K-Theory of Endomorphism Rings and of Rings of Invariants☆

Abstract This paper gives the following description ofK0of the endomorphism ring of a finitely generated projective module. T heorem . Let T be a ring and P a finitely generated,projective T-module. Let I be the trace ideal of P. Then K0(End PT)is isomorphic to a subgroup of K0(T, I).If,further,the natural map K1(T) → K1(T/I)is surjective then K0(End PT)is isomorphic to the subgroup of K0(T)generated by the direct summands of Pn,for n ∈  N . As a corollary we can determineK0of the ring of invariants for many free linear actions. In particular, the following result is proved. T heorem . Let V be a fixed-point-free linear representation of a finite group G over a field k of characteristic zero and let S(V)be the symmetric algebra of V.Let K be any finite-dimensional k-vector space. Then K0(S(V)G ⊗ kS(K)) = 〈[S(V)G ⊗ kS(K)]〉. Similar results are given for suitable noncommutative versions ofS(V).

[1]  William Crawley-Boevey,et al.  NONCOMMUTATIVE DEFORMATIONS OF KLEINIAN SINGULARITIES , 1998 .

[2]  P. Polo,et al.  K-theory of twisted differential operators on flag varieties , 1996 .

[3]  K. Brown,et al.  Grothendieck groups of invariant rings: linear actions of finite groups , 1996 .

[4]  K. Brown,et al.  Grothendieck Groups of Invariant Rings and of Group Rings , 1994 .

[5]  M. P. Holland.  Varieties Which are almost D -Affine , 1993 .

[6]  S. C. Coutinho,et al.  K-theory of twisted differential operators , 1993 .

[7]  D. Yao.  Higher algebraic K-theory of admissible abelian categories and localization theorems , 1992 .

[8]  C. Weibel.  Module structures on the K-theory of graded rings , 1987 .

[9]  T. Goodwillie.  Relative algebraic K-theory and cyclic homology , 1986 .

[10]  I. Reiten,et al.  McKay quivers and extended Dynkin diagrams , 1986 .

[11]  J. Wolf.  Spaces of Constant Curvature , 2010 .

[12]  D. Quillen,et al.  Cyclic homology and the Lie algebra homology of matrices , 1984 .

[13]  G. Gonzalez-Sprinberg,et al.  Construction géométrique de la correspondance de McKay , 1983 .

[14]  V. Srinivas.  Vector bundles on the cone over a curve , 1982 .

[15]  Alekseĭ Ivanovich Kostrikin,et al.  Introduction to algebra , 1982 .

[16]  David F. Anderson.  Projective modules over subrings of $k[X, Y]$ , 1978 .

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

[18]  M. Raynaud.  Anneaux locaux henséliens , 1970 .

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

[20]  Hyman Bass,et al.  K-Theory and stable algebra , 1964 .

[21]  Maurice Auslander,et al.  The Brauer group of a commutative ring , 1960 .