Some Remarks on Nil Groups in Algebraic K-theory

This note explains consequences of recent work of Frank Quinn for computations of Nil groups in algebraic K-theory, in particular the Nil groups occurring in the K-theory of polynomial rings, Laurent polynomial rings, and the group ring of the infinite dihedral group. 1. Statement of Results Let R be a ring with unit. For an integer q, let KqR be the algebraic K-group of Bass and Quillen. Bass defines the NK-groups NKq(R) = ker(KqR[t] → KqR) where the map on K-groups is induced by the ring map R[t] → R, f(t) 7→ f(0). The NK-groups are often called Nil-groups because they are related to nilpotent endomorphisms of projective R-modules. Let G be a group. Let OrG be its the orbit category; objects are G-sets G/H where H is a subgroup of G and morphisms are G-maps. Davis-Lück [8] define a functor K : OrG → Spectra with the key property πqK(G/H) = Kq(RH). The utility of such a functor is to allow the definition of an equivariant homology theory, indeed for a G-CW-complex X, one defines H q (X;K) = πq(mapG(−, X)+ ∧OrG K(−)) (see [8, section 4 and 7] for basic properties). Note that mapG(G/H,X) = X is the fixed point functor and that the “coefficients” of the homology theory are given by H q (G/H ;K) = Kq(RH). A family F of subgroups of G is a nonempty set of subgroups closed under subgroups and conjugation. For such a family, EF (short for EFG) is the classifying space for G-actions with isotopy in F . It is characterized up to G-homotopy type as a G-CW-complex so that E F is contractible for subgroups H ∈ F and is empty for subgroups H 6∈ F . Partially supported by a grant from the National Science Foundation.