Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$.

References [Enhancements On Off] (What's this?)

  • [1] Kamal Bahmanpour and Reza Naghipour, Cofiniteness of local cohomology modules for ideals of small dimension, J. Algebra 321 (2009), no. 7, 1997-2011. MR 2494753 (2010f:13018),
  • [2] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956 (95h:13020)
  • [3] Donatella Delfino and Thomas Marley, Cofinite modules and local cohomology, J. Pure Appl. Algebra 121 (1997), no. 1, 45-52. MR 1471123 (98g:13015),
  • [4] Robin Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1969/1970), 145-164. MR 0257096 (41 #1750)
  • [5] Ken-Ichiroh Kawasaki, On the finiteness of Bass numbers of local cohomology modules, Proc. Amer. Math. Soc. 124 (1996), no. 11, 3275-3279. MR 1328354 (97a:13025),
  • [6] Ken-ichiroh Kawasaki, On a category of cofinite modules which is Abelian, Math. Z. 269 (2011), no. 1-2, 587-608. MR 2836085 (2012h:13026),
  • [7] K.-I. Kawasaki, The category of cofinite modules for ideals of dimension one and codimension one, Actes des rencontres du Centre international de recontres mathematiques 2 (2010),
  • [8] Ken-ichiroh Kawasaki, On the category of cofinite modules for principal ideals, Nihonkai Math. J. 22 (2011), 67-71. MR 2952818
  • [9] Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273 (88h:13001)
  • [10] Leif Melkersson, On asymptotic stability for sets of prime ideals connected with the powers of an ideal, Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 2, 267-271. MR 1027779 (90k:13014),
  • [11] Leif Melkersson, Modules cofinite with respect to an ideal, J. Algebra 285 (2005), no. 2, 649-668. MR 2125457 (2006i:13033),
  • [12] Helmut Zöschinger, Minimax-moduln, J. Algebra 102 (1986), no. 1, 1-32 (German). MR 853228 (87m:13019),

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 13D45, 14B15, 13E05

Retrieve articles in all journals with MSC (2010): 13D45, 14B15, 13E05

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

American Mathematical Society