Math 254B. Pull-Backs of Line Bundles and Rational Sections

This handout discusses pulling back of quasi-coherent sheaves, regular sections of them, and nonzero rational sections of line sheaves.

Throughout this handout, ϕ: X → Y is a morphism of schemes.

Pull-Backs of Quasi-Coherent Sheaves and Their Regular Sections We start by recalling the definition of the pull-back of a quasi-coherent sheaf.

Definition 1. Let 𝒢 be a quasi-coherent sheaf on Y . Then its pull-back via ϕ is the

sheaf

ϕ^{∗}𝒢 = ϕ^{−1}𝒢 ⊗_{ϕ^{−1}𝒪_{Y}} 𝒪_{X} .  (1.1)

Recall that ϕ^{−1}𝒢 is the sheaf associated to the presheaf U ↦ \underrightarrow{lim}_{V ⊇ϕ(U)} 𝒢 (V) .

This can be a bit difficult to understand. Hartshorne (II, Prop. 5.2e) asserts that, if X = Spec A, Y = Spec B, and N is a B-module, then ϕ^{∗}(Ñ) ≅ (N ⊗_{B} A)^{∼}. As proof of this statement, he says that it “follow[s] directly from the definitions.” Frankly, this escapes me, but EGA I 1.6.5 provides more details.

In any case, for all n ∈ N it makes sense to say that ϕ^{∗}n (as a global section of ϕ^{∗}(Ñ) ) is n ⊗ 1 in the above (affine) case.

Let ϕ_{pre}^{−1}𝒢 be the presheaf used to define ϕ^{−1}𝒢 , and let V be an open subset of Y . Then the direct limit simplifies to (ϕ_{pre}^{−1}𝒢)(ϕ^{−1}(U)) = 𝒢 (V) , so this would lead to a natural definition of pull-back of sections of 𝒢 : if s ∈ 𝒢 (V) , then also s ∈ (ϕ_{pre}^{−1}𝒢)(ϕ^{−1}(U)) , so we can map that to s^{+} ∈ (ϕ^{−1}𝒢)(ϕ^{−1}(U)) , and let ϕ^{∗}s be the image of s^{+} ⊗ 1 in (ϕ^{∗}𝒢)(ϕ^{−1}(U)) .

When 𝒢 = 𝒪_{Y} , we can in fact see directly from the definition that

ϕ^{∗}𝒪_{Y} = ϕ^{−1}𝒪_{Y} ⊗_{ϕ^{−1}𝒪_{Y}} 𝒪_{X} = 𝒪_{X} ,

and that if V ⊆ Y is an open set and s ∈ 𝒪_{Y} (V) , then ϕ^{∗}s ∈ 𝒪_{X}(ϕ^{−1}(V)) is the image of s under the map ϕ^{#} : 𝒪_{Y} → ϕ_{∗}𝒪_{X} ; i.e., ϕ^{∗}s = ϕ^{#}(s) ∈ (ϕ_{∗}𝒪_{X})(V) = 𝒪_{X}(ϕ^{−1}(V)) .

So, if f^{∗}ℳ is a line sheaf on Y , then the above definition of f^{∗}s for sections s ∈ ℳ (V) is compatible with the earlier definition (from class) of f^{∗}ℳ using local trivializations and transition functions.

Pull-Backs of Rational Sections Assume from now on that the schemes X and Y are integral.

We recall the definition from class.

Definition 2. Let ℳ be a line sheaf on Y , and let s be a nonzero rational section of ℳ . Assume that there exists an open subset V ⊆ Y such that (i). s is represented by a regular section s_{V} ∈ ℳ (V) , and (ii). the pulled-back section ϕ^{∗}(s_{V}) ∈ (ϕ^{∗}ℳ)(ϕ^{−1}(V)) is nonzero.

Then ϕ^{−1}(V) ≠ ∅ (since ℱ (∅) = 0 for all sheaves ℱ ), and we define the pullback ϕ^{∗}s of s (as a nonzero rational section) to be the nonzero rational section of ϕ^{∗}ℳ on X represented by ϕ^{∗}(s_{V}) ∈ (ϕ^{∗}ℳ)(ϕ^{−1}(V)) .

We can then show that pull-backs of rational sections are related to their divisors, as follows.

Proposition 3. Let s be a nonzero rational section of a line sheaf ℳ on Y , and let D = div(s) . Then: (a). ϕ^{∗}s is defined (as a nonzero rational section of ϕ^{∗}ℳ ) if and only if the

Cartier divisor ϕ^{∗}D is defined, and (b). if so, then div(ϕ^{∗}s) = ϕ^{∗}D.

Proof. (a). Let ξ be the generic point of X, and let y = ϕ(ξ) . Then \overline{ϕ(X)} = \overline{\{y\}}. Also

ϕ^{∗}D is defined ⟺ ϕ(X) ⊈ Supp D ⟺ \overline{\{y\}} = \overline{ϕ(X)} ⊈ Supp D ⟺ y ∉ Supp D .

If this is true, then there is an open neighborhood V of y in Y such that V ∩Supp D = ∅ and ℳ |_{V} ≅ 𝒪_{V} . Moreover, s extends to a regular section s_{V} ∈ ℳ (V) , necessarily nonzero. Choose s_{0} ∈ ℳ (V) that generates ℳ |_{V} ; then 0 = D|_{V} = (s_{V} /s_{0})|_{V} , so s_{V} /s_{0} ∈ 𝒪(V)^{∗}; therefore s_{V} is also a generator of ℳ |_{V} . Moreover, the open set U = ϕ^{−1}(V) is nonempty, and ϕ^{∗}s_{V} generates ϕ^{∗}ℳ |_{U} , so ϕ^{∗}s is defined as a nonzero rational section of ϕ^{∗}ℳ .

(Also, from the point of view of nonzero rational sections as nonzero elements of the stalk of the sheaf at the generic point, we have that y ∉ Supp D implies that the stalk (s_{V})_{y} of s_{V} exists and is a nonzero element of ℳ_{y}, so we can map it to a nonzero element of (ϕ^{∗}ℳ)_{ξ} as with ϕ_{ξ}^{#} : 𝒪_{Y,ϕ(ξ)} → 𝒪_{X,ξ} on pages 72–73 of Hartshorne.)

Conversely, if ϕ^{∗}s is defined (as in Definition 2), then there exists an open subset V ⊆ Y such that s is represented by s_{V} ∈ ℳ (V) and ϕ^{∗}(s_{V}) ∈ (ϕ^{∗}ℳ)(ϕ^{−1}(V)) is nonzero. Since y ∈ V , (s_{V})_{y} is defined, and since ϕ^{∗}(s_{V}) is nonzero (and X is reduced), (s_{V})_{y} ∉ 𝔪_{y}ℳ_{y}. Therefore y ∉ Supp D, so ϕ^{∗}D is defined.

(b). The proof of this part is left as an exercise. □