The Stacks project

Lemma 31.23.8. Let $f : X \to Y$ be a morphism of locally ringed spaces. Assume that pullbacks of meromorphic functions are defined for $f$ (see Definition 31.23.4).

  1. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ Y$-modules. There is a canonical pullback map $f^* : \Gamma (Y, \mathcal{K}_ Y(\mathcal{F})) \to \Gamma (X, \mathcal{K}_ X(f^*\mathcal{F}))$ for meromorphic sections of $\mathcal{F}$.

  2. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ Y$-module. A regular meromorphic section $s$ of $\mathcal{L}$ pulls back to a regular meromorphic section $f^*s$ of $f^*\mathcal{L}$.

Proof. Omitted. $\square$

Comments (4)

Comment #8691 by on

Typo in (2): it should be “let be an invertible -module.”

Part (1) can be stated more generally as:

Let be a morphism of locally ringed spaces. Assume pullbacks of meromorphic functions are defined for . Let , and suppose we have a morphism of -modules. Then there is a canonical morphism of -modules. (In particular, setting , the map to be the unit of and taking global sections gives the pullback map pullback map .)

The construction is the composite of canonical maps:

I regarded this more general form of part (1) useful for instance if one is working with the sheaf of meromorphic differentials on a -variety , i.e., . Pullbacks of meromorphic functions are defined for the normalization . Hence one obtains a morphism . When is a curve, this last morphism is used by Serre in Algebraic Groups and Class Fields to define the “sheaf of regular meromorphic differentials.”

Comment #8692 by on

Also, in my reformulation of part (1), instead of "canonical" one can say that:

The map is the unique morphism of -modules that makes the following diagram commute:

The fact that the constructed morphism indeed makes the square commute can be seen leveraging the factorization .

Uniqueness follows from the fact that by precomposing the -linear map with the unit we are obtaining the adjunct -linear morphism of ; thus, it is uniquely determined.

Comment #8703 by on

Regarding #8691: Ignore last paragraph. Serre does not use that map in his book. Also, the map from the case I explained is just the identity, as and are the sheaves constantly , where , is the generic point.

There are also:

  • 6 comment(s) on Section 31.23: Meromorphic functions and sections

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