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The Stacks project

Lemma 20.47.9. Let (X, \mathcal{O}_ X) be a ringed space. Let K be an object of D(\mathcal{O}_ X). Let m \in \mathbf{Z}.

  1. If K is m-pseudo-coherent and H^ i(K) = 0 for i > m, then H^ m(K) is a finite type \mathcal{O}_ X-module.

  2. If K is m-pseudo-coherent and H^ i(K) = 0 for i > m + 1, then H^{m + 1}(K) is a finitely presented \mathcal{O}_ X-module.

Proof. Proof of (1). We may work locally on X. Hence we may assume there exists a strictly perfect complex \mathcal{E}^\bullet and a map \alpha : \mathcal{E}^\bullet \to K which induces an isomorphism on cohomology in degrees > m and a surjection in degree m. It suffices to prove the result for \mathcal{E}^\bullet . Let n be the largest integer such that \mathcal{E}^ n \not= 0. If n = m, then H^ m(\mathcal{E}^\bullet ) is a quotient of \mathcal{E}^ n and the result is clear. If n > m, then \mathcal{E}^{n - 1} \to \mathcal{E}^ n is surjective as H^ n(E^\bullet ) = 0. By Lemma 20.46.5 we can locally find a section of this surjection and write \mathcal{E}^{n - 1} = \mathcal{E}' \oplus \mathcal{E}^ n. Hence it suffices to prove the result for the complex (\mathcal{E}')^\bullet which is the same as \mathcal{E}^\bullet except has \mathcal{E}' in degree n - 1 and 0 in degree n. We win by induction on n.

Proof of (2). We may work locally on X. Hence we may assume there exists a strictly perfect complex \mathcal{E}^\bullet and a map \alpha : \mathcal{E}^\bullet \to K which induces an isomorphism on cohomology in degrees > m and a surjection in degree m. As in the proof of (1) we can reduce to the case that \mathcal{E}^ i = 0 for i > m + 1. Then we see that H^{m + 1}(K) \cong H^{m + 1}(\mathcal{E}^\bullet ) = \mathop{\mathrm{Coker}}(\mathcal{E}^ m \to \mathcal{E}^{m + 1}) which is of finite presentation. \square


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