A criterion for cofiniteness of modules

2022/01/12 by Mohammad Khazaei, Khazaei, Mohammad, Reza Sazeedeh +1
Mathematics · #Commutative Algebra and Its Applications #Rings, Modules, and Algebras #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2201.04251

Abstract

Let A be a commutative noetherian ring, \frak a be an ideal of A, m,n be non-negative integers and let M be an A-module such that \Extⁱ_A(A/\frak a,M) is finitely generated for all i≤ m+n. We define a class \cS_n(\frak a) of modules and we assume that H_{\frak a}^s(M)∈\cS_n(\frak a) for all s≤ m. We show that H_{\frak a}^s(M) is \frak a-cofinite for all s≤ m if either n=1 or n≥ 2 and \Ext_Aⁱ(A/\frak a,H_{\frak a}^t+s-i(M)) is finitely generated for all 1≤ t≤ n-1, i≤ t-1 and s≤ m. If A is a ring of dimension d and M∈\cS_n(\frak a) for any ideal \frak a of dimension ≤ d-1, then we prove that M∈\cS_n(\frak a) for any ideal \frak a of A.

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