The Stacks project

Lemma 75.19.4. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. The functor $DQ_ X$ of Lemma 75.19.1 has the following boundedness property: there exists an integer $N = N(X)$ such that, if $K$ in $D(\mathcal{O}_ X)$ with $H^ i(U, K) = 0$ for $U$ affine étale over $X$ and $i \not\in [a, b]$, then the cohomology sheaves $H^ i(DQ_ X(K))$ are zero for $i \not\in [a, b + N]$.

Proof. We will prove this using the induction principle of Lemma 75.9.3.

If $X$ is affine, then the lemma is true with $N = 0$ because then $RQ_ X = DQ_ X$ is given by taking the complex of quasi-coherent sheaves associated to $R\Gamma (X, K)$. See Lemma 75.11.3.

Let $(U \subset W, f : V \to W)$ be an elementary distinguished square with $W$ quasi-compact and quasi-separated, $U \subset W$ quasi-compact open, $V$ affine such that the lemma holds for $U$, $V$, and $U \times _ W V$. Say with integers $N(U)$, $N(V)$, and $N(U \times _ W V)$. Now suppose $K$ is in $D(\mathcal{O}_ X)$ with $H^ i(W, K) = 0$ for all affine $W$ étale over $X$ and all $i \not\in [a, b]$. Then $K|_ U$, $K|_ V$, $K|_{U \times _ W V}$ have the same property. Hence we see that $RQ_ U(K|_ U)$ and $RQ_ V(K|_ V)$ and $RQ_{U \cap V}(K|_{U \times _ W V})$ have vanishing cohomology sheaves outside the interval $[a, b + \max (N(U), N(V), N(U \times _ W V))$. Since the functors $Rj_{U, *}$, $Rj_{V, *}$, $Rj_{U \times _ W V, *}$ have finite cohomological dimension on $D_\mathit{QCoh}$ by Lemma 75.6.1 we see that there exists an $N$ such that $Rj_{U, *}DQ_ U(K|_ U)$, $Rj_{V, *}DQ_ V(K|_ V)$, and $Rj_{U \cap V, *}DQ_{U \times _ W V}(K|_{U \times _ W V})$ have vanishing cohomology sheaves outside the interval $[a, b + N]$. Then finally we conclude by the distinguished triangle of Remark 75.19.3. $\square$


Comments (0)


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Your email address will not be published.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).

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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CSS. Beware of the difference between the letter 'O' and the digit '0'.