Lemma 59.96.6. Let $f : X \to Y$ be a proper morphism of schemes. Let $\Lambda $ be a torsion ring. Let $E \in D(X_{\acute{e}tale}, \Lambda )$ and $K \in D(Y_{\acute{e}tale}, \Lambda )$. Then

in $D(Y_{\acute{e}tale}, \Lambda )$.

Lemma 59.96.6. Let $f : X \to Y$ be a proper morphism of schemes. Let $\Lambda $ be a torsion ring. Let $E \in D(X_{\acute{e}tale}, \Lambda )$ and $K \in D(Y_{\acute{e}tale}, \Lambda )$. Then

\[ Rf_*E \otimes _\Lambda ^\mathbf {L} K = Rf_*(E \otimes _\Lambda ^\mathbf {L} f^{-1}K) \]

in $D(Y_{\acute{e}tale}, \Lambda )$.

**Proof.**
There is a canonical map from left to right by Cohomology on Sites, Section 21.50. We will check the equality on stalks at $\overline{y}$. By the proper base change (in the form of Lemma 59.92.3 where $Y' = \overline{y}$) this reduces to the case where $Y$ is the spectrum of an algebraically closed field. This is shown in Lemma 59.96.5 where we use that $\text{cd}(X) < \infty $ by Lemma 59.92.2.
$\square$

Your email address will not be published.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)