Cofiniteness of local cohomology modules for ideals of dimension one

Abstract

Abstract. In this paper, we prove that for any ideal / of dimension one H}(M) is /-cofinite for all i and for any finite A-module M. Furthermore, for any ideal / over any regular local ring A, we investigate the relationship between /-cofiniteness and vanishing for local cohomology modules H}(M). §1. Main TheoremLet A be a Noetherian ring. For an ideal / of A and an yl-module Tkf, we setIn general, a functor H}(-) is the i right derived functor of the functor Γ / (-) = lim fc Hom(A// fe , -). In this note, refining their proof, we prove the following theorem. Let M be a finite A-module. Then for any finite A-module N such that Supp^(iV) C V(I), we have Ext^(7V, Hj-(M)) is of finite type for all i, j.In order to prove Theorem 1.1, we need fundamental two lemmas. (1) Ext^(τ4//, V) is a finite A-module for all t < p.(2) Ext^(A/\/7 5 ^0 is a finite A-module for all t < p.

On the finiteness properties of extension and torsion functors of local cohomology modules

“…By working in the derived category, he also showed that if M is a finitely generated R-module, where R is a complete regular local ring, then H n a (M ) is a-cofinite in two cases: (i) a is a non-zero principal ideal [9,Corollary 6.3]; (ii) a is a prime ideal with dimension one [9,Corollary 7.7]. There are several papers devoted to the extension of Hartshorne's second result to more general situations: We refer the reader to the papers of Huneke and Koh [10], Delfino [4], Delfino and Marley [5], and Yoshida [19].…”

Finiteness Properties of Extension Functors of Cofinite Modules

“…This question has been studied by several authors; see, for example, Hartshorne [10], Chiriacescu [6], Huneke-Koh [11], Delfino [7], Delfino and Marley [8], Yoshida [15], Bahmanpour and Naghipour [2], Abazari and Bahmanpour [1], Bahmanpour, Naghipour and Sedghi [3].…”

Artinian cofinite modules over complete Noetherian local rings

“…Huneke and Koh [9], Delfino [5], Delfino and Marley [6], Yoshida [16], Bahmanpour and Naghipour [2], Abazari and Bahmanpour [1], Kawasaki [11], [12], Bahmanpour, Naghipour and Sedghi [3], Melkersson [15], [14]. More recently, using the main result of [3], in [10] Irani and Bahmanpour have proved that for any ideal I of a Noetherian ring R and any I-cofinite R-module M of dimension d 1, the R-modules Ext i R (M, N ) are finitely generated, for all integers i 0 and all finitely generated R-modules N with support in V (I).…”