Lemma 17.10.7 (01BJ)—The Stacks project

Processing math: 100%

Table of contents
Part 1: Preliminaries
Chapter 17: Sheaves of Modules
Section 17.10: Quasi-coherent modules
Lemma 17.10.7 (cite)

numberstags

Lemma 17.10.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Set $R = \Gamma (X, \mathcal{O}_ X)$ . Let $M$ be an $R$ -module. Let $\mathcal{F}_ M$ be the quasi-coherent sheaf of $\mathcal{O}_ X$ -modules associated to $M$ . If $g : (Y, \mathcal{O}_ Y) \to (X, \mathcal{O}_ X)$ is a morphism of ringed spaces, then $g^*\mathcal{F}_ M$ is the sheaf associated to the $\Gamma (Y, \mathcal{O}_ Y)$ -module $\Gamma (Y, \mathcal{O}_ Y) \otimes _ R M$ .

Proof. The assertion follows from the first description of $\mathcal{F}_ M$ in Lemma 17.10.5 as $\pi ^*M$ , and the following commutative diagram of ringed spaces

$\xymatrix{ (Y, \mathcal{O}_ Y) \ar[r]_-\pi \ar[d]_ g & (\{ *\} , \Gamma (Y, \mathcal{O}_ Y)) \ar[d]^{\text{induced by }g^\sharp } \\ (X, \mathcal{O}_ X) \ar[r]^-\pi & (\{ *\} , \Gamma (X, \mathcal{O}_ X)) }$

(Also use Sheaves, Lemma 6.26.3.) $\square$

Comments (2)

Comment #2037 by Matthieu Romagny on May 11, 2016 at 11:18

Typo in 1st sentence of staatement : a ringed space

Comment #2075 by Johan on June 16, 2016 at 13:02

Thanks, fixed here.

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 (2)

Post a comment