Lemma 17.17.6 (05NI)—The Stacks project

Processing math: 100%

Table of contents
Part 1: Preliminaries
Chapter 17: Sheaves of Modules
Section 17.17: Flat modules
Lemma 17.17.6 (cite)

numberstags

Lemma 17.17.6. Let $(X, \mathcal{O}_ X)$ be a ringed space.

Any sheaf of $\mathcal{O}_ X$ -modules is a quotient of a direct sum $\bigoplus j_{U_ i!}\mathcal{O}_{U_ i}$ .
Any $\mathcal{O}_ X$ -module is a quotient of a flat $\mathcal{O}_ X$ -module.

Proof. Let $\mathcal{F}$ be an $\mathcal{O}_ X$ -module. For every open $U \subset X$ and every $s \in \mathcal{F}(U)$ we get a morphism $j_{U!}\mathcal{O}_ U \to \mathcal{F}$ , namely the adjoint to the morphism $\mathcal{O}_ U \to \mathcal{F}|_ U$ , $1 \mapsto s$ . Clearly the map

$\bigoplus \nolimits _{(U, s)} j_{U!}\mathcal{O}_ U \longrightarrow \mathcal{F}$

is surjective, and the source is flat by combining Lemmas 17.17.4 and 17.17.5. $\square$

Comments (0)

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