The Stacks project

Lemma 59.96.5. Let $X$ be a quasi-compact and quasi-separated scheme such that $\text{cd}(X) < \infty $. Let $\Lambda $ be a torsion ring. Let $E \in D(X_{\acute{e}tale}, \Lambda )$ and $K \in D(\Lambda )$. Then

\[ R\Gamma (X, E \otimes _\Lambda ^\mathbf {L} \underline{K}) = R\Gamma (X, E) \otimes _\Lambda ^\mathbf {L} K \]

Proof. There is a canonical map from left to right by Cohomology on Sites, Section 21.50. Let $T(K)$ be the property that the statement of the lemma holds for $K \in D(\Lambda )$. We will check conditions (1), (2), and (3) of More on Algebra, Remark 15.59.11 hold for $T$ to conclude. Property (1) holds because both sides of the equality commute with direct sums, see Lemma 59.96.3. Property (2) holds because we are comparing exact functors between triangulated categories and we can use Derived Categories, Lemma 13.4.3. Property (3) says the lemma holds when $K = \Lambda [k]$ for any shift $k \in \mathbf{Z}$ and this is obvious. $\square$

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Your email address will not be published.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 0F12. Beware of the difference between the letter 'O' and the digit '0'.