This course is a two-semester introduction to the foundations of algebraic geometry in the language of schemes, along with techniques, examples and applications. The theory of schemes was developed by Alexander Grothendieck and collaborators in the 1960's. It has come to be universally accepted as a flexible and powerful replacement for more limited and unwieldy foundational systems previously used in algebraic geometry.
In the lectures I will give a fairly self-contained presentation of the basic concepts and main results in the subject. The fall term and the first part of the spring will mostly be devoted to setting up foundations. We will turn to applications in the later part of the spring term.
My main goal is to impart a working knowledge of algebraic geometry and some intuition for it. I will try to concentrate on what makes scheme theory 'geometric,' taking a good deal of time, for instance, to explain the precise sense in which classical varieties over an algebraically closed field k can be identified with reduced schemes locally of finite type over k. I will also try to include lots of examples. Often I will just outline the proofs of theorems, although I will sometimes give more detail when something about a proof is conceptually illuminating, or when the reference texts give proofs that I don't like.
In many ways, the most lucid exposition of scheme theory is still Grothedieck's Éléments de géométrie algébrique (EGA), along with various volumes of the Séminaire de Géométrie Algébrique (SGA) for more advanced material. The main problem for English-speaking students is that these are mostly only available in French.
The reference texts for this class will be English language synopses of results from EGA, which I will post on this page as needed, and Ravi Vakil's lecture notes The Rising Sea: Foundations Of Algebraic Geometry. Some other useful references are listed below. I don't plan to follow any of the texts very closely. I will be presenting much of the material in the lectures in my own way.
To follow this class you will need a good grounding in commutative algebra, equivalent to Math 250A and parts of 250B. I will, however, review various results from commutative algebra as we need them.
It is also helpful to have some familiarity with the basics of algebraic topology and differential geometry as a source of perspective on the subject. Classical complex algebraic varieties have an analytic topology as well as the Zariski topology, and are also complex analytic varieties or, in the smooth case, manifolds. An important motivating principle in modern algebraic geometry has been to try to build analogs of these extra features of classical varieties using purely algebraic techniques, which can then be applied in other contexts, such as to varieties over a field of characteristic p.
The Rising Sea: Ravi Vakil, The Rising Sea: Foundations Of Algebraic Geometry, version of Nov. 18, 2017, available here, or along with earlier versions, here. Vakil makes the notes available free for non-commercial use under a Creative Commons License. The notes are book length, so I recommend downloading a copy to your computer rather than reading them online.
Vakil would like you to send him any comments you may have, however minor, that might be helpful for improving the next version.
EGA: Alexander Grothendieck, Éléments de géométrie algébrique, I-IV, Publications Mathématiques de l'IHES, vols. 4, 8, 11, 17, 20, 24, 28, 32 (1960-1967). EGA is no longer available in print, but it is available online as part of the collections of IHES publications hosted by Springer and NUMDAM. See below for synopses in English.
SGA: All volumes except SGA 2 were published in the Springer Lecture Notes in Mathematics series. They are available online through the UC libraries. The easiest way to access them is from the list of references on the SGA Wikipedia page. Click on the numeric string portion of links of the form "doi:10.1007/BFb0058656" there. The Wikipedia page also has links to typeset versions of SGA Volumes 1 and 2 on the arXiv.
Other references: You may find some of the following helpful.
Topic | EGA | Vakil | ||
---|---|---|---|---|
1 | 1/23 | Motivation for Proj(R) via k^{×} action on the affine cone over a projective variety. Graded rings and ideals. Equivalence of ℤ-grading with 𝔾_{m} action. Definition of Proj(R) as a space: { graded prime ideals P ∈ Spec(R) − V(R_{+}) }. | II:2.1-2.4 | 4.4.9-10; 4.5 |
2 | 1/25 | Scheme structure of Proj(R). Example: Proj(k[x_{0},...,x_{n}]) = projective space ℙ^{n}_{k}. | ||
3 | 1/28 | Proj(A[x_{0},...,x_{n}]) = Spec(A) ×_{k} ℙ^{n}_{k}. Morphism Proj(R) − V(φ(S_{+})) → Proj(S) induced by a graded ring homomorphism φ: S → R. Isomorphism Proj(R) ≅ Proj(R_{nℤ}). How closed subscheme V(I) ⊂ Proj(R) depends on graded ideal I. Examples. | II:2.8-2.9 | 6.4, 8.2 |
4 | 1/30 | Weighted projective spaces. Proj(R) as a quotient (Spec(R) − V(R_{+})) / 𝔾_{m}. | 8.2.11 | |
5 | 2/1 | More on (Spec(R) − V(R_{+})) → Proj(R) as a 𝔾_{m} quotient: (i) it's always a ‘coarse’ quotient; (ii) it's a principal 𝔾_{m} bundle and locally a functorial quotient if V(R_{1}) = V(R_{+}); (iii) it's hardly ever a global functorial quotient (a principal bundle with this property is trivial). | ||
6 | 2/4 | Sheaf M^{~} on Proj(R) associated to a graded R-module M. Sheaves 𝒪(d). Examples: Γ(𝒪(d)) on ℙ^{n}, and on V(xy) = {0, ∞} ⊂ ℙ^{1}. | II:2.5 | |
7 | 2/6 | Quasi-coherent (qco) sheaves. Adjointness of (−)^{~} and Γ(X,−), and equivalence Qco(X) = R-mod, on affine X = Spec(R). Some examples of non-qco 𝒪_{X} module sheaves on affine schemes X. | I:1.4, 1.6 | 13.3-4 |
8 | 2/8 | Direct image functor φ_{*} on qco sheaves for a morphism φ of affine schemes. Example showing that ℳ qco does not imply φ_{*}(ℳ) qco for a morphism of arbitrary schemes. Quasicompact, separated and quasi-separated morphisms. | I:5.3-5.5, 9.2; IV:1.1-1.2, 1.7 | 7.3.1, 10.1 |
9 | 2/11 | Intersection of affines in (quasi)-separated schemes. Examples. Theorem: ℳ qco implies φ_{*}(ℳ) qco if φ is quasicompact and quasi-separated. Tensor product of sheaves of modules. | ||
10 | 2/13 | Left adjoint φ^{*} of φ_{*} for a ringed space morphism φ. Tensor and φ^{*} for qco sheaves on schemes. Invertible sheaves. Condition for 𝒪(d) to be invertible on Proj(R). | 0:4.2-4.3 | 16.1-3 |
11 | 2/15 | Constructing and tensoring invertible sheaves. Picard group. Functor Γ_{*}(X,ℳ) = ⊕_{d} Γ(X,ℳ(d)) from 𝒪_{X} modules to graded R modules on X = Proj(R) when V(R_{1}) = ∅. | II:2.6 | |
12 | 2/20 | Theorem 1: Assume X = Proj(R) satisfies V(R_{1}) = ∅. Then Γ_{*} : 𝒪_{X}-modules → graded R-modules is right adjoint to (−)^{~}. Theorem 2: Assume in addition that X is quasi-compact. If ℳ is a qco 𝒪_{X}-module, then ℳ ≅ Γ_{*}(ℳ)^{~}. | ||
13 | 2/22 | Finish proofs from Lecture 12. Criteria for different modules M to localize to the same sheaf M^{~}. | ||
14 | 2/25 | Functor represented by Proj(R): statement. Example: maps to ℙ^{n}. | 16.4 | |
15 | 2/27 | More examples of the functor represented by Proj(R). Outline of the proof. | ||
16 | 3/1 | Classical Segre embedding ℙ^{ l} × ℙ^{ m} → ℙ^{(l+1)(m+1)-1}. General version Proj(R) ×_{A} Proj(S) = Proj(T), where T is the diagonal degrees subring of R ⊗_{A} S. | II:4.3 | 9.6 |
17 | 3/4 | Affine morphisms X → Y. Theorem: the definition is local relative to open covers of Y. Examples: closed embeddings; vector bundles. | II:1 | 7.3.3-7.3.7, 17.1 |
18 | 3/6 | Various definitions of geometric vector bundle. Correspondence E = Spec(S(ℰ)) between vector bundles and locally free sheaves. Invertible sheaf = line bundle. | ||
19 | 3/8 | Sheaf ℋom(ℳ, 𝒩 ). Dual ℰ^{∨} of a locally free sheaf ℰ. Sheaf of sections of E = Spec(S(ℰ)) identified with ℰ^{∨}. Example: relating the tautological line bundle L on ℙ^{n}_{k} to ℒ = 𝒪(1). | 2.3.C-2.3.D | |
20 | 3/11 | Invertible ℒ on X and graded ring homomorphism R → Γ_{+}(ℒ) = ⊕_{d≥0} ℒ^{⊗d} give open W ⊂ X and morphism φ: W → Y = Proj(R): two ways to construct this; criteria to have W = X and φ a locally closed embedding. | II:3.1-2, 3.7 | 16.6 |
21 | 3/13 | Criteria for embedding X ↪ Y = Proj(Γ_{+}(ℒ)) when X is quasi-compact. Ample sheaves. Relative ample sheaves for a morphism f: X → T. Local nature on T of ℒ being ample for f. Examples: (a) ℒ = 𝒪(1) on X = Proj(R) (if quasi-compact); (b) when is 𝒪_{X} ample? | II:3.8, 4.1-2, 4.4-6 | |
22 | 3/15 | Some properties of ample sheaves, the ample cone, and qco sheaves on X when X has an ample sheaf. Examples (described without proof): flag variety, curves. | ||
23 | 3/18 | Base change and composition of morphisms that have an ample sheaf. Quasi-projective morphisms (= finite type with an ample sheaf). Embedding X ↪ ℙ^{n}_{A} for X quasi-projective over Spec(A). | II:5.3 | |
24 | 3/20 | Basic properties of quasi-projective morphisms. Very ample bundles. Theorem: assume q: X → T quasi-projective and T quasi-compact. Then ℒ is ample for q iff some ℒ^{⊗n} is very ample for q. Proposition: if T is a quasi-compact and quasi-separated scheme, then every qco sheaf on T is the direct limit of its locally finitely generated subsheaves. | ||
25 | 3/22 | Finer characterizations of quasi-projective morphisms q: X → T when (a) T is quasi-compact, (b) T is quasi-compact and quasi-separated, (c) T has an ample sheaf. The category of quasi-projective schemes over T=Spec(A) (or any T which has an ample sheaf). Example of elimination theory, to motivate projective and proper morphisms. | ||
26 | 4/1 | More motivating examples for proper and projective morphisms. Nakayama's Lemma. Finite morphisms are closed. Theorem: X = Proj(ℛ) → Y is closed if ℛ is a locally finitely generated qco sheaf of 𝒪_{Y} algebras such that ℛ_{0} is a locally finitely generated 𝒪_{Y} module (start on the proof). | II:5.4-5 | 10.3 |
27 | 4/3 | Finish proof of the theorem on X = Proj(ℛ) → Y. Universally closed and proper morphisms. Morphism yoga. Every Z-morphism from a scheme proper over Z to a scheme separated over Z is proper. Projective morphisms. | I:5.5 | |
28 | 4/5 | Properties of the category of schemes projective over a nice (e.g., affine) base scheme. Classical Grassmann varieties G(n,r). Plücker coordinates and Plücker relations. | 6.7 | |
29 | 4/8 | Grassmann functor G(ℳ,r) on schemes over S, for a given qco 𝒪_{S} module ℳ. Start on representability of the Grassmann functor. | 16.7 | |
30 | 4/10 | Grassmann schemes. Top exterior power of the tautological quotient bundle is very ample for a (closed) projective embedding. Plücker coordinates revisited. | ||
31 | 4/12 | Partial flag varieties for GL_{n} as iterated Grassmann schemes. Components R_{d} of the homogeneous coordinate ring of G(n,m) as GL_{n} representations; calculation of their dimensions; surprising symmetry in d, m and n−m. | ||
32 | 4/15 | Classical and combinatorial-topological definitions of dimension. Preview of dimension theory for Noetherian local rings. Every non-empty closed subset of a locallly Noetherian scheme has a closed point. | IV:4.1 | 11 |
33 | 4/17 | Dimension of Noetherian local rings: ways to characterize dim(A)=0; Krull's theorem (via length and Artin-Rees lemma); prime avoidance and systems of parameters. | ||
34 | 4/19 | Fiber dimension inequality for morphisms of locally Noetherian schemes. Behavior of dimension under a finite morphism. Example of a nice Noetherian domain (localization of a polynomial ring) with closed points of differing codimensions. | ||
35 | 4/22 | Going-down theorem for A⊂B domains, B a finite A-module, and A integrally closed. Brief proof sketch; geometric counterexample with A not integrally closed. Dimension theorem dim_{x}(X) = tr.deg._{k}K(X) for closed points of an irreducible algebraic scheme, using Noether normalization and going-down. | ||
36 | 4/24 | Fiber dimension identity for flat morphisms. Applications of dimension theory for algebraic schemes: algebraic schemes are catenary; example of using fiber dimensions to show that an algebraic scheme is irreducible. | ||
37 | 4/26 | General remarks on cohomology theories (de Rham, simplicial, sheaf). Abelian categories (e.g., modules, sheaves), complexes, H^{i} of a complex, quasi-isomorphisms, definition of derived category D(𝒜)=Q^{-1}C(𝒜). | Notes 2 | 23(†) |
38 | 4/29 | Mapping cones. Long exact sequence of a mapping cone triangle. Triangle in D(𝒜) associated to an exact sequence in C(𝒜). Homotopy. Homotopic maps are equal in D(𝒜), hence induce the same maps on H^{i}. | ||
39 | 5/1 | Deligne's definition of derived functor. Computing RF(A) using F-acyclic resolutions. | ||
40 | 5/3 | Cech complexes, objects acyclic with respect to a base of the topology, Cech cohomology. Theorem: Qco sheaves on an affine scheme are acyclic for the global sections functor. Computation of H^{i}(ℙ_{k}^{n},𝒪 (d)) using Cech complex of the standard affine covering. | Notes 3 | 18.1-3 |
Topic | EGA | Vakil | ||
---|---|---|---|---|
1 | 8/22 | What is algebraic geometry? Classical affine varieties. Simple examples. | 3.1 | |
2 | 8/24 | Unions and intersections. Irreducibility. Dimension. Varieties in k^{1} and k^{2}. Algebraic groups SL_{n} and GL_{n}. k^{2} − {0} | ||
3 | 8/27 | ℙ^{1} and ℙ^{2}. Conics in ℙ^{2}. Elliptic curves and their degenerations. | ||
4 | 8/29 | Ideal of X ⊂ k^{n}. Coordinate ring 𝒪(X). Consequences of Nullstellensatz. Morphisms of affine varieties. | ||
5 | 8/31 | Correspondence between morphisms and k-algebra homomorphisms. Simple examples. Parametrization of nodal curve y^{2} = x^{2}(x+1). | ||
6 | 9/5 | Parametrization of cuspidal curve y^{2} = x^{3}. How to make X_{f} = X − V(f) into a variety. | ||
7 | 9/7 | Presheaves and sheaves. | Ch. 0:3.1, 3.6 | 2.1-2.3 |
8 | 9/10 | Sheaf of regular funcitons 𝒪_{X} on a classical affine variety. Serre varieties. Simple examples. Open subvarieties. | ||
9 | 9/12 | ℙ^{n}. Graded ideals and projective varieties. | ||
10 | 9/14 | Projective quadrics. Functor φ_{*} for sheaves. Morphisms of varieties. | 0:3.4 | Ex. 2.2H |
11 | 9/17 | Examples: k^{n+1} − {0} → ℙ^{n}; projection of a conic from a point. | ||
12 | 9/19 | Schemes: X = Spec(R) as a space. Localization. Open sets X_{f} = Spec(R_{f}). | 0:1; I:1.1 | 1.3.3-1.3.4, 3.1-3.5 |
13 | 9/21 | Stalks. Constructing a sheaf from stalks. Sheaf M^{~} on X = Spec(R) associated to an R-module. Structure sheaf 𝒪_{X}. | 0:3.2; I:1.3 | 2.4-2.5, 4.1 |
14 | 9/24 | Radical ideals and
closed subsets of X = Spec(R). Irreducibitily and primes.
Generic points. Compare |
I:1.1 | 3.6-3.7 |
15 | 9/26 | Classical varieties as Spec(R)_{cl} for R = 𝒪(X) (assuming nullstellensatz). Picture of Spec(ℤ). | 3.2-3.3 | |
16 | 9/28 | Quasi-compactness of X = Spec(R). Theorem M^{~}(X_{f}) = M_{f} via exactness of the Cech complex. | I:1.3 | 4.1 |
17 | 10/1 | Schemes. Examples: affine schemes, open subschemes, classical varieties as X_{cl}. Functor φ^{−1} for sheaves. | 0:3.5, 3.7; I:2.1 | 2.7, 4.3 |
18 | 10/3 | Adjoint functors. Adjunction (φ^{−1},φ_{*}). | 1.5, 2.7 | |
19 | 10/5 | Local rings and local ring homomorphisms. The category of locally ringed spaces. Spec(–) as a functor. | I:2.2 | 4.3.6, 6.1-6.3 |
20 | 10/8 | Morphism of schemes corresponding to a morphism of classical affine varieties. Open embedding X_{f} → X from R → R_{f}. Closed embedding Spec(R → R/I). | I:4.1 | 7.1, 8.1 |
21 | 10/10 | Sheafification. Quotient sheaves. Sheaves as an abelian category; exactness of stalks. | 2.4, 2.6 | |
22 | 10/12 | Closed subschemes. Qco ideal sheaves (discuss; definition later). Reduced subscheme on a closed subset. Example: V(y) ∩ V(y-x^{2}+a) as subscheme of k^{2} varying with a. | I:4.1-4.2, 5.1-5.2 | 8.1, 8.3.9-8.3.11 |
23 | 10/15 | Closed, open and locally closed embeddings and subschemes. Disjoint unions; finite disjoint union of affines is affine. Gluing: sets, spaces, sheaves. | 0:3.3; I:2.3, 3.1, 4.1-4.2 | 8.1; Exx. 2.5D, 4.3D, 4.4A |
24 | 10/17 | Projective space ℙ^{n}_{A} by gluing. | 4.4 | |
25 | 10/19 | Functor ℙ^{n}_{A}(T) = Hom_{A}(T, ℙ^{n}_{A}) = {rank 1 locally free quotients ℒ = 𝒪_{T}^{n+1}/𝒩 }. Case T = Spec(B) with B = ℤ or a field. | 6.3.E, 6.3.M, 6.3.11-12, 15.3.5, 15.3.F, 16.4.1 | |
26 | 10/22 | Adjointness of Spec: Rings^{op} → (Locally Ringed Spaces) to Γ(–, 𝒪). Significance and corollaries of this theorem. Construction and uniqueness of φ: T → Spec(R) from R → 𝒪_{T}(T). | I:1.8 | 6.3.1-6.3.7 |
27 | 10/24 | Functor represented by a scheme. Examples: affine schemes, fiber products. Towards representability theorem: (1) Yoneda lemma; (2) representable functors are 'sheaves.' | 1.3.10, 9.1.1-9.1.4, 9.1.6 | |
28 | 10/26 | Open subfunctors and open coverings. Representability theorem completed. | ||
29 | 10/29 | Constructing X×_{S}Y. Product of varieties over k. What reducedness of the scheme X×_{k}Y says about equations for the classical variety X×Y. | I:3.2 | 9.1.5, 9.5.15-9.5.21 |
30 | 10/31 | A condition for X×_{k}Y to be reduced. Base extension by open/closed embeddings and by 𝔸^{n}_{A}, ℙ^{n}_{A} over Spec(A). | I:3.3, 4.3; IV:4.3, 4.6 | 9.2 |
31 | 11/2 | (Topological X×_{T}Y) → (scheme X×_{T}Y) is surjective; examples. Monomomorphisms. Examples: open, closed embeddings; localization morphisms. Monos are (universally) injective. | I:3.4-3.5 | 9.3-9.5 |
32 | 11/5 | Fibers of a morphism. Example: fibers of 𝔸^{2}_{k} → 𝔸^{1}_{k} | I:3.6 | |
33 | 11/7 | Sober spaces. Jacobson spaces and Jacobson rings. Theorem on algebraic schemes (stated). | IV:10.1-10.4; Notes 1 | |
34 | 11/9 | Morphisms locally of finite type. Local nature of being locally of finite type. | I:6.3, 6.6 | 7.3.12 |
35 | 11/14 | Properties of morphisms locally of finite type relative to composition. Local nature of being Jacobson. Main theorem on schemes locally of finite type over a Jacobson scheme. | ||
36 | 11/16 | Classes cancelled because of smoky air. | ||
37 | 11/19 | Classes cancelled again. | ||
38 | 11/26 | Proof of the main theorem on Jacobson schemes. | ||
39 | 11/28 | Finish proof from Lecture 38. Comparison of classical affine variety X and scheme Spec(𝒪(X)). Corollary: Hilbert's nullstellensatz. | ||
40 | 11/30 | Equivalence (classical varieties over k)↔(reduced algebraic schemes over k). Examples. |
I will post problem sets here from time to time. I plan to assign more problems than I actuallly expect you to do. Grades will be based on your doing a reasonable fraction of them. Of course you are welcome to tackle all the problems if you feel ambitious.
Foundations, Part I
Foundations, Part II
Foundations, Part III
Applications