Lemma 21.39.6 (08Q8)—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 21.39.6 (cite)

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Lemma 21.39.6. Notation and assumptions as in Example 21.39.1. Let $B \to B'$ be a ring map. Consider the commutative diagram of ringed topoi

$\xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \underline{B}) \ar[d]_\pi & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \underline{B'}) \ar[d]^{\pi '} \ar[l]^ h \\ (*, B) & (*, B') \ar[l]_ f }$

Then $L\pi _! \circ Lh^* = Lf^* \circ L\pi '_!$ .

Proof. Both functors are right adjoint to the obvious functor $D(B') \to D(\underline{B})$ . $\square$

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