The Stacks project

Lemma 73.19.4. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. The functor $DQ_ X$ of Lemma 73.19.1 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 ├ętale 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 73.9.3.

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 73.11.3.

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 $W$ ├ętale 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 inverval $[a, b + \max (N(U), N(V), N(U \times _ W V))$. Since the functors $Rj_{U, *}$, $Rj_{V, *}$, $Rj_{U \times _ W V, *}$ have finite cohomological dimension on $D_\mathit{QCoh}$ by Lemma 73.6.1 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 73.19.3. $\square$


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