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. 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Math. 117 (19* *95), 263-278 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