## Tag `0098`

Chapter 6: Sheaves on Spaces > Section 6.26: Morphisms of ringed spaces and modules

Lemma 6.26.4. Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_Y$-modules. Let $x \in X$. Then $$ (f^*\mathcal{G})_x = \mathcal{G}_{f(x)} \otimes_{\mathcal{O}_{Y, f(x)}} \mathcal{O}_{X, x} $$ as $\mathcal{O}_{X, x}$-modules where the tensor product on the right uses $f^\sharp_x : \mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$.

Proof.This follows from Lemma 6.20.3 and the identification of the stalks of pullback sheaves at $x$ with the corresponding stalks at $f(x)$. See the formulae in Section 6.23 for example. $\square$

The code snippet corresponding to this tag is a part of the file `sheaves.tex` and is located in lines 3174–3188 (see updates for more information).

```
\begin{lemma}
\label{lemma-stalk-pullback-modules}
Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$
be a morphism of ringed spaces.
Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_Y$-modules.
Let $x \in X$. Then
$$
(f^*\mathcal{G})_x =
\mathcal{G}_{f(x)}
\otimes_{\mathcal{O}_{Y, f(x)}}
\mathcal{O}_{X, x}
$$
as $\mathcal{O}_{X, x}$-modules where the tensor product on the right
uses $f^\sharp_x : \mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$.
\end{lemma}
\begin{proof}
This follows from Lemma \ref{lemma-stalk-tensor-sheaf-modules}
and the identification of the stalks of pullback sheaves
at $x$ with the corresponding stalks at $f(x)$. See the
formulae in Section \ref{section-presheaves-structures-functorial}
for example.
\end{proof}
```

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