Lemma 61.27.5. Let $X$ be a scheme. Let $\Lambda $ be a Noetherian ring. Let $K, L \in D_ c^-(X_{pro\text{-}\acute{e}tale}, \Lambda )$. Then $K \otimes _\Lambda ^\mathbf {L} L$ is in $D_ c^-(X_{pro\text{-}\acute{e}tale}, \Lambda )$.

**Proof.**
Note that $H^ i(K \otimes _\Lambda ^\mathbf {L} L)$ is the same as $H^ i(\tau _{\geq i - 1}K \otimes _\Lambda ^\mathbf {L} \tau _{\geq i - 1}L)$. Thus we may assume $K$ and $L$ are bounded. In this case we can apply Lemma 61.27.4 to reduce to the case of the étale site, see Étale Cohomology, Lemma 59.76.6.
$\square$

## 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Your email address will not be published.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)