G-Groups of Cohen-Macaulay Rings with n-Cluster Tilting Objects

  • Zachary FloresEmail author


Let \((R, \mathfrak {m}, k)\) denote a local Cohen-Macaulay ring such that the category of maximal Cohen-Macaulay R-modules mcmR contains an n-cluster tilting object L. In this paper, we compute the Quillen K-group G1(R) := K1(modR) explicitly as a direct sum of a finitely generated free abelian group and an explicit quotient of AutR(L)ab when R is a k-algebra and k is algebraically closed with characteristic not two. Moreover, we compute AutR(L)ab and G1(R) for certain hypersurface singularities.


Cohen-Macaulay n-cluster tilting K-theory Hypersurface singularities Automorphism groups 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The author would like to thank Hailong Dao and Jeanne Duflot for their useful comments in the preparation of this manuscript. We would also like to thank the anonymous referee for greatly improving the quality of this manuscript.


  1. 1.
    Bass, H.: Algebraic K-Theory. W.A. Benjamin, Inc. New York, New York (1968)zbMATHGoogle Scholar
  2. 2.
    Dao, H., Huneke, C.: Vanishing of ext, cluster tilting modules and finite global dimension of endomorphism rings. Am. J. Math. 135, 561–578 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dao, H., Faber, E., Ingalls, C.: Noncommutative (crepant) desingularizations and the global spectrum of commutative rings. Algebras Represent. Theory 18, 633–664 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Doherty, B., Faber, E., Ingalls, C.: Computing global dimension of endomorphism rings via ladders. J. Algebra 458, 307–350 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dugger, D., Shipley, B.: K-theory and derived equivalences. Duke Math J. 124, 587–617 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gersten, S.M.: Algebraic k-theory I. In: Proceedings of the Conference Held at the Seattle Research Center of Battelle Memorial Institute, August 28 - September 8, 1972 Higher K-theory of Rings, vol. 3–42. Springer, Berlin (1973)Google Scholar
  7. 7.
    Holm, H.: K-groups for rings of finite Cohen-Macaulay type. Forum Mathematicum 27, 2413–2452 (2015)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Iyama, O.: Trends in Representation Theory of Algebras and Related Topics, Auslander-Reiten Theory Revisited, pp 349–398. European Mathematical Society, Zürich (2008)CrossRefzbMATHGoogle Scholar
  9. 9.
    Iyama, O.: Higher-dimensional Auslander Reiten theory on maximal orthogonal subcategories. Adv. Math. 210, 22–50 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lam, T.Y.: A First Course in Noncommutative Rings. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  11. 11.
    Leuschke, G.J., Wiegand, R.: Cohen-Macaulay Representations. American Mathematical Society, Providence (2012)CrossRefzbMATHGoogle Scholar
  12. 12.
    Leuschke, G.J.: Endomorphism rings of finite global dimension. Can. J. Math. 59, 332–342 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Navkal, V: K -theory of a Cohen-Macaulay Local Ring with n-Cluster Tilting Object. (Doctoral dissertation), Retrieved from ProQuest Dissertations and Theses (Order No. 3563356) (2013)Google Scholar
  14. 14.
    Peng, Y., Guo, X: The K 1-group of tiled orders. Commun. Algebra 41, 3739–3744 (2013)CrossRefzbMATHGoogle Scholar
  15. 15.
    Quillen, D.: Algebraic K-Theory I. In: Proceedings of the Conference Held at the Seattle Research Center of Battelle Memorial Institute, August 28 - September 8, 1972 Higher Algebraic K-theory I, vol. 85–147. Springer, Berlin (1973)Google Scholar
  16. 16.
    Rosenberg, J.: Algebraic K-theory and its Applications. Springer, New York (1994)CrossRefzbMATHGoogle Scholar
  17. 17.
    Sherman, C.: Algebraic K-Theory. In: Proceedings of a Conference Held at Oberwolfach, June 1980. Part 1 Group Representations and Algebraic K-theory, vol. 208–243. Springer, Berlin (1982)Google Scholar
  18. 18.
    Srinivas, V.: Algebraic K-theory. Birkhäuser, Boston (1996)CrossRefzbMATHGoogle Scholar
  19. 19.
    Swanson, I., Huneke, C.: Integral Closure of Ideals, Rings and Modules. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  20. 20.
    Vaserstein, L.: On the whitehead determinant for semi-local rings. J. Algebra 283, 690–699 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Yoshino, Y.: Cohen-Macaulay Modules Over Cohen-Macaulay Rings. Cambridge University Press, Cambridge (1990)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsFort CollinsUSA

Personalised recommendations