Lemma 48.4.1 (0A9K)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 48: Duality for Schemes
Section 48.4: Right adjoint of pushforward and restriction to opens
Lemma 48.4.1 (cite)

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Lemma 48.4.1. In diagram (48.4.0.1) assume that $g$ is flat or more generally that $f$ and $g$ are Tor independent. Then $a \circ Rg_* \leftarrow Rg'_* \circ a'$ is an isomorphism.

Proof. In this case the base change map $Lg^* \circ Rf_* K \longrightarrow Rf'_* \circ L(g')^*K$ is an isomorphism for every $K$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ by Derived Categories of Schemes, Lemma 36.22.5. Thus the corresponding transformation between adjoint functors is an isomorphism as well. $\square$

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