ON MORAVA K-THEORIES OF AN S-ARITHMETIC GROUP
MARIAN F. ANTON
Abstract.We completely describe the Morava K-theories with respectpto_the
prime p for the 'etale model of the classifying space of GLm (Z[ p1, 1=p*
*]) when
p is an odd regular prime. For p = 3 and m = 2 (and conjecturally for m *
*= 1)
these cohomologies are the same as those of the classifying space itself.
1.Introduction
By using an Eilenberg-Moore type spectral sequence, Tanabe calculated the
Morava K-theories for the classifying spaces of certain Chevalley groups. In pa*
*r-
ticular, if K(n) is the n-th Morava K-theory with the ring of coefficients
K(n)*(pt) = Fp[vn, v-1n]
where p is a prime and vn has degree 2(pn - 1), and if q is a power of a prime
different from p, then [12]
*(pt)[[c1, ..., cm ]]
(1.1) K(n)*BGLm (Fq) K(n)*BGLm (~Fq)_q __K(n)________________(c q *
* q
1 - _ c1, .*
*.., cm - _ cm )
i.e. a ring of formal power series in certain "Chern classes" c1,...,cm modul*
*o an
ideal given in terms of generators. Here _q is the Ä dams operation" induced fr*
*om
the Frobenius automorphism x 7! xq of the algebraic closure ~Fqof the field Fq
with q elements. The same formula (1.1)holds for the p-adic version ^K(n) of K(*
*n)
obtained by replacing K(n)*(pt) with K^(n)*(pt) = Zp[vn, v-1n] where Zp denotes
the ring of p-adic integers [12].p_
On the other hand, if A = Z[ p1, 1=p] and p is a regular prime in the sense of
number theory, then Dwyer and Friedlander [5, 6] calculated the mod p cohomol-
ogy of a space BGLm (A'et) which is naturally associated to the classifying spa*
*ce
BGLm (A) of the S-arithmetic group GLm (A). We call the space BGLm (A'et) the
'etale model at p for the classifying space BGLm (A) and recall that it is endo*
*wed
with a natural map [4, 2.5]
(1.2) fA : BGLm (A) ! BGLm (A'et)
The goal of this article is to show how we can use these two calculations in *
*order
to completely describe the Morava K-theories with respect to the prime p of the
'etale model above. Thepmain_result is
Theorem 1.1. If A = Z[ p1, 1=p] with p an odd regular prime, then the n-th
Morava K-theory with respect to the prime p of the 'etale model BGLm (A'et) is *
*an
____________
Date: August 29, 2001.
1991 Mathematics Subject Classification. 55N20,19F27,11F75.
Key words and phrases. Morava K theory, 'etale homotopy, unitary bundles.
1
2 MARIAN F. ANTON
exterior algebra given by the formula
K(n)*BGLm (A'et) K(n)*BGLm (Fq) (p-1)=2
where q is a prime 1 mod p but 6 1 mod p2, the tensor product is over the
ring K(n)*BGLm (Fq), and oei has degree 2i - 3 (1 i m). Moreover, the same
formula holds for the p-adic version ^K(n).
In particular, if m = 1 and n = 1 (and conjecturally for n > 1) the above
theorem and (1.1)give K(n) and ^K(n) theories of the classifying space BGL1 (A)
itself for p odd and regular, according to [7]. Here GL1 denotes the union of *
*all
GLn for n 1 with respect to the block inclusions.
Also, if p = 3 and m = 2, then we showed that the natural map (1.2)is a mod
p equivalence [1]. Hence we deduce the following
Corollary 1.2.pThe_n-th Morava K-theory at the prime 3 of the S-arithmetic
group GL2(Z[ 31, 1=3]) is given by
p _ F3[vn, v-1][[a, c2]]
K(n)*BGL2(Z[ 31, 1=3]) ___________n________n(7n+1)=2
(a7 , c2 mod a)
where the degrees of the generators are |vn| = 2(3n-1), |a| = 2, |c2| = 4, |oe1*
*| = -1,
|oe2| = 1, and the second generator of the ideal is up to an indeterminacy mod*
* a.
Moreover, a similar formula holds for the 3-adic version ^K(n).
Notation 1.3. In what follows p is an odd regular prime when not otherwise stat*
*ed
and A = Z[ip, 1=p] where ip = exp(2ßi=p) is a prescribed p-th root of unity in *
*the
field C of complex numbers.
2. 'Etale models for classifying spaces
2.1. The original definition. Let R = Z[1=p], G a group scheme over Spec(R),
and BG the classifying simplicial scheme obtained by a bar construction as in [*
*8,
1.2]. Then the classifying space BG(D) of the group G(D) of the D-points of G
where D is any finitely generated R-algebra can be thought of as the connected
component of a simplicial function complex [6, 1.4]
(2.1) BG(D) = Map0(Spec(D), BG)Spec(R)
containing the natural base point induced by the unit map Spec(R) ! G of G over
Spec(R). We recall that Map(X, Y )Z is a simplicial set given in dimension i by*
* the
set of simplicial scheme maps X [i] ! Y over Z where X and Y are simplicial
schemes over Z (a scheme is regarded as a constant simplicial scheme) and [i] *
*is
the standard simplicial i-simplex. The tensor product between a simplicial sche*
*me
and a simplicial set is defined in [8, 1.1].
Also, we recall that the 'etale topological type X'etin the sense of Friedlan*
*der
[8, 4.4] is a pro-space (i.e. inverse system of simplicial sets) which is natu*
*rally
associated to a noetherian simplicial scheme X and reflects the 'etale cohomolo*
*gy
of X. For any finitely generated R-algebra D, let D'etdenote the 'etale topolog*
*ical
type Spec(D)'et. By replacing Spec(D), BG, and Spec(R) in (2.1)by their 'etale
topological types D'et, (BG)'et, and R'et, the space BG(D'et) is defined in [6,*
* 1.2] as
the connected component of the simplicial complex of p-adic functions over R'et
(2.2) BG(D'et) = Hom0p(D'et, (BG)'et)R'et
ON MORAVA K-THEORIES OF AN S-ARITHMETIC GROUP 3
containing the corresponding natural base point. This construction is similar *
*to
(2.1)and we can associate with each i-simplex of BG(D) an i-simplex of BG(D'et)
regarded by definition as a map of pro-spaces over R'etfrom D'etx [i] to the
fibrewise p-adic completion of (BG)'etover R'etdenoted by (Z=p)o(BG)'et[4, 2.4].
This assignment is natural in both G and D and gives a map [4, 2.5]
fGD: BG(D) ! BG(D'et)
from the classifying space of the group G(D) to its 'etale model BG(D'et) at p.*
* In
the case when G = GLm is the group scheme over SpecR corresponding to the
general linear group G(R) = GLm (R) and D = A, we obtain the map (1.2). These
definitions actually hold for any prime p.
2.2. A model structure definition. For convenience we will give an alternative
way of thinking of (2.2)pointed out by Isaksen and based on its model structure.
Namely, if pro-SS is the category of pro-spaces then there is a proper simplic*
*ial
model structure on pro-SS introduced in [9]. This means that there are three
classes of morphisms in pro-SS called weak equivalences, cofibrations, and fibr*
*ations
subject to various axioms. Also there is a notion of simplicial function comple*
*x i.e.
a natural assignment to each two pro-spaces X and Y of a simplicial set Map(X, *
*Y )
interacting appropriately with the model structure [9, 16.2].
For the purpose of this paper we will use the induced proper simplicial model
structure on the over-category pro-SS V of pro-spaces over a fixed pro-space V .
With respect to this model structure there is a relativesimplicial function com*
*plex
Map(X, Y )V naturally associated with every pair of objects X, Y in pro-SS V.
Keeping the same notations as in the previous subsection we have the following
Proposition 2.1. For any finitely generated R-algebra D, the space BG(D'et) is
weakly equivalent to the connected component of the natural base point of the s*
*im-
plicial function complex Map(D'et, Tp(BG)'et)R'etin the over-category of pro-sp*
*aces
over R'et,
BG(D'et) ' Map0(D'et, Tp(BG)'et)R'et
Here Tp(BG)'etis a fibrant replacement of (Z=p)o(BG)'etover R'etin the sense of
the simplicial model structure of [9].
Proof.Let X = D'et= {Xff}, Y = (BG)'et, and V = R'et. Then Y ! V is a
(strict) map of pro-spaces and let Tp0(Y ) be the level-space Moore-Postnikov t*
*ower
naturally associated to the fibrewise p-adic completion of Y over V . Then we c*
*an
think of Tp0(Y ) = {Tp0(Y )ffi} as a pro-space over V = {Vffi} and by definitio*
*n [4, 2.3]
(2.3) Homp(X, Y )V = holimfficolimffMap(Xff, Tp0(Y )ffi)Vffi
where Map is the usual relative simplicial function complex of simplicial sets *
*and
holim denotes the homotopy inverse functor from pro-spaces to spaces [2, x6]. By
[9, 10.6], the pro-space Tp(Y ) is the fibrant replacement of Tp0(Y ) in the mo*
*del
structure of Edwards-Hastings. By standard arguments, the space (2.3)is weakly
equivalent to
Map(X, Tp(Y ))V = limfficolimffMap(Xff, Tp(Y )ffi)Vffi
and the conclusion follows from (2.2).
4 MARIAN F. ANTON
3.A homotopy fibre square
3.1. Preliminaries. We collect here a couple of known facts which will be used *
*in
the construction of a computable model for BGLm (A'et) given in the next subsec-
tion. This model is naturally associated to the action of ß1(R'et) on the p-pri*
*mary
roots of unity.
Let D be a finitely generated normal R-algebra and pt : Spec(k) ! Spec(D) a
geometric point corresponding to a homomorphism from D to a separable closed
field k. Then pt determines a base point of D'etand we recall that ß1(D'et, pt)*
* is the
pro-finite Grothendieck fundamental group of D pointed by pt [8, x5]. This group
classifies finite 'etale covering spaces of Spec(D).
Let ~p be the set of all complex numbers z such that zp = 1 and ~p1 the
union of all ~p for 0. Let R1 denote the ring obtained from R by adjoining
the set ~p1 of all p-primary roots of unity,
1p_
R1 = R[ p 1] = Z[1=p, ~p1 ],
and the Galois (pro-)group
= Gal(R1 , R) = {Aut(~p ), 1} {(Z=p )*, 1}
In this context, observe that ß1(R'et) is the Galois group of the maximal unr*
*am-
ified extension of R and let
(3.1) ` : ß1(R'et) !
be the homomorphism given by the action of this Galois group on the p-primary
roots of unity. In other words, R'etis provided with the natural structure map
R'et! K( , 1) which "classifies" the finite 'etale extensions R ! R[~p ]. Also,*
* A'et
is provided with a natural structure map
A'et! R'et! K( , 1)
If k is a field, then k'etis a pro-space of type K(ß, 1), where ß is the Galo*
*is group
over k of the separable algebraic closure of k. In particular, C'etis contracti*
*ble and
(Fq)'etis equivalent to the pro-finite completion of a circle. If R ! Fq is a r*
*esidue
field map, then (Fq)'etis provided with a natural structure map
(Fq)'et! R'et! K( , 1)
as well. This structure map sends the Frobenius element of the Galois group of *
*~Fq
over Fq identified with ß1((Fq)'et) to q 2 Aut(~p ) ~=(Z=p )* in [6, 3.2].
3.2. A homotopy fibre square. Let Um be the Lie group of mxm unitary matri-
ces and ^BUm the p-completion of its classifying space. The following propositi*
*on is
the unstable analogue of [5, 4.5] and its proof is almost the same. For conveni*
*ence,
we review here the main arguments.
Proposition 3.1. Let p be an odd regular prime, A = Z[ip, 1=p], and q a rational
prime 1 mod p but 6 1 mod p2. Then there is a homotopy fibre square
BGLm (A'et)----! B^UWm
?? ?
y ?y
^BGLm (Fq)----! B^Um
ON MORAVA K-THEORIES OF AN S-ARITHMETIC GROUP 5
where W is the wedge of (p - 1)=2 circles, B^UWm denotes the simplicial function
complex of unpointed maps from W to ^BUm , and the right-hand vertical map is t*
*he
evaluation at the base-point.
Proof.As in [5, p. 145] we construct a map
(Fq)'et_ W ! K( , 1)
by sending the first summand via the natural structure map and mapping the other
summand trivially. By a class-field argument (assuming the properties of q from
hypothesis), there exists a map
g : (Fq)'et_ W ! A'et
over K( , 1) which is a mod p cohomology equivalence [5, p. 145]. In other words
g is a ög od mod p model" for A in the sense of [6, 1.9]. This means that by a
spectral sequence argument [4, 2.11] and using 2.1 for G = GLm the map g induces
a homotopy equivalence
Map0(A'et, Tp(BGLm )'et)R'et' Map0((Fq)'et_ W, Tp(BGLm )'et)R'et
which can be reformulated by saying that we get a homotopy fibre square
BGLm (A'et) ----! Map0(W, Tp(BGLm )'et)R'et
?? ?
y ?y
BGLm ((Fq)'et)----! Map0(pt, Tp(BGLm )'et)R'et
where the right-hand vertical map is the evaluation at the base-point (the map
pt ! R'etis induced from R C recalling that C'etis contractible). To finish t*
*he
proof, we need only to identify the appropriate corners of this square.
For the two right-hand corners, we start with the fibration sequence [3, 2.3]
(3.2) {(Z=p)s(BGLm,~Fq)'et}s ! (Z=p)o(BGLm )'et! R'et
where {(Z=p)s(-)}s denotes the Bousfield-Kan p-completion tower and BGLm,~Fq
is the classifying object of GLm over ~Fq. Hence we get that
Map0(pt, Tp(BG)'et)R'et' holim{(Z=p)sBGLm,~Fq)'et}s ' ^BUm
where the last equivalence is proved in [8, 8.8]. Because the composite map
ß1(W ) ! ß1(A'et) ! ß1(R'et) `-!
is trivial by construction, as in [4, p. 146] we get a homotopy equivalence
Map0(W, Tp(BGLm )'et)R'et' ^BUWm
where B^UWm denotes the function complex of unpointed maps from W to B^Um
(basically ß1(R'et) acts on the fibre of (3.2)via `).
Finally, for the lower left-hand corner, there is a homotopy equivalence
B^GLm (Fq) ' BGLm ((Fq)'et)
given in [3, 2.11] by exploiting the action of the Frobenius element on the fib*
*re of
(3.2)via the composite
ß1((Fq)'et) ! ß1(R'et) `-!
and Quillen's homotopy fix point description of ^BGLm (Fq) [10].
6 MARIAN F. ANTON
4. The proof of the main theorem and its corollary
4.1. Proof of 1.1. The proof of the main theorem is based on Strickland's analy*
*sis
of unitary bundles in [11] applied to the homotopy fibre square 3.1.
Let V be a complex vector bundle over a space X and write P V for the associa*
*ted
bundle of projective spaces and U(V ) for the associated bundle of unitary grou*
*ps
U(V ) = {(x, g)|x 2 X and g 2 U(Vx)}
Let EU(V ) denote the geometric realization of the simplicial space {U(V )n+1}n*
* 0
and put BU(V ) = EU(V )=U(V ) the usual simplicial model for the classifying sp*
*ace
of U(V ).
Let E* be an even periodic cohomology theory with complex orientation x 2
~E0CP 1. We are interested in describing E*U(V ) as a Hopf algebra over E*X (us-
ing the group structure on U(V )). The main result involves the exterior algebr*
*a over
the ring E*X generated by the module E*P V which we denote by ~*E*XE*-1P V
and which is a Hopf algebra over E*X by declaring E*P V to be primitive.
Proposition 4.1 ([11, 4.4]). There is a natural isomorphism of Hopf algebras ov*
*er
E*X
~*E*XE*-1P V E*U(V )
We apply this proposition to the tautological bundle
V = flm = EUm xUm Cm
over X = BUm . In this case, we have
*BUm [x]
E*P flm ______E_______________(xm +Ec*Bm-1Um {1, ..., xm-1 }
1x + ... + cm )
where ci is the i-th Chern class of flm and the last isomorphism indicates that
E*P flm is a free module over E*BUm with basis 1, x, ..., xm-1 . In particular,
(4.1) E*U(flm ) ~*E*BUmE*-1P flm E*BUm
where oe lowers the degree by 1.
Going back to the homotopy fibre square 3.1, we observe that B^UWm is the
(p - 1)=2-fold fibre product of ^BUS1m= ^U(flm ) over ^BUm and for E* = K(n) or
^K(n) and any space X we have E*X^ = E*X. In this case, if we apply E* to ^BUWm
and use (4.1)we obtain
E*(B^UWm) (~*E*BUmE*-1P flm ) (p-1)=2
E*BUm (p-1)=2
where the tensor product is over E*BUm . In particular, E*(B^UWm) is a free mod*
*ule
over E*B^Um = E*BUm and therefore 1.1 follows from the above formula by a base-
change induced from 3.1:
E*BGLm (A'et) E*B^GLm (Fq) E*^BUmE*(B^UWm)
where q can be always chosen with the prescribed properties (by Dirichlet's den*
*sity
theorem for instance).
ON MORAVA K-THEORIES OF AN S-ARITHMETIC GROUP 7
4.2. Proof of 1.2. This has been already explained in the Introduction, except *
*for
the analysis of the formula (1.1)in the case p = 3, m = 2, and q = 7. The goal *
*of
this subsection is to complete this analysis.
Proposition 4.2. Let p be an odd rational prime and Fq a finite field with q
elements such that q 1 mod pr but 6 1 mod pr+1 for some integer r > 0. Then
*(pt)[[a, c2]]
K(n)*BGL2(Fq) ____K(n)______________nr(pnr+1)=2
(ap , c2 mod a)
Proof.According to Tanabe's formula
(4.2) K(n)*BGL2(Fq) (K(n)*BU2)_
where the co-invariants are calculated with respect to the q-th Adams operation*
* _
[12]. Recall that
(4.3) K(n)*BU2 K(n)*(pt)[[c1, c2]]
where c1 = x + y and c2 = xy are expressed in terms of the generators of the ri*
*ng
K(n)*(CP 1 x CP 1) K(n)*(pt)[[x, y]]
which are induced by a complex orientation on K(n)*(CP 1) [12, 2.12]. It is eas*
*y to
see that we can replace c1 in (4.3)by the formal group sum a = x +K(n)y of x and
y (induced from the tensor product of complex line bundles). Then the propositi*
*on
follows from (4.2)and the following lemmas
Lemma 4.3. a - _(a) = (unit) x apnr
nr+1)=2
Lemma 4.4. c2 - _(c2) (unit) x c(p2 mod a
where ü nit" means invertible element in (4.3).
Proof of 4.3.Let us expand q in the ring Zp of p-adic integers as
1X
q = ffkpk
k=0
where the coefficients ffk 2 Z are subject to 0 ffk < p, ff0 = 1, ffr 6= 0, a*
*nd
ffk = 0 for 0 < k < r. Then for t = x or y we have
X K(n) nk nr
_(t) = [q](t) = [ffk](tp ) = t + ffrtp + ...
where [q](t) means the formal group q-multiple of t. Hence,
X K(n) nk nr
_(a) = _(x) +K(n)_(y) = [ffk](ap ) = a + ffrap + ...
and the conclusion follows.
Proof of 4.4.With the same notations as in the previous proof, we have
x [-1](y) -y mod a
and hence
nr+1 (pnr+1)=2
c2 - _(c2) -x2 + _(x)_(x) (unit) x xp (unit) x c2 mod a
8 MARIAN F. ANTON
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Department of Pure Mathematics, University of Sheffield, Hicks Building, Shef*
*field
S3 7RH, UK and IMAR, P.O.Box 1-764, Bucharest, RO 70700
E-mail address: Marian.Anton@imar.ro