Lemma 48.3.6. Let f : X \to Y be a morphism of quasi-compact and quasi-separated schemes. Let a be the right adjoint to Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y). Let L \in D_\mathit{QCoh}(\mathcal{O}_ X) and K \in D_\mathit{QCoh}(\mathcal{O}_ Y). Then the map (48.3.5.1)
Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, a(K)) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(Rf_*L, K)
becomes an isomorphism after applying the functor DQ_ Y : D(\mathcal{O}_ Y) \to D_\mathit{QCoh}(\mathcal{O}_ Y) discussed in Derived Categories of Schemes, Section 36.21.
Proof.
The statement makes sense as DQ_ Y exists by Derived Categories of Schemes, Lemma 36.21.1. Since DQ_ Y is the right adjoint to the inclusion functor D_\mathit{QCoh}(\mathcal{O}_ Y) \to D(\mathcal{O}_ Y) to prove the lemma we have to show that for any M \in D_\mathit{QCoh}(\mathcal{O}_ Y) the map (48.3.5.1) induces an bijection
\mathop{\mathrm{Hom}}\nolimits _ Y(M, Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, a(K))) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _ Y(M, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(Rf_*L, K))
To see this we use the following string of equalities
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _ Y(M, Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, a(K))) & = \mathop{\mathrm{Hom}}\nolimits _ X(Lf^*M, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, a(K))) \\ & = \mathop{\mathrm{Hom}}\nolimits _ X(Lf^*M \otimes _{\mathcal{O}_ X}^\mathbf {L} L, a(K)) \\ & = \mathop{\mathrm{Hom}}\nolimits _ Y(Rf_*(Lf^*M \otimes _{\mathcal{O}_ X}^\mathbf {L} L), K) \\ & = \mathop{\mathrm{Hom}}\nolimits _ Y(M \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L, K) \\ & = \mathop{\mathrm{Hom}}\nolimits _ Y(M, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(Rf_*L, K)) \end{align*}
The first equality holds by Cohomology, Lemma 20.28.1. The second equality by Cohomology, Lemma 20.42.2. The third equality by construction of a. The fourth equality by Derived Categories of Schemes, Lemma 36.22.1 (this is the important step). The fifth by Cohomology, Lemma 20.42.2.
\square
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