Lemma 69.14.4 (07US)—The Stacks project

Loading web-font TeX/Math/Italic

Table of contents
Part 4: Algebraic Spaces
Chapter 69: Cohomology of Algebraic Spaces
Section 69.14: Devissage of coherent sheaves
Lemma 69.14.4 (cite)

numberstags

Lemma 69.14.4. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ . Let $\mathcal{P}$ be a property of coherent sheaves on $X$ . Assume

For any short exact sequence of coherent sheaves

$0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0$

if $\mathcal{F}_ i$ , $i = 1, 2$ have property $\mathcal{P}$ then so does $\mathcal{F}$ .
For every reduced closed subspace $Z \subset X$ with $|Z|$ irreducible and every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ Z$ we have $\mathcal{P}$ for $i_*\mathcal{I}$ .

Then property $\mathcal{P}$ holds for every coherent sheaf on $X$ .

Proof. First note that if $\mathcal{F}$ is a coherent sheaf with a filtration

$0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_ m = \mathcal{F}$

by coherent subsheaves such that each of $\mathcal{F}_ i/\mathcal{F}_{i - 1}$ has property $\mathcal{P}$ , then so does $\mathcal{F}$ . This follows from the property (1) for $\mathcal{P}$ . On the other hand, by Lemma 69.14.3 we can filter any $\mathcal{F}$ with successive subquotients as in (2). Hence the lemma follows. $\square$

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