The Stacks project

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

  1. 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 $\geq 1$.

  2. 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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