Lemma 59.77.7. Let $X$ be a connected scheme. Let $\Lambda$ be a Noetherian ring. Let $K \in D_{ctf}(X_{\acute{e}tale}, \Lambda )$ have locally constant cohomology sheaves. Then there exists a finite complex of finite projective $\Lambda$-modules $M^\bullet$ and an étale covering $\{ U_ i \to X\}$ such that $K|_{U_ i} \cong \underline{M^\bullet }|_{U_ i}$ in $D(U_{i, {\acute{e}tale}}, \Lambda )$.

Proof. Choose an étale covering $\{ U_ i \to X\}$ such that $K|_{U_ i}$ is constant, say $K|_{U_ i} \cong \underline{M_ i^\bullet }_{U_ i}$ for some finite complex of finite $\Lambda$-modules $M_ i^\bullet$. See Cohomology on Sites, Lemma 21.53.1. Observe that $U_ i \times _ X U_ j$ is empty if $M_ i^\bullet$ is not isomorphic to $M_ j^\bullet$ in $D(\Lambda )$. For each complex of $\Lambda$-modules $M^\bullet$ let $I_{M^\bullet } = \{ i \in I \mid M_ i^\bullet \cong M^\bullet \text{ in }D(\Lambda )\}$. As étale morphisms are open we see that $U_{M^\bullet } = \bigcup _{i \in I_{M^\bullet }} \mathop{\mathrm{Im}}(U_ i \to X)$ is an open subset of $X$. Then $X = \coprod U_{M^\bullet }$ is a disjoint open covering of $X$. As $X$ is connected only one $U_{M^\bullet }$ is nonempty. As $K$ is in $D_{ctf}(X_{\acute{e}tale}, \Lambda )$ we see that $M^\bullet$ is a perfect complex of $\Lambda$-modules, see More on Algebra, Lemma 15.74.2. Hence we may assume $M^\bullet$ is a finite complex of finite projective $\Lambda$-modules. $\square$

## 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Your email address will not be published.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.

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 09BI. Beware of the difference between the letter 'O' and the digit '0'.