Lemma 38.29.5. In Situation 38.29.1 let $K$ be as in Lemma 38.29.2. For any quasi-compact open $U \subset X$ we have
\[ R\Gamma (U, K) \otimes _ A^\mathbf {L} A_ n = R\Gamma (U_ n, K_ n) \]
in $D(A_ n)$ where $U_ n = U \cap X_ n$.
Proof. Fix $n$. By Derived Categories of Schemes, Lemma 36.33.4 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
\begin{align*} R\Gamma (U_ n, K_ n) & = R\Gamma (U, K_ n) \\ & = \text{hocolim}_ m R\Gamma (X, K_ n \otimes ^\mathbf {L} E_ m) \\ & = \text{hocolim}_ m R\Gamma (X_ n, K_ n \otimes ^\mathbf {L} E_ m|_{X_ n}) \end{align*}
Using Lemma 38.29.3 and the fact that $- \otimes _ A^\mathbf {L} A_ n$ commutes with homotopy colimits we obtain the result. $\square$
Post a comment
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)