Lemma 38.29.4. In Situation 38.29.1 let $K$ be as in Lemma 38.29.2. Then $K$ is pseudo-coherent on $X$.

**Proof.**
Combinging Lemma 38.29.3 and Derived Categories of Schemes, Lemma 36.34.3 we see that $R\Gamma (X, K \otimes ^\mathbf {L} E)$ is pseudo-coherent in $D(A)$ for all pseudo-coherent $E$ in $D(\mathcal{O}_ X)$. Thus it follows from More on Morphisms, Lemma 37.62.4 that $K$ is pseudo-coherent relative to $A$. Since $X$ is of flat and of finite presentation over $A$, this is the same as being pseudo-coherent on $X$, see More on Morphisms, Lemma 37.52.18.
$\square$

## 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Your email address will not be published.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)