Lemma 21.39.8 (08QA)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.39: Homology on a category
Lemma 21.39.8 (cite)

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Lemma 21.39.8. Notation and assumptions as in Example 21.39.3. If there exists a cosimplicial object $U'_\bullet $ of $\mathcal{C}'$ such that Lemma 21.39.7 applies to both $U'_\bullet $ in $\mathcal{C}'$ and $u(U'_\bullet )$ in $\mathcal{C}$, then we have $L\pi '_! \circ g^{-1} = L\pi _!$ as functors $D(\mathcal{C}) \to D(\textit{Ab})$, resp. $D(\mathcal{C}, \underline{B}) \to D(B)$.

Proof. Follows immediately from Lemma 21.39.7 and the fact that $g^{-1}$ is given by precomposing with $u$. $\square$

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