THE TATE-FARRELL COHOMOLOGY OF THE MORAVA STABILIZER
GROUP Sp-1 WITH COEFFICIENTS IN Ep-1
PETER SYMONDS
Abstract. We calculate the Tate-Farrell cohomology of the Morava stabiliz*
*er group Sp-1
with coefficients in the moduli space Ep-1for odd primes p.
1. Introduction
We present a calculation motivated by homotopy theory, although our methods a*
*re alge-
braic and involve the Tate cohomology of a profinite group with compact coeffic*
*ients. As
a reference to the background in homotopy theory we suggest [4, 5]. For the Ta*
*te-Farrell
cohomology of profinite groups with coefficients in compact module we refer to *
*[12], although
most of the results are analogues of one for discrete groups, for which see [2].
Let p be an odd prime and n 2 N and let R be the ring of integers of the unra*
*mified
extension of ^Qpof degree n, so k ~=Fpn. Let Ø be the Frobenius automorphism an*
*d Gal= <Ø>
the Galois group. Let Sn denote the (full) nth Morava stabilizer group: this is*
* the group of
units in the R-algebra M generated by S subject to the relations Sn = p and rS *
*= SØ(r)
for r 2 R. The Galois group Gal acts on Sn simply by Ø(rSi) = Ø(r)S.
Thus Sn is virtually a pro-p group of virtual cohomological dimension n2 and *
*type FP 1.
Ifn n denotes the commutative one-dimensional p-typical formal group law with*
* p-series
xp , then Sn is isomorphic to the group of automorphisms of n over Fp. It ther*
*efore acts on
the ring of functions on the Lubin-Tate moduli space of ?-isomorphism classes o*
*f lifts of n,
which is En,0= R[[u1, . .,.un-1]], a profinite RSn-module. We denote the catego*
*ry of such
modules by CR(Sn). There is also an action of Sn on a graded version En,*= En,0*
*[u 1]. This
is graded by the power of u, normalized so that u has degree -2 (called the int*
*ernal degree).
This combines with the action of Gal on En,*via its action on the coefficient*
*s to give an
action of the semi-direct product Sn o Gal on En,*, and so each En,r2 CR(SnLo G*
*al).
We would like to calculate the ring H*(Sn, En,*)Gal, by which we mean r,sHr*
*(Sn o
Gal, En,s), since this is the initial term of a spectral sequence which converg*
*es to ß*LK(n),
the homotopy groups of the localization of the sphere spectrum at the nth Morav*
*a K-theory
(all at the prime p). What we will actually do is to calculate the Tate-Farrell*
* cohomology
in the case n = p - 1: this is equal to the ordinary cohomology in degrees grea*
*ter than n2.
Theorem 1.1. For odd p and n = p - 1
^H*(Sn, En,*)Gal= ^H*(G, En,*)Gal (x0, . .,.xn-1)
= ^H*(Sn, ^Zp) (ff) Fp[ 1]
= Fp[ 1, fi 1] (ff, x0, . .,.xn-1).
The generators will be defined in the course of the calculation.
1
THE TATE-FARRELL COHOMOLOGY OF THE MORAVA STABILIZER GROUP Sp-1WITH COEFFICIENT*
*S IN Ep-12
L *
* L
Remark. It would be natural to regard En,*as ^ sEn,s, the sum in CR(Sn), but Hr*
*(Sn, ^ sEn,s) ~=
sHr(Sn, En,s). Since only the homogeneous parts appear in the spectral sequenc*
*e, the dif-
ference is immaterial, but we conform to the conventional usage.
We will need the following corollary of [12] 7.3 and the remark following it.*
* It is what we
would expect from the theory for discrete groups in [2]. Similar results for pr*
*ofinite groups
but with discrete coefficients also appear in [11] and [10].
Theorem 1.2. Let G be a profinite group of finite virtual cohomological dimensi*
*on over
R. Suppose that G has no subgroup isomorphic to Z=p x Z=p and only a finite num*
*ber of
conjugacy classes of subgroups isomorphic to Z=p, which we denote by S(p). Let*
* M be a
module in CR(G). Then the Tate-Farrell cohomology satisfies
M
H^*(G, M) ~= H^*(NG(P ), M).
P2S(p)
2. Trivial Coefficients
From now on n = p - 1. The group Sn contains an element a of order p and *
* is a
maximal finite p-subgroup, unique up to conjugacy. The centralizer C = CSn(a) c*
*orresponds
to the units in the ring of cyclotomic integers ^Zp[a] M so, written additive*
*ly, has the form
^Znpx Z=p x Z=n. There is an element b of order n2 which normalizes , with b*
*n generating
the Z=n in the centralizer: let e be an integer such that b-1ab = be. Then G = *
* is the
maximal finite subgroup of order divisible by p.
The subgroup N0 = NSnoGal() fits into a short exact sequence ^Znp! N0 ! T *
*, where
|T | = n3 and T fits into a short exact sequence ** ! T ! Gal. The second gene*
*rator c of
T can be chosen to centralize a.
The action of T on ^Znpis via a cyclic quotient of order n generated by the i*
*mage of b, and
as a module for this it is free of rank 1 or, equivalently, a sum of rank 1 R-l*
*attices, one for
each possible eigenvalue.
First we calculate the Tate-Farrell cohomology with trivial coefficients. We *
*use to denote
an exterior algebra over Fp.
Proposition 2.1. For p odd and n = p - 1:
H^*(Sn, ^Zp) = ^H*(G, ^Zp) (x0, . .,.xn-1) = Fp[fi 1] (x0, . .,.*
*xn-1),
H^*(Sn, Fp) = ^H*(G, Fp) (x0, . .,.xn-1) = Fp[fi 1] (ff, x0, . .,*
*.xn-1),
where |fi| = 2n, |xi| = 1 - 2i and |ff| = -1.
Proof.By Theorem 1.2 we find that H^*(Sn, ^Zp) ~= H^*(N, ^Zp), where N = NSn(<*
*a>) ~=
(^Znpx ****) o ****. Notice that ^H*(N, ^Zp) ~=^H*(^Znpx ****, ^Zp)****, since b ha*
*s order coprime to
p.
By the Künneth Theorem in Tate-Farrell cohomology ([2] X 3 ex. 4), H^*(C, ^Zp*
*) ~=
H^*(****, ^Zp) H*(^Znp, Fp).
It is well known that H^*(, ^Zp) = Fp[i 1], where |i| = 2. To find the act*
*ion of b on i
use dimension shifting to see that H2(, R) ~=H1(, k) ~=Hom (, k). Then*
* b acts on
the latter by sending f to (x 7! bf(b-1xb)), so b(i) = ei.
THE TATE-FARRELL COHOMOLOGY OF THE MORAVA STABILIZER GROUP Sp-1WITH COEFFICIENT*
*S IN Ep-13
Now a basis y0, . .,.yn-1 of H1(^Znp, Fp) can be chosen so that b(yi) = eiyi.*
* We finish by
calculating the invariants under b using the last part of Lemma 4.1 and setting*
* xi= i-i yi.
The calculation for Fp coefficients is almost identical.
3. Coefficients in En
Next we calculate ^H*(, En,*) following the method of Nave [7, 8], which i*
*n turn is based
on unpublished work of Hopkins and Miller. This is also treated in detail for t*
*he prime 3 in
[6].
First we need a change of basis.
Lemma 3.1. ([7, 8]) There are elements z, z1, . .,.zn-1 2 En,0such that, where *
*m denotes
the ideal (p, u1, . .,.un-1) in En,0:
(1) z cu mod (p, m2) for some c a unit in R,
(2) zi ciuui mod (p, u1, . .,.ui-1, m2) for some ci a unit in R.
(3) (1 + a + . .+.ap-1)z = 0,
(4) b(z) = jz for j 2 R a primitive n2 root of unity such that jp-1 = e (mod*
* p),
(5) (a - 1)z = zn-1 and (a - 1)zi+1= zi for 1 i < n - 1.
It follows from (1) and (2) that En,*= R[[z-1z1, . .,.z-1zn-1]][z 1].
Let V be the R-submodule of En,-2spanned by {z, z1, . .,.zn-1}. It follows fr*
*om (3),(4)
and (5) that V is an RG-submodule. Let ffi = p-1i=0ai(z): then a(ffi) = ffi an*
*d b(ffi) = jpffi = ejffi.
Consider the symmetric algebra S[V ] En,*. We claim that, as RG-modules,
8
>>0 r odd,
>< 0
ffi-r R (proj)r = 2pr0 0,
(y) R[V ]r = 0
>>ffi-r V (proj)r = 2(pr0- 1) 0,
>:
(proj) otherwise.
and
R[V ]r-2p= ffiR[V ]r (proj) forr < 0.
Here (proj) indicates a projectiveLsummand. We will write this in the condense*
*d form
R[V ] = B (proj), where B = iffii(R V ).
Recall that if G is a finite group of order not divisible by p2 and M 2 CR(G)*
* is projective
in CR then the isomorphism class of M is uniquely determined by its reduction m*
*odulo p,
k R M. This is true for a cyclic group of order p by the classification of RZ*
*=p-lattices,
(see [9], [3] 34.31), and this classification extends to CR(Z=p). The general c*
*ase follows by a
transfer argument.
Thus we only have to check the claim over kG. But it is true over k from t*
*he calculation
of the symmetric algebra by Almkvist and Fossum [1]. The general case follows b*
*ecause both
0 -r0
ffi-r R and ffi V are defined over G, and the quotients by them must still be*
* projective over
G since this depends only on the restriction to the Sylow p-subgroup. Being pro*
*jective they
force the extension to split, and our claim is proved.
If we invert ffi we obtain a dense subset R[V ][ffi-1] En,*. As an RG-modul*
*e this still has
the same form B (proj), by the second identity in y. In fact this form is pr*
*eserved by
completion:
Proposition 3.2. As a sum of compact modules for RG, En,*= B (proj).
THE TATE-FARRELL COHOMOLOGY OF THE MORAVA STABILIZER GROUP Sp-1WITH COEFFICIENT*
*S IN Ep-14
Proof.Let (z-1z1, . .,.z-1zn-1) denote the ideal generated (topologically) by t*
*he given ele-
ments in En,0. It is easy to check that
En,-2r= R[V ]-2r (z-1z1, . .,.z-1zn-1)r+1En,-2r, r > 0
and also
R[V ]-2r= B-2r P-2r, r > 0
for some projective P-2r. Thus, for r + pt > 0,
En,-2r= ffitEn,-2(r+pt)
= ffitR[V ]-2(r+pt) ffit(z-1z1, . .,.z-1zn-1)r+1En,-2(r+pt)
= B-2r P-2(r+pt) (z-1z1, . .,.z-1zn-1)r+1En,-2r.
Now R[V ][ffi-1]-2r = B-2r lim-!P-2(r+pt)and En,2r= B-2r lim-P-2(r+pt)as t ! *
*1. As
a consequence, if lim-!P-2(r+pt)= iQi, as a sum of indecomposable projective R*
*G-modules
then En,2r= B2r iQi.
We say that x 2 ^Hr(-, En,s) has bidegree |x| = (r, s).
Corollary 3.3. ([7]) The Tate cohomology is given by
H^*(, En,*) = k[ffi 1, i 1] ( ),
where |ffi| = (0, -2p), |i| = (2, 0), |ff| = (1, -2) and b acts by
b(ffi) = ejffi, b(i) = ei, b( ) = ej .
As a consequence
^H*(G, En,*)= k[ 1, fi 1] (ff),
^H*(G, En,*)Gal= Fp[ 1, fi 1] (ff).
where | | = (-2, 2n), |fi| = (2n, 0) and |ff| = (1, 2n).
Proof.The first calculation is an easy consequence of 3.2 (we identify ffi with*
* its image in
H^0(, En,-2p)).
H^*(, En,*)= ^H*(, B)
M
= H^*(, ffir(R V ))
r2Z
M
= ffirk[i 1] ( ) (a well-known calculation)
r2Z
= k[ffi 1, i 1] ( ).
The action of b on ffi is from the definition of ffi and e and that on i was fo*
*und in the proof
of 2.1.
For the action on 2 H1(, V ) it is easy to verify that the quotient ma*
*p V !
V= rad(V ) ~= kz induces an isomorphism on H1, so H1(, V ) ~= zH1(, k) as*
* a ****-
module, and this combines the action on z with that found in calculating the ac*
*tion on i in
2.1.
Thus ^H*(G, En,*) ~=^H*(****, En,*)****and the invariants can be calculated usi*
*ng lemma 4.1
below. They are generated by fi = in, = ffi-ni-1 and their inverses and ff = *
*ffi-1 .
THE TATE-FARRELL COHOMOLOGY OF THE MORAVA STABILIZER GROUP Sp-1WITH COEFFICIENT*
*S IN Ep-15
Finally, notice that c acts on the R-module ^Hr(G, En,s) according to the for*
*mula c(`x) =
Ø(`)c(x), ` 2 R, x 2 ^Hr(G, En,s). This cohomology group is either k or 0, so t*
*he invariants
under c are either Fp or 0. Since the generators , fi, ff can be replaced by *
*any non-zero
element of the ^Hr(G, En,s) that they appear in, we may assume that they are al*
*l invariant
under c and hence generate the invariants under c.
Proof.of 1.1. As before we use Theorem 1.2 to see that ^H*(Sn, En,*)Gal~=H^*(N0*
*, En,*).
Recall that, for any short exact sequence of profinite groups of finite virtu*
*al cohomological
dimension I ! J ! K with K torsion-free, there is a spectral sequence H*(K, ^H**
*(I, M)) )
H^*(J, M) ([2] X 3 ex. 5).
Apply this to C = ^Znpx **** to obtain H*(^Znp, ^H*(, En,*)) ) ^H*(C, En,*)*
*. If we fix both
r and s then ^H*(, En,r) is either k or 0 so ^Znp, being a pro-p group, must*
* act trivially. Thus
the E2-term is isomorphic to ^H*(, En,*) H*(^Znp, Fp) ~=^H*(, En,*) *
*(y0, . .,.yn-1).
We claim that this spectral sequences collapses, so that H^*(C, En,*) ~= ^H*(*
*, En,*)
Fp(y0, . .,.yn-1). To see this notice that, from the proof of 3.3, that the m*
*ap En,r!
En,r=mEn,r~= k induces an injection on H^*(, -). The corresponding spectral*
* sequence
with coefficients k collapses, by the Künneth Theorem, so ours must too.
Now compute the invariants under b using Lemma 4.1. The result is H^*(, E*
*n,*)****
Fp(x0, . .,.xn-1), where the xiare as in 2.1.
Finally, c acts only on the first factor, so taking the invariants under c ju*
*st replaces
H^*(G, En,*) by ^H*(G, En,*)Gal.
4.Invariants
The following lemma is elementary, but systematic use of it simplifies the in*
*variant calcula-
tions above. For example in the proof of 3.3, first calculate k[ffi 1, i 1]****=*
* (k[i 1] k[ 1])****
and then (k[ffi 1, i 1] ( ))****.
Lemma 4.1. Let H be a finite abelian group and let R be a commutative integral *
*domain
such that |H| is invertible in R and R contains a root of unity of order the ex*
*ponent of H.
Suppose that A and B are two RH-modules such that A is a graded-commutative R a*
*lgebra
and the action of H is compatible with this structure. Let H act on A R B diag*
*onally.
Let C be the set of isomorphism classes of homomorphisms from H to Rx. (This *
*can be
identified, perhaps not canonically, with the characters of H.) Then there are *
*decompositions
of RH-modules A = c2CAc and B = c2CBc, where Ac = {a 2 A|ha = c(h)a, h 2 H} a*
*nd
similarly for B. Let CA = {c 2 C|Ac 6= 0}.
Suppose that for each c 2 CA there is a homogeneous element ac 2 Ac that is i*
*nvertible in
A. Then
M
(A B)H = AH ad-1 Bd,
d2CA
M
(A B)c= AH ad-1 Bcd.
d2CA
Suppose that B is also a graded commutative R-algebra and that H acts compatibl*
*y with this
structure. Then A B is also a graded-commutative R-algebra in the usual way, *
*and H acts
as a group of automorphisms.
THE TATE-FARRELL COHOMOLOGY OF THE MORAVA STABILIZER GROUP Sp-1WITH COEFFICIENT*
*S IN Ep-16
(1) If, for each c 2 CA \ CB, there is a homogeneous element bc 2 Bc that is*
* invertible
in B, then (A B)H is a free AH BH -module with basis {ac-1 bc : c 2*
* C0}.
Furthermore if the monomials in c1, . .,.cr 2 CA \ CB yield all the c *
*2 CA \ CB
then (A B)H is generated as a ring by AH , BH and the ac-1i bci.
(2) If B is generated as an R-algebra by d1, . .,.ds, where di 2 Bcdifor som*
*e cdi 2
CA \ CB, then (A B)H is generated as a ring by AH and the ac-1d di.
i
If the di freely generate B as a graded-commutative R-algebra then the*
* ac-1d di
*
* i
freely generate (A B)H over AH . (So if B = R(d1, . .,.ds) then (A *
* B)H =
AH R R(ac-1d d1, . .,.ac-1 ds).)
1 ds
Proof.This is left as an exercise for the reader. Notice that (A B)H = c2C0A*
*c-1 Bc and
Acac0= Acc0.
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[7]Nave, L.S., The Cohomology of Finite Subgroups of Morava Stabilizer Groups *
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*, preprint at
http://www.ma.umist.ac.uk/pas/preprints.
Department of Mathematics, U.M.I.S.T., P.O. Box 88, Manchester M60 1QD, Engla*
*nd
E-mail address: Peter.Symonds@umist.ac.uk
**