Lemma 37.63.4. 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 37.63.2. 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 37.63.3 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 37.53.18. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)