The Stacks project

Lemma 63.10.1. Let $f : X \to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let $\Lambda $ be a ring.

  1. Let $K_ i \in D^+_{tors}(X_{\acute{e}tale}, \Lambda )$, $i \in I$ be a family of objects. Assume given $a \in \mathbf{Z}$ such that $H^ n(K_ i) = 0$ for $n < a$ and $i \in I$. Then $Rf_!(\bigoplus _ i K_ i) = \bigoplus _ i Rf_!K_ i$.

  2. If $\Lambda $ is torsion, then the functor $Rf_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda )$ commutes with direct sums.

Proof. By construction it suffices to prove this when $f$ is an open immersion and when $f$ is a proper morphism. For any open immersion $j : U \to X$ of schemes, the functor $j_! : D(U_{\acute{e}tale}) \to D(X_{\acute{e}tale})$ is a left adjoint to pullback $j^{-1} : D(X_{\acute{e}tale}) \to D(U_{\acute{e}tale})$ and hence commutes with direct sums, see Cohomology on Sites, Lemma 21.20.8. In the proper case we have $Rf_! = Rf_*$ and we get the result from Étale Cohomology, Lemma 59.52.6 in the bounded belo case and from Étale Cohomology, Lemma 59.96.4 in the case that our coefficient ring $\Lambda $ is a torsion ring. $\square$

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Your email address will not be published.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 0G29. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 63.10.1 as pdf