Lemma 15.75.4. Let $R' \to R$ be a surjective ring map whose kernel is a nilpotent ideal. Let $K' \in D(R')$ and set $K = K' \otimes _{R'}^\mathbf {L} R$. Then $K$ is pseudo-coherent if and only if $K'$ is pseudo-coherent.

**Proof.**
One direction follows from Lemma 15.64.12. For the other direction, assume $K$ is pseudo-coherent. Then by Lemma 15.64.5 we can represent $K$ by a bounded above complex $E^\bullet $ of finite free $R$-modules. By Lemma 15.75.3 we can represent $K'$ by a bounded above complex $P^\bullet $ of projective $R'$-modules such that $P^ n \otimes _{R'} R = E^ n$. By Nakayama's lemma we see that $P^ n$ is finite free and we conclude that $K'$ is pseudo-coherent as well.
$\square$

## 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Your email address will not be published.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)