Lemma 91.5.6. Let $A \to B$ be a ring map. Let $\pi$, $\mathcal{O}$, $\underline{B}$ be as in (91.4.0.1). For any $\mathcal{O}$-module $\mathcal{F}$ we have

$L\pi _!(\mathcal{F}) = L\pi _!(Li^*\mathcal{F}) = L\pi _!(\mathcal{F} \otimes _\mathcal {O}^\mathbf {L} \underline{B})$

in $D(\textit{Ab})$.

Proof. It suffices to verify the assumptions of Cohomology on Sites, Lemma 21.39.12 hold for $\mathcal{O} \to \underline{B}$ on $\mathcal{C}_{B/A}$. We will use the results of Remark 91.5.5 without further mention. Choose a resolution $P_\bullet$ of $B$ over $A$ to get a suitable cosimplicial object $U_\bullet$ of $\mathcal{C}_{B/A}$. Since $P_\bullet \to B$ induces a quasi-isomorphism on associated complexes of abelian groups we see that $L\pi _!\mathcal{O} = B$. On the other hand $L\pi _!\underline{B}$ is computed by $\underline{B}(U_\bullet ) = B$. This verifies the second assumption of Cohomology on Sites, Lemma 21.39.12 and we are done with the proof. $\square$

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