The Stacks project

Lemma 6.30.15. Let $(f, f^\sharp ) : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_ Y$-modules. Let $\mathcal{B}$ be a basis for the topology on $Y$. Suppose given for every $V \in \mathcal{B}$ a $\mathcal{O}_ Y(V)$-module map

\[ \varphi _ V : \mathcal{G}(V) \longrightarrow \mathcal{F}(f^{-1}V) \]

(where $\mathcal{F}(f^{-1}V)$ has a module structure using $f^\sharp _ V : \mathcal{O}_ Y(V) \to \mathcal{O}_ X(f^{-1}V)$) compatible with restriction mappings. Then there is a unique $f$-map (see discussion of $f$-maps in Section 6.26) $\varphi : \mathcal{G} \to \mathcal{F}$ recovering $\varphi _ V$ for $V \in \mathcal{B}$.

Proof. Same as the proof of the corresponding lemma for sheaves of algebraic structures above. $\square$

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