A Note on Abelian Categories of Cofinite Modules

Abstract

Let R be a commutative Noetherian ring and I be an ideal of R. In this article we answer affirmatively a question raised by the present author in [1]. Also, as an immediate consequence of this result it is shown that the category of all I-cofinite Rmodules C (R, I) cof is an Abelian subcategory of the category of all R-modules, whenever q(I, R) ≤ 1. These assertions answer affirmatively a question raised by R. Hartshorne in [Affine duality and cofiniteness, Invent. Math. 9(1970), 145-164], in some special cases. Hartshorne in [14] defined an R-module X to be I-cofinite, if Supp X ⊆ V (I) and Ext i R (R/I, X) is a finitely generated R-module for each integer i ≥ 0. Then he posed the following question: Question 1: Whether the category C (R, I) cof of I-cofinite modules is an Abelian subcategory of the category of all R-modules? That is, if f : M −→ N is an R-homomorphism