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 20.50.5. 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 20.54.3) or the more general Derived Categories of Schemes, Lemma 36.22.1. Hence the result by the Yoneda lemma. \square
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