Lemma 48.17.8. In Situation 48.16.1 let f : X \to Y be a morphism of \textit{FTS}_ S. Let K be a dualizing complex on Y. Set D_ Y(M) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(M, K) for M \in D_{\textit{Coh}}(\mathcal{O}_ Y) and D_ X(E) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(E, f^!K) for E \in D_{\textit{Coh}}(\mathcal{O}_ X). Then there is a canonical isomorphism
f^!M \longrightarrow D_ X(Lf^*D_ Y(M))
for M \in D_{\textit{Coh}}^+(\mathcal{O}_ Y).
Proof.
Choose compactification j : X \subset \overline{X} of X over Y (More on Flatness, Theorem 38.33.8 and Lemma 38.32.2). Let a be the right adjoint of Lemma 48.3.1 for \overline{X} \to Y. Set D_{\overline{X}}(E) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{\overline{X}}}(E, a(K)) for E \in D_{\textit{Coh}}(\mathcal{O}_{\overline{X}}). Since formation of R\mathop{\mathcal{H}\! \mathit{om}}\nolimits commutes with restriction to opens and since f^! = j^* \circ a we see that it suffices to prove that there is a canonical isomorphism
a(M) \longrightarrow D_{\overline{X}}(L\overline{f}^*D_ Y(M))
for M \in D_{\textit{Coh}}(\mathcal{O}_ Y). For F \in D_\mathit{QCoh}(\mathcal{O}_ X) we have
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\overline{X}}( F, D_{\overline{X}}(L\overline{f}^*D_ Y(M))) & = \mathop{\mathrm{Hom}}\nolimits _{\overline{X}}( F \otimes _{\mathcal{O}_ X}^\mathbf {L} L\overline{f}^*D_ Y(M), a(K)) \\ & = \mathop{\mathrm{Hom}}\nolimits _ Y( R\overline{f}_*(F \otimes _{\mathcal{O}_ X}^\mathbf {L} L\overline{f}^*D_ Y(M)), K) \\ & = \mathop{\mathrm{Hom}}\nolimits _ Y( R\overline{f}_*(F) \otimes _{\mathcal{O}_ Y}^\mathbf {L} D_ Y(M), K) \\ & = \mathop{\mathrm{Hom}}\nolimits _ Y( R\overline{f}_*(F), D_ Y(D_ Y(M))) \\ & = \mathop{\mathrm{Hom}}\nolimits _ Y(R\overline{f}_*(F), M) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\overline{X}}(F, a(M)) \end{align*}
The first equality by Cohomology, Lemma 20.42.2. The second by definition of a. The third by Derived Categories of Schemes, Lemma 36.22.1. The fourth equality by Cohomology, Lemma 20.42.2 and the definition of D_ Y. The fifth equality by Lemma 48.2.5. The final equality by definition of a. Hence we see that a(M) = D_{\overline{X}}(L\overline{f}^*D_ Y(M)) by Yoneda's lemma.
\square
Comments (0)