Skip to main content



We introduce what is meant by an AC-Gorenstein ring. It is a generalized notion of Gorenstein ring that is compatible with the Gorenstein AC-injective and Gorenstein AC-projective modules of Bravo–Gillespie–Hovey. It is also compatible with the notion of $n$ -coherent rings introduced by Bravo–Perez. So a $0$ -coherent AC-Gorenstein ring is precisely a usual Gorenstein ring in the sense of Iwanaga, while a $1$ -coherent AC-Gorenstein ring is precisely a Ding–Chen ring. We show that any AC-Gorenstein ring admits a stable module category that is compactly generated and is the homotopy category of two Quillen equivalent abelian model category structures. One is projective with cofibrant objects that are Gorenstein AC-projective modules while the other is an injective model structure with fibrant objects that are Gorenstein AC-injectives.

Hide All
[BG16] Bravo, D. and Gillespie, J., ‘Absolutely clean, level, and Gorenstein AC-injective complexes’, Comm. Algebra 44(5) (2016), 22132233.
[BGH14] Bravo, D., Gillespie, J. and Hovey, M., ‘The stable module category of a general ring’, Preprint, 2014, arXiv:1405.5768.
[BP17] Bravo, D. and Pérez, M. A., ‘Finiteness conditions and cotorsion pairs’, J. Pure Appl. Algebra 221(6) (2017), 12491267.
[RFD79] Damiano, R. F., ‘Coflat rings and modules’, Pacific J. Math. 81(2) (1979), 349369.
[DC93] Ding, N. and Chen, J., ‘The flat dimensions of injective modules’, Manuscripta Math. 78(2) (1993), 165177.
[DC96] Ding, N. and Chen, J., ‘Coherent rings with finite self-FP-injective dimension’, Commun. Algebra 24(9) (1996), 29632980.
[EJ01] Enochs, E. and Jenda, O., Relative Homological Algebra, De Gruyter Expositions in Mathematics, 30 (Walter De Gruyter, New York, 2000).
[EG18] Estrada, S. and Gillespie, J., ‘The projective stable category of a coherent scheme’, Proc. Roy. Soc. Edinburgh Sect. A, to appear. Published online (26 February 2018).
[JG10] Gillespie, J., ‘Model structures on modules over Ding–Chen rings’, Homology, Homotopy Appl. 12(1) (2010), 6173.
[JG16a] Gillespie, J., ‘Gorenstein complexes and recollements from cotorsion pairs’, Adv. Math. 291 (2016), 859911.
[JG16b] Gillespie, J., ‘Hereditary abelian model categories’, Bull. Lond. Math. Soc. 48(6) (2016), 895922.
[JG18] Gillespie, J., ‘Gorenstain AC-projective chain complexes’, J. Homotopy Relat. Struct., to appear. Published online (20 March 2018).
[GH10] Gillespie, J. and Hovey, M., ‘Gorenstein model structures and generalized derived categories’, Proc. Edinb. Math. Soc. (2) 53(3) (2010), 675696.
[GL89] Glaz, S., Commutative Coherent Rings, Lecture Notes in Mathematics, 1371 (Springer, Berlin, 1989).
[GT06] Goebel, R. and Trlifaj, J., Approximations and Endomorphism Algebras of Modules, de Gruyter Expositions in Mathematics, 41 (Walter de Gruyter & Co., Berlin, 2006).
[MH99] Hovey, M., Model Categories, Mathematical Surveys and Monographs, 63 (American Mathematical Society, Providence, RI, 1999).
[MH02] Hovey, M., ‘Cotorsion pairs, model category structures, and representation theory’, Math. Z. 241 (2002), 553592.
[YI79] Iwanaga, Y., ‘On rings with finite self-injective dimension’, Comm. Algebra 7(4) (1979), 393414.
[YI80] Iwanaga, Y., ‘On rings with finite self-injective dimension II’, Tsukuba J. Math. 4(1) (1980), 107113.
[WLY17] Wang, J., Liu, Z. and Yang, X., ‘A negative answer to a Gillespie’s question’, Preprint.
[ZP17] Zhao, T. and Pérez, M. A., ‘Relative FP-injective and FP-flat complexes and their model structures’, Preprint, 2017, arXiv:1703.10703.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed