Lemma 86.3.1. 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.
This is almost the same as [Example 4.2, Neeman-Grothendieck].
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 75.5.3. The category D_\mathit{QCoh}(\mathcal{O}_ X) is compactly generated by Derived Categories of Spaces, Theorem 75.15.4. Since X and Y are quasi-compact and quasi-separated, so is f, see Morphisms of Spaces, Lemmas 67.4.10 and 67.8.9. Hence the functor Rf_* commutes with direct sums, see Derived Categories of Spaces, Lemma 75.6.2. This finishes the proof. \square
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