Lemma 30.11.2 (0EBC)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 30: Cohomology of Schemes
Section 30.11: Depth
Lemma 30.11.2 (cite)

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Lemma 30.11.2. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ , $\mathcal{G}$ be coherent $\mathcal{O}_ X$ -modules and $x \in X$ .

If $\mathcal{G}_ x$ has depth $\geq 1$ , then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})_ x$ has depth $\geq 1$ .
If $\mathcal{G}_ x$ has depth $\geq 2$ , then $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})_ x$ has depth $\geq 2$ .

Proof. Observe that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is a coherent $\mathcal{O}_ X$ -module by Lemma 30.9.4. Coherent modules are of finite presentation (Lemma 30.9.1) hence taking stalks commutes with taking $\mathop{\mathcal{H}\! \mathit{om}}\nolimits$ and $\mathop{\mathrm{Hom}}\nolimits$ , see Modules, Lemma 17.22.4. Thus we reduce to the case of finite modules over local rings which is More on Algebra, Lemma 15.23.10. $\square$

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