E′ ×_{E} E′′) ≃ \begin{cases}E′′, & if i = 0 \\ I^{i}E′/(I^{i+1}E′) & if i ≠ 0 \end{cases}

and

gr_{I×\{0\}}^{i}(A′ ×_{A} B) ≃ \begin{cases}B, & if i = 0 \\ I^{i}/I^{i+1}, & if i ≠ 0. \end{cases}

Thus checking γ is an isomorphism amounts to checking that

I^{i}E′/I^{i+1}E′ ≃ E ⊗_{A} (I^{i}/I^{i+1}) ≃ (E′′ ⊗_{B} A) ⊗_{A} (I^{i}/I^{i+1}) ≃ E′′ ⊗_{B} (I^{i}/I^{i+1}).

(2) There is a natural map L → E′ ×_{E} E′′ which is an isomorphism when tensored with B. Since the kernel of A′ ×_{A} B → B is nilpotent and E′ ×_{E} E′′ is flat this implies that the map is an isomorphism. For the cokernel C of L → E′ ×_{E} E′′ must be zero since it becomes zero when tensored with A′ ×_{A} B/(I × {0}) by Nakayama’s lemma. We therefore have an exact sequence

0 → K → L → E′ ×_{E} E′′ → 0

and since E′×_{E} E′′ is flat this sequence remains exact when tensored with A′×_{A}B/(J ×{0}) and hence K = 0 by applying Nakayama’s lemma again.

(3) This follows from (1) and (2) and the fact that if f : \underline{X} → \underline{Y} is a flat morphism locally of finite type of schemes with \underline{Y} locally noetherian, then f is étale at a point x ∈ \underline{X} if and only if the map \underline{X} ×\underline{Y} k(f(x)) → Spec(k(f(x))) is étale at x ([1], VI Proposition 4.8). If L is an étale A′ ×_{A} B-algebra let C′ = L ⊗ A′, C′′ = L ⊗ B, and C = L ⊗ A. Then by (1) C′ ×_{C} C′′ is an étale A′ ×_{A} B-algebra and the natural map

L → C′ ×_{C} C′′

is an isomorphism.

Proposition 3.2.3. Suppose given a deformation situation and logarithmic morphisms of monoids α_{1} : M_{1} → A′, α_{2} : M_{2} → B, α_{0} : M → A together with isomorphisms of monoids

M_{1} ⊕_{α_{1}^{−1}(A^{′∗})} A^{∗} ≃ M ≃ M_{2} ⊕_{α_{2}^{−1}(B^{∗})} A^{∗}

compatible with the induced maps to A.

1. The map M_{1} ×_{M} M_{2} → A′ ×_{A} B is logarithmic.

2. The natural map

B^{∗} ⊕_{M_{1}×_{M} α_{2}^{−1}(B^{∗})} (M_{1} ×_{M} M_{2}) → M_{2}

is an isomorphism.

3. The natural map

A^{′∗} ⊕_{A^{′∗}×_{M} M_{2}} (M_{1} ×_{M} M_{2}) → M_{1}

is an isomorphism.

Proof. For any element m of M_{2} or M_{1} let \overline{m} denote the image in M.

(1) This is immediate from the fact that the α_{i} are assumed logarithmic.

(2) The map M_{1} → M is surjective, and from this it follows that the map in (2) is also surjective. As for injectivity, let (b, m_{1}, m_{2}), (b′, m′_{1}, m′_{2}) ∈ B^{∗} ⊕ M_{1} ×_{M} M_{2} be two triples which map to the same element in M_{2}. Let a ∈ A^{′∗} be an element such that \overline{λ(a)} = \overline{λ(b)} − \overline{λ(b′)}, and such that

m′_{1} = λ(a) + m_{1}.

This is possible since the map A^{′∗} → A^{∗} is surjective. Then (a, λ(b) − λ(b′)) ∈ A^{′∗} ×_{M} M_{2} and the equations

λ(b) = λ(b′) + λ(b) − λ(b′), m′_{2} = λ(b) − λ(b′) + m_{2}

together with the choice of a imply that the triples (b, m_{1}, m_{2}), (b′, m′_{1}, m′_{2}) define the same element of B^{∗} ⊕_{M_{1}×_{M} α_{2}^{−1}(B^{∗})} (M_{1} ×_{M} M_{2}).

(3) Since M_{2} induces M on A, for every m_{1} ∈ M_{1} there exists a ∈ A^{∗} such that λ(a)+\overline{m_{1}} is in the image of M_{2} → M. Surjectivity of the map now follows from the fact that A^{′∗} → A^{∗} is surjective. For injectivity, let (a, m_{1}, m_{2}), (a′, m′_{1}, m′_{2}) ∈ A^{′∗} ⊕ M_{1} ×_{M} M_{2} be triples which map to the same element of M_{1}. Let ã = a/a′ ∈ A^{′∗}. Then we have the equation

m′_{1} = λ(ã) + m_{1}.

Since M = A^{∗} ⊕_{α_{2}^{−1}(A^{∗})} M_{2}, we can choose s, t ∈ α_{2}^{−1}(A^{∗}) such that

\overline{λ(ã)} + \overline{s} = \overline{t}

and

m′_{2} + s = m_{2} + t.

This follows from the fact that (\overline{λ(ã)}, m_{2}) and (1, m′_{2}) both map to \overline{m′_{1}} ∈ M. \overline{s} is a unit in A so there exists b ∈ A^{′∗} mapping to \overline{s}. Then (ãb, t), (b, s) ∈ A^{′∗} ×_{M} M_{2} and the equations

a′ãb = ab, (m_{1}, m_{2})(ãb, t) = (m′_{1}, m′_{2})(b, s)

imply that the triples (a, m_{1}, m_{2}), (a′, m′_{1}, m′_{2}) define the same element of A^{′∗} ⊕_{A^{′∗}×_{M} M_{2}} (M_{1} ×_{M} M_{2}).

3.2.4. If R → A is a ring homomorphism, and if M_{A} is a S-log structure on Spec(A), let LOG_{S,M_{A}}(R) be the category whose objects are S-log structures N_{R} on Spec(R) together with an isomorphism between M_{A} and the pullback of M_{R} to Spec(A). Morphisms in LOG_{S,M_{A}}(R) are morphisms of S-log structures inducing the identity on M_{A} when pulled back to Spec(A).

Theorem 3.2.5. Suppose given a deformation situation and an S-log structure M_{A} on Spec(A). Then the natural functor

LOG_{S,M_{A}}(A′ ×_{A} B) → LOG_{S,M_{A}}(A′) × LOG_{S,M_{A}}(B)

is an equivalence of categories.

Proof. By applying the theorem to both objects and morphisms it suffices to consider the case when M_{S} is trivial. The functor is essentially surjective since the scheme

Spec(A′ ×_{A} B) ≃ (|Spec(B)|, f_{∗}O_{A′} ×_{f_{∗}O_{A}} O_{B})

and by the preceding proposition if M_{A′} ∈ LOG_{S,M_{A}}(A′) and M_{B} ∈ LOG_{S,M_{A}}(B) then

f_{∗}M_{A′} ×_{f_{∗}M_{A}} M_{B} → f_{∗}O_{A′} ×_{f_{∗}O_{A}} O_{B}

is a log structure on A′ ×_{A} B (note here we are also using 3.2.2.3). 3.2.3 implies that this lifting of M_{A′} and M_{B} is unique up to unique isomorphism since any lifting N comes with a canonical map

N → f_{∗}M_{A′} ×_{f_{∗}M_{A}} M_{B}

which must be an isomorphism since it is an isomorphism over Spec(B). Any morphism

f_{∗}M_{A′} ×_{f_{∗}M_{A}} M_{B} → f_{∗}M′_{A′} ×_{f_{∗}M_{A}} M′_{B}

is by the definition of fibre product uniquely determined by the map M_{B} → M′_{B} and M_{A′} → M′_{A′}. From this the theorem follows.

3.2.6. It follows from 2.1.5 that if I is an A_{0}-module, then the set \overline{LOG_{S,M_{A_{0}}} (A_{0}[I])} has a natural A_{0}-module structure. We denote this module by

Ext_{A_{0}^{♯}}(A_{0}, I)

for reasons we now explain.

Recall the definition of an extension ([17], III.2.1.4):

Definition 3.2.7. Let f : X → S be a morphism of fine log schemes and let I be a quasi-coherent sheaf of O_{X}-modules on \underline{X}. An S-extension of X by I is a strict closed immersion

i : X ↪ T

of fine log schemes defined by a square zero ideal together with an isomorphism between I and ker(i^{#}). A morphism of extensions

(i_{1} : X → T_{1}) → (i_{2} : X → T_{2})

is a morphism g : T_{1} → T_{2} such that g ∘ i_{1} = i_{2} such that g induces the identity map on I. We denote the set of isomorphism classes of S-extensions of X by I by

Ext_{S}(X, I).

3.2.8. With notation as in 3.2.5, let X be (Spec(A_{0}), M_{A_{0}}) and X^{♯} the log scheme

(Spec(A_{0}), f^{∗}M_{S}).

Then

Ext_{X^{♯}}(X, I)

is in natural bijection with

\overline{LOG_{S,M_{A_{0}}} (A_{0}[I])}.

This is immediate from the definitions.

The following theorem records the basic facts about Ext_{S}(X, I) which we will need:

Theorem 3.2.9 ([17], III.2.3.3). 1. The set Ext_{S}(X, I) has a natural structure of a Γ(X, O_{X})-module.

2. If f : X → Y, g : Y → S are morphisms of log schemes, then there is a natural exact sequence of modules

0 → Der_{X/Y} (I) → Der_{X/S}(I) → Der_{Y/S}(f_{∗}I) → Ext_{Y} (X, I).

3. If in addition f is an affine morphism, then the sequence in (2) can be extended by the sequence

Der_{Y/S}(f_{∗}I) ⟶ Ext_{Y} (X, I) ⟶ Ext_{S}(X, I) ⟶ Ext_{S}(Y, f_{∗}I).

Proposition 3.2.10. Let \underline{X} = Spec(A) be an affine scheme and let P be a fine monoid. Let X[P] be the log scheme over \underline{X} given by Spec(P → A[P]). Then for all quasi-coherent I on X[P] there is a natural isomorphism of A-modules

Ext^{1}(P^{gp}, I) ≃ Ext\underline{X}(X[P], I)

where Ext^{1}(P^{gp}, I) denotes isomorphism classes of extensions of abelian groups.

Proof. Given an extension E of abelian groups

0 ⟶ 1 + I ⟶^{j} G ⟶^{π} P^{gp} ⟶ 0

define P′ = P ×_{P^{gp}} G and let R_{E} be the ring

A[I][P′]

modulo the relations

(1 + i)e(p′) = e(j(1 + i) + p′), p′ ∈ P′, i ∈ I

and

ie(p′) = π(p′)i, p′ ∈ P′, i ∈ I

where π(p′)i denote the action of π(p′) on the element i ∈ I. There is a natural commutative diagram

P′ ⟶ R_{E}
↓ ↓
P ⟶ A[P].

It follows from the definition of R_{E} that

I ≃ Ker(R_{E} → A[P])

and that P′ induces a lifting of M_{A[P]} to Spec(R_{E}). Conversely given a lifting M_{R} → R of M_{A[P]} → A[P] we get an extension E by pullback

0 ⟶ 1 + I ⟶ M_{R}^{gp} ⟶ M_{A[P]}^{gp} ⟶ 0
=↑ ↑ ↑
0 ⟶^{j} 1 + I ⟶ G ⟶_{π} P^{gp} ⟶ 0.

Applying the above construction to E gives a monoid P′ and a commutative diagram

P′ ⟶ M_{R}
↓ ↓
P ⟶ M_{A}.

From this it follows that there is a natural isomorphism of liftings

(R_{E}, M_{R_{E}}) ≃ (R, M_{R}).

3.2.11. Suppose X = Spec(A) and that I is a quasi-coherent A-module. Let M_{A[I]} be a lifting of an S-log structure M_{A} to A[I]. From the diagram

X  ⟶^{i} (Spec(A[I]), M_{A[I]})
i↓  ↓
(Spec(A[I]), M_{A[I]}) ⟶ ((Spec(A[I]), g^{∗}M_{S})

we see that the set of infinitesimal automorphisms of M_{A[I]} is a torsor under

Der_{A[I]/A[I]^{♯}}(i_{∗}I)

which is a quasi-coherent sheaf. It follows that the functor on A-algebras

R ↦ Ext_{R^{♯}}(R, I ⊗ R)

is a sheaf for the fppf-topology since

H^{i}(Spec(A), Der_{A[I]/A[I]^{♯}}(i_{∗}I)) = 0, i > 0

because Spec(A) is affine.

Theorem 3.2.12. 1. Ext_{A^{♯}}(A, I) is a quasi-coherent sheaf in the fppf-topology and is coherent if I is.

2. Suppose A is reduced, 𝔪 ⊂ A a maximal ideal, Â the completion of A along 𝔪. Suppose in addition that I is coherent. Then the natural map

Ext_{A^{♯}}(A, I) ⊗ Â → \underleftarrow{lim} Ext_{A^{♯}}(A, I/𝔪^{n}I)

is an isomorphism.

3. If A is reduced and I is coherent then there exists a dense set of points p ∈ Spec(A) for which the natural map

Ext_{A^{♯}}(A, I) ⊗ k(p) → Ext_{A^{♯}}(A, I ⊗ k(p))

is an isomorphism.

Proof. Let X (resp. X^{♯}) be the log scheme with underlying scheme Spec(A) and log structure M_{A} (resp. f^{∗}M_{S}). To prove the theorem it suffices to consider the case when f^{∗}M_{S} = O_{X}^{∗}. For by applying 3.2.9 to the sequence of morphisms

X → X^{♯} → \underline{X}

we get an exact sequence

\begin{aligned}0 & → & Der_{X/X^{♯}}(I) & → & Der_{X/\underline{X}}(I) & → & Der_{X^{♯}/\underline{X}}(I) & → \\ & → & Ext_{X^{♯}}(X, I) & → & Ext\underline{X}(X, I) & → & Ext\underline{X}(X^{♯}, I). \end{aligned}

A diagram chase combined with standard facts about the modules of derivation shows that it is enough to prove the theorem for Ext\underline{X}(X, I) and Ext\underline{X}(X^{♯}, I). Thus we may assume that X^{♯} = \underline{X} and by replacing X by an fppf-cover we can also assume that we have a chart

P → M_{X}.

Applying 3.2.5 to

A[P]
↓
A[I] ⟶ A

and observing that A[P] ×_{A} A[I] ≃ A[P][I] where I is viewed as an A[P]-module via A[P] → A, we see that it is enough to consider the case when X = X[P]. From the diagram

X[P] → \underline{X[P]} → \underline{X}

we get a sequence

\begin{aligned}0 & → & Der_{X[P]/\underline{X[P]}}(I) & → & Der_{X[P]/\underline{X}}(I) & → & Der_{\underline{X[P]}/\underline{X}}(I) & → \\ & → & Ext\underline{X[P]}(X[P], I) & → & Ext\underline{X}(X[P], I) & → & Ext\underline{X}(\underline{X[P]}, I). \end{aligned}

By ([15], III.1.2.3) we have an isomorphism

Ext\underline{X}(\underline{X[P]}, I) ≃ Ext^{1}(L_{\underline{X}[P]/\underline{X}}, I)

where L_{\underline{X}[P]/\underline{X}} denotes the cotangent complex of \underline{X}[P]/\underline{X}, and by 3.2.10 we have

Ext\underline{X}(X[P], I) ≃ Ext^{1}(P^{gp}, I).

It follows from standard facts that the statements of the theorem hold for these modules and hence from a diagram chase it follows that the theorem holds for

Ext\underline{X[P]}(X[P], I).

This concludes the proof of the theorem.

The following related result will also be used in what follows:

Theorem 3.2.13. If X = Spec(A) and I as a quasi-coherent sheaf on \underline{X} then Ext_{S}(X, I) is a quasi-coherent sheaf in the fppf-topology on \underline{X} and is coherent if I is.

Proof. Applying 3.2.9 to the sequence

X → S → \underline{S}

we see that it suffices to consider the case when S = \underline{S} to prove the quasi-coherence. In addition to this assumption we can assume that we have a chart P → M_{X}. Then we get a diagram

X → S[P] → \underline{S}

and hence a sequence

Ext_{S[P]}(X, I) → Ext\underline{S}(X, I) → Ext\underline{S}(S[P], I).

Since the map X → S[P] is strict we can determine the cokernel of

Ext\underline{S}(X, I) → Ext\underline{S}(S[P], I).

Given a an S-extension

S[P] ↪ T

of S[P] the obstruction to lifting T to an S-extension of X is the same as the obstruction to lifting the extension of schemes

\underline{S[P]} ↪ \underline{T}

to \underline{X}. For if we have such a lifting of \underline{T} we get a commutative diagram

\underline{X} ⟶ \underline{T}′
↓ ↓
\underline{S[P]} ⟶ \underline{T}

and if we give \underline{T}′ the pullback log structure from T we get a class in Ext\underline{S}(X, I). By ([15], 1.2.5.3) this obstruction lies in

Ext_{A}^{2}(L_{A/S[P]}, I)

where L_{A/S[P]} is the cotangent complex. We therefore have an exact sequence

Der_{X/S[P]}(I) → Ext_{S[P]}(X, I) → Ext\underline{S}(X, I) → Ext\underline{S}(S[P], I) → Ext_{A}^{2}(L_{A/S[P]}, I)

and from this it follows that Ext\underline{S}(X, I) is quasi-coherent. To prove that Ext_{S}(X, I) is coherent when I has this property consider the sequence

X → X^{♯} → S.

This gives a sequence

Der_{X^{♯}/S}(I) → Ext_{X^{♯}}(X, I) → Ext_{S}(X, I) → Ext_{S}(X^{♯}, I)

and all the terms are known to be coherent except Ext_{S}(X, I).

3.3 A second construction of LOG_{S}

Using the results of the preceding section we now give a different proof of 3.1.1 using Artin’s theorem 2.1.11. 3.3.1 (Second proof of 3.1.1). Throughout this proof we think of LOG_{S} as classifying log structures on the étale site. We define an obstruction theory for LOG_{S} as follows. If A → A_{0} is as in 2.1.2 and if M_{A} is a log structure on A and I is an A_{0}-module of finite type we define O_{a}(I) to be

Coker(Ext_{S}(A, I) → Ext\underline{S}(A, I)).

If A′ → A is a surjection with kernel I then A′ defines a class in Ext\underline{S}(A, I) and we define o_{a}(A′) ∈ O_{a}(I) to be the image of this class. This has the right properties by 3.2.13. In light of the results of the preceding section, the only conditions in 2.1.11 which remain to be verified are 2.1.11.2 and 2.1.11.4. 3.3.2. We claim that the functor in 2.1.11.2 is an equivalence of categories. For later use we prove a slightly more general statement:

Proposition 3.3.3. Let Â be a complete noetherian local ring with residue field of finite type over \underline{S}, and let R be a finite Â-algebra. Then the natural functor

(fine log structures on Spec(R)) → \underleftarrow{lim}(fine log structures on Spec(R ⊗ Â/𝔪^{n}))

is an equivalence of categories (here we do not require that morphisms between log structures are isomorphisms).

Proof. Since Â is complete and R/Â is finite, R is a product of complete local Â-algebras. Because R is local and finite over Â, the filtrations induced by 𝔪Â and 𝔪_{R} are cofinal and therefore we may assume that R = Â.

Let A_{n} = Â/𝔪^{n+1}, j_{n} : Spec(A_{n}) → Spec(Â) the natural map. The key observation is the following: if M is a log structure on Spec(Â), then the natural map

H^{0}(Spec(Â), M) → \underleftarrow{lim} H^{0}(Spec(A_{n}), j_{n}^{∗}M)

is an isomorphism. For this it suffices to show that the map

π : H^{0}(Spec(Â), M^{gp}) → \underleftarrow{lim} H^{0}(Spec(A_{n}), j_{n}^{∗}M^{gp})

is an isomorphism. Since Â is a local ring

H^{1}(Spec(Â), O∗ ; Â) = 0.

It follows that we have a commutative diagram

0  0
↓  ↓
Â^{∗}  ⟶^{π_{1}}  \underleftarrow{lim} A_{n}^{∗}
↓  ↓
H^{0}(Spec(Â), M^{gp}) ⟶^{π} \underleftarrow{lim} H^{0}(Spec(A_{n}), j_{n}^{∗}M^{gp})
↓ ↓
H^{0}(Spec(Â), \overline{M}^{gp}) ⟶^{π_{2}} \underleftarrow{lim} H^{0}(Spec(A_{n}), j_{n}^{−1}\overline{M}^{gp})
↓ ↓
0  0.

Here the exactness of the right column follows from the fact that the maps A_{n+1}^{∗} → A_{n}^{∗} are surjective since A_{n} is artinian. By ([6], VIII.8.6) π_{2} is an isomorphism, and since Â is complete, the map π_{1} is also an isomorphism. We conclude that π is an isomorphism. Note that for each n, the map

H^{0}(Spec(A_{n+1}), j_{n+1}^{∗}M) → H^{0}(Spec(A_{n}), j_{n}^{∗}M)

is surjective. This follows from the exact sequence

0 ⟶ 1 + 𝔪^{n}/𝔪^{n+1} ⟶ j_{n+1}^{∗}M^{gp} ⟶ j_{n}^{∗}M^{gp} ⟶ 0

and the fact that

H^{1}(Spec(A_{n}), 𝔪^{n}/𝔪^{n+1}) = 0

since Spec(A_{n}) is affine. Using the isomorphism π we conclude that the map

H^{0}(Spec(Â), M) → H^{0}(Spec(A_{0}), j_{0}^{∗}M)

is surjective and identifies H^{0}(Spec(A_{0}), j_{0}^{∗}M) with the quotient of H^{0}(Spec(Â), M) by the subgroup consisting of those elements in Â^{∗} which reduce to 1 modulo 𝔪. This has the following implication. Suppose M is a fine log structure such that the map

H^{0}(Spec(A_{0}), j_{0}^{∗}M) → j_{0}^{∗}M

is a chart. Then the map

H^{0}(Spec(Â), M) → M

is also a chart. To see this let N be the log structure associated to H^{0}(Spec(Â), M), K the elements in Â whose reduction modulo 𝔪 is equal to 1. Then we have an isomorphism

H^{0}(Spec(Â), M)/K ≃ H^{0}(Spec(A_{0}), j_{0}^{∗}M).

By the construction of the log structure associated to a chart, it follows that the map

H^{0}(Spec(Â), M) → j_{0}^{∗}M

is a chart for j_{0}^{∗}M. Therefore, if x̄ → Spec(Â) is a geometric point of the closed fiber, the natural map N → M induces an isomorphism

\overline{N}x̄ ≃ \overline{j_{0}^{∗}M}x̄ ≃ \overline{M}x̄.

By 1.1.2 the map

Nx̄ → Mx̄

is also an isomorphism. Now since every point of Spec(Â) specializes to a point of the closed fiber, any map between quasi-coherent log structures which is an isomorphism at every point in the closed fiber must be an isomorphism everywhere. Therefore, N = M. To see that the functor is fully faithful, suppose M_{1} and M_{2} are two fine log structures on Spec(Â) and that we are given a compatible family of isomorphisms

ψ_{n} : j_{n}^{∗}M_{1} → j_{n}^{∗}M_{2}.

We have to show that the ψ_{n} are induced from a unique map

ψ : M_{1} → M_{2}.

By the uniqueness, it suffices to construct ψ étale locally.

Let us show that after passing to a finite étale extension of Â we can assume that the maps

\underleftarrow{lim} H^{0}(Spec(A_{n}), j_{n}^{∗}M_{i}) → M_{i}

are charts. If we show this, then the existence and uniqueness of ψ is clear. To prove that the map

\underleftarrow{lim} H^{0}(Spec(A_{n}), j_{n}^{∗}M_{i}) → M_{i}

is a chart it suffices by the above discussion to show that

H^{0}(Spec(A_{0}), j_{0}^{∗}M_{i}) → j_{0}^{∗}M_{i}

is a chart. If B is any artin local ring, M a fine log structure on B, then after passing to a finite étale extension \overline{M} becomes constant and we have an exact sequence

0 ⟶ B^{∗} ⟶ H^{0}(Spec(B), M) ⟶ H^{0}(Spec(B), \overline{M}) ⟶ 0.

Since B is local and artinian, any element m ∈ H^{0}(Spec(B), M) which maps to a unit in some stalk O_{B,x̄} must be an element of B^{∗}. Therefore, if N denotes the log structure associated to H^{0}(Spec(B), M) we have

\overline{N} ≃ H^{0}(Spec(B), \overline{M})_{Spec(B)} ≃ \overline{M}.

This implies that N → M is an isomorphism, and hence also the full faithfulness. To prove essential surjectivity, let M_{n} be a compatible family of fine log structures on the Spec(A_{n}). We wish to construct M on Spec(Â) inducing the M_{n}. By the full faithfulness it suffices to construct M étale locally on Spec(Â). Replacing Spec(Â) by an étale cover we may therefore assume that the map

H^{0}(Spec(A_{0}), M_{0}) → M_{0}

is a chart. As explained above, it follows that the map

H^{0}(Spec(A_{n}), M_{n}) → M_{n}

is a chart for all n. Thus we take for M the log structure associated to

\underleftarrow{lim} H^{0}(Spec(A_{n}), M_{n}) → \underleftarrow{lim} A_{n} = Â.

This is a quasi-coherent log structure which induces the family M_{n}. It remains to see that it is coherent. We claim that any quasi-coherent log structure M on Spec(Â) for which j_{0}^{∗}M is coherent is actually coherent. To see this, let x̄ → Spec(Â) be a geometric point. Then

\overline{M}x̄ ≃ \overline{j_{0}^{∗}M}x̄

is a finitely generated monoid. Therefore, there exists in some étale neighborhood of x̄ a map

P → M

from a fine monoid P which induces a chart at x̄. After choosing a perhaps even smaller neighborhood of x̄ we can assume that P induces a chart for M_{0} on all of Spec(A_{0}). It follows that if N is the log structure associated to P, then the map

N → M

is an isomorphism at every point in the closed fiber. Therefore it is an isomorphism everywhere since they are both quasi-coherent. Hence M is fine.

2.1.11.4 follows from the following lemma which we state in greater generality for later use:

Lemma 3.3.4. Let A_{0} be a reduced ring, f : \underline{X} → Spec(A_{0}) a proper flat morphism of reduced schemes, M_{X} a log structure on \underline{X}, ϕ an automorphism of M_{X} which induces the identity over Spec(k) for a dense set of points A_{0} → k. Then there exists a dense open set \underline{U} ⊂ Spec(A_{0}) such that ϕ restricted to \underline{X}_{U}/\underline{U} is the identity.

Proof. Let ϕ̄ : \overline{M}_{X}^{gp} → \overline{M}_{X}^{gp} be the induced map. \overline{M}_{X}^{gp} is a sheaf of finitely generated abelian groups which locally is a quotient of a constant sheaf, and hence the set of points where ϕ ≠ id is a closed set \underline{Z} ⊂ \underline{X}. Since f is proper, f(\underline{Z}) ⊂ Spec(A_{0}) is a closed set, and by assumption f(\underline{Z})^{c} is dense. Thus we may replace Spec(A_{0}) by f(Z)^{c} and assume that ϕ̄ is the identity. From the exact sequence

0 → O_{X}^{∗} → M_{X}^{gp} → \overline{M}_{X}^{gp} → 0

we deduce that the automorphism ϕ must be of the form m ↦ λ(η(m)) + m where

η : \overline{M}_{X}^{gp} → O_{X}^{∗}

is a group homomorphism. Consider the set of points x ∈ \underline{X} such that η_{x} : \overline{M}_{X}^{gp} → k(x̄) is the trivial map. This is a closed set since \overline{M}_{X}^{gp} is locally a quotient of a finitely generated constant sheaf. Hence the complement is an open set \underline{V} ⊂ \underline{X}. Since f is flat the image of \underline{V} in Spec(A_{0}) is open and its complement is dense by assumption. Hence \underline{V} is the empty set, and η_{x} is the trivial map for all x. Since x is reduced it follows that η is trivial.

This concludes the proof of the theorem.

3.4 Basic properties of LOG_{S} and variants

Proposition 3.4.1. 1. If f : X → S is a morphism of log schemes then the morphism

of stacks

LOG_{X} → LOG_{S}

induced from the map f^{∗}M_{S} → M_{X} is relatively representable.

2. If g : Y → S is a second morphism of log schemes then there is a canonical isomorphism of algebraic stacks

LOG_{X} ×_{LOG_{S}} LOG_{Y} ≃ LOG_{X×_{S}Y}

where X ×_{S} Y denotes the fiber product in the category of integral log schemes.

Proof. (1) follows from the statement that the functor LOG_{X} → LOG_{S} is faithful. For this implies that for any \underline{T} → LOG_{S} the product

LOG_{X} ×_{LOG_{S}} \underline{T}

is a functor and hence an algebraic space.

(2) follows from the universal property of fiber product.

Remark 3.4.2. The functor LOG_{X} → LOG_{S} is locally of finite presentation if and only if \underline{X}/\underline{S} has this property. This can be checked on a flat cover \underline{T}′ → LOG_{X}. For any T → LOG_{S} the functor \underline{T}′ ×_{LOG_{S}} \underline{T} is the functor I over \underline{X} ×\underline{S} \underline{T} corresponding to the diagram

M_{S}|_{\underline{X}×\underline{S} \underline{T}} ⟶ p_{1}^{∗}M_{X}
↓
p_{2}^{∗}M_{T}

as in 3.1.2. From the construction of I we see that I is locally of finite presentation over \underline{X} ×_{S} \underline{T} and hence LOG_{X} → LOG_{S} is locally of finite presentation if and only if \underline{X}/\underline{S} is.

We propose the following definition for translating classical definitions into the logarithmic language:

Definition 3.4.3. Let P be a property of morphisms f : \underline{X} → \underline{Y} between algebraic spaces over \underline{S} which is stable under base change \underline{Y}′ → \underline{Y} and is local for the fppf-topology on \underline{Y}. Then if f : X → S is a morphism of log schemes we say that f has property P if the induced relatively representable morphism of stacks

LOG_{X} → LOG_{S}

has property P.

Proposition 3.4.4. Let f : X → S be a morphism of log schemes.

1. f is formally log smooth (resp. formally log étale) if and only if the map

LOG_{X} → LOG_{S}

is a formally smooth (resp. formally étale) morphism of stacks. This is also equivalent to the morphism

\underline{X} → LOG_{S}

being formally smooth (resp. formally étale).

2. There is a natural isomorphism

Ω_{X/S}^{1} ≃ Ω_{\underline{X}/LOG_{S}}^{1}

where Ω_{X/S}^{1} denotes the logarithmic differentials.

3. The morphism LOG_{X} → LOG_{S} is flat if and only if locally in the fppf topology on

\underline{X} and \underline{S} there exists a chart

P ⟶ M_{X}
θ↑ ↑
Q ⟶ M_{S}

with θ injective and the map

\underline{X} → \underline{S} ×_{Spec(ℤ[Q])} Spec(ℤ[P])

flat.

Proof. That f : X → S is formally log smooth (resp. formally log étale) if and only if LOG_{X} → LOG_{S} has the corresponding property is immediate from the definitions and the infinitesimal lifting property. The statement that X/S is formally log smooth (resp. formally log étale) if and only if \underline{X} → LOG_{S} is formally smooth (resp. formally étale) follows from the fact that when verifying the infinitesimal lifting property for a morphism of log schemes it suffices to consider diagrams

\begin{matrix}Z_{0} & ⟶^{z_{0}} & X \\ ↓ & & ↓ \\ Z & ⟶ & S \end{matrix}

where z_{0} is strict. For given an arbitrary such diagram let Z_{0}^{♯} = (\underline{Z}_{0}, z_{0}^{∗}M_{X}) and let Z^{♯} be the log scheme \underline{Z} together with the sheaf of monoids

M_{Z} ×_{M_{Z_{0}}} z_{0}^{∗}M_{X}.

Then a lifting of z_{0} is equivalent to a lifting of z_{0}^{♯} filling in the diagram

Z_{0}^{♯} ⟶^{z_{0}^{♯}} X
↓ ↓
Z^{♯} ⟶ S.

The second statement in the proposition follows from the universal property of differentials. To prove statement (3) we may replace \underline{S} by an fppf-cover and hence can assume that we have a chart Q → M_{S}. Let S̃ → LOG_{S} be the flat cover given by 3.1.12. Suppose LOG_{X} → LOG_{S} if flat. By (1) the canonical map \underline{X} → LOG_{X} is étale and hence we obtain a diagram

\underline{X} ⟵^{g} S̃ ×_{LOG_{S}} \underline{X}
h↓
S̃

where g is an fppf cover and h is flat. By definition of S̃ this means that locally in the fppf-topology on \underline{X} we have a cover as in (3). Coversely suppose we locally have charts as in (3). The statement that LOG_{X} → LOG_{S} is flat is fppf local and hence we may assume that in fact we have a global such chart. For any injective morphism of fine monoids P → P′ let X[P′] be as in 3.2.10. The X[P′] form a flat cover of LOG_{X} and hence it suffices to verify that each of the X[P′] are flat over LOG_{S}. But we have a commutative diagram

LOG_{X} ⟵ X[P′]
↓ ↓
LOG_{S} ⟵ S[P′]

where S[P′] is the scheme obtained from Q → P → P′. The map S[P′] → LOG_{S} is flat and so is the map

X[P′] = X ×_{Spec(ℤ[P])} Spec(ℤ[P′]) → S[P′] = S ×_{Spec(ℤ[Q])} Spec(ℤ[P]) ×_{Spec(ℤ[P])} Spec(ℤ[P′])

and consequently their composite is also flat.

Corollary 3.4.5. If f : X → S is a morphism of log schemes, then the set of points x ∈ \underline{X} where f is log smooth (resp. log étale) is open.

Proof. The set in question is the same as the set of points where the morphism

\underline{X} → LOG_{S}

is smooth.

Next we derive some standard results on log smoothness using the stacks LOG_{S}.

Proposition 3.4.6. Suppose a chart Q → M_{S} is given and that Q → P is an injective morphism between fine monoids. Then the induced map

\underline{S}[P] → LOG_{S}

is formally smooth if and only if #(P^{gp}/Q^{gp})_{tor} is invertible on \underline{S}.

Proof. The infinitesimal lifting property in this case amounts to the following: Let A be an artinian local S-ring and let M_{A} be an S-log structure on A with a chart

P ⟶^{η} M_{A}
↑ ↑
Q ⟶ M_{S}.

Then for any surjection A′ → A defined by a square zero ideal I and a lifting of M_{A} to a log structure M_{A′} on A′ does there exist a lifting of the chart to M_{A′}?

Let us prove the ”if” direction. Write Ĩ for the quasi-coherent sheaf on the étale site of Spec(A) associate to I. We have a diagram

Q ⟶ M_{A′} ⟶ O_{A′}
↓ ↓ ↓
P ⟶^{η} M_{A} ⟶ O_{A}

and we wish to lift the map η to M_{A′}. In order to fit the diagram on the page, write Hom(M, N) = [M, N] (homomorphisms of sheaves). We have a commutative diagram

[P^{gp}/Q^{gp}, Ĩ] ⟶ [P^{gp}/Q^{gp}, M_{A′}] ⟶ [P^{gp}/Q^{gp}, M_{A}^{gp}] ⟶ Ext^{1}(P^{gp}/Q^{gp}, Ĩ)
↓ ↓ ↓ ↓
[P^{gp}, Ĩ]  ⟶ [P^{gp}, M_{A′}^{gp}] ⟶ [P^{gp}, M_{A}^{gp}] ⟶ Ext^{1}(P^{gp}, Ĩ)
↓ ↓ ↓ ↓
[Q^{gp}, Ĩ]  ⟶ [Q^{gp}, M_{A′}^{gp}] ⟶ [Q^{gp}, M_{A}^{gp}] ⟶ Ext^{1}(Q^{gp}, Ĩ)
↓
Ext^{1}(P^{gp}/Q^{gp}, Ĩ)  .

By our assumptions, Ext^{1}(P^{gp}/Q^{gp}, Ĩ) = 0. Thus since η ∈ [P^{gp}, M_{A}^{gp}] maps to zero in Ext^{1}(Q^{gp}, Ĩ) it must map to zero in Ext^{1}(P^{gp}, Ĩ). Thus there exists an element η̃ ∈ [Q^{gp}, M_{A′}^{gp}] mapping to η. Since

[P^{gp}, Ĩ] → [Q^{gp}, Ĩ]

is surjective we can choose η̃ so as to agree with the given map Q → M_{A′}. Thus locally there exists a lifting of P → M^{gp} to M_{A′}^{gp}. But from the exact sequence,

0 → 1 + I → M_{A′} → M_{A} → 0

we see that in fact that image of this lifting of P lies in M_{A′}. This gives a lifting of the chart étale locally. From the diagram, it follows that the set of liftings of η is a torsor under

Hom(P^{gp}/Q^{gp}, Ĩ).

Since this is a quasi-coherent sheaf, and Spec(A) is affine the obstruction to finding a global lifting vanishes. To prove the other direction, it suffices to show that we can find a point Spec(k) → S[P] such that

I_{k} = Spec(k) ×_{LOG_{S}} S[P] → Spec(k)

is not smooth. The point we will use is any point Spec(k) such that P̄ → \overline{M}_{k} is an isomorphism and for which the characteristic divides the order of (P^{gp}/Q^{gp})_{tor}. In addition by replacing k by a field extension we can choose an isomorphism

M_{k} ≃ k^{∗} ⊕ P_{2}

where P_{2} is a fine sharp monoid. Now in this case we can write down a scheme which represents I_{k}. Namely, for each isomorphism σ : \overline{P} → P_{2} let

T_{σ} = Spec(k[P^{gp}/Q^{gp}]).

Then define a map

T_{σ} → I_{k}

by sending an R-valued point ϕ ∈ T_{σ}(R) corresponding to a map

ϕ : P^{gp}/Q^{gp} → R^{∗}

to the map

P → R^{∗} ⊕ P_{2}, p ↦ (ϕ(p), σ(p̄)).

It is straightforward to verify that this defines an isomorphism

∐_{σ} T_{σ} → I

and certainly T_{σ} is not smooth over k.

3.4.7. Fix a chart Q → M_{S}. Let LOG_{S}^{gt} ⊂ LOG_{S} be the full substack of S-log structures for which LOG_{S}^{gt}(\underline{X}) consists of those log structures for which M_{X/S}^{gp,tor} ⊗ k(x̄) = 0 for all x̄ → \underline{X}. This is an open condition so LOG_{S}^{gt} is an algebraic stack. In addition, we can interpret the result of ([29], 2.13) as saying that the union of the maps

S[P]

with Q → P satisfying the conditions of the proposition form a smooth cover of LOG_{S}^{gt}. As a consequence we get the following theorem originally due to K. Kato:

Theorem 3.4.8. Let f : X → S be a smooth morphism of fine log schemes. Then

1. The morphism \underline{X} → LOG_{S} factors through LOG_{S}^{gt}.

2. Locally in the smooth topology on \underline{X} and \underline{S} there exists a chart

P ⟶ M_{X}
θ↑ ↑
Q ⟶ M_{S}

with θ injective and #(P^{gp}/Q^{gp})^{tor} invertible on \underline{X} such that the induced map

\underline{X} → \underline{S} ×_{Spec(ℤ[Q])} Spec(ℤ[P])

is smooth.

Proof. It suffices to prove the first statement for then the second statement follows from the fact that the maps S[P] → LOG_{S}^{gt} of the preceding paragraph are smooth and cover LOG_{S}^{gt}. Thus what has to be shown is that if \underline{X} → LOG_{S} is smooth then M_{X/S,x̄}^{gp,tor} ⊗ k(x̄) = 0 for all x̄ → \underline{X}. For this we may as well assume that \underline{X} = Spec(R) where R is strictly henselian and local. Let Spec(k) → \underline{X} be a point mapping to the closed point with k algebraically closed and let M = Γ(\underline{X}, M_{X}), N = Γ(\underline{X}, f^{∗}M_{S}). Let J ⊂ R be the ideal generated by the images of the elements m ∈ M − M^{∗}. It follows from the definitions that

I = \underline{X} ×_{LOG_{S}} Spec(k)

is isomorphic to a Hom(M^{gp}/N^{gp}, 𝔾_{m})-torsor over R/J ⊗_{O_{S}} k. In particular if p = char(k) and there exists an element of order p in M^{gp}/N^{gp} then I is not reduced and hence cannot be smooth over k.

3.4.9. One can consider variants on the stack LOG_{S}. We denote by LOG_{S}^{sat} the full substack of LOG_{S} consisting of S-log structures which are saturated. The morphism

LOG_{S}^{sat} → LOG_{S}

is an open immersion of algebraic stacks, because the condition of a log structure being saturated is an open condition.

Corollary 3.4.10. Let S be a fine log scheme and let S^{sat} be the associated saturated log scheme. Then

1. There is a canonical equivalence of stacks

LOG_{S}^{sat} ≃ LOG_{S^{sat}}^{sat}

compatible with the maps to LOG_{S}.

2. If f : X → S is a morphism with X saturated, then there is a canonical isomorphism

Ω_{X/S^{sat}}^{1} ≃ Ω_{X/S}^{1}.

Proof. Statement (1) follows from the universal property of S^{sat}. The second statement follows from the fact that

LOG_{sat}^{sat} → LOG_{S}

is an open immersion.

3.4.11. A second variant of LOG_{S} which is often used is the stack of idealized log structures LOG_{S}^{id}. Recall that an idealized log scheme consists of a fine log scheme X and a coherent sheaf of ideals K ⊂ α^{−1}(0) ⊂ M_{X}. Let S be an idealized log scheme, and let LOG_{S}^{id} be the fibered category over the category of \underline{S}-schemes for which LOG_{S}^{id}(f : \underline{X} → \underline{S}) is the category of idealized log structures (M_{X}, K_{X}) together with a morphism

f^{∗}(M_{S}, K_{S}) → (M_{X}, K_{X}).

Since LOG_{S}^{id} classifies sheaves on the étale site LOG_{S}^{id} is a stack with respect to the étale topology. Given a diagram

(\underline{X}) / ↓

\underline{Y} ⟶ LOG_{S}^{id}

there is a natural injective morphism of functors

j : \underline{X} ×_{LOG_{S}^{id}} \underline{Y} → \underline{X} ×_{LOG_{S}} \underline{Y}

where the map

LOG_{S}^{id} → LOG_{S}

is obtained by forgetting the ideal. Now if X is any log scheme and K_{X}, K′_{X} are two coherent sheaves of ideals in M_{X} (not necessarily in α^{−1}(0)) then the condition that K_{X} = K′_{X} is an open condition by the coherence assumption. It follows from this that

\underline{X} ×_{LOG_{S}^{id}} \underline{Y}

is representable by an open subset of \underline{X} ×_{LOG_{S}} \underline{Y}.

Suppose P is a fine monoid and that K ⊂ P is a finitely generated ideal. Then we can form the idealized log scheme

D(P, K) = Spec(P → ℤ[P]/ℤ[K])

(where ℤ[K] denotes the ideal generated by K). If (P′, K′) is a second such pair then the projection

D(P, K) ×_{LOG_{S}^{id}} D(P′, K′) → D(P, K)

is flat. For this product is representable by an open subset of

Spec(P → ℤ[P]/ℤ[K]) ×_{LOG_{S}} Spec(P′ → ℤ[P′])

and so the result follows from 3.1.11. It follows that if

S̃ = ∐_{(P,K)} D(P, K)

then

S̃ ×_{LOG_{S}^{id}} S̃ ⇉ S̃

is a flat groupoid and LOG_{S}^{id} is represented by the associated algebraic stack.

3.5 A log version of Artin’s theorem

3.5.1. In this section we apply the results of 2.2 to the case when B = LOG_{S}. Fix a fine log schemes S with \underline{S} locally of finite type over an excellent Dedekind ring. Suppose F → F Log_{S} is a limit preserving stack over F Log_{S} in the sense of 1.3.12. We wish to give conditions for F to be a log algebraic stack. 3.5.2. If F^{♯} is a stack over the category of \underline{S}-schemes with a morphism of stacks m : F^{♯} → LOG_{S} (m is equivalent to a log structure on F^{♯} if F^{♯} is algebraic) then F^{♯} defines in a natural way a stack F over F Log_{S}. Namely if X is an S-log schemes, the category F_{X} has as objects pairs (f, f^{b}) where f : \underline{X} → F is a 1-morphism and f^{b} is a morphism of S-log structures

f^{b} : m ∘ f → M_{X}.

A morphism (f′, f^{′b}) → (f, f^{b}) is a morphism h : f′ → f in F ♯ ; \underline{X} such that f^{′b} = f^{b} ∘ m(h). If F is a log algebraic stack, then there is a tautological log structure M_{F} on \underline{F}, and we get a map

m : \underline{F} → LOG_{S}.

This is by definition if we define log algebraic stack using 1.3.1 and by theorem 1.3.8 if we use definition 1.3.12, and the category F/F Log_{S} is just the stack associated to (\underline{F} , m). 3.5.3. This suggests that one may prove representability by a log algebraic stack for F by proceeding in two steps. First show that F is equivalent to the stack over F Log_{S} defined by a stack in the usual sense F^{♯} with a morphism of stacks

m : F^{♯} → LOG_{S}.

Then apply the results of the preceding section to conclude that F^{♯} is algebraic. In section 4.3 we prove a theorem useful for showing that F comes from a pair (F^{♯}, m) as above. This theorem suffices for our applications, so for the remainder of this section let us assume that we are given such a pair (F^{♯}, m) and work out the conditions for F^{♯} to be algebraic.

Definition 3.5.4. Suppose F → F Log_{S} is a stack which is obtained from a pair (F^{♯}, m). If X is a log scheme and a ∈ F_{X} we call a solid if the corresponding pair

(f : \underline{X} → F^{♯}, f^{b} : m ∘ f → M_{X})

has the property that f^{b} is an isomorphism.

3.5.5. The solid objects of F form a category with morphisms being isomorphisms f′ ≃ f (not necessarily compatible with f^{b}). It is immediate that the category of solid objects is naturally equivalent to F^{♯}. An important property of solid objects is that if X_{0} ↪ X is a strict closed immersion of log schemes defined by a nilpotent ideal and if a ∈ F_{X} is an object then a is solid if and only if the pullback of a to F_{X_{0}} is solid. This follows from 1.1.2. 3.5.6. Define the category of noetherian log rings C as follows. C^{op} is the subcategory of F Log of log schemes whose underlying schemes are isomorphic to the spectrum of a noetherian ring. The morphisms in C^{op} are strict morphisms of log schemes. Note that by 3.2.5, co-products exist in C^{op} and hence products exist in C. We denote the objects of C by caligraphic letters and the underlying rings by unadorned letters (i.e. an object A ∈ C has underlying scheme Spec(A)). If A ∈ C, and if I is an A-module of finite type, then we denote by A[I] the object of C with underlying ring A[I] and log structure obtained from A by pullback via the retraction A → A[I]. There is a canonical morphism A[I] → A in C. By construction there is a contravariant functor

Spec : C → F Log.

Definition 3.5.7. A logarithmic deformation situation is a diagram in C B

↓
A′ ⟶ A ⟶ A_{0}

such that

1. A_{0} is reduced.

2. A′ → A and A → A_{0} are surjections with nilpotent kernel.

3. The composite B → A_{0} is surjective.

4. The kernel I = Ker(A′ → A) is an A_{0}-module of finite type.

Condition 2.2.8 becomes: 3.5.8 (S1^{log}). Suppose given a deformation situation and a ∈ F_{A}. Then the canonical functor

F_{a,A′×_{A}B} → F_{a,A′} × F_{a,B}

is an equivalence of categories. 3.5.9. Suppose a logarithmic deformation situation is given, together with an object a ∈ F_{A}. If the condition (S1^{log}) hold, then for any finite A_{0}-module I, the set

\overline{F}_{a_{0},A_{0}[I]}

has a natural A_{0}-module structure as before. Similarly, if Aut_{a_{0}}(A_{0}[I]) denotes the set of automorphisms of a_{0} in F_{A_{0}[I]} which induce the identity over A_{0} then Aut_{a_{0}}(A_{0}[I]) has a natural A_{0}-module structure. Condition 2.2.10 becomes: 3.5.10 (S2^{log}). The modules \overline{F}_{a_{0},A_{0}[I]} and Aut_{a_{0}}(A_{0}[I]) are A_{0}-modules of finite type. 3.5.11 (Obstruction Theory). By an obstruction theory for F we will mean data as follows:

A. For every morphism A → A_{0} in C, with A → A_{0} a surjection with nilpotent kernel and A_{0} reduced, and a ∈ F_{A} a solid object a functor

O_{a} : (A_{0}-modules of finite type) → (A_{0}-modules of finite type).

B. For each diagram A′ → A → A_{0} as in 3.5.7, an element o_{a}(A′) ∈ O_{a}(I) which is zero if and only if \overline{F}_{a,A′} is not empty.

Let A → A_{0} be a morphism in C such that Ker(A → A_{0}) is nilpotent and A_{0} is reduced. Suppose A′ → A is a surjection with kernel an A_{0}-module of finite type I and that a ∈ F_{A}. Then for each lifting of M_{A} to A′, we get a diagram as in (B) and hence via o_{a} an element of O_{a}(I). In other words, o_{a} defines a set map

(liftings of M_{A} to A′) → O_{a}(I).

In particular, there is a natural map

ρ_{I} : Ext_{A^{♯}}(A, I) → O_{a}(I).

We make the following addition requirement of the obstruction theory:

C. ρ_{I} is an A_{0}-module homomorphism.

3.5.12 (Additional Conditions). If F is to be algebraic, the following conditions must also hold. Let a ∈ F_{A} be a solid object with A of finite type over \underline{S}, and let the subscript A_{0}, B_{0}, etc. denote the objects obtained by tensor product with A_{red}.

A. Compatibility with étale localization: If ϕ : A → B is a morphism in C whose underlying morphism of rings is étale, and if b ∈ F_{B} is the image of a ∈ F_{A}, then

\overline{F}_{b_{0},B_{0}[I⊗B_{0}]} ≃ \overline{F}_{a_{0},A_{0}[I]} ⊗ B_{0},
Aut_{b_{0}}(B_{0}[I ⊗ B_{0}]) ≃ Aut_{a_{0}}(A_{0}[I]) ⊗ B_{0},

and

O_{b}(I ⊗ B_{0}) ≃ O_{a}(I) ⊗ B_{0}.

B. Effectivity of formal objects: If 𝔪 ⊂ A_{0} is a maximal ideal and Â the completion of

A_{0} along 𝔪, then

\overline{F}_{a_{0},A_{0}[I]} ⊗ Â ≃ \underleftarrow{lim} \overline{F}_{a_{0},A_{0}[I/𝔪^{n}I]},
Aut_{a_{0}}(A_{0}[I]) ⊗ Â ≃ \underleftarrow{lim} Aut_{a_{0}}(A_{0}[I/𝔪^{n}I]),

and

O_{a_{0}}(I) ⊗ Â ↪ \underleftarrow{lim} O_{a_{0}}(I/𝔪^{n}I).

C. Contructibility: There exists an open dense set of points p ∈ Spec(A_{0}) with k(p) of finite type over \underline{S} such that

\overline{F}_{a_{0},A_{0}[I]} ⊗ k(p) ≃ \overline{F}_{a_{0},A_{0}[I⊗k(p)]},

Aut_{a_{0}}(A_{0}[I]) ⊗ k(p) ≃ Aut_{a_{0}}(A_{0}[I ⊗ k(p)]),

and

O_{a}(I) ⊗ k(p) ⊂ O_{a}(I ⊗ k(p)).

Rewriting theorem 2.2.13 we obtain the following:

Theorem 3.5.13. Let F → F Log_{S} be a limit preserving stack for the strict étale topology, and suppose F is equivalent to the stack defined by a pair (F^{♯}, m) as in 3.5.2. Suppose further that an obstruction theory O as in 3.5.11 is given and satisfies the conditions in 3.5.12. Then F is a log algebraic stack if the following conditions hold:

1. The conditions (S1^{log}), and (S2^{log}) hold and the associated modules satisfy the conditions of 3.5.12.

2. For every log ring Â such that Â is complete and local with residue field of finite type

over \underline{S}, the natural functor

FÂ → \underleftarrow{lim} F_{Â/𝔪^{n}}

is faithful and has dense image.

3. Suppose given a reduced object A_{0} of C of finite type over \underline{S}, a solid object a_{0} ∈ F_{A_{0}}, and an automorphism ϕ of a_{0}. If ϕ induces the identity in F_{k} (k viewed as a log ring with the induced log structure) for a dense set of points A_{0} → k with k of finite type over \underline{S}, then ϕ is the identity over some dense open subset of Spec(A_{0}).

Remark 3.5.14. As in 2.2.14 it suffices to construct local obstruction theories O_{a}. For example, one can always assume that a chart is given for the log structure on A_{0} and the obstruction theory may depend on a choice of chart.

Chapter 4

Technicalities on log structures

In this chapter we gather together various facts about log structures which will be used in the chapters that follow.

4.1 A theorem about integral morphisms

Theorem 4.1.1. Let Y = (Spec(A), M_{Y}) where A is a noetherian local ring, and let Y_{0} be the reduction modulo the maximal ideal. Let f : X → Y be an integral morphism locally of finite type such that the map on schemes \underline{X} → \underline{Y} is proper and flat. Then if the reduction f_{0} : X_{0} → Y_{0} is smooth, f is also smooth.

Proof. Let U ⊂ LOG_{Y} be the maximal open substack flat over \underline{Y}. U can be described as follows. If \underline{Ỹ} → LOG_{Y} is a flat cover, let \underline{Ũ} ⊂ \underline{Ỹ} be the maximal open set flat over \underline{Y}. Then an object of U over a scheme \underline{T} is an object of LOG_{Y} over \underline{T} such that the induced map

\underline{T} ×_{LOG_{Y}} \underline{Ỹ} → \underline{Ỹ}

factors through \underline{Ũ}. The fact that f is integral implies that the induced map

\underline{X} → LOG_{Y}

factors through U. This can be checked locally in the flat topology on \underline{X} and \underline{Y}. We claim that locally in the flat topology on \underline{X} and \underline{Y} we can find a chart

P ⟶ M_{X}
↑ ↑
Q ⟶ M_{Y}

such that ℤ[Q] → ℤ[P] is flat and such that Q → P is injective. In fact if y ∈ \underline{Y} is a point then there exists a flat neighborhood \underline{Y}′ → \underline{Y} and a point y′ ∈ \underline{Y} lying over y and a chart Q → M_{Y} such that the induced map Q → \overline{M}_{Y,ȳ′} is bijective. Replacing \underline{X} by \underline{X} ×\underline{Y} \underline{Y}′ we may therefore assume that we have such a chart. Then by ([29], 2.13 and 2.14) there exists fppf-locally on \underline{X} a point x ∈ X and a chart P → M_{X} such that the map P → \overline{M}_{X,x̄} is bijective. This chart has the required properties by the integrality of f. Since

Y ×_{Spec(ℤ[Q])} Spec(ℤ[P])

is flat over Y and over LOG_{Y} by theorem 3.1.11 the map

Y ×_{Spec(ℤ[Q])} Spec(ℤ[P]) → LOG_{Y}

and hence \underline{X} → LOG_{Y} factors through U. Now to prove that X/Y is smooth it suffices to show that the map

\underline{X} → U

is smooth. Since both \underline{X} and U are flat over Y to check smoothness at points in the closed fiber it suffices to check it after reduction. Therefore f is smooth at all points in the closed fiber. On the other hand, the set of points where f is not smooth is closed and hence has closed image in Y. But Y has no closed sets which do not contain the closed point and hence the set must be empty.

4.2 Log versions of the Grothendieck existence theorem.

In applications of Artin’s method, the Grothendieck Existence Theorem ([10], III.5) usually plays an important role in verifying condition 2.1.11.2. In this section we prove two theorems which may be viewed as versions of the Grothendieck existence theorem for log structures. Recall ([20], Appendix) K. Kato’s definition of a DF log structure (after Deligne and Faltings):

Definition 4.2.1. A DF log structure on a scheme \underline{X} is a pair (M_{X}, t) where M_{X} is a fine log structure on \underline{X}, and t : ℕ^{r} → \overline{M}_{X} is a homomorphism which lifts locally to a chart for M_{X}.

4.2.2. We make the collection of DF log structures into a category DF_{X} by defining a morphism (M_{X}, t) → (M′_{X}, t′) to be a pair (ϕ, ψ) where ϕ : M_{X} → M′_{X} is a morphism of log structures and ψ : ℕ^{r} → ℕ^{r′} is a morphism of monoids such that the diagram

ℕ^{r′} ⟶^{t′} \overline{M′}_{X}
ψ↑ ↑^{ϕ}
ℕ^{r} ⟶^{t} \overline{M}_{X}

commutes. 4.2.3. Define a second category DF′_{X} as follows. Objects are collection (L_{i}, s_{i})_{i=1,...,r} where the L_{i} are invertible sheaves on \underline{X}, and the s_{i} : L_{i} → O_{X} are O_{X}-module homomorphisms. A morphism (L_{i}, s_{i})_{i=1,...,r} → (L′_{j}, s′_{j})_{j=1,...,l} is a collection of pairs (ϕ_{i}, n_{i})_{i=1,...r} where n_{i} ∈ ℕ^{l} and ϕ_{i} is an isomorphism of O_{X}-modules

ϕ_{i} : L_{i} → ⊗(L′_{j})^{n_{ij}}.

Composition of morphisms is defined in the natural way. Note that since the definition of DF′_{X} involves only invertible sheaves and isomorphisms between them, it makes sense to think of the L_{i} as O_{X}^{∗}-torsors instead of invertible sheaves. 4.2.4. There is a functor

G : DF_{X} → DF′_{X}.

Given (M_{X}, t) ∈ DF_{X}, define L_{i} to be the O_{X}^{∗}-torsor of liftings of t(e_{i}) to M_{X}, and let s_{i} be the map coming from α_{X} : M_{X} → O_{X}. Given a morphism ξ = (ϕ, ψ) : (M, t) → (M′, t′) in DF_{X}, define n_{i} = ψ(e_{i}). Then ϕ defines a map from liftings of t(e_{i}) to liftings of t′(ψ(e_{i})) compatible with the maps to O_{X}. That is, we get a map

L_{i} → ⊗_{j}(L′_{j})^{ψ(e_{i})_{j}}.

Proposition 4.2.5. G is an equivalence of categories.

Proof. A quasi-inverse F to G can be constructed as follows. Given (L_{i}, s_{i})_{i=1,...,r} let F ((L_{i}, s_{i})_{i=1,...,r}) be the log structure associated to the pre-log structure

M = {(n, a) : n ∈ ℕ^{r}, a a local generator for ⊗_{i} L_{i}^{n_{ij}}}

together with the natural map ℕ^{r} → M/O_{X}^{∗}. Given a morphism

(ϕ_{i}, n_{i})_{i=1,...r} : (L_{i}, s_{i})_{i=1,...,r} → (L′_{j}, s′_{j})_{j=1,...,l}

in A_{X}, we define ψ by e_{i} ↦ n_{i} and ϕ by

(n, a) ↦ (ψ(n), s_{i}(a)).

It is straightforward to check that F and G are quasi-inverses.

Corollary 4.2.6. Let A be a complete local ring with maximal ideal 𝔪 and let \underline{X}/Spec(A) be a proper scheme of finite type. Denote by X_{n} = X ×_{A} (A/𝔪^{n}). Then the natural functor

DF_{X} → \underleftarrow{lim} DF_{X_{n}}

is an equivalence of categories.

Proof. This follows from 4.2.5 and the usual version of the Grothendieck Existence theorem ([10], III.5).

From this we deduce the following theorem which is originally due to K. Kato (though we have not seen his proof):

Theorem 4.2.7. With notation as in the previous corollary, let C_{n} be the category of fine saturated log structures on \underline{X}_{n} for which the natural map of sheaves

Γ(\underline{X}_{n}, \overline{M}) → \overline{M}

is surjective. Let C_{∞} be the category of such log structures on \underline{X}. Then the natural functor

C_{∞} → \underleftarrow{lim} C_{n}

is an equivalence of categories.

Proof. Essential surjectivity. Say M_{n} is a family of log structures on the \underline{X}_{n}. Let

P = Γ(\underline{X}_{1}, \overline{M}_{1}) = Γ(\underline{X}_{n}, \overline{M}_{n}).

Choose an étale cover \{U_{i}\} over \underline{X}_{1} such that each U_{i} is affine and assume that we have chosen a section c_{i} of

M_{1} → \overline{M}_{1}

over each U_{i}. Since M_{1} is fine and saturated this is possible. Moreover, by the argument used in the proof of 3.4.6 we can choose a compatible collection of sections c_{i,n} over U_{i} of the projections

M_{n} → \overline{M}_{n}

(here we view \{U_{i}\} as an étale cover of \underline{X}_{n}). Given a map of monoids,

ℕ^{r} → P

for some r we now construct a DF log structure N together with a map of log structures

N → M_{n}

on \underline{X}_{n}. We first construct a pre-log structure N. On U_{i} define N_{i} to be

O_{X_{n}}^{∗} ⊕ ℕ^{r}.

N_{i} comes with a map β_{i} : N_{i} → M_{n} coming from the composite

ℕ^{r} ⟶ P ⟶ \overline{M}_{n} ⟶^{c_{i,n}} M_{n}.

On U_{i} ×_{X} U_{j}, there exists a unique element u_{ji}(e) ∈ O_{X}^{∗} for each e ∈ ℕ^{r} such that

β_{i}(e) = λ(u_{ji}(e)) + β_{j}(e)

in M_{n}. Define an isomorphism

ϕ_{ji} : N_{i} → N_{j}, (f, e) ↦ (fu_{ji}(e), e).

Then β_{i} = ϕ_{ji} ∘ β_{j} and on U_{i} ×_{X} U_{j} ×_{X} U_{k} we have

ϕ_{ki} = ϕ_{kj} ∘ ϕ_{ji}.

Define N to be the pre-log structure obtained by gluing the N_{i} along the ϕ_{ji} and let N be the associated log structure. N comes equipped with a map

N → M_{n}

and a surjection ℕ^{r} → \overline{N} which locally lifts to a chart. In addition the induced map

ℕ^{r} → P

is the map we started with. Furthermore, for each n we can apply the above construction and since the c_{i,n} are compatible we get a compatible family of DF log structures N_{n} together with morphisms of log structures

N_{n} → M_{n}.

If

ℕ^{r} ⇉ ℕ^{s} → P

is a presentation for P, then we can apply the above construction to both ℕ^{r} and ℕ^{s} to get a diagram of compatible families of log structures

N′_{n} ⇉ N_{n} → M_{n}

where the morphisms

N′_{n} ⇉ N_{n}

are morphisms of DF log structures. In fact M_{n} is the coequalizer of these two maps. To see this observe that the question is local so we may interpret the above sequence as being obtained from

ℕ^{r} ⇉ ℕ^{s} → P

viewed as a sequence of pre-log structures by taking the associated log structure. But taking the associated log structure preserves equalizers. This proves the essential surjectivity since we can realize any log structure in C_{∞} as the coequalizer of morphisms between DF log structures. Full faithfulness. If O is a strictly henselian local ring and if M → O is a log structure, then the natural map

M → M̂ = Ô^{∗} ⊕_{O} M

is injective. This implies that the functor is faithful. To prove that it is full, let M_{n} and M_{n}′ be two compatible collections of log structures on the \underline{X}_{n} induced from log structures M and M′ on \underline{X}, and say we are given a sequence of maps

f_{n} : M_{n} → M_{n}′.

Let P = Γ(X_{n}, \overline{M}_{n}) and let P′ = Γ(X_{n}, \overline{M}′_{n}). Choose surjections ℕ^{r} → P and ℕ^{r′} → P′ and a map f̃: ℕ^{r} → ℕ^{r′} such that the diagram

ℕ^{r} ⟶ P ⟶ \overline{M}_{n}
f̃↓ ^{f}↓ ^{f}↓
ℕ^{r′} ⟶ P′ ⟶ \overline{M}′_{n}

commutes. Let N_{n} and N′_{n} be the associated DF log structures. We construct a compatible collection of morphisms of DF log structures N_{n} → N′_{n} such that the diagrams

N_{n} ⟶ M_{n}
↓ ↓
N′_{n} ⟶ M′_{n}

commute. Etale locally we have diagrams

O_{X_{n}}^{∗} ⊕ ℕ^{r} ⟶^{β} M_{n}
↓
O_{X_{n}}^{∗} ⊕ ℕ^{r′} ⟶^{β′} M′_{n}.

If e ∈ ℕ^{r} there exists a unique u(e) ∈ O_{X_{n}}^{∗} such that

f(β(e)) = λ(u(e)) + β′(f̃(e)).

Define a map

O_{X_{n}}^{∗} ⊕ ℕ^{r} → O_{X_{n}}^{∗} ⊕ ℕ^{r′}, e ↦ (u(e), f̃(e)).

This defines maps of DF log structures N_{n} → N′_{n}. It follows that we can find a diagram in C_{∞}

N_{1} ⇉ N_{2} → M
↓
N′_{1} ⇉ N′_{2} → M′

such that the maps f_{n} are induced as the maps on co-equalizers from the reductions of the middle arrow. By the faithfullness the two arrows

N_{1} ⇉ M′

are equal and hence we get a map

M → M′

inducing the f_{n}.

The second main result of this section is the following:

Theorem 4.2.8. Suppose A is a complete noetherian local ring with perfect residue field k, and suppose A has the structure of a flat W [[t_{1}, …, t_{s}]]-algebra where W denotes the ring of Witt vectors of k if char(k) > 0 and k if char(k) = 0. We view A and its reductions A_{n} = A/𝔪_{A}^{n} as having a log structure coming from the variables t_{1}, … t_{s}. Suppose \underline{X}/A is a proper scheme of finite type and that M_{n} is a compatible family of M_{A_{n}}-log structures on \underline{X}_{n} = \underline{X} ⊗_{A} A_{n}. Suppose further that étale locally the log structure M_{n} is induced from a smooth map

\underline{X}_{n} → Spec(A_{n}[X_{1}, …, X_{r}]/X_{1} ⋯ X_{r} − t_{i})

for some i. Then there exists a unique log structure M on \underline{X} inducing the M_{n}

Lemma 4.2.9. Let O be a strictly henselian local ring isomorphic to the strict henselization at some point of an \underline{S}-algebra of finite type, x_{1}, …, x_{d} ∈ O elements which induce an isomorphism

Ô ≃ k[[x_{1}, …, x_{d}]]/x_{1} ⋯ x_{r}

for some r. Suppose y_{1}, …, y_{d′} ∈ O is a second collection of elements which induce an isomorphism

Ô ≃ k[[y_{1}, …, y_{d′}]]/y_{1} ⋯ y_{s}.

Then d = d′, s = r, and there exists a unique permuation σ of {1, …, s} such that

y_{σ(i)} = u_{i}x_{i}, 1 ≤ i ≤ s

where u_{i} ∈ O^{∗}.

Proof. By looking at dimensions we must have d = d′. View the y_{i} as elements of k[[x_{1}, …, x_{d}]]/x_{1} ⋯ x_{r} via

Ô ≃ k[[x_{1}, …, x_{d}]]/x_{1} ⋯ x_{r}.

Write

y_{i} = L_{i}(x) + H_{i}(x)

where L_{i}(x) consists of linear terms in the x_{i} and H_{i}(x) consists of terms of degree ≥ 2. Since the y_{i} map to a basis for 𝔪/𝔪^{2} the terms L_{i}(x) are not zero. By looking at degrees we see that

∏ L_{i}(x) = 0

and since none of the L_{i}(x) are zero we must have s ≥ r. By symmetry we therefore have s = r. In addition, by inspection of the ring k[[x_{1}, …, x_{d}]]/x_{1} ⋯ x_{r} we see that we must have L_{i}(x) = a_{i}x_{i} with a_{i} ∈ k^{∗} after possible having to permute the y_{i}.

By induction on n, we can write

y_{i} = x_{i}u_{i} + H_{i}^{>n}(x)

where u_{i} ∈ Ô^{∗} consists of monomials of degree less than or equal to n − 1 and H_{i}^{>n}(x) consists of monomials of degree > n. Let H_{i}^{n+1}(x) denote the terms of degree n + 1. Then by looking at terms of degree n + s in y_{1} ⋯ y_{s} we see that

x_{1} ⋯ x̂_{i} ⋯ x_{s}H_{i}^{n+1}(x) = 0.

It follows that

H_{i}^{n+1}(x) = x_{i}(element of Ô).

By induction we conclude that

y_{i} = u_{i}x_{i}, u_{i} ∈ Ô^{∗}.

Finally to pass from Ô to O apply the Artin approximation theorem ([4]).

Lemma 4.2.10. Let A be a an artinian local ring, t ∈ 𝔪_{A}, and suppose

ϕ : A[[x_{1}, …, x_{n}]]/(x_{1} ⋯ x_{s} − t) ≃ A[[y_{1}, …, y_{n}]]/(y_{1} ⋯ y_{s} − t)

is an isomorphism. Then after a unique permutation of the y_{1}, …, y_{s}

ϕ(x_{i}) = u_{i}y_{i}, 1 ≤ i ≤ s

where u_{i} is a unit.

Proof. We proceed by induction on m such that 𝔪_{A}^{m} = 0. The case m = 1 is 4.2.9. Therefore assume true for m and suppose 𝔪^{m+1} = 0. We view

A[[y_{1}, …, y_{n}]]/(y_{1} ⋯ y_{s} − t)

as a module which is the direct product of free modules with basis

∏ y_{i}^{n_{i}}

where at least one n_{i}, 1 ≤ i ≤ s, is zero. Call these basis elements monomials. By the induction hypothesis we can write

ϕ(x_{i}) = u_{i}y_{i} + H_{i}(y), 1 ≤ i ≤ s

where {u_{i}} are units and H_{i}(y) has coefficients in 𝔪^{m} and no monomial terms with a y_{i}. We show that every coefficient appearing in H_{i}(y) can be written as

t(something) = y_{i}(y_{1} ⋯ ŷ_{i} ⋯ y_{s}(something)).

This will complete the induction step. Let k be the smallest total degree of any monomial appearing in any of the H_{i}(y). Let H_{i}^{=k}(y) be the monomial terms appearing in H_{i}(y) with total degree equal to k. Then we get

t = ∏ ϕ(x_{i}) =  (∏ u_{i})t + (unit in A)y_{1} ⋯ ŷ_{i} ⋯ y_{s}H_{i}^{=k}(y)
+ monomial terms with y_{i}’s and terms of degree ≥ k + s.

Since H_{i}^{=k} has no monomial terms with y_{i}’s this implies that the coefficients of H_{i}^{=k} can be written as multiples of t. Induction on k now completes the induction step for n.

4.2.11. We can now prove theorem 4.2.8. Let Z ↪ X be the closed subscheme defined by the ideal (t_{1}, …, t_{s}). Z defines a log structure N_{X} ⊂ O_{X}. If \underline{U} → \underline{X} is a finite étale morphism, we can define

M̃_{X}(\underline{U}) = \underleftarrow{lim} M_{n}(\underline{U}_{n}) ∩ O_{X}(U) ⊂ \underleftarrow{lim} O_{X}(\underline{U}_{n}).

We also denote by M̃_{X} the associated sheaf on the finite étale site of \underline{X}. Here we use the flatness of A to conclude that \underleftarrow{lim} M_{n}(\underline{U}_{n}) ⊂ \underleftarrow{lim} O_{X}(\underline{U}_{n}). By the previous two lemmas, for every point in the closed fiber of \underline{X}, the inclusion into O_{X} induces an isomorphism M̃_{X} ≃ N_{X}. Hence these two sheaves are isomorphic. We therefore get a map of sheaves

N_{X}|_{\underline{X}_{n}} → M_{X_{n}}

which lemma 4.2.9 shows is an isomorphism.

4.3 Log structures on normal crossing schemes.

4.3.1. Let f : X → S be a smooth, proper, integral morphism of noetherian log schemes, and suppose that M_{X/S} is a sheaf of torsion free groups. Suppose that for every geometric point s̄ → \underline{S}, the underlying scheme of Xs̄ is étale locally isomorphic to

k(s̄)[X_{1}, …, X_{n}]/(X_{1} ⋯ X_{r})

for some natural numbers r and n. The purpose of this section is to show that under these assumptions the log structure M_{X} on \underline{X} is induced by base change in a canonical manner from a smooth morphism

f^{♯} : X^{♯} → S^{♯}

of a special form whose underlying morphism of schemes is that of the original f.

Definition 4.3.2. A log smooth morphism f : X → S is essentially semi-stable if for each geometric point x̄ → \underline{X} the monoids (f^{−1}\overline{M}_{S})x̄ and \overline{M}_{X,x̄} are free monoids and there exist isomorphisms (f^{−1}\overline{M}_{S})x̄ ≃ ℕ^{r} and \overline{M}_{X,x̄} ≃ ℕ^{r+s} such that if we denote the standard generators by e_{i} the map

(f^{−1}\overline{M}_{S})x̄ → \overline{M}_{X,x̄}

is of the form

e_{i} ↦ \{ \begin{matrix}e_{i}, & if i ≠ r \\ e_{r} + e_{r+1} + ⋯ + e_{r+s}, & if i = r \end{matrix}

Remark 4.3.3. If f : X → S is essentially semi-stable, then étale locally on X and S there exist charts ℕ^{r} → M_{S}, ℕ^{r+s} → M_{X} such that the map ℕ^{r} → ℕ^{r+s} given by

e_{i} ↦ \{ \begin{matrix}e_{i}, & if i ≠ r \\ e_{r} + e_{r+1} + ⋯ + e_{r+s}, & if i = r \end{matrix}

is a chart for f and such that the map

O_{S} ⊗_{ℤ[ℕ^{r}]} ℤ[ℕ^{r+s}] → O_{X}

is smooth. To see this observe that if s ∈ \underline{S} is a point then the stalk \overline{M}_{S,s̄} is a free monoid and hence in some étale neighborhood of s there exists a chart ℕ^{r} → M_{S} such that the induced map ℕ^{r} → \overline{M}_{S,s̄} is bijective. If x ∈ \underline{X} is a point lying over s, then by ([29], 2.25) there exists in some étale neighborhood of x a chart

P ⟶ M_{X}
↑ ↑
ℕ^{r} ⟶ M_{S}

such that the induced map

O_{S} ⊗_{ℤ[ℕ^{r}]} ℤ[P] → O_{X}

is smooth and such that the map P → \overline{M}_{X,x̄} is bijective. From the bijectivity of P → \overline{M}_{X,x̄} we conclude that P is a free monoid and that the map ℕ^{r} → P has the desired form (after perhaps applying an automorphism of ℕ^{r}).

Remark 4.3.4. An essentially semi-stable morphism f : X → S is integral and vertical (this means that M_{X/S} is a sheaf of groups). To see the integrality we have to check that the map ℕ^{r} → ℕ^{r+s} described above is integral. But the diagram

ℕ  ⟶^{e↦e_{r}} ℕ^{r}
Δ↓  ↓
ℕ^{s+1} ⟶^{e_{i}↦e_{i+r−s}} ℕ^{r+s}

where Δ is the diagonal map is cocartesian. Since the map Δ is integral so is the map ℕ^{r} → ℕ^{r+s} ([20], 4.1 (i)). To see that it is vertical, let Q be the cokernel of ℕ^{r} → ℕ^{r+s}. By definition of cokernel, to give a map Q → M to an integral monoid M is equivalent to giving a map ℕ^{r+s} → M such that the composite ℕ^{r} → ℕ^{r+s} → M is zero. A map ℕ^{r+s} → M is equivalent to giving elements

m_{1}, …, m_{r+s} ∈ M

and the condition that the map ℕ^{r} → M be zero means that

m_{1} = m_{2} = ⋯ = m_{r−1} = m_{r} + ⋯ + m_{r+s} = 0.

Thus giving an arrow Q → M is equivalent to giving elements m_{r}, …, m_{r+s} ∈ M^{∗} such that m_{r} + ⋯ + m_{r+s} = 0. This in turn is equivalent to giving elements m_{r}, …, m_{r+s−1} ∈ M^{∗}. But the functor on the category of integral monoids

M ↦ {r + s − 1-tuples of elements m ∈ M^{∗}}

is represented by ℤ^{r+s−1} and hence Q ≃ ℤ^{r+s−1}. 4.3.5. Suppose \underline{S} = Spec(k) where k is a separably closed field, and that f : X → S is essentially semi-stable. Let x ∈ \underline{X} be a singular point. Then by remark 4.3.3 we have a chart

ℕ^{r+s} ⟶ M_{X} ⟶ O_{X}
↑ ↑ ↑
ℕ^{r} ⟶ M_{S} ⟶^{α} Ω

in an étale neighborhood of x such that

k ⊗_{ℤ[ℕ^{r}]} ℤ[ℕ^{r+s}] ≃ k[x_{r}, …, x_{r+s}]/(x_{r} ⋯ x_{r+s} − α(e_{r})) → O_{X}

is smooth. Since x is a singular point it follows that α(e_{r}) = 0 and hence the map

\overline{M}_{S} → \overline{M}_{X,x̄}

is of the form ℕ^{r′} → ℕ^{r′+s} as in definition 4.3.2 for some r′ ≤ r. It follows that there is a unique element in the set of irreducible elements Irr(\overline{M}_{S}) of \overline{M}_{S} whose image in \overline{M}_{X,x̄} is not irreducible. This defines a canonical map

s_{X} : \{singular points of X\} → Irr(\overline{M}_{S}).

Definition 4.3.6. A morphism of log schemes f : X → S is special if it is essentially semistable, and if for each geometric point s̄ → \underline{S} the induced map

s_{Xs̄} : \{singular points of Xs̄\} → Irr(\overline{M}_{S,s̄})

induces a bijection between the set of connected components of the singular locus of Xs̄ and Irr(\overline{M}_{S,s̄}).

Theorem 4.3.7. Let f : X → S be a smooth, proper, integral morphism of noetherian log schemes such that M_{X/S} is a sheaf of torsion free groups, and such that for every geometric point s̄ → \underline{S} the underlying scheme of Xs̄ is étale locally isomorphic to

k(s̄)[X_{1}, …, X_{n}]/(X_{1} ⋯ X_{r})

for some natural numbers r and n. Then there exists a pair of log structures (M_{S}^{♯}, M_{X}^{♯}) on \underline{S} and \underline{X} respectively and a morphism

f^{♯} : X^{♯} = (\underline{X}, M_{X}^{♯}) → S^{♯} = (\underline{S}, M_{S}^{♯})

which is special, together with morphisms of log structures

ϕ : M_{S}^{♯} → M_{S}, ψ : M_{X}^{♯} → M_{X}

which make the diagram

\begin{matrix}X & ⟶ & X^{♯} \\ ^{f}↓ & & ^{f^{♯}}↓ \\ S & ⟶ & S^{♯} \end{matrix}

cartesian. Moreover, the datum of (M_{S}^{♯}, M_{X}^{♯}, ϕ, ψ) is unique up to unique isomorphism.

4.3.8. The proof occupies the remainder of this section. We will construct (M_{S}^{♯}, M_{X}^{♯}, ϕ, ψ) fppf-locally on \underline{S} and then use the uniqueness to prove the global result. Let us begin by considering the case when S = Spec(O) where O is a strictly henselian local ring, and we have a chart Q → M_{S} such that Q → \overline{M}_{S,s_{0}} is bijective (here s_{0} denotes the closed point). Denote by T the set of connected components of the singular locus of the closed fiber.

Lemma 4.3.9. Let P be a fine sharp monoid, Q ↪ P an integral morphism, and let p̄ ∈ P/Q be an element. Denote by π : P → P/Q the projection map. Then there exists a unique element p_{0} ∈ P such that

π^{−1}(p̄) = {p_{0} + q|q ∈ Q}.

Proof. Since P is fine and sharp, there exists a partial ordering on P^{+} defined by p_{1} ≤ p_{2} if there exists p_{3} ∈ P such that p_{1} + p_{3} = p_{2}. By ([30]) the set π^{−1}(p̄) contains a unique finite set of minimal elements with respect to this partial ordering. Suppose p_{1}, p_{2} ∈ π^{−1}(p̄) and suppose p_{1} is a minimal element. By construction of the quotient, there exists q_{1}, q_{2} ∈ Q such that p_{1} +q_{1} = p_{2} +q_{2}. Since the map is integral, there exists q_{3}, q_{4} ∈ Q and p ∈ P such that

p_{1} = q_{3} + p, p_{2} = q_{4} + p, q_{1} + q_{3} = q_{2} + q_{4}.

By definition of quotient, p ∈ π^{−1}(p̄) and p ≤ p_{1}. Since p_{1} was minimal it follows that p_{1} = p and that p_{2} = p_{1} + q.

Corollary 4.3.10. Let t be a singular point of X, and let n ∈ M_{X/S,t̄} (≃ ℤ^{r}, r ≥ 0). Then there exists a unique element p_{n} ∈ \overline{M}_{X,t̄} such that

π^{−1}(n) = \{p_{n} + q|q ∈ \overline{M}_{S,s_{0}}\}.

Proof. The map \overline{M}_{S,s_{0}} → \overline{M}_{X,t̄} is integral by assumption.

4.3.11. As above let s_{0} denote the closed point of S and let X_{s_{0}} denote the fiber. By ([29], 2.13, 2.14, and 2.25) for every singular closed point t of X_{s_{0}} there exists a chart

P ⟶^{β} M_{X}
↑ ↑
Q ⟶^{β} M_{S}

in some étale neighborhood of t such that

1. Q → P is injective, and P → \overline{M}_{X,t̄} is bijective.

2. The map

O ⊗_{ℤ[Q]} ℤ[P] → O_{X}

is smooth.

Observe that if p ∈ P is an irreducible element not in Q, and if i = π(p) (notation as in 4.3.9) then p = p_{i}. Indeed by definition we can write

p = p_{i} + q

for some q ∈ Q and since p is irreducible we must have q = 0. Let {p_{i_{1}}, …, p_{i_{n}}} be the set of irreducible elements in P which are not in Q (we index these irreducible elements by their images in P/Q). If i ∈ P/Q is any element and p_{i} is as in 4.3.9, then p_{i} can be written as

p_{i} = ∑ m_{j}p_{i_{j}}.

Indeed writing p_{i} as a sum of irreducible elements we have

p_{i} = ∑ m_{j}p_{i_{j}} + q, q ∈ Q

and by definition of p_{i} we must have q = 0. The {p_{i_{j}}} define a map from a polynomial algebra

γ : O[Y_{i_{1}}, …, Y_{i_{n}}] → O ⊗_{ℤ[Q]} ℤ[P], Y_{i_{j}} ↦ 1 ⊗ e(p_{i_{j}})

which is surjective. We denote by J the kernel of γ.

Proposition 4.3.12. 1. There exists q ∈ Q such that ∑_{j=1}^{n} p_{i_{j}} = q.

2. J is generated by the element

∏_{j=1}^{n} Y_{i_{j}} − α(q).

3. The natural map

Q ⊕_{ℕ} ℕ^{n} → P, (q, (\underline{n}_{i_{j}})) ↦ q + ∑ n_{i_{j}}p_{i_{j}}

is an isomorphism.

Proof. We first reduce to the case when O = k(s_{0}). The reduction step is trivial for (1) and (3). Let us show that (2) holds if the proposition is true on the closed fiber. Since

∑ p_{i_{j}} = q, q ∈ Q

we have an inclusion of ideals

(∏ Y_{i_{j}} − α(q)) ⊂ J

which defines a closed immersion

Spec(O[Y_{i_{1}}, …, Y_{i_{n}}]/J) ↪ Spec(O[Y_{i_{1}}, …, Y_{i_{n}}]/(∏ Y_{i_{j}} − α(q)).

Since both of these schemes are flat over O and the reduction is an isomorphism the closed immersion is an isomorphism also by ([1], VII.1.9). Therefore (2) holds and hence it suffices to consider the case when O = k for some separably closed field k.

Lemma 4.3.13. The ideal J = Ker(γ) is generated by elements of the form

∏ Y_{i_{j}}^{n_{i_{j}}} − α(q) ∏ Y_{i_{j}}^{m_{i_{j}}}

where

∑ n_{i_{j}}p_{i_{j}} = ∑ m_{i_{j}}p_{i_{j}} + q, q ∈ Q

in P.

Proof. We have a surjection

Q ⊕ ℕ^{n} → P, (q, (\underline{n}_{j})) ↦ q + ∑_{j} n_{j}p_{i_{j}}

which identifies P with the quotient of Q ⊕ ℕ^{n} by the equivalence relation (q, (\underline{n}_{j})) = (q′, (\underline{m}_{j})) if

∑ n_{i_{j}}p_{i_{j}} + q = ∑ m_{i_{j}}p_{i_{j}} + q′.

It follows that the kernel of the induced map on monoid algebras

ℤ[Q ⊕ ℕ^{n}] → ℤ[P]

is the ideal generated by elements

e(q, (\underline{n}_{j})) − e(q′, (\underline{m}_{j}))

if ∑ n_{i_{j}}p_{i_{j}} + q = ∑ m_{i_{j}}p_{i_{j}} + q′ (here e(q, (\underline{n}_{j})) denotes the element of the monoid algebra corresponding to (q, (\underline{n}_{j}))). Tensoring over ℤ[Q] with k we find that k ⊗_{ℤ[Q]} ℤ[P] is the quotient of k[ℕ^{n}] by the ideal generated by

α(q)e(\underline{n}_{j}) − α(q′)e(\underline{m}_{j})

if ∑ n_{i_{j}}p_{i_{j}} + q = ∑ m_{i_{j}}p_{i_{j}} + q′. If both q and q′ are non-zero this element is zero so contributes nothing to the kernel. Therefore, writing Y_{i_{j}} for the irreducible element of ℕ^{n} mapping to p_{i_{j}} we find the result.

Lemma 4.3.14. 1. The completion Ô_{X,t} is isomorphic to

k[[T_{1}, …, T_{s}, Y_{i_{1}}, …, Y_{i_{n}}]]/(Y_{i_{1}} ⋯ Y_{i_{n}})

where {T_{1}, …, T_{s}} are independent variables.

2. Let L ⊂ P be the submonoid generated by the set {p_{i}| i ∈ P/Q}. Then L is isomorphic to ℕ^{n}.

Proof. (1) Let Y′ be the log scheme with underlying scheme Spec(k ⊗_{ℤ[Q]} ℤ[P]) and log structure induced by P → ℤ[P]. Then we have a diagram of log schemes

X ⟶^{h} Y′ ⟶ S

where h is log smooth and strict. Since h is strict, the map

Ω_{\underline{X}/\underline{S}}^{1} → Ω_{X/Y′}^{1}

is surjective so in some neighborhood of t we can find elements f_{1}, …, f_{s} in the maximal ideal of t such that df_{1}, …, df_{s} map to a basis for Ω_{X/Y′}^{1} ⊗ k(t). This defines a map

h′ : X → Y′ ⊗\underline{S} 𝔸s ; \underline{S}

of log schemes which is strict and for which the map

h^{′∗}Ω_{Y′⊗\underline{S}𝔸s ; \underline{S}/Y′}^{1} ⊗ k(t) → Ω_{X/Y′}^{1} ⊗ k(t)

is an isomorphism. It follows that in some neighborhood of t the map h′ is étale ([29], 2.20). Thus we can extend the map γ to an étale map

X → k[T_{1}, …, T_{s}] ⊗_{k} (k[Y_{i_{1}}, …, Y_{i_{n}}]/J)

sending t to the point T_{1} = ⋯ = T_{s} = Y_{i_{1}} = ⋯ = Y_{i_{n}} = 0. First we claim that the resulting map

k[[T_{1}, …, T_{s}, Y_{i_{1}}, …, Y_{i_{n}}]] → Ô_{X,t}

induces an isomorphism

k[[T_{1}, …, T_{s}, Y_{i_{1}}, …, Y_{i_{n}}]]/(T_{1}, …, T_{s}, Y_{i_{1}}, …, Y_{i_{n}})^{2} ≃ Ô_{X,t}/𝔪^{2}.

To see this it suffices to show that all our defining equations for J are contained in the ideal (Y_{i_{1}}, …, Y_{i_{n}})^{2}. But all our equations arise from non-trivial equalities

∑ n_{i_{j}}p_{i_{j}} = q + ∑ m_{j}p_{i_{j}}

in P. If ∑ n_{i_{j}} = 1 then we have

p_{i_{j}} = q + ∑ m_{j}p_{i_{j}}

which by the irreducibility of p_{i_{j}} implies that our equation comes from an equality

p_{i_{j}} = q or p_{i_{j}} = p_{i_{j′}} .

Since p_{i_{j}} ∉ Q we must have p_{i_{j}} = p_{i_{j′}} and hence our equation is trivial. Fix an isomorphism

Ô_{X,t} ≃ k[[X_{1}, …, X_{l}]]/(X_{1}, ⋯ X_{h})

for some l and h. It follows from the above that

l = dim(𝔪/𝔪^{2}) = n + s.

For each k ≥ 1, the dimension of 𝔪^{k}/𝔪^{k+1} is equal to the number of monomials of degree k in X_{1}, …, X_{l} which do not contain the string X_{1} ⋯ X_{h}. In particular, the dimension of 𝔪^{h}/𝔪^{h+1} is equal to one less than the number of monomials of degree h in l = n + s variables. This implies that we have at least one equation involving monomials in the Y_{i_{j}} of degree h. Fix one such equation f(\underline{Y}) = 0. By lemma 4.3.13, we can choose our equation so that it is either of the form

f(\underline{Y}) = Y_{i_{1}} ⋯ Y_{i_{h}} − Y_{i′_{1}} ⋯ Y_{i′_{m}}

or of the form

f(\underline{Y}) = Y_{i_{1}} ⋯ Y_{i_{h}}.

Let R be the ring

k[[T_{1}, …, T_{s}, Y_{i_{1}}, …, Y_{i_{n}}]]/(f).

There is a natural surjective map

R → Ô_{X,t}

which induces an isomorphism 𝔪_{R}/𝔪_{R}^{2} ≃ 𝔪/𝔪^{2}. For each integer w ≥ 1 let R_{w} = R/𝔪_{R}^{w} and let M_{w} = Ô_{X,t}/𝔪^{w}. We view M_{w} as an R_{w}-module. Note that the map k = R_{1} → M_{w} ⊗_{R_{w}} R_{1} = k is an isomorphism.

For each integer k the dimension of 𝔪_{R}^{k}/𝔪_{R}^{k+1} is equal to the dimension of 𝔪^{k}/𝔪^{k+1}. In fact, if we define N(k, l) to be the number of monomials of degree k in l variables then the dimensions of 𝔪_{R}^{k}/𝔪_{R}^{k+1} and 𝔪^{k}/𝔪^{k+1} are both equal to

N(k, l) − N(k − h, l) = N(k, n + s) − N(k − h, n + s).

Since we have an isomorphism 𝔪_{R}/𝔪_{R}^{2} ≃ 𝔪/𝔪^{2} the map

gr_{𝔪_{R}}^{∗}(R_{w}) → gr_{𝔪_{R}}^{∗}(M_{w})

is surjective and therefore an isomorphism by looking at dimensions. It now follows from the local criterion for flatness ([1], V.3.2) that M_{w} is a flat R_{w}-module. From this and the fact that R_{1} → M_{w} ⊗_{R_{w}} R_{1} is an isomorphism we conclude that R_{w} ≃ M_{w} (see for example the proof of proposition 3.2.2). It follows that the map

R = \underleftarrow{lim}_{w} R_{w} → Ô_{X_{s_{0}},t̄} ≃ \underleftarrow{lim}_{w} M_{w}

is also an isomorphism. We claim that f cannot be of the form

f(\underline{Y}) = Y_{i_{1}} ⋯ Y_{i_{h}} − Y_{i′_{1}} ⋯ Y_{i′_{m}}.

Since P/Q is a group, for every p_{i_{j}} there exists n_{a} ≥ 0 such that

Y_{i_{j}} ∏_{a=1}^{n} Y_{i_{a}}^{n_{a}} = 0.

More precisely, since P/Q is a group there exists an element p_{−i_{j}} such that

π(p_{i_{j}}) + π(p_{−i_{j}}) = 0

in P/Q. Writing p_{−i_{j}} = ∑ n_{a}p_{i_{a}} we conclude that we have

p_{i_{j}} + ∑_{a=1}^{n} n_{a}p_{i_{a}} ∈ π^{−1}(0).

By lemma 4.3.9, π^{−1}(0) = Q and hence there exists an element q ∈ Q such that

p_{i_{j}} + ∑_{a=1}^{n} n_{a}p_{i_{a}} = q.

Since p_{i_{j}} is not a unit, q ≠ 0 and hence we find that

Y_{i_{j}} ∏_{a=1}^{n} Y_{i_{a}}^{n_{a}} = 0.

Since Ô_{X,t} is reduced, if

∏ Y_{i_{j}}^{m_{j}} = 0

with the m_{j} ≥ 1 then

∏ Y_{i_{j}} = 0.

It follows that

∏ Y_{i_{j}} = 0.

This is impossible if

f(\underline{Y}) = Y_{i_{1}} ⋯ Y_{i_{h}} − Y_{i′_{1}} ⋯ Y_{i′_{m}}.

Therefore, we have an isomorphism

k[[T_{1}, …, T_{s}, Y_{i_{1}}, …, Y_{i_{n}}]]/(Y_{i_{1}} ⋯ Y_{i_{h}}) ≃ Ô_{X,t}

for some h. We claim that h = n. To see this note that we have

∑_{j=1}^{h} p_{i_{j}} = p + q

for some p ∈ P and a nonzero q ∈ Q. If p_{i_{a}} is another one of our irreducible elements with a > h, then since P/Q is a group there exists an element p_{−i_{a}} as in 4.3.9 such that p_{i_{a}} + p_{−i_{a}} maps to zero in P/Q. Write

p_{−i_{a}} = ∑_{j=1}^{n} b_{i_{j}}p_{i_{j}}.

Then at least one of the b_{i_{j}} with 1 ≤ j ≤ h must be zero since otherwise we can write

p_{−i_{a}} = ∑_{j=1}^{h} p_{i_{j}} + ∑_{j=1}^{h}(b_{i_{j}} − 1)p_{i_{j}} + ∑_{j>h} b_{i_{j}}p_{i_{j}} = p + q + ∑_{j>h} b_{i_{j}}p_{i_{j}} + ∑_{j=1}^{h}(b_{i_{j}} − 1)p_{i_{j}}

contradicting the minimality of p_{−i_{a}}. It follows that ∏_{j} Y_{i_{j}}^{b_{i_{j}}} ∉ J. On the other hand p_{i_{a}} + p_{−i_{a}} ∈ π^{−1}(0) = Q and since p_{i_{a}} is not a unit p_{i_{a}} + p_{−i_{a}} is a non-zero element in Q. Therefore Y_{i_{a}} ∏_{j} Y_{i_{j}}^{b_{i_{j}}} ∈ J and hence Y_{i_{a}} is a zero divisor in the ring

k[[T_{1}, …, T_{s}, Y_{i_{1}}, …, Y_{i_{n}}]]/(Y_{i_{1}} ⋯ Y_{i_{h}}).

This is a contradiction so we must have h = n.

(2) Suppose we have an equality

∑ m_{j}p_{i_{j}} = ∑ m′_{j}p_{i_{j}}

in P. Since P is cancellative, we can assume that at least one m_{j} = 0 and hence we get

∏ Y_{i_{j}}^{m_{j}} = ∏ Y_{i_{j}}^{m′_{j}} ≠ 0

in the ring

k(s_{0})[[T_{1}, …, T_{s}, Y_{i_{1}}, …, Y_{i_{n}}]]/(∏ Y_{i_{j}}).

It follows that m_{j} = m′_{j} for all j.

We now complete the proof of the proposition. Suppose we have an equality

q + ∑_{n_{i_{j}}} p_{i_{j}} = q′ + ∑_{m_{i_{j}}} p_{i_{j}}

in P. Then by the integrality of Q → P there exists p ∈ P and q_{3}, q_{4} ∈ Q such that

∑_{j} n_{i_{j}}p_{i_{j}} = p + q_{3}, ∑_{j} m_{i_{j}}p_{i_{j}} = p + q_{4}, q + q_{3} = q + q_{4}.

In addition, writing p = ∑ b_{i_{j}}p_{i_{j}} +q_{5} we see that the congruence relation on Q⊕ℕ^{n} defined by the surjection

Q ⊕ ℕ^{n} → P

is generated by equalities in P of the form

∑_{j} n_{i_{j}}p_{i_{j}} = ∑_{j} b_{i_{j}}p_{i_{j}} + q.

In addition since P is cancellative we can assume that for each j either n_{i_{j}} = 0 or b_{i_{j}} = 0. From lemma 4.3.14 we deduce that any non-trivial equality of this form must have n_{i_{j}} ≥ 1 for all j. Therefore, the congruence relation on Q ⊕ ℕ^{n} is generated by equalities

∑_{j} n_{i_{j}}p_{i_{j}} = q

where all the n_{i_{j}} are greater than 0. In addition, from 4.3.14 we see that we must have an equality

∑_{j=1}^{n} p_{i_{j}} = q

for some q ∈ Q. This proves (1) of the proposition. Now suppose we had an equality in P

∑_{j} n_{i_{j}}p_{i_{j}} = q′

for some q′ ∈ Q and all n_{i_{j}} ≥ 1. Then after perhaps reordering the p_{i_{j}} we can assume that n_{i_{1}} is the smallest of the n_{i_{j}}. This gives

q′ = ∑_{j} n_{i_{j}}p_{i_{j}} = n_{i_{1}}q + ∑_{j}(n_{i_{j}} − n_{i_{1}})p_{i_{j}}.

The integrality of Q → P now implies that ∑_{j}(n_{i_{j}} − n_{i_{1}})p_{i_{j}} is in Q which from above is only possible if n_{i_{j}} = n_{i_{1}} for all j. This implies (2). To see (3), note that the map

Q ⊕_{ℕ} ℕ^{n} → P

surjective by the definition of the p_{i_{j}}. To see that it is injective, suppose (q, (\underline{n}_{j})) and (q′, (\underline{m}_{j})) map to the same element. Without loss of generality, we may assume that at least one n_{j} = 0 and at least one m_{j′} = 0 (since there exists q ∈ Q such that (q, 0) = (0, (1, …, 1)) in Q ⊕_{ℕ} ℕ^{n}). Then we get an equality

q + ∑ n_{j}p_{i_{j}} = q′ + ∑ m_{j}p_{i_{j}}

in P and by the integrality we get that there exists p ∈ P and q_{3}, q_{4} ∈ Q such that

∑ n_{j}p_{i_{j}} = p + q_{3}, ∑ m_{j}p_{i_{j}} = p + q_{4}, q_{3} + q = q_{4} + q′.

Since ∑ p_{i_{j}} ∈ Q we may assume that p = ∑ b_{i_{j}}p_{i_{j}} with at least one b_{i_{j}} = 0. If q_{3} is non-zero we get that

∏ Y_{i_{j}}^{n_{j}} = 0

in k[Y_{i_{1}}, …, Y_{i_{n}}]/J which from above is impossible since at least one n_{j} is zero. Thus q_{3} = 0 and hence

∑ m_{j}p_{i_{j}} = ∑ n_{j}p_{i_{j}} + q_{4}.

By the same reasoning using the fact that at least one m_{j′} = 0 we conclude that q_{4} = 0 and hence q = q′ and (\underline{n}_{j}) = (\underline{m}_{j}). This completes the proof of the proposition.

4.3.15. Let L ⊂ P be the submonoid of 4.3.14.2, and define K ⊂ Q to be the submonoid generated by the element

∑ p_{i_{j}}.

Equivalently

K = L ∩ Q ⊂ P.

If t′ ∈ X is another singular point lying over some ζ ∈ Spec(O), then the above shows that the submonoid L_{t′} ⊂ \overline{M}_{X,t̄′} generated by the irreducible elements of \overline{M}_{X,t̄′} not in the image of \overline{M}_{S,ζ̄} is isomorphic to ℕ^{r} for some r and that the intersection K_{t′} = L_{t′} ∩ \overline{M}_{S,ζ̄} is a free monoid of rank 1. To check this one may base change to an algebraic closure k̄ of k(ζ) and hence can assume that we have a chart Q → M_{S} such that Q → \overline{M}_{S} is bijective. If t′ is a closed point the result now follows from the above discussion. If t′ is not a closed point, then choose a specialization t′′ of t′ and note that the stalk at t′ is obtained by the cospecialization map from t′′ ([17], II.2.3).

Lemma 4.3.16. Suppose we have chosen a chart around t as in 4.3.11 over some étale neighborhood of t which we also denote by X. Let L_{t} and K_{t} be the submonoids of P and Q respectively given by the above construction. Then for any other singular point t′ of X lying over some ζ ∈ Spec(O), the submonoids Lt̄′ ⊂ \overline{M}_{X,t̄′} and Kt̄ ⊂ \overline{M}_{S,ζ̄} are equal to the images of L_{t} and K_{t}.

Proof. From proposition 4.3.12, we have an isomorphism

Q ⊕_{K_{t}} L_{t} ≃ P,

and hence an isomorphism

\overline{M}_{X,t̄′} ≃ \overline{M}_{S,ζ̄} ⊕_{K_{t}} (L_{t}/(p_{i_{j}} = 0 if p_{i_{j}} maps to a unit in O_{X,t̄′}).

This implies the lemma.

4.3.17. Let D be a connected component of the singular locus of the closed fiber of X. Define log structures K̃_{D} ⊂ M_{S} and L̃_{D} ⊂ M_{X} as follows. Observe that for any point t ∈ D we get a free monoid of rank 1 K_{t} ⊂ \overline{M}_{S,s_{0}}. Proposition 4.3.16 implies that this monoid is independent of the choice of t. To see this suppose t′ ∈ D is a second singular point. Choose étale covers {U_{i}} of D and singular points t_{i} in the image of U_{i} such that over each U_{i} we have a chart as in 4.3.16 using t_{i}. We can also assume that t_{1} = t and t_{r} = t′ for some r. Since D is connected, there exists a sequence U_{1}, …, U_{r} such that for each i there exists a singular point x_{i} of U_{i} ×_{X} U_{i+1} mapping to D. It follows from 4.3.16 that the submonoid K_{t_{i}} ⊂ \overline{M}_{S} defined by t_{i} is equal to the submonoid defined by x_{i} and x_{i−1}. We conclude that K_{t} = K_{t′}. We define K̃_{D} to be the unique fine sub-log structure of M_{S} whose image in \overline{M}_{S,s_{0}} is equal to K_{t} for some t ∈ D (and hence all). L̃_{D} is the unique fine sub-log structure of M_{X} whose image in \overline{M}_{X,t′} (t′ any point of \underline{X}) is equal to the image of K̃_{D} if t′ is not singular or if t′ does not specialize to a point of D and is equal to L_{t′} (notation as in 4.3.15) otherwise. Note that K̃_{D} and L̃_{D} are uniquely determined if they exist since we have specified their values on stalks. The fact that these log structures exist follows from lemma 4.3.16 and the above discussion. Indeed K̃_{D} exists since S is just the spectrum of a strictly henselian local ring. To show that L̃_{D} exists it suffices to construct L̃_{D} locally by the stalk-wise description of L̃_{D} as a subsheaf of M_{X}. But locally we can choose a chart as in 4.3.11 and then lemma 4.3.16 implies the result. Define M_{S}^{♯} to be the amalgemation over all D

⊕_{O_{S}^{∗}} K̃_{D}

and similarly define M_{X}^{♯} to be

⊕_{O_{X}^{∗}} L̃_{D}.

Then the natural diagram

M_{X}^{♯} ⟶^{ψ} M_{X}
↑ ↑
f^{∗}M_{S}^{♯} ⟶^{ϕ} f^{∗}M_{S}.

is co-cartesian. Since f^{♯} is smooth at every point of the closed fiber the induced morphism f^{♯} is log smooth everywhere. By construction f^{♯} is essentially semi-stable and the pullback to the closed fiber is special . Note also that the datum of

(M_{S}^{♯}, M_{X}^{♯}, ϕ, ψ)

which is essentially semistable and whose pullback to the closed fiber is special is unique. To see the uniqueness note that starting with a collection

(M_{S}^{♯}, M_{X}^{♯}, ϕ, ψ)

we can apply the above construction to (M_{S}^{♯}, M_{X}^{♯}). It follows from chasing through the construction that this returns the same pair and hence M_{S}^{♯} and M_{X}^{♯} have canonical decompositions

M_{S}^{♯} = ⊕_{O_{S}^{∗}} K̃_{D}, M_{X}^{♯} = ⊕_{O_{S}^{∗}} L̃_{D}.

In addition the image of L̃_{D} and K̃_{D} in M_{X} and M_{S} are the sub-log structures obtained by applying the construction with M_{X} and M_{S}. Any isomorphism between two collections of data must preserve this decomposition and since the K̃_{D} and L̃_{D} are sub-log structures of M_{S} and M_{X} respectively it follows that an isomorphism is unique and exists. The following proposition 4.3.18 therefore completes the proof of theorem 4.3.7 in the case when S = Spec(O) where O is a strictly henselian local ring and Q → M_{S} is a chart such that Q → \overline{M}_{S,s_{0}} is bijective.

Proposition 4.3.18. Let f : X → S be as in 4.3.7 (we no longer assume that S is the spectrum of a strictly henselian local ring) and suppose in addition that f is essentially semi-stable. Then the set of points s ∈ \underline{S} for which the fiber Xs̄ is special over Spec(k(s̄)) is open.

Proof. Let V be the set of points s ∈ \underline{S} for which Xs̄ is special over k(s̄). By ([1], V.4.4) it suffices to show that V is a constructible set and that if s ∈ V then every generization of s is also in V.

Lemma 4.3.19. V is constructible.

Proof. The condition is local on S so we may assume that S = Spec(A) for some ring A and that M_{S} is induced from some map

ℕ^{r} → A.

Let t_{1}, …, t_{r} ∈ A be the images of the standard generators. For each integer n and for each choice of n distinct elements i_{1}, …, i_{n} ∈ {1, …, r} let A_{n,i} be the ring

A_{t_{i_{1}}···t_{in}} /(t_{j})_{j∉{i_{1},...,i_{n}}}.

Let S_{n,i} = Spec(A_{n,i}). Then each S_{n,i} is a constructible subset of S and the union of the S_{n,i} is all of S. To prove the lemma it therefore suffices to consider the case of the S_{n,i}. This reduces us to the case when the log structure on S is given by a map

ℕ^{r} → A

sending all elements to zero. In addition, by base changing to the irreducible components with the reduced structure we may assume that A is an integral domain. In this case, X is étale locally isomorphic to

A[X_{1}, …, X_{l}]/X_{1} ⋯ X_{s}

for some l and s. Let X^{sing} be the singular locus of X with the reduced induced structure. From the local description of X we see that the geometric fibers of X^{sing} are all reduced and that X^{sing} is flat over A. It follows that the function

s ↦ number of connected components of Xsing ; s̄

can be identified with the function

s ↦ h^{0}(X^{sing} ×_{S} Spec(k(s)), O_{X^{sing}×_{S}Spec(k(s))}).

By the semi-continuity theorem ([14], III.12.8) we conclude that the function

s ↦ number of connected components of Xsing ; s̄

is upper semi-continous. It follows that the set of points s where this number is equal to r is constructible. On the other hand, the set of points s where the map in 4.3.5 is surjective is open by the description of the charts in remark 4.3.3. Therefore the set of points where the map in 4.3.5 is bijective is constructible.

Lemma 4.3.20. V is stable under generization.

Proof. We can assume that S = Spec(A), where A is a local ring and that we want to prove that if the closed fiber is special then the fiber over some prime 𝔭 ⊂ A is also. Replacing A by A/𝔭 we may assume that A is a local integral domain and that we wish to prove that the generic fiber is special if the closed fiber is special. In addition by replacing A by an étale cover we may assume that we have a global chart

ℕ^{T} → M_{S}

where T is the set of connected components of the singular locus of the closed fiber. Let n_{t} ∈ ℕ^{T} be an irreducible element such that α(n_{t}) = 0. Let Z denote the set of singular points of X which map to n_{t}. Z is a closed set and we view Z as a subscheme with the reduced structure. What has to be shown is that the generic fiber of Z is non-empty and geometrically connected.

Let t_{0} be a point of the closed fiber whose corresponding element of ℕ^{T} is n_{t}. In an étale neighborhood of t_{0} we have an étale morphism

U → Spec(A[X_{1}, …, X_{n}]/X_{1} ⋯ X_{s})

for some n and s. From this it follows that the generic fiber of Z is not empty. From this local description it also follows that for each point s ∈ \underline{S} the fiber Z_{s} is reduced and that Z is flat over A. Therefore, the generic fiber of Z is geometrically connected if and only if

h^{0}(Z ⊗_{A} F rac(A), O_{Z⊗_{A}F rac(A)}) = 1

where F rac(A) denotes the field of fractions of A. This number is at least 1 since Z ⊗_{A} F rac(A) is not empty. On the other hand, the semi-continuity theorem ([14], III.12.8) combined with the fact that the closed fiber of Z is geometrically connected implies that the generic fiber of Z is geometrically connected. This implies the lemma.

Proposition 4.3.18 now follows from the two lemmas.

We can now complete the proof of 4.3.7. By the uniqueness we may assume that S is affine. Note first that if s is a point, and S_{s} = Spec(O_{S,s̄}) then by 1.3.9 there is an equivalence of categories

\underrightarrow{lim}_{U→S}(fine log structures on X ×_{S} U) → (fine log structures on X ×_{S} S_{s})

where the limit is taken over étale neighborhood U → S of s̄. If X is affine then this is exactly the statement of 1.3.9. In general, we can cover X by finitely many affines {V_{i}} and then the categories above can be described as collections of log structures on the V_{i} together with isomorphisms on the V_{i} ∩ V_{j} satisfying the cocycle condition. Since X is separated (it is proper over an affine scheme) the intersections are again affine and hence the statement follows from this case. It follows that if we have two collections of data

(M_{S,1}^{♯}, M_{X,1}^{♯}, ϕ_{1}, ψ_{1}), (M_{S,2}^{♯}, M_{X,2}^{♯}, ϕ_{2}, ψ_{2})

then there exists exactly one isomorphism between them. Indeed if s ∈ S is a point then after replacing S by an fppf-cover we can assume that we have a chart Q → M_{S} such that the map Q → \overline{M}_{S,s̄} is bijective. Over the strict henselization of the local ring of s there is a unique isomorphism, and this isomorphism can by the above discussion be extended to some étale neighborhood of s. It follows that fppf-locally there exists a unique isomorphism and hence by descent theory there exists a global isomorphism. Finally to prove existence, it suffices to prove existence in some fppf-neighborhood of a point s ∈ S. Thus we may assume that we have a chart Q → M_{S} such that Q → \overline{M}_{S,s̄} is bijective. We have shown that there exists data (M_{S}^{♯}, M_{X}^{♯}, ϕ, ψ) over the strict henselization of the local ring at s. From above we see that we can extend the data (M_{S}^{♯}, M_{X}^{♯}, ϕ, ψ) over the henselization to some étale neighborhood of s. After perhaps shrinking S we can assume that the resulting morphism is still log smooth. We claim that after perhaps shrinking S some more the morphism f^{♯} will be essentially semi-stable. We may assume that we have a chart ℕ^{T} → M_{S} where T denotes the set of connected component of the singular locus of Xs̄. Consider the set \underline{V} of points t ∈ X for which the map

(f^{−1}\overline{M}_{S}^{♯})t̄ → \overline{M}_{X,t̄}^{♯}

is as in the definition of essentially semi-stable. By the discussion in 4.3.3 the set \underline{V} is open and by construction \underline{V} contains every point of the fiber over s. Since f is proper the image of \underline{V}^{c} in \underline{S} is a closed set not containing s. Therefore by replacing S by the complement of f(\underline{V}^{c}) we may assume that f^{♯} is essentially semi-stable everyhwere. It was also shown above that the set of points where the morphism f^{♯} is special is open on the base and hence after shrinking S some more we have constructed datum (M_{S}^{♯}, M_{X}^{♯}, ϕ, ψ) satisfying the conditions of 4.3.7 in an fppf-neighborhood of s. This concludes the proof of 4.3.7.

Chapter 5

Moduli of log curves

Throughout this chapter S = Spec(ℤ) with the trivial log structure. F Log is the category of fine log schemes with the strict étale topology.

5.1 Statement of Theorem

Recall the definition of a stable curve:

Definition 5.1.1 ([8], [9]). Let g ≥ 2 (resp. g = 1) be an integer. A stable curve of genus g over a scheme \underline{X} is a proper, flat morphism f : \underline{C} → \underline{X} such that for every geometric point x̄ ∈ \underline{X}, the fiber \underline{C}x̄ is a reduced, connected 1-dimensional scheme satisfying:

1. \underline{C}x̄ has only ordinary double points.

2. If \underline{E} is a non-singular rational component of \underline{C}x̄ then \underline{E} meets the other components of \underline{C}x̄ in more than 2 points (resp. \underline{C}x̄ is a Néron polygon [9]).

3. \underline{C}x̄ has arithmetic genus g.

5.1.2. If g ≥ 2, let \underline{M}_{g} be the stack of stable curves of genus g, and if g = 1 let \underline{M}_{g} be the stack of irreducible pointed stable curves of genus 1. It is an algebraic stack, and the complement of the open substack \underline{M}_{g}^{0} ⊂ \underline{M}_{g} classifying smooth objects is a divisor \underline{Z} with normal crossings ([8], [24]). Hence there is a natural log structure on \underline{M}_{g}. By 1.3.8, \underline{M}_{g} determines a log algebraic stack M_{g} → F Log. The purpose of this chapter is to determine what objects this log algebraic stack classifies.

5.1.3. Let \underline{C} → \underline{M}_{g} be the universal curve. Then the inverse image of the divisor \underline{Z} in \underline{C} is again a divisor and hence determines a log structure on \underline{C}. The resulting morphism of log stacks C → M_{g} satisfies the conditions in the following definition: Definition 5.1.4. Let g ≥ 2 be an integer, and let X ∈ F Log. Then a stable log curve of genus g over X is a morphism f : C → X such that:

1. f is log smooth, proper, and integral.

2. For every geometric point x̄ ∈ \underline{X}, the fiber \underline{C}x̄ is a reduced connected curve of arithmetic genus g.

3. M_{C/X} is a sheaf of saturated and torsion free monoids.

4. For every geometric point x̄ → X

H^{0}(Cx̄, Der(Cx̄/x̄)) = 0.

Here Der(Cx̄/x̄) means the log derivations. Definition 5.1.5. A stable log curve f : C → X is vertical if M_{C/X} is a sheaf of groups. 5.1.6. If g ≥ 2, let M_{g}^{log} denote the stack over F Log of vertical stable log curves of genus g, and if g = 1 we denote by M_{g}^{log} the stack of vertical morphisms f : C → X satisfying 5.1.4.1-5.1.4.3 with a strict section e : X → C such that for every geometric point x̄ → X

H^{0}(Cx̄, Der(Cx̄/x̄)(−e)) = 0.

There is a natural functor

M_{g} → M_{g}^{log}.  (5.1)

The purpose of this chapter is to prove the following: Theorem 5.1.7. Let g ≥ 1 be an integer.

1. M_{g}^{log} is a log algebraic stack.

2. The functor 5.1 is an equivalence of log stacks.

5.2 Basic properties of log curves.

5.2.1. The results of this section are essentially contained in ([19]). Let f : C → X be a stable log curve, and let c ∈ \underline{C} be a point with x = f(c). Define

P = \overline{M}_{C,c̄}, Q = \overline{M}_{X,x̄}.

After perhaps replacing X by an fppf-cover we can find an étale neighborhood of c̄ and a chart for f

h : C → X ×_{Spec(ℤ[Q])} Spec(ℤ[P])

with h smooth ([29], 2.14).

Lemma 5.2.2. Suppose \underline{X} = Spec(k) with k separably closed and that M_{C/X,c̄} ≠ 0. Then h is étale and

M_{C/X,c̄}^{gp} ≃ ℤ.

Proof. Let G = M_{C/X,c̄}^{gp}. G is a free abelian group of positive rank. Since f is integral, the map

Spec(ℤ[P]) → Spec(ℤ[Q])

is flat, and consequently we have

1 = dim(C/X) ≥ dim(ℤ[P]/ℤ[Q]) = rank(G) > 0.

It follows that the rank of G is 1 and that the inequality is an equality.

Corollary 5.2.3. If f : C → X is a stable log curve and if c̄ ∈ C is a geometric point in the support of M_{C/X} then

M_{C/X,c̄} ≃ ℕ or ℤ.

Proof. Any fine saturated submonoid of ℤ is ℕ or ℤ.

Lemma 5.2.4. With notation as above, if c ∈ \underline{C} is a point with M_{C/X,c̄} ≃ ℕ then c is a smooth point.

Proof. Since \underline{C}/\underline{X} is flat we may assume that \underline{X} is the spectrum of a separably closed field k. Choose a chart as in 5.2.1. Then P/Q is isomorphic (canonically) to ℕ. Choose an element p_{1} ∈ P mapping to 1 as in 4.3.10. Then with the notation of 4.3.10 we have

np_{1} = p_{n}, for all n ∈ ℕ.

If not, we have np_{1} = p_{n} + q for some non-zero q ∈ Q and hence e(np_{1}) ⊗ 1 = (e(p_{1}) ⊗ 1)^{n} is zero in the ring

ℤ[P] ⊗_{ℤ[Q]} k.

Therefore, e(p_{1}) ⊗ 1 is nilpotent, which contradicts the assumption that the fibers are reduced. Thus we get a canonical isomorphism

ℕ ⊕ Q ≃ P.

Hence in an étale neighborhood of c,

\underline{C} ≃ Spec(k[t]).

Lemma 5.2.5. If M_{C/X,c̄} ≃ ℤ and if \underline{X} = Spec(k) with k separably closed then c̄ is an ordinary double point.

Proof. Continuing with the notation of 4.3.10, choose an isomorphism M_{C/X,c̄} ≃ ℤ and let p_{n} be the primitive element of P mapping to n. We claim that if n > 0 then p_{n} = np_{1} and if n < 0 then p_{n} = (−n)p_{−1}. If not, we can write

np_{1} = p_{n} + q, q ∈ Q

if n > 0 or

(−n)p_{−1} = p_{n} + q, q ∈ Q

if n < 0. This implies that the image of p_{1} (or p_{−1}) in k ⊗_{ℤ[Q]} ℤ[P] is nilpotent which is a contradiction. We therefore get a co-cartesian diagram

ℕ^{2} ⟶ P
↑ ↑
ℕ ⟶^{1↦p_{1}+p_{−1}} Q

and so

X ×_{Spec(ℤ[Q])} Spec(ℤ[P]) ≃ Spec(k[u, v]/uv).

Corollary 5.2.6. If (f : C → X) ∈ M_{g}^{log} then there exists a cartesian diagram unique up to unique isomorphism

\begin{matrix}C & ⟶^{β̃} & C^{†} \\ ^{f}↓ & & ^{f^{†}}↓ \\ X & ⟶^{β} & X^{†} \end{matrix}

where C^{†}/X^{†} is a special log curve in the sense of 4.3.6, and the morphisms \underline{C} → \underline{C}^{†} and \underline{X} → \underline{X}^{†} are isomorphisms.

Proof. The previous lemma implies that the hypotheses of 4.3.7 are satisfied.

Corollary 5.2.7. M_{g}^{log} is naturally equivalent to the log stack defined by the stack of special log curves with its natural morphism to LOG_{Spec(ℤ)} as in 3.5.2.

In light of definition 3.5.4 we might also refer to the special log curves as solid log curves.

Corollary 5.2.8. If C/X is an object of M_{g}^{log} then \underline{C}/\underline{X} is an object of \underline{M}_{g}.

Proof. It suffices to consider the case when \underline{X} is isomorphic to the spectrum of a separably closed field. In this case it follows from ([19], 1.2) that Ω_{C/X}^{1} is isomorphic to the dualizing sheaf of \underline{C}/\underline{X}. The corollary now follows from the definition in ([24]).

Remark 5.2.9. This last corollary (5.2.8) is the only result in this section which used the assumption 5.1.4.4.

5.3 Cohomological calculations.

5.3.1. The contents of this section are not needed to prove that M_{g}^{log} is a log algebraic stack but are necessary for showing that the functor 5.1 is an equivalence of log stacks. Let f : C → X be a special curve of genus g ≥ 2 where \underline{X} = Spec(A) for an artinian local ring A. Suppose I is an A-module of finite type killed by the maximal ideal of A (and hence a vector space over A_{0} = A/𝔪). Assume for this section that Schlessinger’s conditions (S1^{log}) and (S2^{log}) (3.5.8 and 3.5.10) hold. Then the 8-term exact sequence 2.1 in the proof of 2.2.13 gives in this case a short exact sequence

0 → E_{a}(I) → \overline{\underline{M}}_{g,A[I]}^{log} → Ext_{A^{♯}}(A, I) → 0

where the module E_{a}(I) is the cokernel of

i : Hom(\overline{M}_{A}^{gp}, I) → H^{1}(C, Der_{X}(C, I ⊗ O_{C})).

The map i is defined by identifying Hom(\overline{M}_{A}^{gp}, I) with the infinitesimal automorphisms of M_{A[I]} and sending such an automorphism σ : Spec(A[I]) → Spec(A[I]) to the class of

C ×_{X} Spec(A[I]) ⟶ Spec(A[I]) ⟶^{σ} Spec(A[I]).

Lemma 5.3.2. i is injective.

Proof. We can replace A by an étale cover and hence can assume that the sheaf \overline{M}_{A} is a constant sheaf. Let X′ = Spec(A[I]) and let C′/X′ be a lifting of C, and suppose σ : X′ → X′ is an automorphism coming from an infinitesimal automorphism of the log structure on X′. To prove that i is injective it suffices to show that C′_{σ} = C′ ×_{X′,σ} X′ is not isomorphic over X′ to C′/X′. Suppose

h : C′_{σ} → C′

were such an isomorphisms. The underlying schemes of C′_{σ} and C′ are the same and hence h induces an infinitesimal automorphism of \underline{C}′. On the other hand such infinitesimal automorphisms are classified by ([34], 4.4)

Ext^{0}(Ω_{\underline{C}/\underline{X}}, O\underline{C})

which is zero by ([8], 1.4). Thus the underlying morphism of schemes of h is the identity. On the other hand, if T denotes the singular points of the closed fiber of C, then σ is given by a map

∂ : ℕ^{T} → I

where we identify ℕ^{T} and \overline{M}_{A} via the canonical isomorphism given by the map in 4.3.5. Suppose t ∈ T is such that ∂(e_{t}) is not zero. The log structure on the completion Ô_{C′,t} is given by the pushout via

ℕ^{t} ↪ ℕ^{T}

from

ℕ^{2} ⟶ A[I][[x, y]]/(xy − a)
Δ↑ ↑
ℕ ⟶  A[I]

where the map ℕ^{2} → A[I][[x, y]]/(xy − a) is the standard one. Thus h induces an automorphism h^{b} of Ô_{C,t}^{∗} ⊕ ℕ^{2} compatible with the maps to Ô_{C′,t} and such that h^{b} ∘ Δ = Δ ∘ σ. It follows that h^{b} is of the form

(1, 0) ↦ λ(1 + yf) + (1, 0), (0, 1) ↦ λ(1 + xg) + (0, 1)

where f and g are elements of A[I][[x, y]]/(xy − a) with coefficients in I. Their product must be in the image of M_{A[I]} so f and g are both zero which means that h and hence σ is the identity.

5.3.3. Forgetting the log structure defines a morphism of sheaves

Der_{X}(C, I ⊗ O_{C}) → Der\underline{X}(\underline{C}, I ⊗ O_{C})

which can be shown to be injective by reasoning similar to the proof of the lemma. Let Q be the cokernel. From the exact sequence

0 → Der_{X}(C, I ⊗ O_{C}) → Der\underline{X}(\underline{C}, I ⊗ O_{C}) → Q → 0

we obtain a map

H^{1}(C, Der_{X}(C, I ⊗ O_{C})) → H^{1}(C, Der\underline{X}(\underline{C}, I ⊗ O_{C})).

This map is surjective for Q is a sheaf with support at the singular points of \underline{C} and hence H^{1}(C, Q) = 0.

Proposition 5.3.4. The map

H^{1}(C, Der_{X}(C, I ⊗ O_{C})) → H^{1}(C, Der\underline{X}(\underline{C}, I ⊗ O_{C})).

induces an isomorphism

E_{a}(I) ≃ H^{1}(C, Der\underline{X}(\underline{C}, I ⊗ O_{C})).

Proof. There is an injection

H^{1}(C, Der\underline{X}(\underline{C}, I ⊗ O_{C})) → Ext^{1}(Ω_{\underline{C}/\underline{X}}, I ⊗ O_{C})

coming from the spectral sequence of a composite of functors, and the composite

H^{1}(C, Der_{X}(C, I ⊗ O_{C})) → Ext^{1}(Ω_{\underline{C}/\underline{X}}, I ⊗ O_{C})

is the natural one coming from Ω_{\underline{C}/\underline{X}} → Ω_{C/X}. On the other hand, the module

Ext^{1}(Ω_{\underline{C}/\underline{X}}, I ⊗ O_{C})

is in bijection with flat liftings of the scheme \underline{C}/\underline{X} to \underline{X}′ by ([34], 4.4), and the map

H^{1}(C, Der_{X}(C, I ⊗ O_{C})) → Ext^{1}(Ω_{\underline{C}/\underline{X}}, I ⊗ O_{C})

can also be described as the map obtained by forgetting the log structure. This last statement can be seen as follows. Let {U_{i}} be an affine cover of C and let r : X′ → X be the natural retraction. Then a class in H^{1}(C, Der_{X}(C, I ⊗ O_{C})) can be described by a collection of sections D_{ij} ∈ Der_{X}(C, I ⊗ O_{C})(U_{ij}) over the U_{ij} = U_{i} ∩ U_{j} such that

D_{ij}|_{U_{ijk}} − D_{ik}|_{U_{ijk}} + D_{jk}|_{U_{ijk}} = 0.

The corresponding log smooth lifting of C to X′ is obtained by gluing U_{i} ×_{X,r} X′ and U_{j} ×_{X,r} X′ over U_{ij} via D_{ij} which can be thought of as an isomorphism between U_{ij} ×_{X,r} X′ → U_{ij} ×_{X,r} X′ ([20], 3.14). That is, we glue as in the following diagram

U_{ij} ×_{X,r} X′ ⟶^{D_{ij}} U_{ij} ×_{X,r} X′
↓ ↓
U_{i} ×_{X,r} X′  U_{j} ×_{X,r} X′.

If \underline{D}_{ij} denotes the image of D_{ij} in Der\underline{X}(\underline{C}, I ⊗ O\underline{C})(\underline{U}_{ij}) then the class

[D_{ij}] ∈ H^{1}(C, Der_{X}(C, I ⊗ O_{C}))

maps to the class

[\underline{D}_{ij}] ∈ H^{1}(C, Der\underline{X}(\underline{C}, I ⊗ O\underline{C})).

From the construction given in ([34], section 4) we see that the image of this class in

Ext^{1}(Ω_{\underline{C}/\underline{X}}, I ⊗ O_{C})

corresponds to the deformation of \underline{C} obtained by gluing using the \{\underline{D}_{ij}\}, which is just the deformation of \underline{C} obtained by forgetting the log structures. From this it follows that the map

H^{1}(C, Der_{X}(C, I ⊗ O_{C})) → H^{1}(C, Der\underline{X}(\underline{C}, I ⊗ O_{C})).

factors through E_{a}(I). E_{a}(I) and H^{1}(C, Der\underline{X}(\underline{C}, I ⊗ O_{C})) are both finite dimensional vector spaces over A_{0}. Since the map between them is surjective to prove that it is an isomorphism it suffices to prove that they have the same dimension. Let d be the dimension of I as an A_{0}-vector space, and let |T | be the number of singular points of C ⊗ A_{0}. The dimension of

H^{1}(C, Der_{X}(C, I ⊗ O_{C}))

is by ([13], 3.3.9) d(3g − 3) and the dimension of

Hom(\overline{M}_{X}^{gp}, I))

is d|T |. Thus the dimension of E_{a}(I) is

d(3g − 3 − |T |).

To compute the dimension of H^{1}(C, Der\underline{X}(\underline{C}, I ⊗ O_{C})) we may as well replace C by C ⊗ A_{0}. In fact we may even assume that A is an algebraically closed field. An element of H^{1}(C, Der\underline{X}(\underline{C}, I ⊗ O_{C})) corresponds to a vector field on the normalization ˜ \underline{C} which vanishes at all points lying over singular points of \underline{C}. Let \underline{C}_{i} be an irreducible component of \underline{C} and ˜ \underline{C}_{i} its normalization. Let q_{1}^{i}, …, q_{r_{i}}^{i} be the points of ˜ \underline{C}_{i} mapping to singular points of \underline{C}. Then the dimension of H^{1}(C, Der\underline{X}(\underline{C}, I ⊗ O_{C})) is equal to

∑_{i} h^{1}( ˜\underline{C}_{i}, Der\underline{X}( ˜\underline{C}_{i}, I ⊗ O)(−q_{1}^{i} − ⋯ − q_{r_{i}}^{i}))).

By Serre duality,

h^{1}( ˜\underline{C}_{i}, Der\underline{X}( ˜\underline{C}_{i}, I ⊗ O)(−q_{1}^{i} − ⋯ − q_{r_{i}}^{i}))) = h^{0}( ˜\underline{C}_{i}, Ω^{⊗2}( ˜\underline{C}_{i}, I ⊗ O)(q_{1}^{i} + ⋯ + q_{r_{i}}^{i}))).

which by the Riemann-Roch theorem is equal to

d(3g_{i} − 3 + r_{i}).

On the other hand, the genus g of C is given by the formula

g = (∑ g_{i}) + |T | − ν + 1

where ν is the number of irreducible components. It follows that the dimension of

H^{1}(C, Der\underline{X}(\underline{C}, I ⊗ O_{C}))

is also equal to

d(3g − 3 − |T |).

5.3.5. The cokernel of the map

H^{1}(C, Der\underline{X}(\underline{C}, I ⊗ O)) → Ext^{1}(Ω_{\underline{C}/\underline{X}}, I ⊗ O)

is equal to

H^{0}(C, Ext^{1}(Ω_{\underline{C}/\underline{X}}, I ⊗ O)).

This group also has a natural logarithmic interpretation. We state the following proposition slightly more generally because we found the more general statement interesting:

Proposition 5.3.6. Instead of assuming that I is killed by the maximal ideal of A assume just that I^{2} = 0. Then there is a natural map

β : Der_{A}(A[\overline{M}_{A}^{gp}], I) → H^{0}(\underline{C}, Ext^{1}(Ω_{\underline{C}/A}^{1}, f^{∗}I))

which induces an isomorphism

Coker(Hom(\overline{M}_{A}^{gp}, I) → Der_{A}(A[\overline{M}_{A}^{gp}], I)) ≃ H^{0}(\underline{C}, Ext^{1}(Ω_{\underline{C}/A}^{1}, f^{∗}I)).

Proof. First construct a map of sheaves

β : f^{∗}Der_{A}(A[\overline{M}_{A}^{gp}], I) → Ext^{1}(Ω_{\underline{C}/A}^{1}, f^{∗}I)

as follows. We have maps

C ⟶^{f} X ⟶ \underline{X}

which since f is smooth induces an exact sequence

0 → f^{∗}Ω_{X/\underline{X}}^{1} → Ω_{C/\underline{X}}^{1} → Ω_{C/X}^{1} → 0

and hence a class in

Ext^{1}(Ω_{C/X}^{1}, f^{∗}Ω_{X/\underline{X}}).

Note that there is an isomorphism

Ω_{X/\underline{X}}^{1} ≃ O_{X} ⊗ \overline{M}_{A}^{gp}.

From the natural map Ω_{\underline{C}/\underline{X}}^{1} → Ω_{C/X}^{1} and functoriality the above class induces an element

ξ_{C} ∈ Ext^{1}(Ω_{\underline{C}/\underline{X}}^{1}, O_{C} ⊗ \overline{M}_{A}^{gp}).

Note that in our case, \overline{M}_{A}^{gp} is isomorphic to the free monoid on the elements of T, and hence an element of Der_{A}(A[\overline{M}_{A}^{gp}], I) can be thought of as a T -tuple of elements \{α_{t}\} ∈ I. A section of f^{∗}Der_{A}(A[\overline{M}_{A}^{gp}], I) thus induces a map

O_{C} ⊗ \overline{M}_{A}^{gp} → f^{∗}I

by e_{t} ↦ α_{t}. By looking at the image of ξ_{C} a section of f^{∗}Der_{A}(A[\overline{M}_{A}^{gp}], I) defines a class in

Ext^{1}(Ω_{\underline{C}/A}^{1}, f^{∗}I).

This defines the map β. The sheaf Ext^{1}(Ω_{\underline{C}/A}^{1}, f^{∗}I) is a skyscraper sheaf with support at the singular points T. Choose a chart ℕ^{T} → A. If t ∈ T, then we can choose local coordinates

A[x, y]/(xy − α_{t})

where α_{t} is the image in A of the element e_{t} ∈ ℕ^{T}. The sheaf Ext^{1}(Ω_{\underline{C}/A}^{1}, f^{∗}I) can be described explicitly in terms of these coordinates as the cokernel of ([13] p. 100-101)

Hom(O_{C,t} < dx, dy >, I ⊗ O_{C,t}) → Hom(O_{C,t} < xy − α_{t} >, I ⊗ O_{C,t}).

Here O_{C,t} < dx, dy > denotes the free module on generators dx and dy and O_{C,t} < xy−α_{t} > denotes the free module with generator xy − α_{t}. If dx ↦ ϕ_{x} and dy ↦ ϕ_{y} is an element of Hom(O_{C,t} < dx, dy >, I ⊗ O_{C,t}) then its image is the map which sends xy − α_{t} to ϕ_{y} ⊗ x + ϕ_{x} ⊗ y. It follows from direct calculation that the cokernel is isomorphic to I/α_{t}I and hence

Ext^{1}(Ω_{\underline{C}/A}^{1}, f^{∗}I) ≃ ⊕_{t}(I/α_{t}I)_{t}.

It is straightforward to check that if ∂ ∈ Der_{A}(A[ℕ^{T}], I) then the associated element of Ext^{1}(Ω_{\underline{C}/A}^{1}, f^{∗}I) is the one coming from the map

xy − α_{t} ↦ ∂(e_{t}).

From this the proposition follows.

Now

Coker(Hom(\overline{M}_{A}^{gp}, I) → Der_{A}(A[\overline{M}_{A}^{gp}], I))

is naturally identified with Ext_{A^{♯}}(A, I). For we have a diagram

A → A[\overline{M}_{A}^{gp}] → A

and hence and exact sequence

Der_{A[\overline{M}_{A}^{gp}]/A}(I) → Ext_{A[\overline{M}_{A}^{gp}]}(A, I) → Ext_{A^{♯}}(A, I) → Ext_{A}(A[\overline{M}_{A}^{gp}], I).

By 3.2.10

Ext_{A}(A[\overline{M}_{A}^{gp}], I) = 0

and we can identify

Hom(\overline{M}_{A}^{gp}, I) ≃ Der_{A[\overline{M}_{A}^{gp}]/A}(I).

Finally to give an element in

Ext_{A[\overline{M}_{A}^{gp}]}(A, I)

is equivalent to giving a map

A[\overline{M}_{A}^{gp}] → A[I]

which induces the canonical map to A. This in turn is equivalent to an element of

Der_{A}(A[\overline{M}_{A}^{gp}], I).

Combining the propositions we get the following theorem Theorem 5.3.7. Suppose I is killed by the maximal ideal of A. Then there is a commutative diagram obtained by forgetting the log structures

E_{a}(I)  →  \overline{\underline{M}}_{g,a,A[I]}^{log}  →  Ext_{A^{♯}}(A, I)
↓  ↓  ↓
H^{1}(C, Der\underline{X}(\underline{C}, I ⊗ O)) → Ext^{1}(Ω_{\underline{C}/\underline{X}}, I ⊗ O) → H^{0}(C, Ext^{1}(Ω_{\underline{C}/\underline{X}}, I ⊗ O))

in which all vertical arrows are isomorphisms. 5.3.8. The case when g = 1 is similar. Let (C, e)/X be a special curve. By the same reasoning as in 5.3.2, the map

Hom(\overline{M}_{A}^{gp}, I) → Ext^{1}(Ω_{C/X}(e), I ⊗ O)

is injective. Moreover,

Ext^{1}(Ω_{C/X}(e), I ⊗ O) ≃ H^{1}(C, I ⊗ O(−e)) ≃ I

since Ω_{C/X} is trivial. Thus since the map is injective it is an isomorphism. Consequently,

\overline{\underline{M}}_{g,a,A[I]}^{log} ≃ Ext_{A^{♯}}(A, I).

Similarly, deformations of (\underline{C}, e)/\underline{X} are classified by

Ext^{1}(Ω_{\underline{C}/\underline{X}}(e), I ⊗ O).

We have

h^{1}(C, Der\underline{X}(\underline{C}, I ⊗ O(−e))) = h^{1}(C̃, Der\underline{X}( ˜\underline{C}, I ⊗ O(−e − (o) − (∞))) = 0

and therefore

Ext^{1}(Ω_{\underline{C}/\underline{X}}(e), I ⊗ O) ≃ H^{0}(C, Ext^{1}(Ω_{\underline{C}/\underline{X}}(e), I ⊗ O)) ≃ H^{0}(C, Ext^{1}(Ω_{\underline{C}/\underline{X}}, I ⊗ O))

(note that Ext^{1}(Ω_{\underline{C}/\underline{X}}(e), I ⊗ O) is supported away from e). Combining this with the argument in 5.3.6 we obtain:

Theorem 5.3.9. Forgetting the log structures induces a commutative diagram of isomorphisms

\overline{\underline{M}}_{1,a,A[I]}^{log}  ⟶  Ext_{A^{♯}}(A, I)
↓  ↓
Ext^{1}(Ω_{\underline{C}/\underline{X}}(e), I ⊗ O) ⟶ H^{0}(C, Ext^{1}(Ω_{\underline{C}/\underline{X}}, I ⊗ O)).

5.4 Verification of 3.5.13

5.4.1. Using the results of the preceding sections, we now verify the conditions of 3.5.13 for M_{g}^{log}, and thereby prove that M_{g}^{log} is a log algebraic stack. 5.4.2 (Obstruction Theory). Let A → A_{0} be a morphism of log rings as in 3.5.11 with I = Ker(A → A_{0}) a square zero ideal. Suppose C_{0}/Spec(A_{0}) is a log stable curve. It is immediate from the definition of log stable curve that any log smooth lifting of C_{0} is again a log stable curve. It follows from ([20] 3.14.4) that the obstruction to finding a log smooth lifting of C_{0} lies in

H^{2}(C_{0}, Hom(Ω_{C_{0}/X_{0}}^{1}, IO_{C_{0}}))

which is zero since C_{0}/X_{0} is a curve ([14], III.11.2). In the case of g = 1, it follows from ([20], 3.11) that the section also lifts. Thus we take for our obstruction theory O the zero modules. 5.4.3 (Schlessinger’s Criteria). (S1^{log}). This follows from 3.2.2 and 3.2.5 and the fact that we know that the functor M_{g}^{log} is unobstructed (one could also argue directly from the constructions in 3.2.2 and 3.2.5 that the liftings obtained from these theorems are log smooth). (S2^{log}). By ([20], 3.14.3), the set of liftings of a log stable curve f : C_{0} → Spec(A_{0}) to A_{0}[I] are classified by

H^{1}(C_{0}, Hom_{C_{0}}(Ω_{C_{0}/A_{0}}^{1}, f^{∗}I)).

In the case when C_{0} is pointed, say e : Spec(A_{0}) → C_{0} it follows as in the classical case that the set of liftings of the pair (C_{0}, e) are classified by

H^{1}(C_{0}, Hom_{C_{0}}(Ω_{C_{0}/A_{0}}^{1}, f^{∗}I ⊗ O_{C_{0}}(−e))).

In both cases they are A_{0}-modules of finite type. Again by ([20], 3.14.2), the infinitesimal automorphisms of a lifting of C_{0}/A_{0} to A_{0}[I] are classified by

H^{0}(C_{0}, Hom_{C_{0}}(Ω_{C_{0}/A_{0}}^{1}, f^{∗}I)).

It follows that the module Aut_{a_{0}}(A_{0}[I]) is equal to this cohomology group when g ≥ 2 and when g = 1 is equal to the kernel of

H^{0}(C_{0}, Hom_{C_{0}}(Ω_{C_{0}/A_{0}}^{1}, f^{∗}I)) → H^{0}(Spec(A_{0}), Hom_{A_{0}}(e^{∗}Ω_{C_{0}/A_{0}}^{1}, I)).  (5.2)

The following lemma shows that (S2)^{log} holds.

Lemma 5.4.4. Aut_{a_{0}}(A_{0}[I]) = 0.

Proof. We prove that when g ≥ 2, the module H^{0}(C_{0}, Hom_{C_{0}}(Ω_{C_{0}/A_{0}}^{1}, f^{∗}I)) = 0 and in the case g = 1 that the map in 5.2 is an isomorphism. We first reduce to the case when I ≃ A_{0}. I is finitely generated so there exists a resolution

0 = I_{0} ⊂ I_{1} ⊂ ⋯ ⊂ I_{n} = I

such that I_{i}/I_{i−1} ≃ A_{0}/𝔭_{i} for some prime 𝔭_{i} ⊂ A_{0}. Since Ω_{C_{0}/A_{0}}^{1} is locally free, and f is a flat morphism, the sequence

0 → Hom_{C_{0}}(Ω_{C_{0}/A_{0}}^{1}, f^{∗}I_{i−1}) → Hom_{C_{0}}(Ω_{C_{0}/A_{0}}^{1}, f^{∗}I_{i}) → Hom_{C_{0}}(Ω_{C_{0}/A_{0}}^{1}, f^{∗}(A_{0}/𝔭_{i})) → 0

is exact. Let C_{i} = C_{0} ×_{A_{0}} (A_{0}/𝔭_{i}) and let j : C_{i} ↪ C be the closed immersion. Then

Hom_{C_{0}}(Ω_{C_{0}/A_{0}}^{1}, f^{∗}(A_{0}/𝔭_{i})) ≃ j_{∗}Hom_{C_{i}}(Ω_{C_{i}}^{1}, O_{C_{i}}).

Applying the global section functor and induction now gives the reduction step for g ≥ 2. To handle the case when g = 1, note that since Ω_{C_{0}}^{1} is the dualizing sheaf, it follows that Ω_{C_{0}}^{1} is trivial in this case. Thus the above exact sequence can be rewritten as

0 → f^{∗}I_{i−1} → f^{∗}I_{i} → j_{∗}O_{C_{i}} → 0

and hence taking cohomology gives a diagram

0 ⟶  I_{i−1}  ⟶  I_{i}  ⟶  A_{i}/𝔭_{i}  ⟶ 0
↓  ↓  ↓
0 ⟶ H^{0}(C_{0}, f^{∗}I_{i−1}) ⟶ H^{0}(C_{0}, f^{∗}I_{i}) ⟶ H^{0}(C_{i}, O_{C_{i}}).

Since C_{0}/A_{0} is proper, the right vertical arrow is an isomorphism and by induction the left arrow is also. The five lemma now implies that the middle arrow is an isomorphism. Thus we may assume that I = A. The case g = 1 now follows from the above argument since Ω_{C_{0}/A_{0}}^{1} is trivial. For g ≥ 2, the case when A is a field is given in ([8], 1.4) and the general case follows from this by standard base change results.

5.4.5 (Additional Conditions 3.5.12). These conditions follows from the above descriptions of the modules together with standard results on cohomology. 5.4.6 (Condition 3.5.13.2). In this case the functor is an equivalence of categories. The full faithfulness follows from the following lemma and the usual Grothendieck existence theorem applied to the underlying schemes:

Lemma 5.4.7. Fix a complete noetherian local ring A, and let M_{A} be a log structure on A. Suppose \underline{C}/A is a stable curve and that M_{C}^{1} and M_{C}^{2} are two M_{A}-log structures on \underline{C}/A such that the resulting log schemes C^{1} = (\underline{C}, M_{C}^{1}), C^{2} = (\underline{C}, M_{C}^{2}) over (Spec(A), M_{A}) are special log curves. Then given a compatible collection of isomorphisms

ϕ_{n} : M_{C_{n}}^{1} → M_{C_{n}}^{2}

over C_{n}, there exists a unique isomorphism ϕ : M_{C}^{1} ≃ M_{C}^{2} inducing the ϕ_{n}.

Proof. It suffices to construct ϕ after a finite étale extension of A by the uniqueness. Therefore, we may assume that we have a chart ℕ^{T} → M_{A} where T denotes the set of singular points of the closed fiber. Note that the canonical isomorphism ℕ^{T} → \overline{M}_{A,s_{0}} (where s_{0} denotes the closed point) defined by the map in 4.3.5 depends only on the closed fiber of C, and hence C^{1} and C^{2} define the same isomorphism ℕ^{T} → \overline{M}_{A,s_{0}} which we lift to a chart. Let t ∈ T be a singular point. In some étale neighborhood U of t we have by remark 4.3.3 a map

β^{i} : ℕ^{2} → M_{C}^{i}

such that the diagram

ℕ^{2} ⟶^{β^{i}} M_{C}^{i}
Δ↑ ↑
ℕ^{t} ⟶ ℕ^{T}

commutes (here Δ denotes the diagonal), and such that the resulting map

ℕ^{2} ⊕_{ℕ} ℕ^{T} → M_{C}^{i}

is a chart for M_{C}^{i}. At the stalk at t, we have an isomorphism

ℕ^{2} ⊕_{ℕ} ℕ^{T} ⟶^{≃} \overline{M}_{C,t̄}^{1} ⟶^{ϕ_{n}} \overline{M}_{C,t̄}^{2} ⟶^{≃} ℕ^{2} ⊕_{ℕ} ℕ^{T}

and after perhaps composing β^{1} by the automorphism of ℕ^{2} that switches the generators we may assume that the ϕ_{n} induce the identity map

ℕ^{2} ⊕_{ℕ} ℕ^{T} → ℕ^{2} ⊕_{ℕ} ℕ^{T} .

Let (x, y) be the pair (α_{M_{C}^{1}}(β^{1}(1, 0)), α_{M_{C}^{1}}(β^{1}(0, 1))) of elements of O_{C} and let (u, v) be the pair (α_{M_{C}^{2}}(β^{2}(1, 0)), α_{M_{C}^{2}}(β^{2}(0, 1))). To give an isomorphism ϕ between M_{C}^{1} and M_{C}^{2} over U such that the map

ϕ ∘ β^{1} : ℕ^{2} → \overline{M}_{C}^{1} → \overline{M}_{C}^{2}

equals

β^{2} : ℕ^{2} → \overline{M}_{C}^{2}

is equivalent to giving units a, b ∈ O_{C}^{∗} such that ab = 1 and

x = au, y = bv.

We claim that a solution to these equations is unique if it exists. To see this suppose we have two solutions (a, b) and (a′, b′). The completion of O_{C} at t is isomorphic to

A[[u, v]]/(uv − α(e_{t}))

where α(e_{t}) ∈ 𝔪_{A} is the image if 1 ∈ ℕ^{t}. Therefore, if x = au = a′u we must have

(a − a′)u = 0

and hence a − a′ ∈ (v). Similarly, if y = bv = b′v we must have b − b′ ∈ (u). Thus we can write a′ = a + vf and b′ = b + ug for some f and g. We show that vf and ug are zero modulo 𝔪_{A}^{n} for all n. If n = 1, then α(t) = 0 so we have

1 = a′b′ = (a + vf)(b + ug) = ab + bvf + aug = 1 + bvf + aug

which implies that vf = 0 and ug = 0. Thus assume true for n − 1. Then (vf)(ug) is zero modulo 𝔪^{n} and hence modulo 𝔪^{n} we again have the equation

1 = a′b′ = (a + vf)(b + ug) = ab + bvf + aug = 1 + bvf + aug.

This implies the uniqueness of the pair (a, b). Since we are given ϕ_{n} over C_{n} it follows from the Artin approximation theorem ([4]) that étale locally we have elements a, b as above. By the uniqueness of the solutions to the equations

x = Xu, y = Y v, XY = 1

it follows that the isomorphisms constructed locally glue, and that the resulting global isomorphism M_{C}^{1} ≃ M_{C}^{2} induces the ϕ_{n}.

The essential surjectivity follows from:

Lemma 5.4.8. Let A be a complete local ring with residue field k of finite type and let \underline{C}/A be a stable curve. Suppuse further that a log structure M_{A} on A is given together with a compatible collection of M_{A}-log structures M_{C_{n}} on \underline{C} ⊗ (A/𝔪^{n}) making C_{n}/A_{n} a special log curve. Then there exists a unique log structure on \underline{C} inducing those on the C_{n} making C/A a special log curve.

Proof. By the uniqueness we may assume that the log structure on A is induced by a chart

ℕ^{s} → A.

Let W denote the ring of Witt vectors of k if char(k) > 0 and k if char(k) = 0. Consider the functor F on artinian local W [[t_{1}, …, t_{s}]]-algebras Q with residue field k for which F (Q) consists of isomorphism classes of log smooth liftings of C_{1}/k to Q where Q is given the log structure associated to the images of the t_{i}. We claim that the functor F is prorepresentable. By ([32], 2.11) it suffices to show that given a deformation situation (see 2.1.2)

B ; ↓ ; A′ ⟶ A

of W [[t_{1}, …, t_{s}]]-algebras with residue field k, the natural map

F (A′ ×_{A} B) → F (A′) ×_{F (A)} F (B)

is bijective, and that the tangent space F (k[ϵ]) is finite dimensional (these are conditions (S1) and (S2) of 2.1.4 and 2.1.6 for F). This follows from (S1^{log}) and (S2^{log}) for M_{g}^{log} and the fact that we know that a special log curve admits no infinitesimal automorphisms (5.4.4). Indeed given curves C_{1}/A′ and C_{2}/B defining a class

([C_{1}], [C_{2}]) ∈ F (A′) ×_{F (A)} F (B).

We can view C_{1} in a unique way as a lifting of C_{2,A} to A′ since there are no infinitesimal automorphisms. Thus (S1^{log}) implies that there exists a curve C/A′ ×_{A} B lifting C_{1} and C_{2}. If C̃/A′ ×_{A} B is a second such curve, then there are unique isomorphisms C̃_{A′} ≃ C̃_{A′} and C̃_{B} ≃ C_{B} and hence (S1^{log}) again implies that C̃ ≃ C. That (S2) follows from (S2^{log}) is clear, and hence F is pro-representable. By 5.4.2 F is formally smooth, and hence F is pro-represented by a formally smooth W [[t_{1}, …, t_{s}]]-algebra R. Thus over the reductions R_{n} of R we have a compatible family (\underline{C}_{n}^{u}, M_{C_{n}^{u}}) which is the universal data. By the Grothendieck existence theorem ([10], III.5) and 4.2.8 there exists a log curve (\underline{C}^{u}, M_{C^{u}}) over R inducing the (\underline{C}_{n}^{u}, M_{C_{n}^{u}}). Pulling back (\underline{C}^{u}, M_{C^{u}}) via the map R → A induced by the C_{n} we get a log curve C̃ over A inducing the (\underline{C}_{n}, M_{C_{n}}). Applying the Grothendieck existence theorem again we conclude that there is a unique isomorphism \underline{C̃} ≃ \underline{C} compatible with the isomorphism on reductions. This proves existence of M_{C} on \underline{C} and uniqueness follows from 5.4.7.

5.4.9 (Separation Condition). Condition 3.5.13.3 follows from the following lemma which shows that there is a dense open set where the underlying automorphism of schemes is trivial and lemma 3.3.4. Lemma 5.4.10. Let A_{0} be a reduced ring, and let π : \underline{X} → Spec(A_{0}) be a separated flat morphism of reduced schemes. Then if ϕ : \underline{X} → \underline{X} is an automorphism of \underline{X} over Spec(A_{0}) such that for a dense set of points s ∈ Spec(A_{0}), ϕ ⊗ k(s) = id, then ϕ = id.

Proof. Let \underline{U} = \underline{X} − Γ_{ϕ}^{−1}(Δ_{X}). Then \underline{U} is an open subscheme of \underline{X}. Since π is flat π(\underline{U}) is open in Spec(A_{0}). Moreover, π(\underline{U})^{c} contains a dense set of points. Hence π(\underline{U}) equals the empty set.

5.5 Comparison with [8], [9]

Proposition 5.5.1. Let F_{1} and F_{2} be log algebraic stacks, and suppose G : F_{1} → F_{2} is a functor such that

1. G is fully faithful.

2. If X ∈ F Log has \underline{X} isomorphic to the spectrum of an artinian local ring, then G induces a bijection on isomorphism classes

\overline{F}_{1,X} → \overline{F}_{2,X}.

Then G is an equivalence of log algebraic stacks.

Proof. If x : X → F_{1}, y : Y → F_{1} are 1-morphisms, then we abuse notation and write X ×_{F_{2}} Y for the product G(X) ×_{F_{2}} G(Y). Note that the full faithfulness implies that in this situation there is a natural isomorphism

X ×_{F_{1}} Y ≃ X ×_{F_{2}} Y.

Choose X → F_{1} which is smooth, relatively strict and surjective. Then F_{1} is equivalent to the stack associated to the groupoid X ×_{F_{1}} X ⇉ X. Since G is fully faithful, to prove that G is an equivalence it suffices that the induced 1-morphism X → F_{2} is also smooth, strict, and surjective. Suppose Y → F_{2} is a 1-morphism. To prove that X ×_{F_{2}} Y → Y is strict and surjective it suffices to consider the case when \underline{Y} is the spectrum of a field. But in that case, by assumption 2, the 1-morphism Y → F_{2} arises from a 1-morphism Y → F_{1}. To prove that X ×_{F_{2}} Y → Y is smooth, consider a diagram

X ×_{F_{2}} Y ⟵ Z_{0}
↓ ↓
Y  ⟵ Z

where \underline{Z} is isomorphic to the spectrum of an artinian local ring, and Z_{0} → Z is a closed immersion defined by a nilpotent ideal. From assumption 2, the composite Z → Y → F_{2} is induced from a 1-morphism Z → F_{1} and from this it follows that there exists a lifting filling in the diagram.

5.5.2. In the case of the functor (5.1.6)

G : M_{g} → M_{g}^{log}

the full faithfulness follows from 4.3.7. To verify the second condition proceed as follows. Suppose that A′ → A is a surjective morphism between artinian local log rings with kernel I, an A/𝔪-module of finite type, and that a ∈ M_{g,A} is obtained from a strict morphism to M_{g}. Then we claim that the map

\overline{M}_{g,a,A′} → \overline{M}_{g,G(a),A′}^{log}  (5.3)

is bijective. From the full faithfulness the map is injective. Thus suppose C/A′ is an object of \overline{M}_{g,G(a),A′}^{log}. Certainly there exists a log structure N on A′ lifting M_{A} and a morphism

(Spec(A′), N) → M_{g}

lifting a, such that the induced log curve C′/(Spec(A′), N) has underlying scheme isomorphic to \underline{C} (just pull back the log structure on M_{g} via the map Spec(A′) → \underline{M}_{g} defined by \underline{C}). Now the set of liftings of M_{A} to A′ are a torsor under

Ext_{A^{♯}}(A, I).

Therefore M_{A′} − N defines a class in this group which from 5.3.7 and 5.3.9 is the image of the class of \underline{C} − \underline{C}′ under the map given in those theorems. It follows that the class is zero and that N and M_{A′} are isomorphic as liftings of M_{A}. Fix such an isomorphism and view C′ as a lifting of a to A′. The isomorphism classes of log smooth liftings are a torsor under

H^{1}(C ⊗ A/𝔪, Der_{A/𝔪}(C ⊗ A/𝔪, I ⊗ O_{C⊗A/𝔪}))

and the class of C − C′ maps to zero in

\overline{\underline{M}}_{g,A[I]}^{log}

It follows from this and 5.3.1 that there exists an infinitesimal automorphism σ of A′ such that

C′ ⟶ Spec(A′) ⟶^{σ} Spec(A′)

is isomorphic to C/Spec(A′). This proves the bijectivity of the map (5.3). It follows that it is enough to verify that G induces a bijection on isomorphism classes over separably closed fields. Thus the following lemma concludes the proof of 5.1.7: Lemma 5.5.3. Let k be a separable closed field, M_{k} a log structure on Spec(k), X = (Spec(k), M_{k}), and suppose \underline{C}/\underline{X} is a curve. If M_{C}^{1} and M_{C}^{2} are two M_{k}-log structures on \underline{C} which make C/X a special log curve then there exists a unique automorphism σ : M_{k} → M_{k} and an isomorphism

(\underline{C}, M_{C}^{1}) ≃ (\underline{C}, M_{C}^{2}) ×_{X,σ} X.

Proof. Let T be the set of singular points of \underline{C} and choose a chart

ℕ^{T} → M_{k}.

If t ∈ T let ℕ^{2} → M_{C}^{1} and ℕ^{2} → M_{C}^{2} be charts inducing isomorphisms on the strict henselizations

(k[X, Y]/XY)^{sh} ≃ O_{C,t̄} ≃ (k[U, V]/UV)^{sh}.

By 4.2.9 there exists units ζ_{1}, ζ_{2} ∈ O_{C,t̄}^{∗} such that

X = ζ_{1}U, Y = ζ_{2}V, or X = ζ_{1}V, Y = ζ_{2}U.

Moreover by the proof of 4.2.9 we see that we can choose ζ_{1}, ζ_{2} such that ζ_{1}ζ_{2} = a_{t} ∈ k and that the condition of having their product in k determines the ζ_{i} uniquely. Define σ on M_{k} by sending the generator of ℕ^{T} corresponding to t to λ(a_{t}) + e_{t}. Then we see that for each geometric point x̄ → \underline{C} there exists a unique morphism σ̃x̄ making the diagram

M_{C,x̄}^{1} ⟶^{σ̃x̄} M_{C,x̄}^{2}
↑ ↑
M_{k} ⟶^{σ} M_{k}

commute. Since our log structures are fine, we can extend the σ̃x̄ to étale neighborhoods and therefore obtain maps σ̃_{i} on some étale cover {U_{i}} of C. By the uniqueness the induced maps p_{1}^{∗}σ̃_{i} and p_{2}^{∗}σ̃_{i} agree on U_{i} ×_{C} U_{j} and hence glue to give a global isomorphism.

Chapter 6

Modular Curves: First

Construction

Throughout this chapter N will denote a fixed integer greater than 0, and S denotes Spec(ℤ[1/N]) with the trivial log structure.

6.1 Full level N structure.

6.1.1. Let M_{N}^{log} be the fibered category over F Log such that for X ∈ F Log the category M_{N,X}^{log} consists of data (f : C → X, e, i) such that

A. f is a vertical morphism satisfying conditions 1-3 of 5.1.4.

B. e ∈ C(X) is a strict section.

C. i is an inclusion of groups

i : (ℤ/Nℤ)^{2} ↪ Aut\underline{X}(\underline{C}).

D. For every geometric point x̄ ∈ X, the fiber Cx̄ is either a smooth curve or isomorphic to the Néron N-gon ([9]) and p_{a}(\underline{C}x̄) = 1.

E. If \underline{C}x̄ is smooth then the action of (ℤ/Nℤ)^{2} commutes with the canonical group law on (\underline{C}x̄, e).

F. If \underline{C}x̄ is singular then (ℤ/Nℤ)^{2} acts trivially on P ic^{0}(\underline{C}x̄).

G. Etale locally on X there exists σ ∈ (ℤ/Nℤ)^{2} which extends to an isomorphism of log schemes C → C over X such that for each geometric point x̄ → \underline{X}, the restriction σx̄ of σ to the fiber Cx̄ acts transitively on the irreducible components of \underline{C}x̄.

We hope that the ambiguity of the notation M_{N}^{log} as compared to our earlier notation of M_{g}^{log} for the moduli stack of stable curves does not lead to much confusion. Both unfortunately seem to be standard notations in their respective contexts. The main result of this chapter is the following theorem:

Theorem 6.1.2. M_{N}^{log} is a log algebraic stack, and is naturally equivalent to the stack M_{N} of ([9], III.2.1) endowed with the log structure corresponding to the divisor at infinity.

The proof of this theorem will occupy the following 4 sections. Remark 6.1.3. We will show (6.3.1) that the data of (C) above can be replaced with an inclusion of groups

i : (ℤ/Nℤ)^{2} ↪ Aut_{X}(C)

(automorphisms of C/X as a log scheme).

6.2 Solid objects.

Definition 6.2.1. An object (f : C → X, e, i) is solid if there exists locally on X and C a chart

ℕ^{2} ⟶ Γ(C, M_{C})
Δ↑ ↑
ℕ ⟶ Γ(X, M_{X})

for f such that the induced map

\underline{C} → \underline{X} ×_{ℤ[ℕ]} ℤ[ℕ^{2}]

is étale.

Remark 6.2.2. In 3.5.4 we had a notion of a solid object of a stack over the category of log schemes. Our abuse of language is justified by corollary 6.2.4 below.

Theorem 6.2.3. Given (f : C → X, e, i) ∈ M_{N}^{log} there exists a solid object (f^{♯} : C^{♯} → X^{♯}, e^{♯}, i^{♯}) and a morphism whose underlying morphism of schemes is an isomorphism

X → X^{♯}

such that (f : C → X, e, i) is obtained by base change. This data is unique in the sense of 4.3.7.

Proof. Uniqueness. Let x̄ ∈ X be a geometric point, t ∈ Cx̄ a singular point. Then \overline{M}_{C,t̄}^{♯} is the submonoid of \overline{M}_{C,t̄} generated by the irreducible elements which are not in the image of \overline{M}_{X,x̄}, and \overline{M}_{X,x̄}^{♯} is the inverse image of \overline{M}_{C,t̄}^{♯} in \overline{M}_{X,x̄}. From this the uniqueness follows. Existence. By remark 5.2.9, theorem 4.3.7 applies. Let

ϕ : M′_{X} → M_{X}, ψ : M′_{C} → M_{C}

be the log structures provided by that theorem (we write M′ instead of M^{♯} so as not to confuse the notation). Define M_{X}^{♯} to be the image of ϕ and M_{C}^{♯} to be the pushout

\begin{matrix}f^{∗}M′_{X} & ⟶ & f^{∗}M_{X}^{♯} \\ ↓ & & ↓ \\ M′_{C} & ⟶ & M_{C}^{♯}. \end{matrix}  (6.1)

We claim that the pair (M_{X}^{♯}, M_{C}^{♯}) has the desired properties. To see this we may assume that \underline{X} is isomorphic to the spectrum of a strictly henselian local ring. Let T be the set of singular points of the closed fiber. Then we have decompositions (see 4.3.17)

M′_{X} ≃ ⊕_{t∈T,O_{X}^{∗}} K̃_{t}, M′_{C} ≃ ⊕_{t∈T,O_{C}^{∗}} L̃_{t}.

For each t ∈ T, the map K̃_{t} → M_{X} is injective and induced from a map ℕ → M_{X}. Recall that this decomposition of M′_{X} arose from a canonical isomorphism

ℕ^{T} ≃ \overline{M}′_{X,x_{0}}

where x_{0} denotes the closed point. K̃_{t} is simply the inverse image in M′_{X} of the subsheaf of \overline{M}′_{X} defined by the image of ℕ^{t} ↪ ℕ^{T}. Therefore, if σ is an automorphism of C/X then it follows from the construction of K̃_{t} that

K̃_{t} = K̃_{σ(t)}.

Combining this with 6.1.1.G we see that all the K̃_{t} are equal when viewed as sub-log structures of M_{X}. It follows from 4.3.6 that the diagram (6.1) locally admits a chart

ℕ^{2} ⊕ ℕ^{|T |−1} ⟶ ℕ^{2}
Δ×1↑ ↑^{Δ}
ℕ ⊕ ℕ^{|T |−1} ⟶ ℕ.

This implies that (M_{X}^{♯}, M_{C}^{♯}) has the desired form.

Corollary 6.2.4. M_{N}^{log} is naturally equivalent to the log stack defined by the stack of solid objects with its natural morphism to LOG_{Spec(ℤ[1/N])} as in 3.5.2.

6.3 The action of (ℤ/Nℤ)^{2}.

Theorem 6.3.1. Let (f : C → X, e, i) ∈ M_{N}^{log} be a solid object. Then for every σ ∈ (ℤ/Nℤ)^{2} there exist a unique endomorphism σ_{C} : C → C of C as a log scheme over X whose underlying morphism of schemes is i(σ).

Lemma 6.3.2. Let \underline{X} be a scheme of finite type over \underline{S}, and let

F : (\underline{X} − schemes) → Set

be a relatively representable (1.2.8.2), limit preserving functor such that

1. F is a sheaf for the étale topology.

2. The natural morphism of functors F → \underline{X} is formally étale. That is, given any

diagram

Spec(A_{0}) ⟶^{z_{0}} F
↓ ↓
Spec(A) ⟶ \underline{X}

where A → A_{0} is a surjective morphism between artinian local rings, there exists a unique lifting z : Spec(A) → F of z_{0}.

3. For every complete local ring Â with residue field of finite type over \underline{X}, the natural

map

F (Â) → \underleftarrow{lim} F (Â/𝔪^{n})

is bijective.

4. If k is a field, then F (k) consists of one element.

Then the natural morphism of functors F → \underline{X} is an isomorphism.

Proof. We verify the conditions of ([2], 3.4). Conditions (2) and (3) imply that the functor F is effectively pro-representable.

Condition (2) implies that for any

y_{1} : \underline{Y}_{1} → F, y_{2} : \underline{Y}_{2} → F

with \underline{Y}_{1} and \underline{Y}_{2} locally of finite type over \underline{X}, the map

\underline{Y}_{1} ×_{F} \underline{Y}_{2} → \underline{Y}_{1} ×\underline{X} \underline{Y}_{2}

is an open immersion. Now since the map F → \underline{X} is formally étale, a map \underline{Y} → F is formally étale at a point p ∈ \underline{Y} if and only if the composite \underline{Y} → \underline{X} is formally étale at p. Therefore the set of points where \underline{Y} → F is formally étale is open. This implies the lemma.

6.3.3. With notation as in the theorem, consider the functor F on \underline{X}-schemes which to any \underline{T} /\underline{X} associates the set of automorphisms σ_{C} as in 6.3.1 for C ×_{X} T/T. Since F classifies morphism between étale sheaves, F is a sheaf with respect to this topology. By 1.3.9 F is also limit preserving. If

σ_{C} : \underline{Z}_{1} → F, σ′_{C} : \underline{Z}_{2} → F

are two elements of F, then the product \underline{Z}_{1} ×_{F} \underline{Z}_{2} is the functor over \underline{Z}_{1} ×\underline{X} \underline{Z}_{2} expressing the condition that the two maps

p_{1}^{∗}σ_{C}^{b}, p_{2}^{∗}σ_{C}^{′,b} : σ^{∗}M_{C} → M_{C}

are equal. By 3.1.5 applied to the diagram

σ^{∗}M_{C} ⟶^{p_{1}^{∗}σ_{C}^{b}} M_{C}
p_{2}^{∗}σ_{C}^{′,b}↓
M_{C}

we see that \underline{Z}_{1} ×_{F} \underline{Z}_{2} is represented by a locally closed subscheme of \underline{Z}_{1} ×\underline{X} \underline{Z}_{2}. 6.3.4. In order to understand the deformation theory of objects of M_{N}^{log} we need to make sense of quotients by finite group actions. Suppose X is a log scheme and that G is a finite group acting on X such that the orbit of any point is contained in an affine open set of \underline{X}. Then the scheme-theoretic quotient \underline{X}/G exists. Moreover, if the action of G is free, then the map

π : \underline{X} → \underline{X}/G

is étale. Suppose we are in this situation. Then the G-action defines descent datum on M_{X} relative to the morphism π and hence we get a fine log structure M_{X/G} on \underline{X}/G. Explicitly, the log structure is given by

\underline{U} ↦ (M_{X}(\underline{U} ×_{\underline{X}/G} \underline{X}))^{G}.

In what follows we denote by X/G the quotient \underline{X}/G with this log structure, and π : X → X/G is the natural morphism of log schemes.

Lemma 6.3.5. Let C/X be a solid object of M_{N}^{log} and let σ ∈ (ℤ/Nℤ)^{2} be an element satisfying 6.1.1.G. Choose an extension of σ to an automorphism of C/X. Then the log scheme C/ < σ > is a special object of M_{1}^{log} (see 5.2.6 for the definition of special object).

Proof. The action of σ on \underline{C} is free and hence the map

\underline{C} → \underline{C}/ < σ >

is finite and étale. The extension of σ to C/X defines descent datum for the X-log structure on C to \underline{C}/ < σ >. The question of whether the induced object is special is étale local and hence follows from the fact that C/X is solid.

Lemma 6.3.6. The map F → \underline{X} is formally étale.

Proof. The question is étale local on \underline{X} so we may assume that we have an element σ ∈ (ℤ/Nℤ)^{2} as in 6.1.1.G. Choose an extension of σ to an automorphism σ : C → C over X. Suppose now that A → A_{0} is a surjective morphism of artinian local rings with kernel I killed by the maximal ideal of A. Denote by (f_{0} : C_{0} → A_{0}, e_{0}, i_{0}) the induced object over A_{0} and by (f : C → A, e, i) the induced object over A. We suppose that F (A_{0}) is not empty. Fix an element g ∈ (ℤ/Nℤ)^{2}. Since g commutes with σ it induces a map which we denote by the same letter

g : \underline{C}/ < σ >→ \underline{C}/ < σ > .

By assumption we have a map between objects of M_{1}^{log}

g_{C_{0}} : (C_{0}/ < σ >, e_{0}) → (C_{0}/ < σ >, g(e)_{0}).

Using g_{C_{0}} we can view (C/ < σ >, e) as a log smooth lifting of (C_{0}/ < σ >, g(e)_{0}).

It follows from the fact that M_{1}^{log} is algebraic that there exists a unique infinitesimal automorphism ϵ of M_{A} and an isomorphism h such that the diagram

(C/ < σ >, e) ⟶^{h} (C/ < σ >, g(e))
f↓  ^{f}↓
Spec(A) ⟶^{ϵ}  Spec(A)

commutes and h reduces to g_{C_{0}} modulo the maximal ideal. Since an irreducible pointed nodal curve of genus 1 admits no infinitesimal automorphisms, the underlying map of schemes associated to h must be equal to g. It follows that for all i ∈ ℤ, h^{i} defines a map

(C/ < σ >, e) ⟶^{h^{i}} (C/ < σ >, (ig)(e))
f↓  ^{f}↓
Spec(A) ⟶^{ϵ^{i}}  Spec(A).

In particular, for i = N we have a commutative diagram

(C/ < σ >, e) ⟶^{h^{N}} (C/ < σ >, e)
f↓  ^{f}↓
Spec(A) ⟶^{ϵ^{N}} Spec(A).

From 5.3.2 we conclude that ϵ^{N} is the identity. On the other hand, the set of infinitesimal automorphisms of M_{A_{0}} is an A_{0} module and hence multiplication by N is injective. It follows that ϵ is the identity. We now have a diagram

(C, e)  (C, g(e))
↓ ↓
(C/ < σ >, e) ⟶^{h} (C/ < σ >, g(e)).

The vertical maps are étale and strict, and hence to find a map

C → C

filling in the diagram, it suffices to construct a map of schemes. On the other hand, the étale sites of C and C_{0} are equivalent so such a map is obtained from the one given over A_{0}. This proves that F → \underline{X} is formally étale.

6.3.7. Now let Â be a complete local ring and consider the map

F (Â) → \underleftarrow{lim} F (Â/𝔪^{n})

Suppose g ∈ (ℤ/Nℤ)^{2}, and we are given a compatible system of isomorphisms

g_{n} : C_{n} → C_{n}

extending the action of g on \underline{C}. It follows from 5.4.7 that there exists an isomorphism g^{∗}M_{C} ≃ M_{C} on \underline{C}. It remains to verify that F (k) consists of a single element for every algebraically closed k. Thus suppose k is an algebraically closed field. By ([9], II.3.2) there exists on \underline{C} a unique structure of a generalized elliptic curve such that the action of (ℤ/Nℤ)^{2} is given by translation by N-torsion points. Moreover, the arguments of the proof of 5.5.3 show that the log structure on C is isomorphic to the natural one. In other words, if t is a singular point of C then the completion of the local ring at t is isomorphic to

k[[x, y]]/(xy)

with the standard log structure. Identifying the local rings via the translation under which the log structure is invariant, we see that the action on k[[x, y]]/(xy) of an element in (ℤ/Nℤ)^{2} is given by x ↦ ζx, y ↦ ζ^{−1}y for an N-th rooth of unity ζ. From this it follows that there is a unique extension to a morphism of log schemes over Spec(k). This concludes the proof of theorem 6.3.1.

6.4 Proof that M_{N}^{log} is algebraic.

We verify the conditions of 3.5.13.

Proposition 6.4.1. Let A′ → A be a strict surjection of log rings with nilpotent kernel I such that I^{2} = 0, (C′, e′)/A′ ∈ M_{1}^{log} a special object, and (C, e)/A the reduction of C′. Then for any automorphism (which need not fix e)

g : C → C

such that g^{N} is the identity, there exists a unique lifting

g′ : C′ → C′

of g for which g^{′N} = id.

Proof. The set of liftings of g are a torsor under the infinitesimal automorphisms of C′/C which are given by

H^{0}(C, Der_{A}(C, I)).

If g′ and h′ are two liftings of g, then for all i ∈ ℤ, we get a class

g^{′i} − h^{′i} ∈ H^{0}(C, Der_{A}(C, I)).

The formula

g^{′i}h^{′−i} = g^{′(i−1)}(g′h^{′−1})g^{′(1−i)}g^{′(i−2)}(g′h^{′−1})g^{′(2−i)} ⋯ g′h^{′−1}

implies that this class is equal to

∑_{j=0}^{i−1} Lie(g^{j})(g′ − h′).

Now in our case, the action of Lie(g) is trivial by ([9], II.3.2 and the description of the Tate curve VII.1.12). Therefore, we get the formula

g^{′i} − h^{′i} = i(g′ − h′).

This implies the uniqueness of the lifting. For if g′ and h′ both satisfy the conditions of the proposition, then

N(g′ − h′) = g^{′N} − h^{′N} = 0

which is only possible if g′ − h′ = 0 since N is invertible. As for existence, we observe that the obstruction to lifting g to any map is a class

c ∈ H^{1}(C, Der_{A}(C, I)).

For if α : C′ ⊗_{A′} A → C is the map making C′ a lifting of C, then the existence of a lifting h′ is equivalent to the statement that the two log smooth liftings of C

(C′, α) and (C′, g ∘ α)

are isomorphic. The class c is defined by

c = (C′, g ∘ α) − (C′, α) ∈ H^{1}(C, Der_{A}(C, I)).

It follows from the construction of c that for any i ∈ ℤ the class

(C′, g^{i+1} ∘ α) − (C′, g^{i} ∘ α) ∈ H^{1}(C, Der_{A}(C, I))

is also equal to c. Therefore, for any i ∈ ℤ we have an equality

ic = (C′, g^{i} ∘ α) − (C′, α).

Since g^{N} ∘ α we must have Nc = 0 and hence c = 0 since N is invertible. This shows that we can lift g to some h′. Let

∂ = id − h^{′N} ∈ H^{0}(C, Der_{A}(C, I))

and let g′ = h′ + \frac{1}{N} ∂. Then

id − g^{′N} = (id − h^{′N}) − ∂ = 0

and hence g^{′N} = 0.

Corollary 6.4.2. With notation as above, if C ∈ M_{N,A}^{log} is a solid object, σ as in 6.1.1.G, and C/ < σ >∈ M_{1,A}^{log} the corresponding log elliptic curve, then the natural functor

M_{N,A′,C}^{log} → M_{1,A′,C/<σ>}^{log}

is an equivalence of categories.

Proof. Since the map

C → C/ < σ >

is strict and étale, the functor

(liftings of C to A′) → (liftings of C/ < σ > with (ℤ/Nℤ)^{2}/ < σ >-action)

is an equivalence of categories. The result therefore follows from the proposition.

6.4.3. Let A′ → A be a strict surjection of log rings, with nilpotent kernel, and C/A ∈ M_{N}^{log} a formally solid object. If C′/A′ is a lifting of C/A, then it follows from the corollary that the map

Aut_{M_{N,A′}^{log}} (C′) → Aut_{M_{N,A}^{log}}(C)

is bijective. Indeed both sides are sheaves on the étale site of A′ and hence the assertion can be verified étale locally. Thus we may assume that we have a σ as in 6.1.1.G and in this case the automorphisms of C and C′ are in bijection with the automorphisms of C/ < σ > and C′/ < σ > which respect the (ℤ/Nℤ)^{2}/ < σ >-action.

The abscence of infinitesimal automorphisms implies that the functor on étale A-algebras

L : B ↦ {isomorphism classes of liftings of C/A to B ×_{A} A′}

is a sheaf (here B ×_{A} A′ has the log structure induced from A′). As in 5.3.1, there is a natural action of

Der_{A}(A, I)

on this sheaf which is locally simply transitive by the corresponding result for M_{1}^{log}. Since the sheaf Der_{A}(A, I) is coherent the functor L is smooth. In addition, this implies that given a deformation situation (3.5.7) and C ∈ M_{N,A}^{log} the natural map

\overline{M}_{N,C,A′×_{A}B} → \overline{M}_{N,C,A′} × \overline{M}_{N,C,B}

is bijective. It is injective because any automorphism of C lifts. To prove surjectivity, let

(C_{A′}, C_{B}) ∈ \overline{M}_{N,C,A′} × \overline{M}_{N,C,B}

and choose an element C̃ ∈ \overline{M}_{N,C,A′×_{A}B} whose image in \overline{M}_{N,C,B} is equal to C_{B}. Let C̃_{A′} be the image of C̃ in \overline{M}_{N,C,A′}. Then the difference

∂ = C̃_{A′} − C_{A′} ∈ Der_{A}(A, I) ≃ Der_{B}(B, I).

Therefore, C̃ + ∂ maps to the pair (C_{A′}, C_{B}). The bijectivity of

\overline{M}_{N,C,A′×_{A}B} → \overline{M}_{N,C,A′} × \overline{M}_{N,C,B}

combined with the fact that all automorphisms lift implies that the functor

M_{N,C,A′×_{A}B} → M_{N,C,A′} × M_{N,C,B}

is an equivalence of categories. This verifies 3.5.13.1. 6.4.4. Condition 3.5.13.2 follows from the same argument as in 5.4.7-5.4.9. Suppose {C_{n}} is a compatible family of solid objects of M_{N}^{log} over the reductions Â/𝔪^{n} of a complete noetherian local log ring Â. We show that there exists a unique object in M_{N,Â}^{log} inducing the C_{n}. The uniqueness follows from 5.4.7 and the usual Grothendieck existence theorem. To prove essential surjectivity we may by descent theory and the uniqueness replace Â by an étale cover and therefore can assume that we have σ ∈ (ℤ/Nℤ)^{2} as in 6.1.1.G for C_{0}. In addition we may assume that we have a chart

ℕ → Â

for the log structure on Â. Let W be the ring of Witt vectors of k = Â/𝔪 if char(k) > 0 and k otherwise, and define F to be the functor on artin local W [[ℕ]]-algebras Q with residue field k for which F (Q) is equal to

\overline{M}_{N,C_{0},Spec(ℕ→Q)}

(notation as in 2.3.3). Here the map ℕ → Q is the map induced from W [[ℕ]] → Q. As shown above, the functor F is isomorphic to the deformation functor of C_{0}/ < σ > and hence is pro-representable by a formally smooth W [[ℕ]]-algebra. 3.5.13.2 follows by the usual Grothendieck existence theorem and 4.2.8. Finally condition 3.5.13.3 follows from 3.3.3 and 5.4.10. This concludes the proof of the statement that M_{N}^{log} is a log algebraic stack.

6.5 Comparison with ([9], III.2.1).

The following theorem of Deligne-Rapoport shows that an object of M_{N}^{log} admits a canonical structure of a generalized elliptic curve:

Theorem 6.5.1 ([9], II.3.2). Let C/X be a solid object of M_{N}^{log}. Then there exists a unique structure of a generalized elliptic curve on (\underline{C}/\underline{X}, e) such that the action of (ℤ/Nℤ)^{2} is given by translation by N-torsion points on \underline{C}^{reg}.

6.5.2. Let M_{N} be the classifying stack of generalized elliptic curves with full level Nstructure ([9], IV.2.4), and give M_{N} the log structure corresponding to its divisor at infinity. Denote by E the tautological family of M_{N} endowed with the log structure coming from the inverse image of the divisor at infinity on M_{N}. A concrete description of these log structures is given by the Tate curve ([9], VII.2.4). In particular, for any morphism

\underline{T} → \underline{M}_{N}

we obtain a solid object of M_{N}^{log} by pulling back the log structures and the tautological family E. This defines a functor

G : M_{N} → M_{N}^{log}.

It follows from 6.5.1 that this functor is fully faithful. From the argument in 5.5.3, if \underline{X} is isomorphic to the spectrum of a field then

G : \overline{M}_{N,X} → \overline{M}_{N,X}^{log}

is bijective. By the discussion in 6.4.3, if A → A_{0} is a surjection of artinian local log rings with kernel I (an A_{0}-module of finite type), then for any a ∈ M_{N,A_{0}}^{log} the set

\overline{M}_{N,a,A}^{log}

is a torsor under the set of infinitesimal automorphisms of M_{A} over M_{A_{0}}. It follows that the map

\overline{M}_{N,a,A} → \overline{M}_{N,a,A}^{log}

is surjective and hence bijective by the full faithfulness. This proves that 5.5.1.2 holds, and hence completes the proof of 6.1.2.

Chapter 7

Modular Curves: Second

Construction

As in the preceding chapter, S denotes the scheme Spec(ℤ[1/N]) with the trivial log structure, where N is a fixed integer greater than 0.

7.1 Valuative Log Spaces.

Throughout this section all log schemes will be schemes with log structure on the Zariski site. The results of this section are due to K. Kato (unpublished, outlined in [16]).

Definition 7.1.1. A log space is a pair Z = (\underline{Z}, M_{Z}) consisting of a locally ringed space \underline{Z} together with an integral log structure M_{Z} on \underline{Z}. Z is valuative if for every point z ∈ \underline{Z}, M_{Z,z} has the property that for each a ∈ M_{Z,z}^{gp} either a or −a is in M_{Z,z}.

If M is an abstract integral monoid, we also say that M is valuative if for each a ∈ M^{gp} either a or −a is in M.

Lemma 7.1.2. Let M be an integral monoid.

1. M is valuative if and only if every finitely generated ideal is principal.

2. If M is valuative then M is saturated. M is valuative if and only of M/M^{∗} is valuative.

3. Suppose M is valuative and t, a ∈ M^{gp} are elements such that there exists integers

i, j ∈ ℤ such that

−it + a ∈ M and jt − a ∈ M.

Then there exists an integer n ∈ ℤ such that we can take i = n, j = n + 1. If m is a second integer with this property then either t ∈ M^{∗} or m = n ± 1 and not both n + 1 and n − 1 work.

Proof. (1) Suppose M is an integral monoid such that every ideal is principal. To prove that M is valuative it suffices to show that the set

{±a|a ∈ M} ⊂ M^{gp}

is a group. For this it suffices to show that for any a, b ∈ M either a − b or b − a is in M. Let K be the ideal generated by a and b. By assumption K is principal, say generated by k ∈ M. It follows that there exists natural numbers n_{a}, n_{b} ∈ ℤ not both zero and m ∈ M such that

k = m + n_{a}a + n_{b}b.

Without loss of generality assume n_{a} ≠ 0. Since a ∈ K, there exists m_{2} ∈ M such that

a = m_{2} + k.

Let m_{3} = m + (n_{a} − 1)a + n_{b}b. Then

k = m_{2} + m_{3} + k

and consequently m_{2}, m_{3} ∈ M^{∗}. It follows that K is generated by a which implies that b = a + m for some m ∈ M, and hence b − a ∈ M. The case when n_{b} ≠ 0 gives a − b ∈ M. Suppose now that M is valuative, and let K be a finitely generated ideal; say generated by a_{1}, …, a_{r} ∈ M. For each i, either a_{i} − a_{i−1} or a_{i−1} − a_{i} is in M by the valuative condition, and consequently one of the sets a_{1}, …, â_{i}, …, a_{r} or a_{1}, …, â_{i−1}, …, a_{r} generate K. By induction on the number of generators we conclude that K is principal.

(2) If a ∈ M^{gp}, then either a or −a is in M. If na ∈ M for n > 0, and −a ∈ M then −na ∈ M and hence na ∈ M^{∗}. It follows that a ∈ M^{∗}. This proves that M is saturated if M is valuative. The second statement is immediate.

(3) If it + a ∈ M for all i ∈ ℤ and for some j ∈ ℤ, jt − a ∈ M. Then −jt + a and jt − a are both in M, hence are units. This implies that t and a are both units. Assume this is not the case.

Then there exists i ∈ ℤ such that −it + a ∈ M but (−i − 1)t + a ∉ M. Therefore (i + 1)t − a ∈ M, and we can take n = i. Suppose n and m both have the desired property and that |n − m| > 1. Then

(−nt + a) + ((m + 1)t − a), (−a + (n + 1)t) + (a − mt)

are both in M. This implies that (n − m + 1)t, (m − n + 1)t ∈ M. It follows that t ∈ M^{∗} which we assumed was not the case.

7.1.3. Let X be a fine log scheme, and let K ⊂ M_{X} be a coherent sheaf of ideals. Consider maps

f : Z → X

where Z is a log space with the property that for every z ∈ Z, the ideal generated by K in M_{Z,z} is principal. Lemma 7.1.4. There exists a universal such map

b : B_{K}(X) → X.

Moreover, B_{K}(X) is a fine log scheme.

Proof. The question is local on X so we may assume that X is given as Spec(ℤ[P]) where P is a fine monoid, and that the sheaf of ideals is induced from some ideal K ⊂ P. Denote by P^{·} the graded monoid

P^{·} = ∐_{n=0}^{∞} K^{n},

and let S^{·} be the graded ring ℤ[P^{·}]. For each k ∈ K let P_{k}^{·}[0] be the monoid consisting of the degree zero elements of the monoid P_{k}^{·} obtained by inverting k ∈ K^{1}. Let \underline{B}_{K}(X) be the scheme

P roj(S^{·}).

There is a natural cover \{\underline{U}_{k}\}_{k∈K} where

\underline{U}_{k} = Spec(ℤ[P_{k}^{·}[0]]).

Each \underline{U}_{k} has a natural log structure, and there are natural isomorphisms between these log structures on the \underline{U}_{k} ∩ \underline{U}_{k′}. Thus the log structures glue to make B_{K}(X) a log scheme. The natural map P → P^{·} induces a morphism of log schemes

b : B_{K}(X) → X.

It follows from the construction that this is the desired map.

7.1.5. Now suppose X is a log space. Associated to X are two functors X^{val} and B(X) defined as follows. X^{val} is the functor on valuative log spaces

Z ↦ Hom(Z, X).

B(X) is the functor on the category of log spaces which to any Z associates the set of morphisms f : Z → X such that for all z ∈ Z and finitely generated ideal, K ⊂ \overline{M}_{X,f(z)}, the ideal generated by K in \overline{M}_{Z,z} is principal.

Lemma 7.1.6. Suppose X is locally isomorphic to Spec(P → ℤ[P]) where P is a fine monoid. Then B(X) is representable by a log space locally of the form \underleftarrow{lim} X_{i} where each X_{i} is a log scheme locally isomorphic to Spec(P_{i} → ℤ[P_{i}]) where P_{i} is a fine monoid.

Proof. The question is local on X so without loss of generality we may assume that X = Spec(P → ℤ[P]). For every ideal K ⊂ P let B_{K}(X) be the log blowup defined in 7.1.4. The B_{K}(X) form a projective system, and we let

X̃ = \underleftarrow{lim} B_{K}(X).

If f : Z → X defines an object of B(X)(Z), then for every ideal K ⊂ P and point z ∈ Z, the ideal generated by K in M_{Z,z} is principal. It follows that there exists a unique factorization of the map f through X̃. On the other hand, let x̃ ∈ X̃ be a point. Then

\overline{M}_{X̃,x̃} = \underrightarrow{lim} \overline{M}_{B_{K}(X),x_{K}}.

If K_{x} ⊂ \overline{M}_{X,x} is an ideal, then the inverse image of K in P is an ideal whose image generates K. It follows that the image of K in \overline{M}_{X̃,x̃} generates a principal ideal. From this it follows that X̃ represents B(X).

Theorem 7.1.7. Suppose X is a fine log scheme. Then the functor X^{val} is representable.

Proof. By the universal property, it follows that the question is local on X. Moreover, choosing a chart locally we may assume that

X = Spec(P → ℤ[P]).

We now construct log spaces B^{r}(X) for r ∈ ℤ inductively together with morphisms B^{r}(X) → B^{r−1}(X) which represent B(B^{r−1}(X)) → B^{r−1}(X). Moreover, each B^{r}(X) will be of the form \underleftarrow{lim} B^{r−1}(X_{i}) where X_{i} is a log scheme locally isomorphic to Spec(Q → ℤ[Q]) for a fine monoid Q. Let B^{0}(X) = X, B^{1}(X) = B(X). To construct B^{r}(X) from B^{r−1}(X), write B^{r−1}(X) = \underleftarrow{lim} B^{r−2}(X_{i}) where X_{i} is of the above form. The collection

{B(B^{r−2}(X_{i}))}_{i}

form a projective system by functoriality and we let B^{r}(X) = \underleftarrow{lim} B(B^{r−2}(X_{i})). There is a natural map B^{r}(X) → B^{r−1}(X) by the universal property of inverse limit. It is straightforward to verify that this map has the correct properties. From this construction we obtain a projective system {B^{r}(X)} and we let

B^{·}(X) = \underleftarrow{lim} B^{r}(X).

This space together with the natural map B^{·}(X) → X represents X^{val}. To prove this recall the characterization of valuative monoids 7.1.2.1. Suppose x̃ ∈ B^{·}(X) is a point. Then

\overline{M}_{B^{·}(X),x̃} = \underrightarrow{lim}_{r} \overline{M}_{B^{r}(X),x̃_{r}}.

If K ⊂ \overline{M}_{B^{·}(X),x̃} is a finitely generated ideal then K is induced from some ideal K_{r} ⊂ \overline{M}_{B^{r}(X),x̃_{r}} for some r. But K_{r} induces a principal ideal in \overline{M}_{B^{r+1}(X),x̃_{r+1}} and hence K is principal. It follows that B^{·}(X) is valuative. Suppose f : Z → X is a morphism where Z is valuative. Then by the universal property and 7.1.2.1 there exists a unique lifting of f to B(X). Again by the universal property this map lifts to B^{2}(X). By induction we obtain a compatible system of maps Z → B^{r}(X) lifting f and hence a unique map

Z → B^{·}(X).

It follows that B^{·}(X) represents X^{val}.

From this point on we do not distinguish between the functor X^{val} and the space which represents it. 7.1.8. Suppose X, Y, and Z are fine log schemes. Then it follows that the valuative log space

(X ×_{Z} Y)^{val}

represents the functor on valuative log spaces

X^{val} ×_{Z^{val}} Y^{val}.

Therefore, the full subcategory of the category of valuative log spaces consisting of spaces which are locally of the form X^{val} for some fine log schemes X has fiber products. We refer to this subcategory as the category of algebraic valuative log spaces. It follows that a group object in the category of algebraic valuative log spaces is equivalent to a valuative log space X together with morphisms

+ : X × X → X

and

− : X → X

such that the usual diagrams commute.

7.2 The Kato-Tate Curve.

7.2.1. Let \underline{T} be a scheme, and let t ∈ Γ(\underline{T} , O_{T}) be a global section. Let T denote the log scheme obtained by giving \underline{T} the log structure coming from ℕ → O_{T}, 1 ↦ t. Construct a log scheme 𝔾_{m}^{t} over T as follows. 𝔾_{m}^{t} is covered by open sets \{U_{i+ \frac{1}{2}} \}_{i∈ℤ} given by

U_{i+ \frac{1}{2}} = \underline{T} [x_{i}, y_{i+1}]/(x_{i}y_{i+1} − t)

with the log structure coming from ℕ^{2} → O_{U_{i+ \frac{1}{2}}} , (1, 0) ↦ x_{i}, (0, 1) ↦ y_{i+1}. U_{i+ \frac{1}{2}} and U_{i− \frac{1}{2}} are glued along

U_{i+ \frac{1}{2}} [1/x_{i}] ≃ \underline{T} [x_{i}^{±1}] ≃ \underline{T} [y_{i}^{±1}] ≃ U_{i− \frac{1}{2}} [1/y_{i}]

via x_{i} = 1/y_{i}. This isomorphism extends to an isomorphism of log schemes, and 𝔾_{m}^{t} is by definition the log scheme obtained by gluing. On U_{i+ \frac{1}{2}} let α_{i} ∈ Γ(U_{i+ \frac{1}{2}} , M_{𝔾_{m}^{t}}^{gp}) be the section corresponding to (−i − 1, −i) ∈ ℤ^{2} (i.e. t^{−1−i}y_{i+1}). From the definitions it follows that α_{i} and α_{i−1} define the same section over

U_{i+ \frac{1}{2}} ∩ U_{i− \frac{1}{2}} .

Let α ∈ Γ(𝔾_{m}^{t}, M_{𝔾_{m}^{t}}) be the resulting global section.

Proposition 7.2.2. If Z is a valuative log space, then there is a natural bijection between 𝔾_{m}^{t}(Z) and the set of a ∈ Γ(Z, M_{Z}^{gp}) such that for each point z ∈ Z, there exists integers i ≤ j such that −a − it and jt + a are both in M_{Z,z}. This bijection is given by

(f : Z → 𝔾_{m}^{t}) ↦ image of α in Γ(Z, M_{Z}^{gp}).

Proof. To give a map Z → U_{i+ \frac{1}{2}} is equivalent to giving a pair (g_{1}, g_{2}) ∈ Γ(Z, M_{Z}), where g_{1} is the image of x_{i} and g_{2} is the image of y_{i+1}. For such a map the global section α maps to (−i − 1)t + g_{2} ∈ Γ(Z, M_{Z}^{gp}). Note that −it − α = t − g_{2} = g_{1} and hence this defines a morphism of functors from U_{i+ \frac{1}{2}} to the functor which to any Z associates the set of elements a ∈ Γ(Z, M_{Z}^{gp}) such that (i+1)t+a and −it−a are both in Γ(Z, M_{Z}). On the other hand, given such an a ∈ Γ(Z, M_{Z}^{gp}) we can recover the pair (g_{1}, g_{2}) as (−it − a, (i + 1)t + a), and so U_{i+ \frac{1}{2}} represents this functor. Note that a ∈ Γ(Z, M_{Z}^{gp}) maps to

U_{i+ \frac{1}{2}} ∩ U_{i− \frac{1}{2}}

if and only if both i and i + 1 have the property of 7.1.2.3. The gluing of the U_{i+ \frac{1}{2}} is done exactly so that the intersection represents the functor of a’s such that both i and i + 1 work. Suppose now that a ∈ Γ(Z, M_{Z}^{gp}) and that for each point z ∈ Z there exists integers i ≤ j such that −it + a and jt − a are both in M_{Z,z}. By 7.1.2.3 we may assume that j = i + 1. Moreover, since M_{Z} is a sheaf we can cover Z by open sets V such that over V the same integer i works for each z ∈ V. From the above there exists a unique map

V → U_{(−i−1)+ \frac{1}{2}}

such that a|V is the image of α. By the uniqueness these maps glue to give a map

V → 𝔾_{m}^{t}.

This proves the proposition.

7.2.3. If Z is a valuative log space, then there is a natural group structure on the set of elements a ∈ Γ(Z, M_{Z}^{gp}) which have the property described in proposition 7.2.2. This gives 𝔾_{m}^{t} the structure of a group functor. Denote by

(𝔾_{m}^{t})^{val} → 𝔾_{m}^{t}

the universal valuative log space associated to the 𝔾_{m}^{t} by 7.1.7. By the universal property it is a group object in the category of algebraic valuative log spaces. If Z is a valuative log space, then there is an action of ℤ on 𝔾_{m}^{t}(Z) given by a ↦ a + t. This action can be described on the log scheme by the map

U_{i+ \frac{3}{2}} → U_{i+ \frac{1}{2}} , x_{i} ↦ x_{i+1}, y_{i+1} ↦ y_{i+2}.

Also the automorphism a ↦ −a on the functor 𝔾_{m}^{t} can be described on the log scheme by

U_{i+ \frac{1}{2}} → U_{−i− \frac{1}{2}} , x_{−i−1} ↦ y_{i+1}, y_{−i} ↦ x_{i}.

7.2.4. There are certain functors associated to (𝔾_{m}^{t})^{val} which will be used. To define these functors, we need to be more careful about the distinction between the element of t ∈ Γ(T, O_{T}) and the section of M_{T} which we have also been denoting by t. Therefore, for the purposes of this paragraph, let us denote by t^{†} the section of M_{T} and reserve t for the element in Γ(T, O_{T}). If Z is a valuative log space, define

(𝔾_{m}^{t})^{val}[N](Z) = {a ∈ Γ(Z, M_{Z}^{gp})| for all z ∈ Z ∃ j ∈ ℤ s.t. Na = jt^{†} in M_{Z,z}^{gp}},
(E^{t})^{val}[N](Z) = (𝔾_{m}^{t})^{val}[N](Z)/ℤ

where the action of ℤ is given by a ↦ a + t^{†}. If j is an integer, we also define

[N, j](Z) = {a ∈ Γ(Z, M_{Z}^{gp})| Na = jt^{†}}.

Suppose now that our chosen t ∈ Γ(T, O_{T}) is not a unit in O_{T,t} for any t ∈ T. That is, t is locally nilpotent. In this case, if a ∈ (𝔾_{m}^{t})^{val}[N](Z) for some connected valuative log space Z, then there exists a unique integer j such that

Na = jt^{†}

globally. The point is that if V_{1}, V_{2} ⊂ Z are open sets such that Na = j_{1}t^{†} on V_{1} and Na = j_{2}t^{†} on V_{2}, then on V_{1} ∩ V_{2}, (j_{1} − j_{2})t^{†} ∈ M_{Z}^{∗} and hence j_{1} = j_{2}. It follows that under this hypothesis there is an isomorphism of functors

(𝔾_{m}^{t})^{val}[N] ≃ ∐_{j∈ℤ}[N, j].

Also if a ∈ (𝔾_{m}^{t})^{val}[N](Z) then there exists a unique integer

j ∈ {0, 1, …, N − 1}

and s ∈ ℤ such that N(a + st) = jt^{†}. It follows that there is an isomorphism of functors

(E^{t})^{val}[N] ≃ ∐_{j=0}^{N−1}[N, j].

The following explicit description of [N, j] will be important in what follows:

Proposition 7.2.5. Let j be an integer, and let r = gcd(N, j). Choose integers α and β such that r = αN + βj. The functor [N, j] is represented by the fine, saturated, valuative log scheme

\underline{T} [w_{1}, w_{2}]/(w_{1}^{r} = 1, w_{2}^{N/r}w_{1}^{β} = t)

with log structure induced from ℕ via 1 ↦ w_{2}. The composite

[N, j] → (𝔾_{m}^{t})^{val} → 𝔾_{m}^{t}

maps [N, j] into the intersection of those U_{i+ \frac{1}{2}} with

−iN ≥ j ≥ −Ni − N

and for those i the map

\underline{T} [x_{i}, y_{i+1}]/(x_{i}y_{i+1} − t) → \underline{T} [w_{1}, w_{2}]/(w_{1}^{r} = 1, w_{2}^{N/r}w_{1}^{β} = t)

is given by

x_{i} ↦ w_{1}^{α−iβ}w_{2}^{(−j−iN)/r}, y_{i+1} ↦ w_{1}^{−α+β(i+1)}w_{2}^{(j+iN+N)/r}.

Proof. Suppose Z → U_{i+ \frac{1}{2}} is a morphism such that α ↦ a ∈ Γ(Z, M_{Z}^{gp}) satisfies

Na = jt

(here we have returned to our earlier abuse of notation and think of t as a section of M_{T}). By 7.2.2 a is equal to the image of y_{i+1} minus (i + 1)t. It follows that (j + Ni + N)t ∈ Γ(Z, M_{Z}) and hence

j + Ni + N ≥ 0.

Similarly, the image of x_{i} is −it−a and hence (−iN −j)t ∈ Γ(Z, M_{Z}) from which it follows that

−iN − j ≥ 0.

From this we obtain the inequalities

−iN ≥ j ≥ −iN − N.

Let P_{i+ \frac{1}{2}} be the free monoid on generators x_{i} and y_{i+1}, t ∈ P_{i+ \frac{1}{2}} the element x_{i} + y_{i+1}. U_{i+ \frac{1}{2}} is obtained as the fiber product

Spec(P_{i+ \frac{1}{2}} → ℤ[P_{i+ \frac{1}{2}} ])
↓
T ⟶ Spec(ℕ → ℤ[ℕ]).

The condition that Na = jt is represented by the subscheme corresponding to the monoid

M = P_{i+ \frac{1}{2}} /Ny_{i+1} = (j + N(i + 1))t.

To calculate the saturation of M, observe that there is an isomorphism

M^{gp} → ℤ/rℤ ⊕ ℤ

given by

x_{i} ↦ (α − iβ, (j + iN)/r), y_{i+1} ↦ (−α + β(i + 1), (−j − (i + 1)N)/r).

It follows that

M^{sat} ≃ ℤ/rℤ ⊕ −ℕ ⊂ ℤ/rℤ ⊕ ℤ.

Since

T ×_{ℤ[ℕ]} ℤ[M^{sat}]

is valuative it represents [N, j].

7.2.6. From this description, it follows that the product in the category of fine saturated log schemes

(E^{t})^{val}[N] ×_{T} (E^{t})^{val}[N]

is valuative and hence is also equal to the product in the category of algebraic valuative log schemes. In particular, (E^{t})^{val}[N] is a group object in the category of fine saturated log schemes. It is not however a group object in the category of log schemes. If it were, then the underlying scheme would be a finite flat group scheme of rank N^{2}. If N is invertible in \underline{T} this would imply that the underlying scheme is étale which it is not.

Lemma 7.2.7. There are no automorphisms of [N, j] over U_{i+ \frac{1}{2}} .

Proof. Denote by w̃_{2} the element of M_{[N,j]} mapping to w_{2}. \overline{M}_{[N,j]} is generated by the image of w̃_{2} and hence an automorphism must be of the form

w̃_{2} ↦ λ(η) + w̃_{2}, w_{1} ↦ λ(ζ) + w_{1}

where η, ζ ∈ O_{[N,j]}^{∗}. By the explicit description of the map

[N, j] → U_{i+ \frac{1}{2}}

it follows that η and ζ are subject to the relations

ζ^{r} = 1, η^{N/r}ζ^{β} = 1, ζ^{α−iβ}η^{(−j−iN)/r} = 1, ζ^{−α+β(i+1)}η^{(j+iN+N)/r} = 1

where r, α, β are as in 7.2.5. Combining the second and third equations gives

η^{N/r} = ζ^{−β}, η^{j/r} = ζ^{α}.

This in turn gives

1 = (ζ^{−β})^{α}(ζ^{α})^{β} = η^{(αN/r)+(βj/r)} = η.

From this we get the equations

ζ^{β} = 1, ζ^{α} = 1

and hence

1 = (ζ^{β})^{j/r}(ζ^{α})^{N/r} = ζ.

7.2.8. Under the hypothesis that t is locally nilpotent, the quotient of 𝔾_{m}^{t} by ℤ exists ([9], VII.1.6) and we denote it by E^{t}. By 7.2.3 this action of ℤ is compatible with the action of ℤ on the functor (𝔾_{m}^{t})^{val} and hence there is an induced map

(E^{t})^{val}[N] → E^{t}.

E^{t} admits one nontrivial automorphism

− : E^{t} → E^{t}

which on the underlying pointed curve with its canonical structure of a generalized elliptic curve is given by negation ([9], VII.1.3). If − : (E^{t})^{val}[N] → (E^{t})^{val}[N] denotes negation as a group functor then it follows from the above explicit description that the induced diagram of morphisms of log schemes

(E^{t})^{val}[N] ⟶ E^{t}
−↓ ↓^{−}
(E^{t})^{val}[N] ⟶ E^{t}

commutes. In terms of the description

(E^{t})^{val}[N] ≃ ∐_{j=0}^{N−1}[N, j]

the negation map is given by the maps

[N, 0] → [N, 0], [N, j] → [N, N − j]
a ↦ −a, a ↦ −a + t.

Lemma 7.2.9. 1. For any integer j, there are no non-trivial automorphisms of [N, j] over E^{t}.

2. Suppose j, j′ ∈ {0, 1, …, N − 1} are distinct integers, and x_{1} → [N, j], x_{2} → [N, j′] are points endowed with the pullback log structure. Then there are no isomorphisms of log points

x_{1} → x_{2}

over E^{t}.

3. There are no non-trivial automorphisms of (E^{t})^{val}[N] over E^{t}.

Proof. (1). If j = 0, then [N, 0] mapsto U_{\frac{1}{2}} ∩ U_{\frac{3}{2}} . For any integer i, the map

U_{i+ \frac{1}{2}} → E^{t}

is étale ([9], proof of 1.6). On the other hand, the intersection of U_{\frac{1}{2}} ∩ U_{\frac{3}{2}} with the orbit of any point under the action of ℤ is either empty or consists of one point. It follows that the map

U_{\frac{1}{2}} ∩ U_{\frac{3}{2}} → E^{t}

is an open immersion into the regular locus of E^{t}. For j ≠ 0, the images of x_{i} and y_{i+1} under the map

[N, j] → U_{i+ \frac{1}{2}}

are both nilpotent. Hence the image of [N, j] in E^{t} has support in the singular locus. Let M be an integer sufficiently large such that [N, j] → U_{i+ \frac{1}{2}} factors through the closed subscheme Z_{i+ \frac{1}{2}} defined by the M-th power of the ideal of the singular locus in U_{i+ \frac{1}{2}} . If J denotes the ideal of the singular locus in E^{t}, then the following diagram is cartesian

Z_{i+ \frac{1}{2}} ⟶ U_{i+ \frac{1}{2}}
↑ ↑
Z(J^{M}) ⟶ E^{t}.

Since the orbit of any point intersects Z_{i+ \frac{1}{2}} in at most on point, it follows that

Z_{i+ \frac{1}{2}} → Z(J^{M})

is an isomorphism. It follows from this that any automorphism of [N, j] over E^{t} induces an automorphism of [N, j] over U_{i+ \frac{1}{2}} . From 7.2.6 there are no non-trivial such automorphisms.

(2) By the above discussion, it suffices to prove that there are no isomorphisms of log points x_{1} → x_{2} over U_{\frac{−1}{2}} . If such an isomorphism existed, we would get an isomorphism

\overline{M}_{[N,j],x_{1}} ≃ \overline{M}_{[N,j′],x_{2}}.

Looking at the image of t it follows that gcd(N, j) = gcd(N, j′) and by looking at the image of y_{0} we get

j + (i + 1)N = j′ + (i + 1)N.

From this the assertion follows.

(3) By (2), any automorphism of (E^{t})^{val}[N] must preserve the decomposition

(E^{t})^{val}[N] ≃ ∐_{j=0}^{N−1}[N, j]

and hence induces an automorphism of [N, j]. By (1) there are no non-trivial such automorphisms.

7.2.10. Let X be a log scheme, and let E/X be a special object of M_{1}^{log} (4.3.6). Consider the functor

F : (\underline{X} − schemes) → Set

which to any \underline{Y} → \underline{X} associates the isomorphism classes of pairs (E_{N}, i) having the following properties (let Y denote the scheme \underline{Y} with the pullback log structure):

1. E_{N} is a group object in the category of fine saturated log schemes over Y such that \underline{E}_{N} is a finite flat scheme of rank N^{2}.

2. i : E_{N} → E ×_{X} Y is a morphism of log schemes.

3. If \underline{Y}_{0} is isomorphic to the spectrum of an artinian local ring and \underline{Y}_{0} → \underline{Y} is a morphism, then either E ×_{X} Y_{0} is smooth and E_{N} are the usual N-torsion points, or there exists étale locally on \underline{Y}_{0} a global section t ∈ Γ(\underline{Y}_{0}, M_{Y_{0}}) which generates \overline{M}_{Y_{0}} and a commutative diagram

E_{N} ×_{Y} Y_{0} ⟶^{≃} (E^{t})^{val}[N]
i↓ ↓
E ×_{X} Y_{0} ⟶^{≃}  E^{t}.

4. Two pairs (E_{N}, i) and (E′_{N}, i′) are isomorphic if there exists an isomorphisms E_{N} ≃ E′_{N} of log schemes over E ×_{X} Y, which induces an isomorphism of group objects in the category of fine saturated log schemes over Y.

Theorem 7.2.11. The functor F is representable by \underline{X}. In other words, there exists a unique pair (E_{N}, i) satisfying conditions 1-3 over X.

Proof. We apply 6.3.2 to F. Observe first that given two pairs (E_{N}, i) and (E′_{N}, i′) over Y, there exists at most one isomorphism between them. Two such isomorphisms would define a non-trivial morphism E_{N} → E_{N} over E ×_{X} Y. By passage to the local ring of a point, and then to the completion we may assume that \underline{Y} is isomorphic to the spectrum of a complete local ring. It follows from 7.2.9.3 and 7.2.10.3 that the automorphism induces the trivial automorphism modulo every power of the maximal ideal. By the Grothendieck existence theorem (E_{N} is proper over E) and 3.3.3 it follows that the automorphism is trivial globally. From this we conclude that the functor F is a sheaf for the étale topology. Suppose now that \underline{Y} is isomorphic to the spectrum of an artinian local ring and that E×_{X} Y is not smooth, and that we have a diagram

(E^{t})^{val}[N] ⟵^{≃} E_{N}  E′_{N}  ⟶^{≃} (E^{t′})^{val}[N]
↓  ^{i}↓  ↓_{i′}  ↓
E^{t}  ⟵^{≃} E ×_{X} Y ⟶_{id} E ×_{X} Y ⟶^{≃}  E^{t′}.

By the following lemma we must have t = t′:

Lemma 7.2.12. Let A be an artinian local ring, ℕ → A a monoid morphism such that the image of 1 is in the maximal ideal of A. Denote by t the image of 1, and let X be the associated log scheme. Then for any unit u ∈ A^{∗} the Kato-Tate curves E^{t} and E^{ut} are isomorphic over X if and only if u = 1.

Proof. We can without loss of generality assume that A is strictly henselian. Let J be the maximal ideal of A, and let l be the smallest integer such that J^{l} = 0. We proceed by induction on l. If l = 1, then A is an algebraically closed field k. We can view E^{ut} as being obtained by base change from E^{t} by an automorphism

u : X → X.

Suppose we are given an isomorphism ϕ filling in the diagram

\begin{matrix}E^{t} & ⟶^{ϕ} & E^{t} \\ ↓ & & ↓ \\ X & ⟶^{u} & X. \end{matrix}

The automorphism group of \underline{E}^{t} as a pointed curve is \{±1\} and hence using 7.2.3 we may assume that ϕ is the identity map on schemes. Choosing a chart at the singular point x_{0} and looking at the completion we see that ϕ^{b} induces an isomorphism

Ô_{E^{t},x_{0}} ⊕ ℕ^{2} ⟶^{ϕ^{b}} Ô_{E^{t},x_{0}} ⊕ ℕ^{2}
↓ ↓
k[[x, y]]/(xy) ⟶^{id} k[[x, y]]/(xy)

where x and y are the images of the generators x̃ and ỹ of ℕ^{2}. It follows that ϕ^{b} is given by

x̃ ↦ λ(1 + yf_{x}) + x̃, ỹ ↦ λ(1 + xf_{y}) + ỹ

where f_{x}, f_{y} ∈ k[[x, y]]/(xy). This implies that

x̃ + ỹ ↦ λ(1 + yf_{x} + xf_{y}) + x̃ + ỹ.

By assumption 1 + yf_{x} + xf_{y} is the image of u ∈ k^{∗} and hence f_{x} = f_{y} = 0. It follows that u = 1. This proves the case l = 1. For general l, note that if u ∈ A is such that u ↦ 1 ∈ A/J^{l−1} then E^{ut} is obtained from E^{t} by the infinitesimal automorphism defined by 1 ↦ λ(u) + 1 of X over X ⊗ A/J^{l−1}. The induction step now follows from 5.3.8.

Since the automorphism group of E^{t}/Y is {±1}, it follows from 7.2.8 and 7.2.9 that there exists a unique isomorphism E_{N} ≃ E′_{N} filling in the diagram. This implies that F (\underline{Y}) for every \underline{Y} isomorphic to the spectrum of an artinian local ring consists of one element. By passage to the limit using 3.3.3 and the Grothendieck existence theorem, it follows that F (\underline{Y}) consists of one element also in the case when \underline{Y} is isomorphic to the spectrum of a complete local ring. This implies all the conditions of 6.3.2 except relative representability. Thus suppose we are given two collections of data a, a′ : \underline{Y} → F. We apply 6.3.2 to the functor I = \underline{Y} ×_{F} \underline{Y}. Since the E_{N} are finite over \underline{Y}, I is a sheaf for the étale topology. In addition, since there exists at most one isomorphism between a and a′, I is relatively representable. The remaining conditions of 6.3.2 follow from the fact that F (A) consists of exactly one element for any artinian local ring A combined with the fact that any isomorphism between two collections of data is unique. It follows that the map

I → \underline{Y} ×\underline{X} \underline{Y}

is an isomorphism. This concludes the verification of 6.3.2 and the proof of the theorem.

Definition 7.2.13. Let X be a log scheme over S, and let E/X be an object of M_{1}^{log}. The N-torsion points on E is the pair (E_{N}, i) obtained in the case when E/X is solid by theorem 7.2.11. In the general case, it is the pair (E_{N}^{♯}, i^{♯}) ×_{X^{♯}} X (notation as in 4.3.7).

7.2.14. E_{N} is a log étale group log scheme (in the category of fine saturated log schemes) since we assume that N is invertible on the base. Note however that \underline{E}_{N} is neither a group object nor an étale scheme.

7.3 Full Level Structure.

Lemma 7.3.1. Let E/X ∈ M_{1}^{log} be an object with X saturated. Then for any morphism X′ → X of connected saturated log schemes, the natural map

E_{N}(X) → E_{N}(X′)

is injective, and after a log étale base extension X′′ → X independent of X′ the induced map

E_{N}(X′′) → E_{N}(T_{i})

is bijective where T_{i} is any connected component of (X′ ×_{X} X′′)^{sat}.

Proof. Let

X′ → X^{′♯} → X

be the canonical factorization of the map X′ → X, and suppose f_{1}, f_{2} ∈ E_{N}(X) map to the same element in E_{N}(X′). We claim that then f_{1} = f_{2} in E_{N}(X^{′♯}). To see this note that the underlying morphisms of schemes of f_{1} and f_{2} are equal. Let us denote this map by f so we have a diagram

M_{X^{′♯}} → f^{∗}M_{E_{N}} ⇉ M_{X^{′♯}} → M_{X′}.

By construction of E_{N}, if m ∈ f^{∗}M_{E_{N}} then Nm is in the image of M_{X^{′♯}}. Since X is saturated this implies that for all sections m ∈ M_{X^{′♯}} there exists ζ ∈ μ_{N} such that

f_{1}^{b}(m) = f_{2}^{b}(m) + λ(ζ).

Now if the images of f_{1}^{b}(m) and f_{2}^{b}(m) in M_{X′} are equal we must have ζ = 1 and hence they are already equal in M_{X^{′♯}}. Thus it suffices to consider the case when X′ → X is strict.

It follows from the fact that E_{N} is finite over \underline{X} that the condition that the underlying morphisms of schemes of f_{1} and f_{2} are equal is represented by a closed subscheme of \underline{X}. By the above argument, we see that over this closed subscheme the two maps

f_{1}^{b}, f_{2}^{b} : f^{∗}\overline{M}_{E_{N}} → \overline{M}_{X}

are equal. From the proof of 3.1.5 it follows that the condition f_{1}^{b} = f_{2}^{b} is also represented by a closed subscheme (not just a locally closed subscheme). Therefore, the condition f_{1} = f_{2} is represented by a closed subscheme \underline{S} ⊂ \underline{X}. On the other hand, the fact that E_{N} → X is log étale implies that \underline{S} → \underline{X} is also étale. Hence \underline{S} is also open and since \underline{X} is connected must be either empty or all of \underline{X}. From this the first part of the lemma follows. To prove the second statement, it suffices to consider the case when X = X^{♯}. Moreover, the question is étale local so we may assume that \underline{X} is isomorphic to the spectrum of a strictly henselian local ring A. In this case the log structure on A is given by ℕ → A. Consider the log étale base change induced from

×N : ℕ → ℕ

and replace X by the result. Then E_{N}(X ⊗ A/𝔪_{A}^{k}) consists of N^{2} elements for all k. Now since \underline{E}_{N} is a finite affine scheme over A, to give a section of \underline{E}_{N} → \underline{X} is equivalent to solving some polynomial equations in A. By assumption we have N^{2} solutions over Â, and it then follows from the Artin approximation theorem ([4]) that we get N^{2} sections over A. Given one such section i : \underline{X} → \underline{E}_{N}, we want to find a map

i^{∗}M_{E_{N}} → M_{X}.

If x̄ → \underline{X} denotes the closed point of \underline{X}, then since \underline{X} is isomorphic to the spectrum of a strictly henselian local ring it suffices to construct a map

i^{∗}M_{E_{N} ,x̄} → M_{X,x̄}.

It follows from the construction of E_{N} that there exists an isomorphism

i^{∗}M_{E_{N} ,x̄} ≃ ℕ ⊕_{ℕ} M_{X,x̄}

where the map ℕ → ℕ is given by multiplication by some integer l which divides N, and ℕ → M_{X,x̄} is the composite

ℕ → M_{X^{♯}} → M_{X,x̄}.

Therefore to give a morphism i^{∗}M_{E_{N} ,x̄} → M_{X,x̄} of monoids such that the composite

M_{X,x̄} → i^{∗}M_{E_{N} ,x̄} → M_{X,x̄}

is equal to the identity is equivalent to giving an element of μ_{l}(A). The map μ_{l}(A) → μ_{l}(Â) is bijective and hence there exists a unique isomorphism inducing the given one over Â.

Definition 7.3.2. Let E/X be an object of M_{1}^{log} with X connected and saturated. A full level-N-structure on E is an isomorphism

ϵ : (ℤ/Nℤ)^{2} → E_{N}(X).

In general, a full level-N-structure is given by a full level-N-structure on each connected component of X.

Remark 7.3.3. It follows from from the lemma that there is a natural notion of pullback of a full level-N-structure for a morphism of saturated log schemes f : X′ → X. 7.3.4. Denote by F Log^{sat} ⊂ F Log the full subcategory of saturated log schemes. The strict étale topology on F Log restricts to give a topology on F Log^{sat}. Thus it makes sense to speak of stacks over F Log^{sat}. Let M′_{N} be the fibered category over F Log^{sat} whose objects over a log scheme X consists of pairs (E, ϵ) where E is an object in M_{1,X}^{log} and ϵ is a full level-N-structure on E.

Theorem 7.3.5. M′_{N} is canonically isomorphic to M_{N}^{log} (defined in chapter 6) restricted to F Log^{sat}.

Forgetting the level structure defines a functor

\begin{matrix}M′_{N} \\ π & ↓ \\ M_{1}^{log}. \end{matrix}

If E/X is an object of M_{1}^{log} and X_{0} ↪ X is a strict closed immersion defined by a nilpotent ideal, then it follows from the fact that E_{N} is log étale that

E_{N}(X) → E_{N}(X_{0})

is bijective. This implies that the functor π is log étale.

Lemma 7.3.6. Let X, Y ∈ F Log^{sat} be log schemes, x : X → M′_{N}, y : Y → M′_{N} objects corresponding to log elliptic curves E_{X} and E_{Y} with full level structure. Let X ×_{M′_{N}} Y be the functor on F Log which to any Z ∈ F Log associates the set of triples (f, g, α) where f : Z → X, g : Z → Y and α is an isomorphism

E_{X} ×_{X} Z ≃ E_{Y} ×_{Y} Z

such that the diagram

(ℤ/Nℤ)^{2} ⟶ E_{X,N}(X)
↓ ↓
E_{Y,N}(Y) ⟶ E_{X,N}(Z).

commutes. Then X ×_{M′_{N}} Y is representable by an object of F Log.

Proof. Let X → M′_{N}, Y → M′_{N} be two objects, and let

P = π(X) ×_{M_{1}^{log}} π(Y).

Denote by

E_{X,N} ⟶^{≃} E_{Y,N}
↓ ↓
E_{X} ⟶^{≃} E_{Y}

the tautological isomorphism over P. For each a ∈ (ℤ/Nℤ)^{2} we get two sections a_{X}, a_{Y} : P → E_{X,N}. Let P_{a} denote the fiber product

P_{a} ⟶  P
↓  ^{a_{X}×a_{Y}} ↓
E_{X,N} ⟶^{Δ} E_{X,N} ×_{P} E_{X,N}.

P_{a} represents the functor of isomorphisms between E_{X} and E_{Y} such that the induced sections a_{X} and a_{Y} are the same. It follows that the functor

X ×_{M′_{N}} Y

is represented by the log scheme obtained by taking the product over P of all the P_{a}.

Lemma 7.3.7. An object X → M′_{N} is relatively strict (i.e. X ×_{M′_{N}} Y → Y is strict for all Y) if and only if étale locally there exists a chart for the morphism X → X^{♯} (4.3.7)

ℕ  ⟶^{×N}  ℕ
↓  ↓
Γ(X^{♯}, M_{X^{♯}}) ⟶ Γ(X, M_{X}).

Proof. Let Y → M′_{N} be an object. To prove that

X ×_{M′_{N}} Y → Y

is strict it suffices to consider the case when X and Y both have underlying scheme isomorphic to the spectrum of a separably closed field. Therefore we can assume that isomorphisms between the objects over X and Y with the Kato-Tate curve have been chosen. We compute the schemes P_{a} of the preceding lemma. The scheme \underline{P} consists of a finite set of points (two in fact), and so since the construction of P_{a} is local, we may replace \underline{P} by a single point. It suffices to construct P_{a} for two generators of (ℤ/Nℤ)^{2}, and we can choose these generators such that one has corresponding section in [N, 0] and the other has section in [N, 1]. Suppose first the section is in [N, 0]. In this case, the map

[N, 0] → [N, 0] ×_{P} [N, 0]

is strict and so the map P_{a} → P is also strict. In the case when a_{X} is a section to [N, 1], either a_{Y} maps to some other [N, j] with j ≠ 1 and P_{a} is empty, or a_{Y} also maps to [N, 1]. In the latter case, the diagram

P
↓^{a_{X}×a_{Y}}
E_{X,N} ⟶ E_{X,N} ×_{P} E_{X,N}

admits a chart

M_{P}
↑
M_{P} ⊕_{ℕ} ℕ ⟵ M_{P} ⊕_{ℕ} Q

where Q is the pushout of the diagram

\begin{aligned}& ℕ & ⟶^{×N} & ℕ \\ ×N & ↓ \\ & ℕ & & . \end{aligned}

The log structure M_{P} is given by

M_{X} ⊕_{ℕ} M_{Y}

and it follows from this that the log structure on P_{a} is given by

M_{X} ⊕_{ℕ} M_{Y}

modulo the relation (a, 0) = (0, b) where a ∈ M_{X}, b ∈ M_{Y} are elements such that Na and Nb are the images of the generator of ℕ. This proves the lemma.

Lemma 7.3.8. M′_{N} is naturally equivalent to the stack defined by the stack of relatively strict objects with its natural morphism to LOG_{Spec(ℤ[1/N])} in the sense of 3.5.2.

Proof. Given a morphism s : X → E_{N} over X^{♯}, we get a diagram of inclusions of log structures on X

M_{X^{♯}} ↪ s^{∗}M_{E_{N}} ↪ M_{X}.

From the construction of the log schemes E_{N} we see that if we take the union over sections s giving X a full level N-structure

⋃_{s} s^{∗}M_{E_{N}} ⊂ M_{X}

then we get a sublog structure which satisfies the assumptions in lemma 7.3.7. This amounts to saying that for every geometric point x̄ → \underline{X}

⋃_{s} s^{−1}\overline{M}_{E_{N} ,x̄} ⊂ \overline{M}_{X,x̄}

is a monoid isomorphic to ℕ such that N times the generator is the image of the generator of M_{X^{♯}}. To prove this it suffices to check it in the case when \underline{X} is isomorphic to the spectrum of a separably closed field. In this case it follows from the structure of the log schemes [N, j] described in 7.2.5.

Proof of 7.3.5 We have to show that the stack of solid objects with its natural morphism to LOG_{ℤ[1/N]} is algebraic. It follows from 7.3.7 that it is relatively representable. Thus it suffices to exhibit a smooth cover. Let {U_{i}} be an étale strict cover of M_{1}^{log} such that each U_{i} admits a chart ℕ → M_{U_{i}}. Let {U′_{i}} be the family of log schemes obtained by taking the fibred products

U′_{i} ⟶ Spec(ℤ[ζ_{N}, 1/N][ℕ])
↓  ^{×N}↓
U_{i} ⟶  Spec(ℤ[ℕ]).

Then by lemma 7.3.1 after perhaps replacing the U′_{i} be a strict étale cover E_{N}(U′_{i}) consists of N^{2} elements. Thus for each isomorphism

(ℤ/Nℤ)^{2} ≃ E_{N}(U′_{i})

we get a solid object U′_{i} → M′_{N}. Let {Ũ_{i}} be the collection of objects of M′_{N} obtained from the U′_{i} by considering all such isomorphisms. Then we claim that

U = ∐ Ũ_{i} → M′_{N}

defines an étale surjective cover. For any solid object Y → M′_{N} the induced morphism

U ×_{M′_{N}} Y → Y

is strict by 7.3.7. It is also étale since the extensions U′_{i} → U_{i} are log étale. Therefore to check that the map is surjective we can assume that \underline{Y} is isomorphic to the spectrum of a field. In this case it is immediate. This proves that M′_{N} is algebraic. To construct the canonical isomorphism between M′_{N} and M_{N}^{log} we first construct an isomorphism between the underlying stacks. By ([9], IV.3.5), \underline{M}_{N}^{log} is the normalization of \underline{M}_{1}^{log} in the substack of \underline{M}_{N}^{log} where the log structure is trivial. The morphism

\underline{π} : \underline{M}′_{N} → \underline{M}_{1}^{log}

is quasi-finite. If x : Spec(k) → \underline{M}′_{N} is an object with k separably closed, then it follows from 7.2.8 that Aut(x) is trivial unless we are in the case of N = 2 in which case Aut(x) = Aut(\underline{π}(x)). Since objects of \underline{M}′_{N} do not admit infinitesimal automorphisms we conclude that \underline{π} is relatively representable. If \underline{X} → \underline{M}_{1}^{log} is a 1-morphism then the product

P = \underline{M}′_{N} ×_{\underline{M}}′_{N} \underline{X} → \underline{X}

is flat since it is the underlying morphism of algebraic spaces of a log étale and integral morphism

M′_{N} ×_{M_{1}^{log}} ×X → X

where X is given the pullback log structure from M_{1}^{log}. (ℤ/Nℤ)^{2} acts on P and fiber by fiber the action is simply transitive. It follows that P is a (ℤ/Nℤ)^{2}-torsor. This implies that the morphism \underline{π} is finite, and that \underline{M}′_{N} is normal. This in turn implies that \underline{M}′_{N} is the normalization of \underline{M}_{1}^{log} in the open substack of M′_{N} where the log structure is trivial. Since the substacks where the log structure is trivial of M′_{N} and M_{N}^{log} are canonically isomorphic this gives a canonical isomorphism between the underlying stacks. To extend this to an isomorphism of log stacks, simply observe that both M_{N}^{log} and M′_{N} are smooth and hence the log structure is canonically isomorphic to the pushforward of the trivial log structure on the open substack where the log structure is trivial. This concludes the proof of 7.3.5.

7.4 M_{0}(N)^{log} and M_{1}(N)^{log}.

7.4.1. The techniques of 7.3 generalize more easily to the construction of other moduli stacks than those of chapter 6. Since the arguments of this section are essentially the same as the arguments of the preceding one we omit some of the details. Define M_{1}(N)^{log} to be the fibered category over F Log^{sat} whose objects over a saturated log scheme X consists of pairs (E, i) where E/X is an object of M_{1}^{log} and i : X → E_{N} is a section of exact order N. M_{0}(N)^{log} is the fibered category over F Log^{sat} whose objects over a connected log scheme X consists of pairs (E, H) where E is an object of M_{1}^{log} and H is a saturated log subgroup scheme of E_{N} étale locally on \underline{X} isomorphic to the log subgroup scheme defined by a section X → E_{N} of exact order N. By this we mean that if X → E_{N} is a section of exact order N, then we obtain a sub-group object of E_{N} in the category of fine saturated log schemes

∐_{(ℤ/Nℤ)}X ↪ E_{N}.

Theorem 7.4.2. M_{1}(N)^{log} (resp. M_{0}(N)^{log}) is a log algebraic stack, and the underlying algebraic stack \underline{M}_{1}(N)^{log} (resp. \underline{M}_{0}(N)^{log}) is canonically isomorphic to the algebraic stack M_{Γ_{00}(N)}[1/N] (resp. M_{Γ_{0}(N)}[1/N]) of ([9]).

7.4.3. Suppose X is a log scheme with underlying scheme isomorphic to the spectrum of a separably closed field. Let E/X be an object of M_{1}^{log}. To each section a : X → E_{N} we can attach an integer e(a) ∈ {0, 1, …, N − 1} as follows. If E/X is smooth let e(a) be equal to 1. Otherwise, choose an isomorphism of E with the Kato-Tate curve. Such an isomorphism is canonical up to the automorphism “ − ” of 7.2.8. The section a therefore defines a point of [N, j] for some j. Applying the automorphism “ − ” has the effect of switching [N, j] and [N, N − j]. Consequently the integer

e(a) = gcd(N, j)

does not depend on the choices. If X is an arbitrary log scheme and E/X is an object of M_{1}^{log} together with a section a : X → E_{N}, then for each geometric point t → X we get an integer which we denote by e_{a}(t). If a′ is another section which defines the same subgroup of E_{N}(X^{val}) as a then it follows from the construction of E_{N} that the integer e_{a′}(t) is equal to e_{a}(t). This implies that for an object of M_{0}(N)^{log} we also have a well-defined integer e(t) for each geometric point t → \underline{X}.

Definition 7.4.4. If X is a log scheme, x : X → M_{1}(N)^{log} (resp. x : X → M_{0}(N)^{log}) a 1-morphism, then x is solid if in an étale neighborhood of every geometric point t → X, there exists a chart for the morphism X → X^{♯}

ℕ  ⟶^{×e(t)}  ℕ
↓  ↓
Γ(X^{♯}, M_{X^{♯}}) ⟶ Γ(X, M_{X}).

We justify this apparent abuse of the word solid with the following:

Lemma 7.4.5. The stacks M_{1}(N)^{log} and M_{0}(N)^{log} are naturally equivalent to the stacks defined by the stacks of solid objects with the natural morphisms to

LOG_{Spec(ℤ[1/N])}

in the sense of 3.5.2.

Proof. The argument of 7.3.8 also gives this lemma.

The proof of 7.3.5 can with slight modifications be carried out to prove 7.4.2. The shortest method, however, is to apply 3.5.13 using the maps

M_{0}(N)^{log} → M_{1}^{log}

and

M_{1}(N)^{log} → M_{1}^{log}

which are log étale. Then the deformation theories of the stacks are the same as the deformation theory of M_{1}^{log}. Thus condition 3.5.13.1 is immediate. Condition 3.5.13.2 follows from 3.3.3, the usual Grothendieck existence theorem, and 3.5.13.2 for M_{1}^{log}. Condition 3.5.13.3 follows from 3.3.4 and 5.4.10 and hence the conditions of are verified. Remark 7.4.6. The ramification structure of of the maps

\underline{M}_{0}(N)^{log} → \underline{M}_{1}^{log}, \underline{M}_{1}(N)^{log} → \underline{M}_{1}^{log}

is given by lemma 7.4.5. The ramification index at a geometric point t is given precisely by the integer e(t) of 7.4.4. The point is that if X is a log scheme with underlying scheme the spectrum of a separably closed field, and X → M_{1}(N)^{log} a 1-morphism with e(t) = 1, then the group of automorphisms of X over X^{♯} is isomorphic to μ_{e(t)}. For each automorphism we get a different morphism of log stacks

X → M_{1}(N)^{log}

but they all induce the same morphism

\underline{X} → \underline{M}_{1}(N)^{log}.

For example, if N is prime then the description of E_{N} given in 7.2.5 shows that \underline{M}_{1}(N)^{log} has N − 1 points that are unramified corresponding to the sections in [N, 0] and N − 1 points with ramification index N corresponding to the sections in [N, j], j ≠ 0. A similar remark applies to M_{0}(N)^{log}.

Chapter 8

Analytic theory of log elliptic

curves

8.1 Classical theory

We review the classical relationship between abelian varieties over ℂ and families of Hodge structures.

Definition 8.1.1. A pure Hodge structure over ℤ of weight n is a free ℤ module H_{ℤ} together with a filtration F^{·} on the complex vector space H_{ℂ} = H_{ℤ} ⊗ ℂ such that

H_{ℂ} = ⊕_{p+q=n}F^{p} ∩ \overline{F}^{q}

where \overline{F}^{q} denotes the complex conjugate of F^{q} under the action of Aut(ℂ) via the second factor of H_{ℤ} ⊗ ℂ.

8.1.2. If we define

H^{p,q} = F^{p} ∩ \overline{F}^{q}, p, q ∈ ℤ

then we have a decomposition

H_{ℂ} = ⊕_{p+q=n}H^{p,q}.

The type of H is the set of pairs (p, q) for which H^{p,q} ≠ 0. 8.1.3. Suppose R is any ℝ-algebra. We define an action of an element z ∈ (R ⊗_{ℝ} ℂ)^{∗} on H^{p,q} ⊗_{ℂ} (R ⊗_{ℝ} ℂ) by multiplication by z^{p}z̄^{q} where z̄ denotes the element of (R ⊗_{ℝ} ℂ)^{∗} obtained by applying complex conjugation to the second factor.

Because H^{q,p} = \overline{H}^{p,q} this action is invariant under complex conjugation on H_{R⊗_{ℝ}ℂ} and hence defines an action of (R ⊗_{ℝ} ℂ)^{∗} on H_{R}. In other words, if 𝕊 denotes the algebraic group representing the functor on ℝ-algebras

R ↦ 𝔾_{m}(R ⊗_{ℝ} ℂ)

then we have a representation

𝕊 → Aut(H_{ℝ}).

In fact, using this construction one can show that to give a Hodge structure is equivalent to giving a free ℤ-module H_{ℤ} together with a representation

𝕊 → Aut(H_{ℝ}).

There is a norm map

N : 𝕊 → 𝔾_{m}

of real algebraic groups which sends

z ↦ zz̄, z ∈ 𝕊(R)

for any ℝ-algebra R.

Definition 8.1.4. For any n ∈ ℤ, we define a Hodge structure ℤ(n) called the Tate Hodge structure by taking H_{ℤ} = (2πi)^{n}ℤ ⊂ ℂ and letting z ∈ 𝕊(R) act via N(z)^{−n}.

8.1.5. If H is a Hodge structure let C : H_{ℝ} → ℝ denote the action of the element i ∈ 𝕊(ℝ) = ℂ^{∗}.

Definition 8.1.6. A polarization on a Hodge structure H of weight n is a morphism of Hodge structures

ϕ : H × H → ℤ(−n)

such that the induced pairing

H_{ℝ} × H_{ℝ} → ℝ

defined by

(x, y) ↦ (2πi)^{n}ϕ(x, Cy)

is symmetric and positive definite. A Hodge structure H is polarizable if it admits a polarization.

Remark 8.1.7. In the definition, H × H denotes the Hodge structure with lattice H_{ℤ} × H_{ℤ} and action of 𝕊 on H_{ℝ} × H_{ℝ} given by the action on each component. By definition ϕ is a morphism of Hodge structures if ϕ respects the action of 𝕊.

Finally we extend these definitions to families. Let S be a complex manifold.

Definition 8.1.8. A family of Hodge structures of weight n on S consists of a local system H_{ℤ} of free ℤ-modules on S together with a filtration F^{·} on H_{ℤ} ⊗ O_{S} by holomorphic sub-bundles such that for each point s ∈ S, the fiber (H_{ℤ,s}, F_{s}^{·}) is a Hodge structure of weight n. A polarization on H is a morphism of local systems

H_{ℤ} × H_{ℤ} → ℤ(−n)_{S}

which induces a polarization on (H_{ℤ,s}, F_{s}^{·}) for each s ∈ S. H is polarizable if H admits a polarization. For a family of Hodge structures (H_{ℤ}, F) we let

∇ : H_{ℤ} ⊗ O_{S} → H_{ℤ} ⊗ Ω_{S}^{1}

be the canonical connection. We say that (H_{ℤ}, F) is a variation of Hodge structures if for all p ∈ ℤ

∇(F^{p}) ⊂ F^{p−1} ⊗ Ω_{S}^{1}.

Remark 8.1.9. If (H_{ℤ}, F) is a rank 2 family of Hodge structures of weight −1 then H_{ℤ} admits a canonical polarization. For locally there are exactly two pairings ϕ and −ϕ

H_{ℤ} × H_{ℤ} → ℤ(1)

for which the underlying map of local systems is surjective and (2πi)^{−1}ϕ is anti-symmetric. One checks from the definition of polarization that for each point s ∈ S exactly one of ϕ and −ϕ defines a polarization. Since the filtration F varies holomorphically the condition that a pairing ϕ defines a polarization at each point is an open condition. Therefore, by the uniqueness the locally constructed polarizations glue to a global one. 8.1.10. Suppose \underline{S}/ℂ is a smooth scheme of finite type and that f : \underline{A} → \underline{S} is an abelian variety (with trivial log structure). Define R_{1}f_{∗}ℤ to be the dual of the local system

R^{1}f_{∗}ℤ

on \underline{S}^{an} (the complex analytic space associated to \underline{S}). Then R_{1}f_{∗}ℤ carries a canonical structure of a polarizable Hodge structure of type {(−1, 0), (0, −1)} and rank 2g where g is the relative dimension of \underline{A}. The filtration on R_{1}f_{∗}ℤ is induced from the Hodge filtration on

R^{1}f_{∗}ℤ ⊗ O_{S} ≃ R^{1}f_{∗}(Ω_{A/S}^{·})

via the de-Rham isomorphism. The following theorem can be found in ([7]) and it is a version of this theorem for log elliptic curves which is the goal of this chapter:

Theorem 8.1.11. Let \underline{S}/ℂ be a smooth scheme of finite type. The functor

(abelian varieties over \underline{S})

↓

(polarizable variations of Hodge structures of type {(−1, 0), (0, −1)} on S^{an}) is an equivalence of categories.

8.2 Log Hodge structures

We recall some of the theory of log Hodge structures due to K. Kato and S. Usui ([22]). 8.2.1. Let S/ℂ be a fine saturated log scheme whose underlying scheme is locally of finite type over ℂ. Recall that there is a ringed space (S^{log}, O_{S}^{log}) attached to S ([21]). We fix an integer n (the weight) and a collection of integers (h^{p,q})_{p,q∈ℤ} (the Hodge type) such that h^{p,q} = 0 if p + q ≠ n.

Definition 8.2.2. A pre-log Hodge structure on S is a pair (H_{ℤ}, F) where

1. H_{ℤ} is a locally constant sheaf of ℤ-modules on S^{log} whose stalks are free of rank

Σh^{p,q}.

2. F is a descending filtration on the O_{S}^{log} module

O_{S}^{log} ⊗_{ℤ} H_{ℤ}.

3. Each F^{p} is the pullback via

τ_{S} : S^{log} → \underline{S}

of a locally free O_{S}-module of rank Σ_{r≥p}h^{r,w−r}.

4. Each F^{p} is locally on S^{log} an O_{S}^{log} direct summand of O_{S}^{log} ⊗ H_{ℤ}.

8.2.3. The conditions for a pre-log Hodge structure to be a log Hodge structure are defined pointwise. Therefore, assume \underline{S} = Spec(ℂ). For each point y ∈ S^{log} let sp(y) be the set of ℂ-homomorphisms

O_{S,y}^{log} → ℂ.

Note that if the rank of \overline{M}_{S}^{gp} is n then

O_{S,y}^{log} ≃ ℂ[T_{1}, …, T_{n}].

Let sp(S) be the set of all monoid morphisms

M_{S} → ℂ

which induce the identity on ℂ^{∗}. Recall ([29], p. 50) that on S^{log} there is a sheaf L and an exact sequence

0 → ℤ(1) → L → τ_{S}^{−1}M_{S}^{gp} → 0.

Moreover, there is a natural map

L → O_{S}^{log}.

(O_{S}^{log} is constructed using L). For a section a ∈ M_{S} let log(a) denote any lifting of a to L_{y}. Such a lifting is well-defined up to adding multiplies of 2πi. Given a map

s : O_{S,y} → ℂ

defining a point in sp(y) we get a map

M_{S} → ℂ^{∗}

by

a ↦ exp(s(log(a))).

This is well defined since exp(2πi) = 1. We therefore have a canonical map

sp(y) → sp(S).

Choosing generators for M_{S} realizes sp(S) as a closed subset of

ℂ^{n}

for some n. This defines a topology on sp(S) which one can verify is independent of the choice of the generators.

Definition 8.2.4. Suppose \underline{S} = Spec(ℂ). A pre-log Hodge structure (H_{ℤ}, F) on S is a log Hodge structure if it satisfies the following additional conditions:

5. The map

d ⊗ 1 : O_{S}^{log} ⊗ H_{ℤ} → Ω_{S}^{1,log} ⊗ H_{ℤ}

sends F^{p} to Ω_{S}^{1,log} ⊗ F^{p−1}.

6. If y ∈ S^{log}, then there exists a neighborhood U of α : M_{S} → ℂ (log structure morphism) in sp(S) such that for any s ∈ sp(y) which maps to U the pair

(H_{ℤ,y}, F (s))

where F (s) is the induced filtration on

ℂ ⊗ H_{ℤ} ≃ ℂ ⊗_{s,O_{S,y}^{log}} ⊗(O_{S,y}^{log} ⊗ H_{ℤ})

is a Hodge structure in the usual sense.

For general S, a pre-log Hodge structure is a log Hodge structure if the restriction to each fiber is a log Hodge structure.

8.2.5. If H is a variation Hodge structures on \underline{S} in the sense of 8.1.8 then the local system τ^{−1}H_{ℤ} together with the filtration defined by the filtration on H_{ℂ} ⊗ O_{S} is a log Hodge structure. This is immediate from the definition.

Definition 8.2.6. For any integer n, define the Tate log Hodge structure ℤ^{log}(n) to be the log Hodge structure induced from ℤ(n).

Definition 8.2.7. Let (H_{ℤ}, F) be a log Hodge structure on S of weight n. A polarization on (H_{ℤ}, F) is a map of local systems on S^{log}

ϕ : H_{ℤ} × H_{ℤ} → ℤ^{log}(−n)

such that in condition 8.2.4.6 we can choose y and U such that for any s ∈ sp(y) with image in U the triple

(H_{ℤ,y}, F (s), ϕ_{y})

is a polarized Hodge structure in the usual sense. A log Hodge structure is polarizable if it admits a polarization.

The following proposition will be used in what follows:

Proposition 8.2.8. Let \underline{S} be a smooth analytic space and let \underline{Y} ⊂ \underline{S} be a divisor with normal crossings. Denote by S the log analytic space consisting of \underline{S} with the log structure coming from \underline{Y}.

1. Let (H_{ℤ}^{1}, F^{1}) and (H_{ℤ}^{2}, F^{2}) be pre-log Hodge structures on S^{log} and let

ψ : (H_{ℤ}^{1}, F^{1})|_{S^{∗}} → (H_{ℤ}^{2}, F^{2})|_{S^{∗}}

be an isomorphism. Then ψ extends uniquely to all of S^{log}.

2. Suppose (H_{ℤ}, F) is a pre-log Hodge structure on S^{log} and that a map

ϕ : H_{ℤ} × H_{ℤ} → ℤ^{log}(−n)

is given such that (H_{ℤ}, F, ϕ)|_{S^{∗}} is a polarized log Hodge structure. Then

(H_{ℤ}, F, ϕ)

is a polarized log Hodge structure.

Proof. (1) Let j^{log} : S^{∗} ↪ S^{log} be the canonical inclusion. Then by ([29], 5.12) the map j^{log} is locally aspheric and hence there are canonical isomorphisms

H_{ℤ}^{1} ⟶^{≃} j_{∗}^{log}j^{log,∗}H_{ℤ}^{1} ⟶^{j_{∗}^{log}(ψ)} j_{∗}^{log}j^{log,∗}H_{ℤ}^{2} ⟶^{≃} H_{ℤ}^{2}.

It is clear that this isomorphism is the unique isomorphism of local systems extending ψ. We must check that it is an isomorphism of pre-log Hodge structures. Let E^{i} be the locally free sheaf

τ_{S∗}(H_{ℤ}^{i} ⊗_{ℤ} O_{S}^{log}).

Thus from the isomorphism H_{ℤ}^{1} ≃ H_{ℤ}^{2} we get an isomorphism E^{1} ≃ E^{2}. Fix some q ∈ ℤ, and let L^{qi} denote the locally free sheaf τ_{S∗}F^{qi} on S. Using the lemma 8.2.9 below and 8.2.2.3 we see that to verify that the extension of ψ preserves the filtrations it suffices to verify that ψ preserves the L^{qi}. Condition 8.2.2.4 implies that the quotient E^{i}/L^{qi} is locally free. Hence the map

L^{q1} → E^{2}/L^{q2}

is zero since this can be verified on the dense open set S^{∗}. Consequently L^{q1} ⊂ L^{q2} and by symmetry we get the reverse inclusion. This concludes the proof of (1).

(2) It follows from Schmid’s nilpotent orbit theorem ([33], see also [22] 3.2.8) that there exists an extension of (H_{ℤ}, F, ϕ)|_{S^{∗}} to a polarized log Hodge structure on all of S^{log} (note that by [22], 3.2.7 the monodromy of (H_{ℤ}, F, ϕ)|_{S^{∗}} is unipotent). By (1) we get an isomorphism of this extension with (H_{ℤ}, F). Now two maps

ϕ_{1}, ϕ_{2} : H_{ℤ} × H_{ℤ} → ℤ^{log}(−n)

which agree on a dense open set must be equal since all the sheaves involved are locally constant. Thus the isomorphism given by (1) is compatible with the polarizations. This implies (2).

Lemma 8.2.9. With notation as above, let F be a locally free sheaf on \underline{S}. Then the natural map

F → τ_{S∗}τ_{S}^{∗}F

is an isomorphism.

Proof. The question is local on \underline{S} and hence it suffices to prove the lemma for O_{S}. Let s ∈ \underline{S} be a point and let s̃ ∈ S^{log} be a point in the fiber over s. Then since τ_{S} is proper we have

(τ_{S∗}τ_{S}^{∗}O_{S})_{s} ≃ O_{S,s̃}^{log,π_{1}(S_{s}^{log},s̃)}.

We can choose isomorphisms

O_{S,s̃}^{log} ≃ O_{S,s}[T_{1}, …, T_{n}], π_{1}(S_{s}^{log}, s̃) ≃ ℤ^{n}

such that the i-th generator γ_{i} of ℤ^{n} acts by

T_{i} ↦ T_{i} + 2πi, T_{j} ↦ T_{j} j ≠ i.

If f ∈ O_{S,s̃}^{log} we can write

f = ∑_{i} p_{i}(T_{1}, …, T_{n−1})T_{n}^{i}.

By induction we conclude that each p_{i}(T_{1}, …, T_{n_{1}}) ∈ O_{S,s} and hence it suffices to consider the case n = 1. In this case we have

f(0) = f(2πin)

for all n ∈ ℤ and hence f = f(0).

8.3 The log Hodge structure associated to a log elliptic curve

8.3.1. Let S/ℂ be a smooth log scheme of finite type and let f : E → S be an object of M_{1}^{log} (a log elliptic curve). Looking at the associated analytic spaces we get a diagram

E^{log} ⟶^{τ_{E}} E^{an}
f_{log}↓ ↓^{f}
S^{log} ⟶^{τ_{S}} S^{an}.

Theorem 8.3.2. The sheaf R_{1}f_{log,∗}ℤ on S^{log} (the dual of R^{1}f_{log,∗}ℤ) has a unique structure of a log Hodge structure of type {(−1, 0), (0, −1)} inducing the usual Hodge structure over the open set of S where the log structure is trivial.

The proof occupies the remainder of this section. 8.3.3. We reduce the problem to the universal case. If E′/S′ is obtained from a strict étale morphism S′ → M_{1} and E/S is obtained by base change from a map S → S′ then we have a diagram

E^{log} ⟶^{α} E^{′,log}
f_{log}↓ ↓^{f′_{log}}
S^{log} ⟶^{β} S^{′,log}

which is cartesian by definition. Now the map

E^{′,log} → S^{′,log}

is proper. To see this consider the diagram

E^{′,log} ⟶^{τ_{E′}} E^{′,an}
f′_{log}↓ ↓^{f′}
S^{′,log} ⟶^{τ_{S′}} S^{′,an}

and let C ⊂ S^{′,log} be a compact set. Then

f_{log}^{−1}(C) ⊂ τ_{E′}^{−1} ∘ f^{′,−1}(τ_{S′}(C))

and hence is a closed subset of a compact set (since we know that f′ and τ_{E′} are proper). It follows from the proper base change theorem that we have an isomorphism

R_{1}f_{log,∗}ℤ ≃ β^{−1}R_{1}f′_{log,∗}ℤ.

Thus it suffices to prove uniqueness for S and existence for S′.

8.3.4. The uniqueness follows from 8.2.8. For the assumptions on S ensure that the log structure on S is of the form assumed in 8.2.8 ([17], II.2.1.13 and [30]). 8.3.5. From now on we therefore assume that E/S is a universal family. That is, that the map

S → M_{1}

is strict and étale. Consider the de-Rham cohomology with its Hodge filtration

0 ⟶ R^{0}f_{∗}Ω_{E/S}^{1} ⟶ R^{1}f_{∗}(Ω_{E/S}^{·}) ⟶ R^{1}f_{∗}O_{E} ⟶ 0.

Just as in the classical case, the de-Rham cohomology comes with a Gauss-Manin connection

∇ : R^{1}f_{∗}(Ω_{E/S}^{·}) → Ω_{S/ℂ}^{1} ⊗ R^{1}f_{∗}(Ω_{E/S}^{·})

which extends the Gauss-Manin connection on S^{∗}. The explicit description of this connection in ([23], appendix) shows that ∇ has unipotent monodromy around the points S − S^{∗}. Thus (R^{1}f_{∗}(Ω_{E/S}^{·}), ∇) is an object of D_{nilp}(S) in the notation of ([21]). Applying ([21], 0.5) and taking duals we conclude that there exists a unique local system of ℂ-vector spaces H_{ℂ} on S^{log} extending the local system on S^{∗} coming from singular homology. In addition, the Hodge filtration on R^{1}f_{∗}(Ω_{E/S}^{·}) induces a filtration F on H_{ℂ} ⊗ O_{S}^{log}. Now since the map

j_{log} : S^{∗} ↪ S^{log}

is locally aspheric, we have an isomorphism

H_{ℂ} ≃ j_{log∗}j_{log}^{∗}H_{ℂ}.

This implies that H_{ℂ} admits a unique integral structure H_{ℤ} extending that on S^{∗}. For the same reason the canonical polarization on (H_{ℤ}, F)|_{S^{∗}} extends uniquely to a pairing on all of H_{ℤ}. Therefore from 8.2.8 we see that (H_{ℤ}, F, ϕ) is a log Hodge structure of type {(−1, 0), (0, −1)}.

8.4 Elliptic log Hodge structures

8.4.1. Let S/ℂ be a smooth log scheme with \underline{S}/ℂ smooth. In the previous section we defined a functor

(log elliptic curves over S) ↓ (rank 2 log Hodge structures of type {(−1, 0), (0, −1)} on S^{log}).

In this section we describe a subcategory of the category of log Hodge structures on S^{log} which we will show in the next section is the image of the above functor. 8.4.2. Let (H_{ℤ}, F) be a rank 2 log Hodge structure of type

{(−1, 0), (0, −1)}

on S^{log} and suppose s̃ ∈ S^{log} is a point mapping to s ∈ S^{an}. The fundamental group π_{1}(S_{s}^{log}, s̃) of the fiber over s has the following explicit description. On S^{log} there is an exact sequence of sheaves ([29], p.50)

0 ⟶ ℤ(1) ⟶ L ⟶ τ_{S}^{−1}M_{S}^{gp} ⟶ 0

which induces a map

M_{S,s}^{gp} → H^{1}(S_{s}^{log}, ℤ(1)).

By ([29], 5.15) this map induces an isomorphism

\overline{M}_{S,s}^{gp} ≃ H^{1}(S_{s}^{log}, ℤ(1)) ≃ Hom(π_{1}(S_{s}^{log}, s̃), ℤ(1))

and hence an isomorphism

α : Hom(\overline{M}_{S,s}^{gp}, ℤ(1)) ≃ π_{1}(S_{s}^{log}, s̃).

Definition 8.4.3. Let ℤ(1)_{≤0} = (2πi)(−ℕ) ⊂ 2πiℤ. A rank 2 log Hodge structure (H_{ℤ}, F) on S^{log} of type

{(−1, 0), (0, −1)}

is an elliptic log Hodge structure if for each s̃ ∈ S^{log} the image of the map

Hom(\overline{M}_{S,s}, ℤ(1)_{≤0}) → Aut(H_{ℤ,s̃})

is contained in (id + ℚ_{≥0} · N) for some nilpotent endomorphism N of H_{ℤ,s̃}.

Remark 8.4.4. If (H_{ℤ}, F) comes from a log elliptic curve E/S then it is elliptic. For in this case (H_{ℤ}, F) is obtained by pullback from the universal curve, and in the universal case \overline{M}_{S,s} is isomorphic to either ℕ or 0 and has unipotent monodromy by ([23], appendix).

Definition 8.4.5. Let N be a positive integer. A full level N structure on an elliptic log Hodge structure (H_{ℤ}, F) is an isomorphism

α : (ℤ/Nℤ)_{S^{log}} → H_{ℤ}/NH_{ℤ}.

8.4.6. If S/ℂ is log smooth with \underline{S}/ℂ smooth, and if E/S is a log elliptic curve with a full level N-structure then the log Hodge structure associated to E has a level-N structure. Over S^{∗} this follows from the classical theory and since H_{ℤ} is locally constant and S^{∗} → S^{log} is locally aspheric there exists a unique extension of this level structure to all of S^{log}.

The following is a special case of the theory of K. Kato and S. Usui ([22]):

Theorem 8.4.7. The functor

F : (log smooth analytic spaces S/ℂ) → Set

which to any to any S/ℂ associates the isomorphism classes of elliptic log Hodge structures with level N-structure is representable if N ≥ 3 by a smooth log analytic space. In addition, the natural map

(M′_{N})^{an} → F

is an isomorphism.

Proof. We explain how this follows from their results. Let H_{0} be the free ℤ-module with basis e_{0} and e_{1}. Define an anti-symmetric pairing

ϕ_{0} : H_{0} × H_{0} → (2πi)ℤ, ϕ_{0}(e_{0}, e_{1}) = 2πi.

Let Γ(N) be the subgroup of SL_{2}(ℤ) of matrices congruent to the identity modulo N. If (H_{ℤ}, F) is a elliptic log Hodge structure then we claim that there exists a unique polarization ϕ such that the pair (H_{ℤ}, ϕ) is locally isomorphic to (H_{0}, ϕ_{0}). To see this note that on the open dense set S^{∗} where the log structure is trivial (H_{ℤ}, F) admits a polarization by 8.1.9. Since the local system H_{ℤ} is isomorphic to the pushforward of that on S^{∗} we get a pairing ϕ on H_{ℤ}. By 8.2.8 this defines a polarization on (H_{ℤ}, F). Now there are only two anti-symmetric pairings to ℤ(1) which are surjective and they are ϕ and −ϕ. It follows from the definition of polarization that −ϕ is not a polarization and hence the polarization is unique. Therefore, an elliptic log Hodge structure (H_{ℤ}, F) admits a unique polarization ϕ such that (H_{ℤ}, ϕ) is locally isomorphic to (H_{0}, ϕ_{0}). In addition, if (H_{ℤ}, F) comes with a level N-structure then we can locally choose an isomorphism such that the induced map

(ℤ/NH_{ℤ})^{2} ≃ H_{0}/NH_{0} → H_{ℤ}/NH_{ℤ}

is that given by the level structure. It follows that the data of the level structure is equivalent to a global section ι of the sheaf

Γ(N) \ Isom((H_{ℤ}, ϕ), (H_{0}, ϕ_{0}))

over S^{log}. Now let Σ be the set of monoids

Σ = {ℚ_{≥0}N|N ∈ M_{2}(ℤ), N nilpotent}.

Then we see that an elliptic log Hodge structure is equivalent to the data of a polarized log Hodge structure of type Φ (notation as in ([22])). The result now follows from ([22], 3.3.3) and the construction of the moduli space given there.

8.5 The main result

Theorem 8.5.1. Let S/ℂ be a smooth log scheme with \underline{S}/ℂ smooth. Then the functor

(log elliptic curves over S) ↓ (elliptic log Hodge structures on S^{log})

is an equivalence of categories.

The proof occupies the remainder of the section.

Proposition 8.5.2. Let S/ℂ be a log scheme with \underline{S} reduced, and suppose E_{1} and E_{2} are log elliptic curves over S. Denote by I the functor on \underline{S}-schemes Isom\underline{S}(\underline{E}_{1}, \underline{E}_{2}) (ignoring the log structures). Suppose we are given a section s_{U} : \underline{U} → I for some dense open set U ⊂ \underline{S} containing all points where the j-invariants of E_{1} and E_{2} are equal to 0 or 1728. Then s_{U} extends to all of \underline{S}.

Proof. By our assumptions we may as well remove the closed subset where the j-invariants are equal to 0 and 1728. In this case, I is a {±1} pseudo-torsor finite over S. Thus I consists of two connected components I_{1} and I_{2}. Suppose the image of s_{U} is contained in I_{1}. Consider the two maps

\underline{S} ⇉ Spec(ℂ[1/j])

defined by E_{1} and E_{2}. Since \underline{S} is reduced and these maps agree on a dense open set they must be equal. It follows that for all s ∈ \underline{S}(ℂ) the fiber I_{s} consists of exactly two points and that I_{1,s} consists of a single point. Consider the map of coherent sheaves on \underline{S}

η : O_{S} → π_{∗}O_{I_{1}}.

Since η is an isomorphism on a dense open set the map is injective. Let Q be the cokernel. Then Q is a coherent sheaf on S such that for all closed points s ∈ S we have Q ⊗ k(s) = 0 since I_{1}×k(s) ≃ Spec(k(s)). It follows that Q_{s} = 0 by Nakayama’s lemma and hence Q = 0. Thus the map η is an isomorphism and consequently I_{1} → S is an isomorphism.

8.5.3. It follows that the functor in 8.5.1 is fully faithful. Let E_{1} and E_{2} be two log elliptic curves over S and suppose we are given an isomorphism i between the corresponding log Hodge structures. By 5.2.6 we get log structures M_{1}, M_{2} ⊂ M_{S} such that E_{l} is obtained from a special object over (\underline{S}, M_{l}). We claim that M_{1} = M_{2}. In fact we can explicitly determine the image of

\overline{M}_{l} ⊂ \overline{M}_{S}.

Choose a point s̃ ∈ S^{log} and let s ∈ \underline{S} be the image. For each l = 1, 2 the action of π_{1}(S_{s}^{log}, s̃) on Hl ; s̃ (the log Hodge structure associated to E_{i}) factors through the map

π_{1}(S_{s}^{log}, s̃) ≃ Hom(\overline{M}_{S,s}, ℤ(1)_{≤0}) → Hom(\overline{M}_{l,s}, ℤ(1)_{≤0}).

Thus i induces a commutative diagram

π_{1}(S_{s}^{log}, s̃)  ⟶ Hom(\overline{M}_{1,s}, ℤ(1)_{≤0})
↓ ↓
Hom(\overline{M}_{2,s}, ℤ(1)_{≤0}) ⟶  Aut(H1 ; s̃).

For a suitable choice of basis H1 ; s̃ ≃ ℤ^{2} the image of π_{1}(S_{s}^{log}, s̃) is contained in the monoid of matrices

(\begin{matrix}1 & p \\ 0 & 1 \end{matrix})

for p ∈ ℕ. In terms of this basis, the canonical generator of Hom(\overline{M}_{l,s}, ℤ(1)_{≤0}) must act via the matrix

(\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) .

Note that this is true for some basis by ([23], appendix) and one checks by calculation that then it is true for any basis with the above property. We conclude that the image (n_{1}, …, n_{r}) ∈ ℕ^{r} of the canonical generator of \overline{M}_{l} in \overline{M}_{S,s} is determined by the condition that the map

ℕ^{r} → ℤ(1)_{≤0}, e_{i} ↦ −2πi, e_{j} ↦ 0, j ≠ i

acts via the matrix

(\begin{matrix}1 & n_{i} \\ 0 & 1 \end{matrix})

From this we see that the images of \overline{M}_{l,s} in \overline{M}_{S,s} must be equal and hence M_{1} = M_{2}. Let S^{♯} be the corresponding log scheme and let E_{l}^{♯} be the special log elliptic curves inducing the E_{l} by base change. Now to give an isomorphism \underline{E}_{1}^{♯} ≃ \underline{E}_{2}^{♯} over \underline{S} is equivalent to giving an isomorphism between the log schemes E_{1}^{♯} ≃ E_{2}^{♯} over S^{♯}. For the description in 1.3.13 of the functor

Isom\underline{S}(\underline{E}_{1}^{♯}, \underline{E}_{2}^{♯})

in terms of the special objects of the log stack M_{1} shows that for any isomorphism w : \underline{E}_{1}^{♯} ≃ \underline{E}_{2}^{♯} there exists an automorphism α : S^{♯} → S^{♯} which is the identity on \underline{S} and a commutative diagram

E_{1}^{♯} ⟶^{w̃} E_{2}^{♯}
↓ ↓
S^{♯} ⟶^{α} S^{♯}

inducing w. But by our assumptions on S, the morphism α must be the identity and hence we get an isomorphism of log schemes over S^{♯}. It follows that if we are given an isomorphism between \underline{E}_{1} and \underline{E}_{2} then we get an isomorphism between E_{1}^{♯} and E_{2}^{♯} and hence an isomorphism between the E_{i}. Now we get an isomorphism E_{1}|_{S^{♯∗}} ≃ E_{2}|_{S^{♯∗}} from i by 8.1.11. By 8.5.2 this isomorphism extends to an isomorphism \underline{E}_{1}^{♯} ≃ \underline{E}_{2}^{♯} over all of \underline{S}. Let j be the corresponding isomorphism between the log Hodge structures. Then i and j agree on a dense open subset of S^{log} and since they are maps between local systems they must be equal. Conversely, if i and j are two isomorphisms between E_{1} and E_{2} which induce the same map on the associated log Hodge structures, then by 8.1.10 we must have i = j over S^{∗}. But E_{1} and E_{2} are reduced and separated and hence if two maps agree on a dense open set they must be equal. This proves that the functor is fully faithful. 8.5.4. To show that the functor is essentially surjective, let (H_{ℤ}, F) be an elliptic log Hodge structure on S^{log}. By the full faithfulness the question is Zariski local on S. Thus we may assume that we have a smooth morphism ([17], II.2.1.13)

S → Spec(ℂ[ℕ^{r}])

giving a chart for M_{S}. Let N ≥ 3 be an integer and consider the map

×N : Spec(ℂ[ℕ^{r}]) → Spec(ℂ[ℕ^{r}])

given by

ℕ^{r} → ℕ^{r}, \underline{n} ↦ N\underline{n}.

We let S′ be the fiber product of

Spec(ℂ[ℕ^{r}])
×N↓
S ⟶ Spec(ℂ[ℕ^{r}]).

We claim that on S′ the local system H_{ℤ}/NH_{ℤ} on S^{′,log} descends to a local system on S^{′,an}. To see this it suffices to show that the map

τ^{−1}τ_{∗}H_{ℤ}/NH_{ℤ} → H_{ℤ}/NH_{ℤ}

is an isomorphism. Let s ∈ S^{′,log} be a point. Then

(τ^{−1}τ_{∗}H_{ℤ}/NH_{ℤ})_{s} = (H_{ℤ}/NH_{ℤ})^{π_{1}(S_{τ(s)}^{′,log},s)}.

Now the action of π_{1}(S_{s}^{′,log}, s) ≃ Hom(\overline{M}_{S′,τ(s)}^{gp}, ℤ(1)) on (H_{ℤ}/NH_{ℤ})_{s} factors through the map

Hom(\overline{M}_{S′,τ(s)}^{gp}, ℤ(1)) → Hom(\overline{M}_{S,τ(s̄)}^{gp}, ℤ(1))

where s̄ is the image of s in S^{log}. Since the map

\overline{M}_{S,τ(s̄)}^{gp} → \overline{M}_{S′,τ(s)}^{gp}

is multiplication by N the action is trivial. It follows that the local system H_{ℤ}/NH_{ℤ} descends to S′. Therefore by replacing S′ by a finite étale cover we can assume that H_{ℤ}/NH_{ℤ} is trivial. In other words, we have found a log smooth cover S′ → S whose underlying morphism is faithfully flat such that (H_{ℤ}, F) admits a level-N structure. It follows from 8.4.7 that there exists a log elliptic curve E′/S′ whose associated log Hodge structure is isomorphic to (H_{ℤ}, F). Let U′ ⊂ S′ be the open subset where \underline{E}′/\underline{S}′ is smooth. The image of U′ in S is an open set U since S′/S is flat. Now for all s ∈ U^{log} the natural map

(τ^{−1}τ_{∗}H_{ℤ})_{s} → H_{ℤ,s}

is an isomorphism. For as before we have

(τ^{−1}τ_{∗}H_{ℤ})_{s} = (H_{ℤ,s})^{π_{1}(S_{τ(s)}^{log},s)}.

If s′ ∈ S^{′,log} is a point lying over s then the quotient

π_{1}(S_{τ(s)}^{log}, s)/π_{1}(S_{τ(s′)}^{log}, s′)

is killed by multiplication by N. Therefore, if γ ∈ π_{1}(S_{τ(s)}^{log}, s) then Nγ acts as the identity. But γ acts via a unipotent endomorphism so this implies that γ acts as the identity. It follows that (H_{ℤ}, F)|_{U} is a Hodge structure in the usual sense. Therefore by 8.1.10 there exists an elliptic curve E_{U}/U whose associated log Hodge structure is isomorphic to (H_{ℤ}, F)|_{U} such that E_{U} ×_{S} S′ ≃ E′|_{U′} canonically. Now to prove the theorem we may as well remove the closed set in U where the j-invariant of E_{U} is 0 or 1728.

Lemma 8.5.5. \underline{S}′ ×\underline{S} \underline{S}′ is reduced.

Proof. Let T = Spec(ℂ[ℕ^{r}]) and let T′ → T be the map induced from ×N : ℕ^{r} → ℕ^{r}. Then \underline{S}′ ×\underline{S} \underline{S}′ is smooth over \underline{T}′ ×\underline{T} \underline{T}′. Hence it suffices to consider \underline{T}′ ×\underline{T} \underline{T}′. To prove the lemma it therefore suffices to prove the following statement and apply induction. If R is a reduced ring then

R[X, Y]/X^{N} − Y^{N}

is reduced. Since R is reduced it suffices to show that the ring is reduced after reduction mod 𝔭 for each prime 𝔭 ⊂ R. Thus it suffices to consider the case when R is an integral domain and by embedding into the field of fractions we may assume that R is a field extension of ℂ. Let K be an algebraic closure of the field of fractions of R[X]. Then R[X, Y]/(X^{N} − Y^{N}) embeds into

K[Y]/Y^{N} − a

which is isomorphic to the product of N copies of K. It follows that R[X, Y]/(X^{N} − Y^{N}) is reduced.

8.5.6. By the lemma \underline{S}′ ×\underline{S} \underline{S}′ is reduced and hence by the proposition there exists a unique isomorphism p_{1}^{∗}\underline{E}′ ≃ p_{2}^{∗}\underline{E}′ over \underline{S}′ ×\underline{S} \underline{S}′ which agrees with the isomorphism over \underline{U}′ ×\underline{S} \underline{U}′ coming from the isomorphism

E_{U} ×_{S} S′ ≃ E′|_{U′}.

Let us verify that this defines descent datum for \underline{E}′. This amounts to saying that a map

\underline{S}′ ×\underline{S} \underline{S}′ ×\underline{S} \underline{S}′ → Aut(\underline{E}′)

which is the identity on a dense open set is equal to the identity. But Aut(\underline{E}′) ≃ \{±1\} and hence this is clear. Let \underline{E}/\underline{S} be the pointed curve defined by the descent datum. \underline{E} defines a map

e : \underline{S} → \underline{M}_{1}.

Now the inverse image of the log structure on M_{1} is contained in the subsheaf of functions in O_{S} which are invertible on the open set where \underline{E}/\underline{S} is smooth. Now this open set contains S^{∗} since this can be verified by base change to S′. By ([17], 3.4.1) the log structure on S is isomorphic to

j_{∗}^{log}O_{S^{∗}}^{∗}

It follows that

e^{∗}M_{M_{1}} ⊂ M_{S} ⊂ O_{S}

and hence we get a morphism of log stacks

e : S → M_{1}

giving E/S the structure of a log elliptic curve. E/S comes together with an isomorphism between the log Hodge structure associated to E and the log Hodge structure we started with over the open set U ⊂ S. By 8.2.8 this isomorphism extends to all of S^{log}. This proves the essential surjectivity of the functor in 8.5.1 and hence concludes the proof of the theorem.

Bibliography

[1] A. Altman and S. Kleiman, Introduction to Grothendieck duality theory, Springer-Verlag, Berlin, 1970, Lecture Notes in Mathematics, Vol. 146.

[2] M. Artin, Algebraization of formal moduli. I, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 21–71.

[3] , The implicit function theorem in algebraic geometry, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford Univ. Press, London, 1969, pp. 13–34.

[4] , Théorèmes de représentabilité pour les espaces algébriques, Les Presses de l’Université de Montréal, Montreal, Que., 1973, En collaboration avec Alexandru Lascu et Jean-François Boutot, Séminaire de Mathématiques Supérieures, No. 44 (Été, 1970).

[5] , Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165–189.

[6] M. Artin, A. Grothendieck, and J.-L. Verdier, Théorie des topos et cohomologie étale des schémas., Springer-Verlag, Berlin, 1972, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat, Lecture Notes in Mathematics, Vol. 269, 270, 305.

[7] P. Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. (1971), no. 40, 5–57.

[8] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969), no. 36, 75–109.

[9] P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, pp. 143–316. Lecture Notes in Math., Vol. 349.

[10] J. Dieudonné and A. Grothendieck, Éléments de géométrie algébrique, Inst. Hautes Études Sci. Publ. Math. (1961-1967), no. 4, 8, 11, 17, 20, 24, 28, 32.

[11] T. Gómez, Algebraic stacks, http://xxx.lanl.gov, math.AG/9911199, 1999.

[12] M. Hakim, Topos annelés et schémas relatifs, Springer-Verlag, Berlin, 1972, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64.

[13] J. Harris and I. Morrison, Moduli of curves, Springer-Verlag, New York, 1998.

[14] R. Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, No. 52.

[15] L. Illusie, Complexe cotangent et déformations. I, Springer-Verlag, Berlin, 1971, Lecture Notes in Mathematics, Vol. 239.

[16] , Logarithmic spaces (according to K. Kato), Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), Academic Press, San Diego, CA, 1994, pp. 183–203.

[17] L. Illusie and A. Ogus, Lectures on logarithmic algebraic geometry, manuscript notes, 2000.

[18] F. Kato, Log smooth deformation theory, Tôhoku Math. J. (2) 48 (1996), no. 3, 317– 354.

[19] , Log smooth deformation and moduli of log smooth curves, Internat. J. Math. 11 (2000), no. 2, 215–232.

[20] K. Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191–224.

[21] K. Kato and C. Nakayama, Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over c, Kodai Math. J. 22 (1999), no. 2, 161–186.

[22] K. Kato and S. Usui, Logarithmic Hodge structures and classifying spaces, preprint, 1998.

[23] N. Katz, p-adic properties of modular schemes and modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, pp. 69–190. Lecture Notes in Mathematics, Vol. 350.

[24] F. F. Knudsen, The projectivity of the moduli space of stable curves. II. The stacks M_{g,n}, Math. Scand. 52 (1983), no. 2, 161–199.

[25] D. Knutson, Algebraic spaces, Springer-Verlag, Berlin, 1971, Lecture Notes in Mathematics, Vol. 203.

[26] G. Laumon and L. Moret-Bailly, Champs algébriques, Springer-Verlag, Berlin, 2000.

[27] J. S. Milne, Étale cohomology, Princeton University Press, Princeton, N.J., 1980.

[28] Y. Nakkajima, Liftings of simple normal crossing log K3 and log Enriques surfaces in mixed characteristics, J. Algebraic Geom. 9 (2000), no. 2, 355–393.

[29] A. Ogus, Logarithmic de Rham cohomology, preprint, 1999.

[30] , Course taught at UC Berkeley, 2000.

[31] M. Olsson, Crystalline cohomology of schemes over algebraic stacks, in preparation.

[32] M. Schlessinger, Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), 208– 222.

[33] W. Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211–319.

[34] A. Vistoli, The deformation theory of local complete intersections, http://xxx.lanl.gov, math.AG/9703008, 1999.