Lemma 86.3.7. Let S be a scheme. Let f : X \to Y be a morphism of quasi-separated and quasi-compact algebraic spaces over S. For all L \in D_\mathit{QCoh}(\mathcal{O}_ X) and K \in D_\mathit{QCoh}(\mathcal{O}_ Y) (86.3.2.1) induces an isomorphism R\mathop{\mathrm{Hom}}\nolimits _ X(L, a(K)) \to R\mathop{\mathrm{Hom}}\nolimits _ Y(Rf_*L, K) of global derived homs.
Proof. By construction (Cohomology on Sites, Section 21.36) the complexes
R\mathop{\mathrm{Hom}}\nolimits _ X(L, a(K)) = R\Gamma (X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, a(K))) = R\Gamma (Y, Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, a(K)))
and
R\mathop{\mathrm{Hom}}\nolimits _ Y(Rf_*L, K) = R\Gamma (Y, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(Rf_*L, a(K)))
Thus the lemma is a consequence of Lemma 86.3.3. Namely, a map E \to E' in D(\mathcal{O}_ Y) which induces an isomorphism DQ_ Y(E) \to DQ_ Y(E') induces a quasi-isomorphism R\Gamma (Y, E) \to R\Gamma (Y, E'). Indeed we have H^ i(Y, E) = \mathop{\mathrm{Ext}}\nolimits ^ i_ Y(\mathcal{O}_ Y, E) = \mathop{\mathrm{Hom}}\nolimits (\mathcal{O}_ Y[-i], E) = \mathop{\mathrm{Hom}}\nolimits (\mathcal{O}_ Y[-i], DQ_ Y(E)) because \mathcal{O}_ Y[-i] is in D_\mathit{QCoh}(\mathcal{O}_ Y) and DQ_ Y is the right adjoint to the inclusion functor D_\mathit{QCoh}(\mathcal{O}_ Y) \to D(\mathcal{O}_ Y). \square
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