Lemma 76.12.5. In Situation 76.12.1 let $K$ be as in Lemma 76.12.2. 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 76.12.5. In Situation 76.12.1 let $K$ be as in Lemma 76.12.2. 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 74.27.3 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 76.12.3 and the fact that $- \otimes _ A^\mathbf {L} A_ n$ commutes with homotopy colimits we obtain the result. $\square$

Your email address will not be published.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.

## Comments (0)