Lemma 86.4.1 (0E5C)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 86: Duality for Spaces
Section 86.4: Right adjoint of pushforward and base change, I
Lemma 86.4.1 (cite)

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Lemma 86.4.1. In diagram (86.4.0.1) the map $a \circ Rg_* \leftarrow Rg'_* \circ a'$ is an isomorphism.

Proof. The base change map $Lg^* \circ Rf_* K \to Rf'_* \circ L(g')^*K$ is an isomorphism for every $K$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ by Derived Categories of Spaces, Lemma 75.20.4 (this uses the assumption of Tor independence). Thus the corresponding transformation between adjoint functors is an isomorphism as well. $\square$

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