Abelian Category of Weakly Cofinite Modules and Local Cohomology

Cofinite modules and cofiniteness of local cohomology modules

“…If R is a local ring with dim(R/a) ≤ 2 and X is an a-torsion R-module such that Ext i R (R/a, X) is a finite R-module for all i ≤ 2, then X is an a-cofinite R-module by [8, Theorem 3.5] (see also [22,Theorem 2.6] and [19,Theorem 3.3]). In the third main result of this paper, we improve and generalize this result.…”

“…From [8, Theorem 3.5], X is an a-cofinite R-module whenever R is a local ring with dim(R/a) ≤ 2 and X is an a-torsion R-module such that Ext i R (R/a, X) is a finite R-module for all i ≤ 2 (see also [22,Theorem 2.6] and [19,Theorem 3.3]). Assume that dim(R/a) ≤ 2, t is a non-negative integer, and n > 0 or Supp R (X) ∩ Var(a) ∩ Max(R) is a finite set.…”