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 $\geq 2$.

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