Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T08:13:19.488Z Has data issue: false hasContentIssue false
  • English
  • Français

STABILITY OF GORENSTEIN FLAT MODULES

Published online by Cambridge University Press:  09 December 2011

SAMIR BOUCHIBA  and
MOSTAFA KHALOUI
SAMIR BOUCHIBA
Affiliation:
Department of Mathematics, University Moulay Ismail, Meknes 50000, Morocco e-mail: [email protected]; [email protected]
MOSTAFA KHALOUI
Affiliation:
Department of Mathematics, University Moulay Ismail, Meknes 50000, Morocco e-mail: [email protected]; [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Sather-Wagstaff et al. proved in [8] (S. Sather-Wagsta, T. Sharif and D. White, Stability of Gorenstein categories, J. Lond. Math. Soc.(2), 77(2) (2008), 481–502) that iterating the process used to define Gorenstein projective modules exactly leads to the Gorenstein projective modules. Also, they established in [9] (S. Sather-Wagsta, T. Sharif and D. White, AB-contexts and stability for Goren-stein at modules with respect to semi-dualizing modules, Algebra Represent. Theory14(3) (2011), 403–428) a stability of the subcategory of Gorenstein flat modules under a procedure to build R-modules from complete resolutions. In this paper we are concerned with another kind of stability of the class of Gorenstein flat modules via-à-vis the very Gorenstein process used to define Gorenstein flat modules. We settle in affirmative the following natural question in the setting of a left GF-closed ring R: Given an exact sequence of Gorenstein flat R-modules G = ⋅⋅⋅ G2G1G0G−1G−2 ⋅⋅⋅ such that the complex H ⊗RG is exact for each Gorenstein injective right R-module H, is the module M:= Im(G0 → G−1) a Gorenstein flat module?

Keywords

13D0213D0513D0716E0516E10
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 1 , January 2012 , pp. 169 - 175
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Auslander, M., Anneaux de Gorenstein et torsion en algèbre commutative, Secrétariat mathématique, Séminaire d'algèbre commutative dirigé par Pierre Samuel, 1966/1967. Texte rédigé, d'après des exposés de Maurice Auslander, par Marquerite Mangeney, Christian Peskine et Lucien Szpiro, Ecole Normale Supérieure de Jeunes Filles (Paris, 1967).Google Scholar
2.Auslander, M. and Bridger, M., Stable module theory, in Memoirs of the American mathematical society, vol. 94 (American Mathematical Society, Providence, RI, 1969), 146 pp.Google Scholar
3.Bennis, D., Rings over which the class of Gorenstein flat modules is closed under extentions, Comm. Algebra 37 (3) (2009), 855–868.Google Scholar
4.Bennis, D. and Mahdou, N., Strongly Gorenstein projective, injective, and flat modules, J. Pure Appl. Algebra 210 (2007), 437–445.CrossRefGoogle Scholar
5.Christensen, L. W., Frankild, A. and Holm, H., On Gorenstein projective, injective and flat dimensions – a functorial description with applications, J. Algebra 302 (2006), 231–279.CrossRefGoogle Scholar
6.Holm, H., Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), 167–193.Google Scholar
7.Rotman, J., An introduction to homological algebra, in Pure and applied mathematics (Academic Press, New York, 1979).Google Scholar
8.Sather-Wagstaff, S., Sharif, T. and White, D., Stability of Gorenstein categories, J. Lond. Math. Soc. 77 (2) (2008), 481–502.CrossRefGoogle Scholar
9.Sather-Wagstaff, S., Sharif, T. and White, D., AB-contexts and stability for Gorenstein flat modules with respect to semidualizing modules, Algebra Represent. Theory 14 (3) (2011), 403–428.CrossRefGoogle Scholar
10.Yang, G. and Liu, Z., Stability of Gorenstein flat categories, Glasgow Math. J. 54 (2012), 177–191.CrossRefGoogle Scholar