The Stacks project

Lemma 48.12.3. Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \to Y$ be a morphism of schemes which is proper, flat, and of finite presentation. The map ( is an isomorphism for every object $K$ of $D_\mathit{QCoh}(\mathcal{O}_ Y)$.

Proof. By Lemma 48.12.1 we know that $a$ commutes with direct sums. Hence the collection of objects of $D_\mathit{QCoh}(\mathcal{O}_ Y)$ for which ( is an isomorphism is a strictly full, saturated, triangulated subcategory of $D_\mathit{QCoh}(\mathcal{O}_ Y)$ which is moreover preserved under taking direct sums. Since $D_\mathit{QCoh}(\mathcal{O}_ Y)$ is a module category (Derived Categories of Schemes, Theorem 36.18.2) generated by a single perfect object (Derived Categories of Schemes, Theorem 36.15.3) we can argue as in More on Algebra, Remark 15.59.11 to see that it suffices to prove ( is an isomorphism for a single perfect object. However, the result holds for perfect objects, see Lemma 48.8.1. $\square$

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