Lemma 37.66.4. 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 pseudo-coherent $E$ in $D(\mathcal{O}_ X)$, then $K$ is pseudo-coherent relative to $A$.

**Proof.**
Assume $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ and $R\Gamma (X, E \otimes ^\mathbf {L} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent $E$ in $D(\mathcal{O}_ X)$. Let $x \in X$. We will show that $K$ is pseudo-coherent relative to $A$ in a neighbourhood of $x$ and this will prove the lemma.

Choose $U, n, V, Z, z, E$ as in Lemma 37.66.2. Denote $p : X \times \mathbf{P}^ n \to X$ and $q : X \times \mathbf{P}^ n \to \mathbf{P}^ n_ A$ the projections. Then for any $i \in \mathbf{Z}$ we have

by Derived Categories of Schemes, Lemma 36.22.1. By Derived Categories of Schemes, Lemma 36.30.5 the complex $Rq_*(E \otimes ^\mathbf {L} Lq^*\mathcal{O}_{\mathbf{P}^ n_ A}(i))$ is pseudo-coherent on $X$. Hence the assumption tells us the expression in the displayed formula is a pseudo-coherent object of $D(A)$. By Derived Categories of Schemes, Lemma 36.34.2 we conclude that $Rq_*(Lp^*K \otimes ^\mathbf {L} E)$ is pseudo-coherent on $\mathbf{P}^ n_ A$. By Lemma 37.66.3 we have

Since $U \to V$ is a closed immersion into an open subscheme of $\mathbf{P}^ n_ A$ this means $K|_ U$ is pseudo-coherent relative to $A$ by Lemma 37.56.18. $\square$

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