Lemma 77.13.5. In Situation 77.13.1 let $K$ be as in Lemma 77.13.2. For any étale morphism $U \to X$ with $U$ quasi-compact and quasi-separated we have
in $D(A_ n)$ where $U_ n = U \times _ X X_ n$.
Lemma 77.13.5. In Situation 77.13.1 let $K$ be as in Lemma 77.13.2. For any étale morphism $U \to X$ with $U$ quasi-compact and quasi-separated we have
in $D(A_ n)$ where $U_ n = U \times _ X X_ n$.
Proof. Fix $n$. By Derived Categories of Spaces, Lemma 75.27.3 there exists a system of perfect complexes $E_ m$ on $X$ such that $R\Gamma (U, K) = \text{hocolim} R\Gamma (X, K \otimes ^\mathbf {L} E_ m)$. In fact, this formula holds not just for $K$ but for every object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Applying this to $K_ n$ we obtain
Using Lemma 77.13.3 and the fact that $- \otimes _ A^\mathbf {L} A_ n$ commutes with homotopy colimits we obtain the result. $\square$
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