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Pullback of a Cartier divisor computes the pullback of its line bundle

Statement

Let $f : X \to Y$ be a morphism of schemes and let $D$ be a Cartier divisor on $Y$ whose pullback $f^{*} D$ is defined (Pullback of a Cartier divisor). Then there is a canonical isomorphism of $O_{X}$ -modules $O_{X} (f^{*} D) ≅ f^{*} O_{Y} (D),$ where $f^{*}$ is the pullback of modules (Pullback of a module along a morphism of ringed spaces) and $O_{Y} (D)$ , $O_{X} (f^{*} D)$ are the invertible sheaves of Invertible sheaf of cartier divisor. If $D$ is effective, then the constant section $1 \in Γ (Y, O_{Y} (D))$ corresponds under this isomorphism to the constant section $1 \in Γ (X, O_{X} (f^{*} D))$ , and the isomorphism is independent of the admissible local-equation datum used to define $f^{*} D$ .

Facts & Assumptions

Given: A morphism $f : X \to Y$ , a Cartier divisor $D$ on $Y$ , and an $f$ -admissible local-equation datum ${(U_{i}, a_{i} / s_{i})}$ representing $D$ , with pulled-back equations $g_{i}^{'} = f^{#} (a_{i}) / f^{#} (s_{i})$ on $f^{- 1} U_{i}$ (Pullback of a Cartier divisor).

[F1]

$f^{*} D$ is the Cartier divisor on $X$ represented by the local-equation datum ${(f^{- 1} U_{i}, g_{i}^{'})}$ ; it is independent of the admissible datum, and for effective $D$ with regular equations $f_{i} \in S_{Y} (U_{i})$ one may take $g_{i}^{'} = f^{#} (f_{i})$ , the defined pullback then being effective (Pullback of a Cartier divisor).

[F2]

For the datum ${g_{i}}$ of $D$ one has $O_{Y} (D) ∣_{U_{i}} = g_{i}^{- 1} O_{U_{i}}$ , and $O_{X} (f^{*} D) ∣_{f^{- 1} U_{i}} = g_{i}^{' - 1} O_{f^{- 1} U_{i}}$ ; the sheaves are well defined and independent of the datum (Invertible sheaf of cartier divisor).

[F3]

On the overlap $U_{i} \cap U_{j}$ one has $g_{i} / g_{j} \in O_{Y}^{\times} (U_{i} \cap U_{j})$ ; consequently $g_{i}^{'} / g_{j}^{'} = f^{#} (g_{i} / g_{j}) \in O_{X}^{\times} (f^{- 1} (U_{i} \cap U_{j}))$ (Cartier divisor, Pullback of a Cartier divisor).

[F4]

The pullback of modules is $f^{*} G = O_{X} \otimes_{f^{- 1} O_{Y}} f^{- 1} G$ (Pullback of a module along a morphism of ringed spaces), its stalks satisfy $(f^{*} G)_{x} ≅ O_{X, x} \otimes_{O_{Y, f (x)}} G_{f (x)}$ (The stalk of a tensor product sheaf is the tensor product of the stalks, The stalk of an inverse image sheaf is the stalk over the image point), and it is a functor. If $M$ is an invertible $O_{Y}$ -module with generator $e$ over an open $V$ , then $f^{*} M$ is an invertible $O_{X}$ -module with generator $f^{*} (e)$ over $f^{- 1} V$ : at $x$ the stalk $M_{f (x)} = O_{Y, f (x)} e$ is free of rank one, so $O_{X, x} \otimes_{O_{Y, f (x)}} M_{f (x)} ≅ O_{X, x} \cdot (1 \otimes e)$ by the tensor unit isomorphism (The regular module is a tensor unit: $R \otimes_{R} N ≅ N$ and $M \otimes_{R} R ≅ M$ ), and the module map $O_{f^{- 1} V} \to f^{*} M ∣_{f^{- 1} V}$ sending $1$ to $f^{*} (e)$ is an isomorphism on stalks, hence an isomorphism of sheaves (A morphism of sheaves is an isomorphism exactly when it is an isomorphism on every stalk).

[F5]

An isomorphism of sheaves of modules may be presented by a gluing datum of local isomorphisms on a common cover; if two local isomorphisms agree on overlaps, they glue to a global isomorphism (Compatible local sheaves glue uniquely up to unique isomorphism, A gluing datum for sheaves on an open cover, A sheaf on a topological space).

Proof

1.1F2F3

Generators and transitions. On $U_{i}$ the sheaf $O_{Y} (D)$ is freely generated by $e_{i} = g_{i}^{- 1}$ , and on $U_{j}$ by $e_{j} = g_{j}^{- 1}$ ; on the overlap $e_{i} = u_{i j}^{- 1} e_{j}$ with $u_{i j} = g_{i} / g_{j}$ , a unit of $O_{Y} (U_{i} \cap U_{j})$ . On $X$ , the sheaf $O_{X} (f^{*} D)$ is freely generated over $f^{- 1} U_{i}$ by $e_{i}^{'} = g_{i}^{' - 1}$ , and $e_{i}^{'} = v_{i j}^{- 1} e_{j}^{'}$ with $v_{i j} = g_{i}^{'} / g_{j}^{'} = f^{#} (u_{i j})$ , a unit of $O_{X} (f^{- 1} (U_{i} \cap U_{j}))$ .

2.1F4step 1.1

Local isomorphisms. By [F4] the pullback $f^{*} O_{Y} (D)$ is freely generated over $f^{- 1} U_{i}$ by $f^{*} (e_{i})$ . Define an $O_{f^{- 1} U_{i}}$ -linear map $φ_{i} : f^{*} O_{Y} (D) ∣_{f^{- 1} U_{i}} \to O_{X} (f^{*} D) ∣_{f^{- 1} U_{i}}$ by $φ_{i} (f^{*} (e_{i})) = e_{i}^{'}$ . Since source and target are freely generated of rank one by these sections, $φ_{i}$ is an isomorphism.

3.1F4step 1.1step 2.1

Compatibility on overlaps. Over $f^{- 1} (U_{i} \cap U_{j})$ one has $f^{*} (e_{i}) = f^{*} (u_{i j}^{- 1} e_{j}) = f^{#} (u_{i j})^{- 1} f^{*} (e_{j})$ by the functoriality of $f^{*}$ and step 1.1, while $e_{i}^{'} = f^{#} (u_{i j})^{- 1} e_{j}^{'}$ by step 1.1. Hence $φ_{i} (f^{*} (e_{i})) = f^{#} (u_{i j})^{- 1} φ_{j} (f^{*} (e_{j})) = φ_{j} (f^{*} (e_{i}))$ , so $φ_{i}$ and $φ_{j}$ agree on the overlap.

4.1F5step 3.1

Gluing. The local isomorphisms $φ_{i}$ cover $X$ on the opens $f^{- 1} U_{i}$ and agree on all overlaps by step 3.1, so they glue to an isomorphism of $O_{X}$ -modules $Θ : f^{*} O_{Y} (D) \to O_{X} (f^{*} D)$ .

5.1F1F2F4step 4.1

Canonical sections for effective divisors. Suppose $D$ is effective, with regular equations $f_{i}$ on a refined cover, and put $f_{i}^{'} = f^{#} (f_{i})$ and $e_{i} = f_{i}^{- 1}$ , $e_{i}^{'} = f_{i}^{' - 1}$ . The constant section $1 \in Γ (Y, O_{Y} (D))$ satisfies $1 = f_{i} e_{i}$ over $U_{i}$ , and the constant section $1 \in Γ (X, O_{X} (f^{*} D))$ satisfies $1 = f_{i}^{'} e_{i}^{'}$ over $f^{- 1} U_{i}$ . Since $f^{*}$ is a functor and $Θ (f^{*} (e_{i})) = e_{i}^{'}$ , one has $Θ (f^{*} (1)) = Θ (f_{i}^{'} f^{*} (e_{i})) = f_{i}^{'} e_{i}^{'} = 1$ , so the constant sections correspond.

6.1step 4.1step 5.1∎

Conclusion. $Θ$ is a canonical isomorphism $O_{X} (f^{*} D) ≅ f^{*} O_{Y} (D)$ , and it matches the constant sections in the effective case; replacing the admissible datum by another one changes $e_{i}$ and $e_{i}^{'}$ by the same units and hence leaves $Θ$ unchanged, so the isomorphism is independent of the datum.

No choice principle is used: on each chart the isomorphism is determined by the given generators, and the local maps glue because they agree on overlaps. If $X$ or $Y$ is empty both sheaves are the zero sheaf and the isomorphism is the unique one.

Depends on

Used by

Dependency tree · two levels

31 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Divisors, §31.15 Lemma 15.5 and §31.14 Definition 14.12 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, Ch. 15 §§15.2–15.3 (standard reference, not scraped)