Lemma 69.12.3 (07UC)—The Stacks project

Table of contents
Part 4: Algebraic Spaces
Chapter 69: Cohomology of Algebraic Spaces
Section 69.12: Coherent modules on locally Noetherian algebraic spaces
Lemma 69.12.3 (cite)

Lemma 69.12.3. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. The category of coherent $\mathcal{O}_ X$-modules is abelian. More precisely, the kernel and cokernel of a map of coherent $\mathcal{O}_ X$-modules are coherent. Any extension of coherent sheaves is coherent.

Proof. Choose a scheme $U$ and a surjective étale morphism $f : U \to X$. Pullback $f^*$ is an exact functor as it equals a restriction functor, see Properties of Spaces, Equation (66.26.1.1). By Lemma 69.12.2 we can check whether an $\mathcal{O}_ X$-module $\mathcal{F}$ is coherent by checking whether $f^*\mathcal{F}$ is coherent. Hence the lemma follows from the case of schemes which is Cohomology of Schemes, Lemma 30.9.2. $\square$

Comments (2)

Comment #7867 by Anonymous on November 10, 2022 at 16:42

It seems to me that the proof implicitly uses some comparison results between modules on small Zariski and small étale sites. I wonder if adding some links to these results might be good (e.g. Lemma 35.8.10 and Lemma 35.10.2 or Lemma 35.10.3).

Comment #8083 by Stacks Project on January 23, 2023 at 20:57

Dear Anonymous, this is a rare instance where I slightly disagree with you. I think that this lemma and its proof never refers to modules in the Zariski topology. However, in translating to the case of schemes (at the very end), one just needs to know that the categories of (quasi-)coherent modules on a locally Noetherian scheme $U$ are the same as the corresponding categories of (quasi-)coherent modules on $U$ viewed as an algebraic space to be able to conclude. This was discussed for quasi-coherent modules much earlier, but was somehow missing in this section. So I've added that in this commit and I hope that this will help the future readers.

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