The Stacks project

Lemma 61.11.6. Let $f : X \to Y$ be a separated finite type morphism of quasi-separated and quasi-compact schemes. Let $\Lambda $ be a torsion ring. For every $K \in D(Y_{\acute{e}tale}, \Lambda )$ and $L \in D(X_{\acute{e}tale}, \Lambda )$ the map (61.11.4.1) induces an isomorphism

\[ R\mathop{\mathrm{Hom}}\nolimits _ X(L, Rf^!K) \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ Y(Rf_!L, K) \]

of global derived homs.

Proof. By the construction in Cohomology on Sites, Section 21.35 we have

\[ R\mathop{\mathrm{Hom}}\nolimits _ X(L, Rf^!K) = R\Gamma (X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (L, Rf^!K)) = R\Gamma (Y, Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (L, Rf^!K)) \]

(the second equality by Leray) and

\[ R\mathop{\mathrm{Hom}}\nolimits _ Y(Rf_!L, K) = R\Gamma (Y, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (Rf_!L, K)) \]

Thus the lemma is a consequence of Lemma 61.11.5. $\square$


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