The Stacks project

Lemma 20.51.4. Let $X$ be a ringed space. Let $K, M$ be objects of $D(\mathcal{O}_ X)$. Let $x \in X$. The canonical map

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, M)_ x \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X, x}}(K_ x, M_ x) \]

is an isomorphism in the following cases

  1. $K$ is perfect,

  2. $K$ is pseudo-coherent and $M$ is (locally) bounded below.

Proof. Let $Y = \{ x\} $ be the singleton ringed space with structure sheaf given by $\mathcal{O}_{X, x}$. Then apply Lemma 20.51.3 to the flat inclusion morphism $Y \to X$. $\square$

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