Lemma 75.13.11. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $K, L, M$ be objects of $D_\mathit{QCoh}(\mathcal{O}_ X)$. The map

$K \otimes _{\mathcal{O}_ X}^\mathbf {L} R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, K \otimes _{\mathcal{O}_ X}^\mathbf {L} L)$

of Cohomology on Sites, Lemma 21.35.7 is an isomorphism in the following cases

1. $M$ perfect, or

2. $K$ is perfect, or

3. $M$ is pseudo-coherent, $L \in D^+(\mathcal{O}_ X)$, and $K$ has finite tor dimension.

Proof. Checking whether or not the map is an isomorphism can be done étale locally hence we may assume $X$ is an affine scheme. Then we can write $K, L, M$ as $\epsilon ^*K_0, \epsilon ^*L_0, \epsilon ^*M_0$ for some $K_0, L_0, M_0$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ by Lemma 75.4.2. Then we see that Lemma 75.13.9 reduces cases (1) and (3) to the case of schemes which is Derived Categories of Schemes, Lemma 36.10.9. If $K$ is perfect but no other assumptions are made, then we do not know that either side of the arrow is in $D_\mathit{QCoh}(\mathcal{O}_ X)$ but the result is still true because $K$ will be represented (after localizing further) by a finite complex of finite free modules in which case it is clear. $\square$

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