THE GENERALIZED BURNSIDE RING AND THE
K-THEORY OF A RING WITH ROOTS OF UNITY
W. G. Dwyer , E. M. Friedlander and S. A. Mitchell
University of Notre Dame
Northwestern University
University of Washington
x1. Introduction
Determining the algebraic K-theory of rings of integers in number fields hasb*
* eenthe
goal of much research. In [10] D. Quillen showed that the Hurewicz map h : Q0(S*
*0) !
BGL (Z)+ (see 1.1 for the notation) induces an interesting map on homotopy grou*
*ps from
the stable homotopy groups of spheres to the algebraic K-theory of the ring Z o*
*f rational
integers. Quillen observed that if ` is an odd prime and if p 6= ` is another p*
*rime generating
the `-adic units, then the composite map
h ae
Q0(S0)! BGL (Z)+! B GL (Fp)+
induces a surjection
sss(S0; Z=`) ! K (Z; Z=`) ! K (Fp; Z=`)
on mod ` homotopy groups. In geometric terms, one can write Q0(S0) ' ImJ Coker J
where ImJ is the factor of Q0(S0) associated to the J-homomorphism J : O !Q0(S *
*0).
The space ImJ is then equivalent after localization at` to B GL (Fp)+ in such a*
* way that
the composite
h ae
Im J Coker J 'Q0(S0)! BGL (Z)+! B GL (Fp)+
!
!
can!be identified after localization at ` with the projection r : ImJ Coker J! *
* Im J.
!!In [6] the third author complemented Quillen's result by proving that the Hur*
*ewicz
map!h!does not in fact detect anything in stable homotopy theory beyond what Qu*
*illen
!
Partially supported by the N.S.F. Partially supported by theN.S.F. and N*
*SA Grant # MDA904-
90-H-4006.
Typeset by AM S-T*
*EX
2 DWYER, FRIEDLANDER AND MITCHELL
found. He did this by verifying that the localization of h at` factors through *
*Im J, or,
more precisely, that the maps
h; h s r : Q0(S0) ! B GL (Z)+
are homotopic after localization at `,where s : Im J! Q0(S0) is a suitable righ*
*t inverse
after localization at ` for the factor projection r: Q0(S0)! Im J. It follows *
*that Quillen's
map h : sss(S0; Z=`) ! K (Z; Z=`) annihilates the homotopy of CokerJ, and that *
*for any
ring R the structureof K (R; Z=`) as a module over sss(S0; Z=`) extends (unique*
*ly) to a
structure of K (R;Z=`) as a module overss (Im J;Z=`) = K (Fp; Z=`).
Let ia denote a primitive `a-th root of unity and the group of `-primary roo*
*ts of unity
of Z[ia]: The inclusion ae GL1(Z[ia]) leads to an enriched analogue
h : Q0(B+ ) ! BGL (Z[ia])+
of Quillen's Hurewicz map. Let ` and p be primes as before (` is odd and p gene*
*rates the
`-adic units) and let Fqdenote Fp[ia]. In Z[ia] there is a unique prime above p*
* with residue
class field Fq. The main result of B. Harris and G. Segal [3] implies that the *
*composition
h ae
Q0(B+)! BGL (Z[ia])+! B GL (Fq)+
induces a surjection of mod ` homotopy groups. In fact,for a suitable space X *
*there is
after localization at ` a product decomposition Q0(B +) ' BGL (Fq)+ X in such a*
* way
that the (localized) composition
BGL (Fq)+ X ' Q0(B +) ! BGL (Z[ia])+ ! BGL (Fq)+
can be identified with the projection r: BGL (Fq)+ X! B GL (Fq)+ .
The purpose of this paper is to extend the results of the third author to thi*
*s wider
context. Theorem 4.1 assertsthat after localization at ` the maps
h; h s r : Q0(B+) ! B GL (Z[ia])+
are homotopic, where s : B GL (Fq)+! Q0(B+ ) is a suitable right inverse for t*
*he factor
projection r. It follows that for any algebra Rover Z[ia] the structure of K (R*
*;Z=`) as
a module over sss(B+; Z=`) extends (uniquely) to a structure of K(R; Z=`) as a *
*module
over K (Fq;Z=`). We also make some further remarks (4.10) about the relationshi*
*p of our
work to [6], and briefly consider the situation of infinite cyclotomic extensio*
*ns (4.12).
Let G be a finite `-group. To prove Theorem 4.1 we show that h, hs rhave homo*
*topic
compositions with any map BG ! Q0(B+). Ourtechnique is to use the generalized
Burnside ring A(G;) to study maps from BG to Q0(B+) and to use the representati*
*on
ring RRG of G over R to study mapsfrom B Gto B GL (R)+. The key algebraic resul*
*t is
Theorem 3.5, which amounts to an algebraic analogue of (4.1) and asserts that t*
*he map
from A(G; ) to RZ[ia]G naturally factors through a surjection A(G; )! RFqG.
We conjecture that, after localization at `, the composition
h s : B GL (Fq)+ ! BGL (Z[ia])+
extends to an infinite loop map. One can view Proposition 4.9 as evidence that *
*such an
extension exists.