The Stacks project

Lemma 15.65.7. Let $A$ be a Noetherian ring. Let $K \in D(A)$ be pseudo-coherent, i.e., $K \in D^-(A)$ with finite cohomology modules. Let $\mathfrak m$ be a maximal ideal of $A$. If $H^ i(K)/\mathfrak m H^ i(K) \not= 0$, then there exists a finite $A$-module $E$ annihilated by a power of $\mathfrak m$ and a map $K \to E[-i]$ which is nonzero on $H^ i(K)$.

Proof. (The equivalent formulation of pseudo-coherence in the statement of the lemma is Lemma 15.64.17.) Choose $K \to M[-i]$ as in Lemma 15.65.6. By Artin-Rees (Algebra, Lemma 10.51.2) we can find an $n$ such that $H^ i(K) \cap \mathfrak m^ n M \subset \mathfrak m H^ i(K)$. Take $E = M/\mathfrak m^ n M$. $\square$

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