Lemma 48.8.1. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes. The map $Lf^*K \otimes ^\mathbf {L}_{\mathcal{O}_ X} a(L) \to a(K \otimes _{\mathcal{O}_ Y}^\mathbf {L} L)$ defined above for $K, L \in D_\mathit{QCoh}(\mathcal{O}_ Y)$ is an isomorphism if $K$ is perfect. In particular, (48.8.0.1) is an isomorphism if $K$ is perfect.

Proof. Let $K^\vee$ be the “dual” to $K$, see Cohomology, Lemma 20.48.5. For $M \in D_\mathit{QCoh}(\mathcal{O}_ X)$ we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ Y)}(Rf_*M, K \otimes ^\mathbf {L}_{\mathcal{O}_ Y} L) & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ Y)}( Rf_*M \otimes ^\mathbf {L}_{\mathcal{O}_ Y} K^\vee , L) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}( M \otimes ^\mathbf {L}_{\mathcal{O}_ X} Lf^*K^\vee , a(L)) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(M, Lf^*K \otimes ^\mathbf {L}_{\mathcal{O}_ X} a(L)) \end{align*}

Second equality by the definition of $a$ and the projection formula (Cohomology, Lemma 20.52.3) or the more general Derived Categories of Schemes, Lemma 36.22.1. Hence the result by the Yoneda lemma. $\square$

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