The Stacks project

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. $\square$