The Stacks project

Lemma 37.69.5. Let $A$ be a ring. Let $X$ be a scheme separated and of finite presentation over $A$. Let $K \in D_\mathit{QCoh}(\mathcal{O}_ X).$ If $R \Gamma (X, E \otimes ^{\mathbf{L}} K)$ is pseudo-coherent in $D(A)$ for every perfect $E \in D(\mathcal{O}_ X)$, then $K$ is pseudo-coherent relative to $A$.

Proof. In view of Lemma 37.69.4, it suffices to show $R \Gamma (X, E \otimes ^{\mathbf{L}} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent $E \in D(\mathcal{O}_ X)$. By Derived Categories of Schemes, Proposition 36.40.5 it follows that $K \in D^-_\mathit{QCoh}(\mathcal{O}_ X)$. Now the result follows by Derived Categories of Schemes, Lemma 36.34.3. $\square$

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