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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