Lemma 20.48.4. 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 20.48.3 to the flat inclusion morphism $Y \to X$. $\square$

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