Lemma 37.67.3. Let $f : X \to S$ be locally of finite type. Let $\{ S_ i \to S\} $ be an fppf covering of schemes. Denote $f_ i : X_ i \to S_ i$ the base change of $f$ and $g_ i : X_ i \to X$ the projection. Let $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$. Let $m \in \mathbf{Z}$. Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if each $Lg_ i^*E$ is $m$-pseudo-coherent relative to $S_ i$.

**Proof.**
This follows formally from Lemmas 37.67.1 and 37.67.2. Namely, if $E$ is $m$-pseudo-coherent relative to $S$, then $Lg_ i^*E$ is $m$-pseudo-coherent relative to $S$ (by the first lemma), hence $Lg_ i^*E$ is $m$-pseudo-coherent relative to $S_ i$ (by the second). Conversely, if $Lg_ i^*E$ is $m$-pseudo-coherent relative to $S_ i$, then $Lg_ i^*E$ is $m$-pseudo-coherent relative to $S$ (by the second lemma), hence $E$ is $m$-pseudo-coherent relative to $S$ (by the first lemma).
$\square$

## 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Your email address will not be published.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)