The Stacks project

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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