Lemma 18.36.4 (05V5)—The Stacks project

Processing math: 100%

Table of contents
Part 1: Preliminaries
Chapter 18: Modules on Sites
Section 18.36: Stalks of modules
Lemma 18.36.4 (cite)

previous lemma

numberstags

Lemma 18.36.4. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi or ringed sites. Let $p$ be a point of $\mathcal{C}$ or $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and set $q = f \circ p$ . Then

$(f^*\mathcal{F})_ p = \mathcal{F}_ q \otimes _{\mathcal{O}_{\mathcal{D}, q}} \mathcal{O}_{\mathcal{C}, p}$

for any $\mathcal{O}_\mathcal {D}$ -module $\mathcal{F}$ .

Proof. We have

$f^*\mathcal{F} = f^{-1}\mathcal{F} \otimes _{f^{-1}\mathcal{O}_\mathcal {D}} \mathcal{O}_\mathcal {C}$

by definition. Since taking stalks at $p$ (i.e., applying $p^{-1}$ ) commutes with $\otimes$ by Lemma 18.26.2 we win by the relation between the stalk of pullbacks at $p$ and stalks at $q$ explained in Sites, Lemma 7.34.2 or Sites, Lemma 7.34.3. $\square$

Comments (0)

There are also:

2 comment(s) on Section 18.36: Stalks of 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).

Comments (0)

Post a comment