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)

 
 

 

Gorenstein injective modules and local cohomology


Author: Reza Sazeedeh
Journal: Proc. Amer. Math. Soc. 132 (2004), 2885-2891
MSC (2000): Primary 13D05, 13D45
DOI: https://doi.org/10.1090/S0002-9939-04-07461-1
Published electronically: May 21, 2004
MathSciNet review: 2063107
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we assume that $R$ is a Gorenstein Noetherian ring. We show that if $(R,\mathfrak{m})$ is also a local ring with Krull dimension $d$ that is less than or equal to 2, then for any nonzero ideal $\mathfrak{a}$of $R$ , $H_{\mathfrak{a}}^d(R)$ is Gorenstein injective. We establish a relation between Gorenstein injective modules and local cohomology. In fact, we will show that if $R$is a Gorenstein ring, then for any $R$-module $M$ its local cohomology modules can be calculated by means of a resolution of $M$ by Gorenstein injective modules. Also we prove that if $R$ is $d$-Gorenstein, $M$ is a Gorenstein injective and $\mathfrak a$is a nonzero ideal of $R$, then ${\Gamma}_{\mathfrak{a}}(M)$ is Gorenstein injective.


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

  • 1. M. P. Brodmann and R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge University Press, Cambridge, 1998. MR 99h:13020
  • 2. W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge University Press, Cambridge, 1993. MR 95h:13020
  • 3. K. Divaani-Aazar and P. Schenzel, Ideal topology, local cohomology and connectedness, Math. Proc. Cambridge Philos. Soc. 131 (2001), 211-226. MR 2002f:13034
  • 4. E. Enochs and O. M. G. Jenda, $h$-divisible and cotorsion modules over one-dimensional Gorenstein rings, Journal of Algebra 161 (1993), 444-454. MR 94j:13008
  • 5. E. Enochs and O. M. G. Jenda, Mock finitely generated Gorenstein injective modules and isolated singularities, Journal of Pure and Applied Algebra 96 (1994), 259-269. MR 95i:13007
  • 6. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995), 611-633. MR 97c:16011
  • 7. E. Enochs and O. M. G. Jenda, Relative homological algebra, de Gruyter, Berlin, 2000. MR 2001h:16013
  • 8. I. G. Macdonald, Secondary representation of modules over a commutative ring, Symposia Mathematica, Vol. 11, Academic Press, London, 1973, 23-43. MR 49:7252
  • 9. L. Melkersson and P. Schenzel, The co-localization of an Artinian module, Proc. Edinburgh Math. Soc. (2) 38 (1995), 121-132. MR 96a:13020
  • 10. R. Y. Sharp, Artinian modules over commutative rings, Math. Proc. Cambridge Philos. Soc. 111 (1992), 25-33. MR 93a:13009
  • 11. J. Xu, Flat covers of modules, Lecture Notes in Mathematics, Vol. 1634, Springer-Verlag, Berlin, 1996. MR 98b:16003

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13D05, 13D45

Retrieve articles in all journals with MSC (2000): 13D05, 13D45


Additional Information

Reza Sazeedeh
Affiliation: Institute of Mathematics, University for Teacher Education, 599, Taleghani Avenue, Tehran 15614, Iran – and – Department of Mathematics, Urmia University, Iran

DOI: https://doi.org/10.1090/S0002-9939-04-07461-1
Keywords: Cover, Gorenstein injective, Gorenstein projective, local cohomology
Received by editor(s): December 5, 2002
Received by editor(s) in revised form: June 21, 2003
Published electronically: May 21, 2004
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society