Lemma 76.51.5. Let A be a ring. Let X be an algebraic space 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 76.51.4, 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 Spaces, Proposition 75.29.3 it follows that K \in D^-_\mathit{QCoh}(\mathcal{O}_ X). Now the result follows by Derived Categories of Spaces, Lemma 75.25.7. \square
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