NK of nonabelian groups /

Abstract

Let Rng denote the category of associative rings with identity. A functor F: Rng (--->) Z-mod gives rise to another functor NF: Rng (--->) Z-mod defined as follows: NF(A) = ker (F(A{T}) (--->) F(A)) induced by the augmentation homomorphism t (--->) 0. By the work of H. Bass, A. Heller, R. Swan and D. Quillen, NK(, i)A vanishes for a regular ring A, i = 0,1,2.

This technique can be extended to the case where G is a finite group (not necessarily abelian) of square-free order by first observing that NK(, i)ZG is a module over the Forbenium functor G(, 0)ZG. By T. -Y. Lam, NK(, i)ZG can then be computed in principle from NK(, i)ZH for its hyperelementary subgroups H. In particular, NK(, i)ZH vanishes for H a hyperelementary group of square-free order, i = 0,1 and thus NK(, i)ZG vanishes as well.

R. Martin showed that NK(, i)ZG vanishes for G a finite abelian group of square-free order, i = 0,1 by utilizing Mayer-Vietoris exact sequences due to J. Milnor.