The Stacks project

Lemma 21.43.13. Notation and assumptions as in Lemma 21.43.12. Suppose that $K$ is an object of $\mathit{QC}(\mathcal{O})$ and $M$ arbitrary in $D(\mathcal{O}_\tau )$. For every object $U$ of $\mathcal{C}$ we have

\[ \mathop{\mathrm{Hom}}\nolimits _{D((\mathcal{O}_ U)_\tau )}(\epsilon ^*K|_ U, M|_ U) = R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}(U)}(R\Gamma (U, K), R\Gamma (U, M)) \]

Proof. We have

\[ \mathop{\mathrm{Hom}}\nolimits _{D((\mathcal{O}_ U)_\tau )}(\epsilon ^*K|_ U, M|_ U) = \mathop{\mathrm{Hom}}\nolimits _{D((\mathcal{O}_ U)_{\tau '})}(K|_ U, R\epsilon _*M|_ U) \]

by adjunction. Hence the result by Lemma 21.43.5 and the fact that

\[ R\Gamma (U, M) = R\Gamma (U, R\epsilon _*M) \]

by Leray. $\square$

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