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A homological dimension related to AB rings

  • Tokuji Araya
Original Paper
  • 11 Downloads

Abstract

There are many homological dimensions which are closely related to ring theoretic properties. The notion of a AB ring has been introduced by Huneke and Jorgensen. It has nice homological properties. In this paper, we shall define a homological dimension which is closely related to a AB ring, and investigate its properties.

Keywords

Auslander condition AB ring AB-dimension CI-dimension G-dimension 

Mathematics Subject Classification

13D05 13H10 13D07 

Notes

Acknowledgements

The author is indebted to Ryo Takahashi, Yuji Yoshino and Olgur Celikbas for their many useful and helpful comments and suggestions. The author also thank the refree for some comments.

References

  1. Araya, T., Yoshino, Y.: Remarks on a depth formula, a grade inequality and a conjecture of Auslander. Commun. Algebra 26(11), 3793–3806 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Auslander, M., Bridger, M.: Stable module theory. Memoirs of the American Mathematical Society, No. 94 (1969)Google Scholar
  3. Auslander, M., Buchsbaum, D.A.: Homological dimension in Noetherian rings. Proc. Natl. Acad. Sci. USA 42, 36–38 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Auslander, M., Buchsbaum, D.A.: Homological dimension in local rings. Trans. Am. Math. Soc. 85, 390–405 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Auslander, M., Buchweitz, R.-O.: The homological theory of maximal Cohen–Macaulay approximations. (French summary) Mém. Soc. Math. France (N.S.) No. 38 (1989), pp 5–37Google Scholar
  6. Avramov, L.L.: Homological dimensions and related invariants of modules over local rings. In: Representations of Algebra. vol. I, II, pp. 1–39. Beijing Norm. Univ. Press, Beijing (2002)Google Scholar
  7. Avramov, L.L., Gasharov, V.N., Peeva, I.V.: Complete intersection dimension. (English summary). Inst. Hautes Études Sci. Publ. Math. 86(1997), 67–114 (1998)zbMATHGoogle Scholar
  8. Bruns, W., Herzog, J.: Cohen–Macaulay rings. (English summary). In: Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)Google Scholar
  9. Christensen, L.W., Holm, H.: Algebras that satisfy Auslander’s condition on vanishing of cohomology. Math. Z. 265(1), 21–40 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Diveris, K.: Finitistic extension degree. Algebras Represent. Theory 17(2), 495–506 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Gerko, A.A.: On homological dimensions. (Russian) Mat. Sb. 192 (2001), no. 8, 79–94; translation in Sb. Math. 192 (2001), no. 7-8, 1165–1179Google Scholar
  12. Happel, D.: Homological conjectures in representation theory of finite-dimensional algebras, Sherbrook Lecture Notes Series (1991). http://www.math.ntnu.no/~oyvinso/Nordfjordeid/Program/references.html
  13. Hochster, M.: Topics in the homological study of modules over commutative rings, CBMS Regoinal Conf. Ser. in Math. 24, AMS, Providence, RI (1975)Google Scholar
  14. Huneke, C., Jorgensen, D.A.: Symmetry in the vanishing of Ext over Gorenstein rings. Math. Scand. 93(2), 161–184 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Jorgensen, D.A., Şega, L.M.: Nonvanishing cohomology and classes of Gorenstein rings. Adv. Math. 188(2), 470–490 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Peskine, C., Szpiro, L.: Dimension projective finie et cohomologie locale. I.H.E.S. Publ. Math. 42, 47–119 (1973)CrossRefzbMATHGoogle Scholar
  17. Roberts, P.: Le théorème d’intersection, C. R. Acad. Sc. Paris Sér. I 304 (1987), 177–180Google Scholar
  18. Serre, J.-P.: Sur la dimension homologique des anneaux et des modules noetheriens. (French) In: Proceedings of the International Symposium on Algebraic Number Theory, Tokyo & Nikko, 1955, pp. 175–189. Science Council of Japan, Tokyo (1956)Google Scholar

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© The Managing Editors 2018

Authors and Affiliations

  1. 1.Department of Applied Science, Faculty of ScienceOkayama University of ScienceOkayamaJapan

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