Lemma 69.13.4 (07UM)—The Stacks project

Processing math: 100%

Table of contents
Part 4: Algebraic Spaces
Chapter 69: Cohomology of Algebraic Spaces
Section 69.13: Coherent sheaves on Noetherian spaces
Lemma 69.13.4 (cite)

previous lemma

numberstags

Lemma 69.13.4. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ . Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. Let $\mathcal{G}$ be a coherent $\mathcal{O}_ X$ -module. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Denote $Z \subset X$ the corresponding closed subspace and set $U = X \setminus Z$ . There is a canonical isomorphism

$\mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}^ n\mathcal{G}, \mathcal{F}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{G}|_ U, \mathcal{F}|_ U).$

In particular we have an isomorphism

$\mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}^ n, \mathcal{F}) \longrightarrow \Gamma (U, \mathcal{F}).$

Proof. Let $W$ be an affine scheme and let $W \to X$ be a surjective étale morphism (see Properties of Spaces, Lemma 66.6.3). Set $R = W \times _ X W$ . Then $W$ and $R$ are Noetherian schemes, see Morphisms of Spaces, Lemma 67.23.5. Hence the result hold for the restrictions of $\mathcal{F}$ , $\mathcal{G}$ , and $\mathcal{I}$ , $U$ , $Z$ to $W$ and $R$ by Cohomology of Schemes, Lemma 30.10.5. It follows formally that the result holds over $X$ . $\square$

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