The Stacks project

Lemma 36.10.9. Let $X$ be a scheme. 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, Lemma 20.39.6 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. Lemma 36.10.8 reduces cases (1) and (3) to the affine case which is treated in More on Algebra, Lemma 15.97.3. (You also have to use Lemmas 36.10.2, 36.10.7, and 36.10.4 to do the translation into algebra.) 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 we can work locally and reduce to the case that $K$ is a finite complex of finite free modules in which case it is clear. $\square$


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