The Stacks project

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. $\square$

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