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. $\square$

Comment #746 by Anfang Zhou on

Typo. It should be $\mathcal{F}_{f(x)} \otimes_{\mathcal{O}_{Y, f(x)}} \mathcal{O}_{X, x}$

Comment #752 by on

Thanks for the comment. I have fixed this as well as the other typo you pointed out. See this commit.

There are also:

• 3 comment(s) on Section 6.26: Morphisms of ringed spaces and modules

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).