Global Gorenstein dimensions

Bennis, Driss; Mahdou, Najib

doi:10.1090/S0002-9939-09-10099-0

Global Gorenstein dimensions
HTML articles powered by AMS MathViewer

by Driss Bennis and Najib Mahdou PDF

Proc. Amer. Math. Soc. 138 (2010), 461-465 Request permission

Abstract:

In this paper, we prove that the global Gorenstein projective dimension of a ring $R$ is equal to the global Gorenstein injective dimension of $R$ and that the global Gorenstein flat dimension of $R$ is smaller than the common value of the terms of this equality.

References

Driss Bennis and Najib Mahdou, Strongly Gorenstein projective, injective, and flat modules, J. Pure Appl. Algebra 210 (2007), no. 2, 437–445. MR 2320007, DOI 10.1016/j.jpaa.2006.10.010
D. Bennis, N. Mahdou and K. Ouarghi, Rings over which all modules are strongly Gorenstein projective. Accepted for publication in Rocky Mountain Journal of Mathematics. Available from arXiv:0712.0127v2.
Lars Winther Christensen, Gorenstein dimensions, Lecture Notes in Mathematics, vol. 1747, Springer-Verlag, Berlin, 2000. MR 1799866, DOI 10.1007/BFb0103980
Edgar E. Enochs and Overtoun M. G. Jenda, Relative homological algebra, De Gruyter Expositions in Mathematics, vol. 30, Walter de Gruyter & Co., Berlin, 2000. MR 1753146, DOI 10.1515/9783110803662
Robert M. Fossum, Phillip A. Griffith, and Idun Reiten, Trivial extensions of abelian categories, Lecture Notes in Mathematics, Vol. 456, Springer-Verlag, Berlin-New York, 1975. Homological algebra of trivial extensions of abelian categories with applications to ring theory. MR 0389981
Henrik Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), no. 1-3, 167–193. MR 2038564, DOI 10.1016/j.jpaa.2003.11.007
Henrik Holm, Rings with finite Gorenstein injective dimension, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1279–1283. MR 2053331, DOI 10.1090/S0002-9939-03-07466-5
H. Holm, Gorenstein Homological Algebra, Ph.D. thesis, University of Copenhagen, Denmark (2004).
Arun Vinayak Jategaonkar, A counter-example in ring theory and homological algebra, J. Algebra 12 (1969), 418–440. MR 240131, DOI 10.1016/0021-8693(69)90040-4
C. U. Jensen, On the vanishing of $\underleftarrow {\mmlToken {mi}{lim}}^{(i)}$, J. Algebra 15 (1970), 151–166. MR 260839, DOI 10.1016/0021-8693(70)90071-2
W. K. Nicholson and M. F. Yousif, Quasi-Frobenius rings, Cambridge Tracts in Mathematics, vol. 158, Cambridge University Press, Cambridge, 2003. MR 2003785, DOI 10.1017/CBO9780511546525
Joseph J. Rotman, An introduction to homological algebra, Pure and Applied Mathematics, vol. 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 538169