Topological Models for Arithmetic
William G. Dwyer and Eric M. Friedlander
University of Notre Dame
Northwestern University
Abstract. In this paper, we use topological models to compute the
`-adic topological K-theory of certain arithmetic rings A. Our tech-
nique is to exploit class field theory to show that the etale topological
type of A is equivalent in an appropriate sense to something relatively
simple. Calculating the K-theory of this simple "topological model"
provides an explicit determination of the`-adic topological K-theory
of A and, by means of a comparison map, gives information about the
algebraic K-theory of A. For example, we are sometimes able to com-
pute the mod ` cohomology of certain "unstable" topological K-theory
spaces and verify that it injects into the cohomology of the correspond-
ing unstable algebraic K-theory spaces. This gives an explicit lower
bound for H (GL (n; A); Z=`):
x0. Introduction
In[4] we introduced the "etale K-groups" of a ring A as the homotopy
groups of a twisted topological K-theory space associated tothe etale
topological type Aet of A. The construction of these groups involves
choosing a prime ` invertiblein A and working with K-theory completed
at `. Building upon work of C. Soule [19], we described a natural map
from algebraic K-groups (completed at `) to etale K-groups,and showed
that!this map is in general surjective for arithmetic rings. The well-
known!Lichtenbaum-Quillen Conjectures essentially assert that this map
is!an!isomorphism.
! Many of the basic results of class field theory and arithmetic duality
can!be!interpreted as statements about the etale cohomology of various
arithmetic!rings!A [10] or equivalently as statements about the (ordi-
nary)!cohomology of Aet: In[5] we exploited this fact to find, for some
specific!arithmetic!rings A,an elementary topological space XA mapping
to!Aet bya cohomology isomorphism. This construction allowed us not
only!to!compute the etale K-groups of A,but also to identify the under-
lying!twisted topological K-theory space and compute its cohomology
(which,!by!the Lichtenbaum-Quillen conjecture, should be isomorphic
to!the!cohomology of GL (A)).
!
Partially supported by the NationalScience Foundation and by NSA Grant # MDA
904-90-H-4006.
2 Dwyer andFriedlander
Inthe present paper, we extend this program to cover other rings A.
In each case,for a specific prime ` invertible in A, we find a "goo dmod `
model" XA for A; this is an explicit space or pro-space XA which in an
appropriate sense captures the mod ` cohomology of Aet. For example,
there is a very simple mod ` model for A if ` is a regular prime and
A = Z[1=`]. We also find that in fortunate situations (such as the one
just mentioned), the cohomology of the resulting (unstable) K-theory
space of the model XA isnot only computable but also injects into the
cohomology of the (unstable) algebraic K-theory space of A. In some
cases this sheds light on the Lichtenbaum-Quillen conjecture (2.4, 6.4,
6.6).
Inbrief, the contents of our paper are as follows. Section 1 motivates
and presents the definition of a goodmod ` model of an arithmetic ring
A. Such a model captures the etale cohomology of A with coefficients
in "Tate twists" of Z=`. In x2, we provide aparticularly simple model
for A = Z[1=`] whenever ` is an odd regular prime. Verifying this model
is more delicate than verifying themo dels for affine curves and local
fields described in x3. The remainder of the pap eris concerned with an-
alyzing the cohomology of (unstable) K-theory spaces asso ciated with
our good mod ` models and studying the relationship between this co-
homology and that of the general lineargroups of A. In x4, we discuss
the Eilenberg-Moore spectral sequence and apply it to our topological
K-theory spaces. Our comparison of cohomology of algebraic and top o-
logical K-theory spaces is achieved byrestricting to maximal tori: the
actual comparison is carried out in x6, following a check in x5 that corre-
sponding algebraic and topological tori have the same cohomology. We
conclude the paper by mentioning afew open problems in x7.
0.1 Notation: We fix a prime number ` and let R denote the ring
Z[1=`], F the finite field Z=`, i`n the `n'th root of unity e2ssi=`nand `n
the multiplicative group of `n'th roots of unity. The symbol i without
a subscript denotes i`.
Ourfundamental reference for homotopy theoretic completions is [2].
If X is a space or pro-space the `-adic completion tower of X is a pro-
space denoted F! (X); if X is aspace then F! (X) is a tower in the
usual sense whose inverse limit is the `-adic completion F1 (X) of X.
If E ! Bis a fibration the fibrewise `-adic completion tower of E over
B [4, p. 250] is denoted Fffl!(E); if E and B are spaces,the inverse limit
of this tower is the fibrewise `-adic completion Fffl1(E) of E over B.
x1. Good mod ` models
We recall that Xet is a "space" which reflects the etale cohomology
Topological Models 3
coherent discussion of the etale cohomology Het(X; F) of X with coef-
ficients in a sheaf F on the "etale site"; endurance now assured, he or
she can then peruse [1] and [7] for the construction and properties of
the pro-space (i.e., inversesystem of simplicial sets) Xet.
For any finitely generated R-algebra A, let Aetdenote the etale topo-
logical type (Spec A)et. For any connected,noetherian simplicial scheme
X (e.g., Spec A for a noetherian domain A), Xetis an inverse system of
connected simplicial sets [7, 4.4]. If X is normal in each simplicial de-
gree, as is always the case in our arithmetic examples, Xet is homotopy
equivalent to an inverse system in the homotopy category of simplicial
sets each of which has finitehomotopy groups [7, 7.3].
Two key properties of Aet are given in the following
1.1 Proposition. Let a : Spec k !Spec A be a geometric point of the
finitely generated normal R-algebra A, corresponding to a homomor-
phism from A to a separably closed fieldk and determining a base point
of Aet. Then
(1) ss1(Aet;a) = sset1(A; a) , the (profinite) Grothendieckfundamental
group ofA p ointed by a (this group classifies finite etale covering
spaces of Sp ec A).
(2) H (Aet; C) = Het(Spec A; C); the etale cohomology of Spec A
with coefficients in the local coefficient system C (such a local co-
efficient system is an abelian group C provided with a continuous
action of ss1et(A; a)).
Theetale K -theoryspace associated to Aet is constructed using func-
tion complexes. Namely, for each integer n, we consider the group
scheme GL n;R whose coordinatealgebra is the Hopf algebra
R[xi;j; t]=det (xi;j)t = 1 :
The usual bar construction [7, 1.2, x2] (over R) determines a simplicial
scheme BGL n;R with etale topological type (BGL n;R )et.
1.2 Definition: The space BGL n(Aet) is defined to be (see [4, 2.4,
4.5])
BGL n (Aet) j Hom 0`(Aet;(BGL n;R)et)Ret
the connected component of the functioncomplex of maps over Ret from
Aet to the pro-space fibrewise mod ` completion over Ret of (BGL n;R)et.