K-groups for rings of finite Cohen–Macaulay type

For a local Cohen-Macaulay ring R of finite CM-type, Yoshino has applied methods of Auslander and Reiten to compute the Grothendieck group K0 of the category modR of finitely generated R-modules. For the same type of rings, we compute in this paper the first Quillen K-group K1(modR). We also describe the group homomorphism R � ! K1(modR) induced by the inclusion functor projR ! modR and illustrate our results with concrete examples.

[1]  V. Navkal.  K'-Theory of a Local Ring of Finite Cohen-Macaulay Type , 2011, 1108.2000.

[2]  Overtoun M. G. Jenda,et al.  Relative homological algebra , 1956 .

[3]  Graham J. Leuschke.  Endomorphism Rings of Finite Global Dimension , 2005, Canadian Journal of Mathematics.

[4]  秀人 浅芝.  M. Auslander, I. Reiten and S. O. Smalø: Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math., 36, Cambridge Univ. Press, 1995年，xiv + 423ページ． , 2007 .

[5]  A. Kuku,et al.  Higher Algebraic K-Theory , 2006 .

[6]  L. Vaserstein.  On the Whitehead determinant for semi-local rings , 2005 .

[7]  N. Cimen.  One-dimensional rings of finite Cohen-Macaulay type , 1998 .

[8]  J. Rosenberg,et al.  Algebraic K-Theory and Its Applications , 1995 .

[9]  P. H. Kropholler,et al.  REPRESENTATIONS AND COHOMOLOGY I: Basic representation theory of finite groups and associative algebras , 1994 .

[10]  R. Wiegand.  One-Dimensional Local Rings with Finite Cohen-Macaulay Type , 1994 .

[11]  A. Duncan.  COHEN‐MACAULAY MODULES OVER COHEN‐MACAULAY RINGS , 1992 .

[12]  Tsit Yuen Lam,et al.  A first course in noncommutative rings , 2002 .

[13]  D. Benson,et al.  Representations and Cohomology , 1991 .

[14]  V. Srinivas.  Algebraic K-Theory , 1990 .

[15]  Y. Yoshino,et al.  Cohen-Macaulay modules over Cohen-Macaulay rings , 1990 .

[16]  R. Wiegand.  Noetherian Rings of Bounded Representation Type , 1989 .

[17]  R. Buchweitz,et al.  The Homological Theory of Maximal Cohen-Macaulay Approximations , 1989 .

[18]  J. Wunram.  Reflexive modules on quotient surface singularities , 1988 .

[19]  I. Reiten,et al.  Almost split sequences for rational double points , 1987 .

[20]  Maurice Auslander,et al.  Rational singularities and almost split sequences , 1986 .

[21]  I. Reiten,et al.  Grothendieck groups of algebras and orders , 1986 .

[22]  Hélène Esnault,et al.  Reflexive modules on quotient surface singularities. , 1985 .

[23]  C. Sherman.  Group representations and algebraic K-theory , 1982 .

[24]  M. C. R. Butler.  Grothendieck groups and almost split sequences , 1981 .

[25]  J. Herzog.  Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerlegbaren Cohen-Macaulay-Moduln , 1978 .

[26]  I. Reiner,et al.  Integral representations and diagrams. , 1978 .

[27]  Auslander Maurice,et al.  Representation Theory of Artin Algebras I , 1974 .

[28]  Maurice Auslander,et al.  Representation Theory of Artin Algebras II , 1974 .

[29]  D. Quillen,et al.  Higher algebraic K-theory: I , 1973 .

[30]  S. Gersten.  Higher K-theory of rings , 1973 .

[31]  L. Vaserstein.  ON THE STABILIZATION OF THE GENERAL LINEAR GROUP OVER A RING , 1969 .

[32]  Hyman Bass,et al.  Algebraic K-theory , 1968 .

[33]  J. Drozd,et al.  COMMUTATIVE RINGS WITH A FINITE NUMBER OF INDECOMPOSABLE INTEGRAL REPRESENTATIONS , 1967 .

[34]  P. Gabriel,et al.  Des catégories abéliennes , 1962 .

[35]  O. Litoff.  On the commutator subgroup of the general linear group , 1955 .