The Stacks project

Lemma 21.53.4. Let $\mathcal{C}$ be a site. Let $\Lambda $ be a Noetherian ring. Let $K, L \in D^-(\mathcal{C}, \Lambda )$. If the cohomology sheaves of $K$ and $L$ are locally constant sheaves of $\Lambda $-modules of finite type, then the cohomology sheaves of $K \otimes _\Lambda ^\mathbf {L} L$ are locally constant sheaves of $\Lambda $-modules of finite type.

Proof. We'll prove this as an application of Lemma 21.53.1. 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. By Lemma 21.53.1 we may assume that $K$ and $L$ are represented by complexes of locally constant sheaves of $\Lambda $-modules of finite type. Then we can replace these complexes by bounded above complexes of finite free $\Lambda $-modules. In this case the result is clear. $\square$


Comments (0)


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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 094H. Beware of the difference between the letter 'O' and the digit '0'.