# The Stacks Project

## Tag 009Y

Lemma 6.30.17. 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 $\mathcal{B}_Y$ be a basis for the topology on $Y$. Let $\mathcal{B}_X$ be a basis for the topology on $X$. Suppose given for every $V \in \mathcal{B}_Y$, and $U \in \mathcal{B}_X$ such that $f(U) \subset V$ a $\mathcal{O}_Y(V)$-module map $$\varphi_V^U : \mathcal{G}(V) \longrightarrow \mathcal{F}(U)$$ compatible with restriction mappings. Here the $\mathcal{O}_Y(V)$-module structure on $\mathcal{F}(U)$ comes from the $\mathcal{O}_X(U)$-module structure via the map $f^\sharp_V : \mathcal{O}_Y(V) \to \mathcal{O}_X(f^{-1}V) \to \mathcal{O}_X(U)$. Then there is a unique $f$-map of sheaves of modules (see Definition 6.21.7 and the discussion of $f$-maps in Section 6.26) $\varphi : \mathcal{G} \to \mathcal{F}$ recovering $\varphi_V^U$ as the composition $$\mathcal{G}(V) \xrightarrow{\varphi_V} \mathcal{F}(f^{-1}(V)) \xrightarrow{\text{restrc.}} \mathcal{F}(U)$$ for every pair $(U, V)$ as above.

Proof. Similar to the above and omitted. $\square$

The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 4328–4361 (see updates for more information).

\begin{lemma}
\label{lemma-f-map-basis-above-and-below-modules}
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 $\mathcal{B}_Y$ be a basis for the topology on $Y$.
Let $\mathcal{B}_X$ be a basis for the topology on $X$.
Suppose given for every $V \in \mathcal{B}_Y$, and
$U \in \mathcal{B}_X$ such that $f(U) \subset V$ a
$\mathcal{O}_Y(V)$-module map
$$\varphi_V^U : \mathcal{G}(V) \longrightarrow \mathcal{F}(U)$$
compatible with restriction mappings. Here the
$\mathcal{O}_Y(V)$-module structure on $\mathcal{F}(U)$
comes from the $\mathcal{O}_X(U)$-module structure
via the map $f^\sharp_V : \mathcal{O}_Y(V) \to \mathcal{O}_X(f^{-1}V) \to \mathcal{O}_X(U)$.
Then there is a unique $f$-map of sheaves of modules (see
Definition \ref{definition-f-map} and the discussion
of $f$-maps in Section \ref{section-ringed-spaces-functoriality-modules})
$\varphi : \mathcal{G} \to \mathcal{F}$
recovering $\varphi_V^U$ as the composition
$$\mathcal{G}(V) \xrightarrow{\varphi_V} \mathcal{F}(f^{-1}(V)) \xrightarrow{\text{restrc.}} \mathcal{F}(U)$$
for every pair $(U, V)$ as above.
\end{lemma}

\begin{proof}
Similar to the above and omitted.
\end{proof}

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