Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Foxby duality and Gorenstein injective
and projective modules


Authors: Edgar E. Enochs, Overtoun M. G. Jenda and Jinzhong Xu
Journal: Trans. Amer. Math. Soc. 348 (1996), 3223-3234
MSC (1991): Primary 13C10, 13C11; Secondary 13C99
DOI: https://doi.org/10.1090/S0002-9947-96-01624-8
MathSciNet review: 1355071
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In 1966, Auslander introduced the notion of the $G$-dimension of a finitely generated module over a Cohen-Macaulay noetherian ring and found the basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Auslander and Bridger (Mem. Amer. Math. Soc., vol. 94, 1969)). Enochs and Jenda attempted to dualize the notion of $G$-dimensions. It seemed appropriate to call the modules with $G$-dimension 0 Gorenstein projective, so the basic problem was to define Gorenstein injective modules. These were defined in Math. Z. 220 (1995), 611--633 and were shown to have properties predicted by Auslander's results. The way we define Gorenstein injective modules can be dualized, and so we can define Gorenstein projective modules (i.e. modules of $G$-dimension 0) whether the modules are finitely generated or not. The investigation of these modules and also Gorenstein flat modules was continued by Enochs, Jenda, Xu and Torrecillas. However, to get good results it was necessary to take the base ring Gorenstein. H.-B. Foxby introduced a duality between two full subcategories in the category of modules over a local Cohen-Macaulay ring admitting a dualizing module. He proved that the finitely generated modules in one category are precisely those of finite $G$-dimension. We extend this result to modules which are not necessarily finitely generated and also prove the dual result, i.e. we characterize the modules in the other class defined by Foxby. The basic result of this paper is that the two classes involved in Foxby's duality coincide with the classes of those modules having finite Gorenstein projective and those having finite Gorenstein injective dimensions. We note that this duality then allows us to extend many of our results to the original Auslander setting.


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

  • 1. M. Auslander, Anneaux de Gorenstein et torsion en algèbre commutative, séminaire d'algèbre commutative, Ecole Normale Supérieure de Jeunes Filles, Paris 1966/67. MR 37:1435
  • 2. M. Auslander and M. Bridger, Stable Module Theory, Memoirs A.M.S. 94, A.M.S., Providence, R.I., 1969. MR 42:4580
  • 3. M. Auslander and R.O. Buchweitz, Maximal Cohen-Macaulay approximations, Soc. Math. France, Mémoire 38 (1989), 5-37. MR 91h:13010
  • 4. R. Belshoff, E. Enochs and J. Xu, The existence of flat covers, PAMS 122 (1994), 985-991. MR 95b:16001
  • 5. E. Enochs, Flat covers and flat cotorsion modules, Proc. AMS 92 (1984), 179-184. MR 85j:13016
  • 6. E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math 39 (1981), 33-38. MR 83a:16031
  • 7. E. Enochs and O.M.G. Jenda, Gorenstein injective and projective modules, Math. Zeit. 220 (1995), 611--633. CMP 1996:5
  • 8. E. Enochs and O.M.G. Jenda, Gorenstein balance of Hom and Tensor, Tsukuba J. Math. 19 (1995), 1--13. CMP 1995:17
  • 9. E. Enochs and O.M.G. Jenda, On Gorenstein injective modules, Comm. Alg. 2 (10) (1993), 3489-3501. MR 94g:13006
  • 10. E. Enochs and O.M.G. Jenda, Mock finitely generated Gorenstein injective modules and isolated singularities, J. Pure Applied Algebra 96 (1994), 259--269. MR 95i:13007
  • 11. E. Enochs, O.M.G. Jenda and B. Torrecillas, Gorenstein flat modules, J. Nanjing University, 10 (1) (1993), 1-9. MR 95a:16004
  • 12. E. Enochs, O.M.G. Jenda and J. Xu, Covers and envelopes over Gorenstein rings (to appear in Tsukuba J. Math.).
  • 13. H.-B. Foxby, Gorenstein modules and related modules, Math. Scand. 31 (1972), 267-284. MR 48:6094
  • 14. H.-B. Foxby, Duality homomorphisms for modules over certain Cohen-Macaulay rings, Math. Zeit. 132 (1973), 215-226. MR 48:6095
  • 15. H.-B. Foxby, Quasi-perfect modules over Cohen-Macaulay rings, Math. Nachr. 66 (1975), 103-110. MR 51:12838
  • 16. H.-B. Foxby, Gorenstein dimensions over Cohen-Macaulay rings, Proceedings of the international conference on commutative algebra, W. Bruns (editor), Universität Osnabrück, 1994.
  • 17. T. Ishikawa, On injective and flat modules, J. Math. Soc. Japan 17 (1965), 291-296. MR 32:5711
  • 18. C.U. Jensen, On the vanishing of $\varprojlim ^{(i)}$, J. Algebra 15 (1970), 151-169. MR 41:5460
  • 19. M. Raynaud and L. Gruson, Critères de platitude et de projectivité, Invent. Math.13 (1971), 1-89. MR 46:7219
  • 20. J. Xu and E. Enochs, Gorenstein flat covers of modules over Gorenstein rings (to appear in J. Algebra).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 13C10, 13C11, 13C99

Retrieve articles in all journals with MSC (1991): 13C10, 13C11, 13C99


Additional Information

Edgar E. Enochs
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027

Overtoun M. G. Jenda
Affiliation: Department of Discrete Mathematics and Statistics, Auburn University, Auburn, Alabama 36849-5307

Jinzhong Xu
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027

DOI: https://doi.org/10.1090/S0002-9947-96-01624-8
Received by editor(s): September 7, 1994
Received by editor(s) in revised form: October 2, 1995
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society