The Stacks project

Lemma 75.19.2. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. If the right adjoints $DQ_ X$ and $DQ_ Y$ of the inclusion functors $D_\mathit{QCoh}\to D$ exist for $X$ and $Y$, then

\[ Rf_* \circ DQ_ X = DQ_ Y \circ Rf_* \]

Proof. The statement makes sense because $Rf_*$ sends $D_\mathit{QCoh}(\mathcal{O}_ X)$ into $D_\mathit{QCoh}(\mathcal{O}_ Y)$ by Lemma 75.6.1. The statement is true because $Lf^*$ similarly maps $D_\mathit{QCoh}(\mathcal{O}_ Y)$ into $D_\mathit{QCoh}(\mathcal{O}_ X)$ (Lemma 75.5.5) and hence both $Rf_* \circ DQ_ X$ and $DQ_ Y \circ Rf_*$ are right adjoint to $Lf^* : D_\mathit{QCoh}(\mathcal{O}_ Y) \to D(\mathcal{O}_ X)$. $\square$

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