Lemma 77.12.5 (0CX8)—The Stacks project

Lemma 77.12.5. In Situation 77.12.1 let $K$ be as in Lemma 77.12.2. For any étale morphism $U \to X$ with $U$ quasi-compact and quasi-separated we have

\[ R\Gamma (U, K) \otimes _ A^\mathbf {L} A_ n = R\Gamma (U_ n, \mathcal{F}_ n) \]

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 $\mathcal{F}_ n$ we obtain

\begin{align*} R\Gamma (U_ n, \mathcal{F}_ n) & = R\Gamma (U, \mathcal{F}_ n) \\ & = \text{hocolim}_ m R\Gamma (X, \mathcal{F}_ n \otimes ^\mathbf {L} E_ m) \\ & = \text{hocolim}_ m R\Gamma (X_ n, \mathcal{F}_ n \otimes ^\mathbf {L} E_ m|_{X_ n}) \end{align*}

Using Lemma 77.12.3 and the fact that $- \otimes _ A^\mathbf {L} A_ n$ commutes with homotopy colimits we obtain the result. $\square$

The Stacks project

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