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On the category of cofinite modules which is Abelian

Authors: Kamal Bahmanpour, Reza Naghipour and Monireh Sedghi
Journal: Proc. Amer. Math. Soc. 142 (2014), 1101-1107
MSC (2010): Primary 13D45, 14B15, 13E05
Published electronically: January 6, 2014
MathSciNet review: 3162233
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Abstract: Let $ R$ denote a commutative Noetherian (not necessarily local) ring and $ I$ an ideal of $ R$ of dimension one. The main purpose of this paper is to generalize, and to provide a short proof of, K. I. Kawasaki's theorem that the category $ \mathscr {M}(R, I)_{cof}$ of $ I$-cofinite modules over a commutative Noetherian local ring $ R$ forms an Abelian subcategory of the category of all $ R$-modules. Consequently, this assertion answers affirmatively the question raised by R. Hartshorne in his article Affine duality and cofiniteness [Invent. Math. 9 (1970), 145-164] for an ideal of dimension one in a commutative Noetherian ring $ R$.

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Additional Information

Kamal Bahmanpour
Affiliation: Department of Mathematics, Faculty of Mathematical Sciences, University of Mohaghegh Ardabili, 56199-11367, Ardabil, Iran

Reza Naghipour
Affiliation: Department of Mathematics, University of Tabriz, Tabriz, Iran – and – School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran

Monireh Sedghi
Affiliation: Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
Email:, m{\textunderscore}

Keywords: Abelian category, arithmetic rank, cofinite module, Noetherian rings
Received by editor(s): December 6, 2011
Received by editor(s) in revised form: April 25, 2012
Published electronically: January 6, 2014
Dedicated: Dedicated to Professor Robin Hartshorne
Communicated by: Irena Peeva
Article copyright: © Copyright 2014 American Mathematical Society

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