The Stacks project

Lemma 57.9.9. Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$. Let $\mathcal{A}$ be an abelian category. Let $H : D_{perf}(\mathcal{O}_ X) \to \mathcal{A}$ be a homological functor (Derived Categories, Definition 13.3.5) such that for all $K$ in $D_{perf}(\mathcal{O}_ X)$ the object $H^ i(K)$ is nonzero for only a finite number of $i \in \mathbf{Z}$. Then there exists an integer $m \geq 1$ such that $H^ i(\mathcal{F}) = 0$ for any coherent $\mathcal{O}_ X$-module $\mathcal{F}$ and $i \not\in [-m, m]$. Similarly for cohomological functors.

Proof. Combine Lemma 57.9.8 with Derived Categories, Lemma 13.35.8. $\square$

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