Lemma 36.21.4. Let $X$ be a quasi-compact and quasi-separated scheme. The functor $DQ_ X$ of Lemma 36.21.1 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 open in $X$ 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 Cohomology of Schemes, Lemma 30.4.1.
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 Lemmas 36.3.5 and 36.7.3.
Asssume $U, V$ are quasi-compact open in $X$ and the lemma holds for $U$, $V$, and $U \cap V$. Say with integers $N(U)$, $N(V)$, and $N(U \cap V)$. Now suppose $K$ is in $D(\mathcal{O}_ X)$ with $H^ i(W, K) = 0$ for all affine open $W \subset X$ and all $i \not\in [a, b]$. Then $K|_ U$, $K|_ V$, $K|_{U \cap V}$ have the same property. Hence we see that $RQ_ U(K|_ U)$ and $RQ_ V(K|_ V)$ and $RQ_{U \cap V}(K|_{U \cap V})$ have vanishing cohomology sheaves outside the inverval $[a, b + \max (N(U), N(V), N(U \cap V))$. Since the functors $Rj_{U, *}$, $Rj_{V, *}$, $Rj_{U \cap V, *}$ have finite cohomological dimension on $D_\mathit{QCoh}$ by Lemma 36.4.1 we see that there exists an $N$ such that $Rj_{U, *}DQ_ U(K|_ U)$, $Rj_{V, *}DQ_ V(K|_ V)$, and $Rj_{U \cap V, *}DQ_{U \cap V}(K|_{U \cap V})$ have vanishing cohomology sheaves outside the interval $[a, b + N]$. Then finally we conclude by the distinguished triangle of Remark 36.21.3. $\square$
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