Lemma 17.9.6 (01BA)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 17: Sheaves of Modules
Section 17.9: Modules of finite type
Lemma 17.9.6 (cite)

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Over any ringed space, sheaves of modules of finite type have closed support.

Lemma 17.9.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$ -modules. If $\mathcal{F}$ is of finite type then support of $\mathcal{F}$ is closed.

Proof. This is a reformulation of Lemma 17.9.5. $\square$

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Comment #889 by Konrad Voelkel on August 07, 2014 at 21:27

Suggested slogan: Over any ringed space, sheaves of modules of finite type have closed support.

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