The Stacks project

This is almost the same as [Example 4.2, Neeman-Grothendieck].

Lemma 84.3.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism between quasi-separated and quasi-compact algebraic spaces over $S$. The functor $Rf_* : D_\mathit{QCoh}(X) \to D_\mathit{QCoh}(Y)$ has a right adjoint.

Proof. We will prove a right adjoint exists by verifying the hypotheses of Derived Categories, Proposition 13.38.2. First off, the category $D_\mathit{QCoh}(\mathcal{O}_ X)$ has direct sums, see Derived Categories of Spaces, Lemma 73.5.3. The category $D_\mathit{QCoh}(\mathcal{O}_ X)$ is compactly generated by Derived Categories of Spaces, Theorem 73.15.4. Since $X$ and $Y$ are quasi-compact and quasi-separated, so is $f$, see Morphisms of Spaces, Lemmas 65.4.10 and 65.8.9. Hence the functor $Rf_*$ commutes with direct sums, see Derived Categories of Spaces, Lemma 73.6.2. This finishes the proof. $\square$


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