= :
So by Nagao's theorem, as explained above, and generate a projectiv*
*e free
product in GL2(K[t]). Now extend x0 and y0 to (p - 1) x (p - 1) matrices x and y
by adding the other eigenvalues of a or S along the diagonal. Finally, K[t] ca*
*n be
embedded in a p-adic completion of K by sending t to some transcendental over K.
7
4. Generation by torsion elements
The isomorphism class of a crossed product algebra such as D depends only on *
*the
p-adic valuation of Sp-1. We could instead have used a generator T with T p-1= *
*-p.
p-1_
(In fact this new algebra is isomorphic to D by sending T to ! = i 2S. It will*
* be
convenient in this section to use this description of D and the corresponding m*
*ap
ae : Sl ! F+pp-1. (If a = 1 + a1T; a1 2 O, then aeT(a) = a1 mod T .) Note that
1-p_ ! oe
aeT(a) = ae(a)i 2 and aeT(a ) = aeT(a) .
In order to prove the Main Theorem we need to showpthat-we1can find a p-th
root of unity a 2 Sl of such that a; a!; : :;:a! generate Sl=Sl*, i.e. such*
* that
p-1
aeT(a); ::; aeT(a! ) form a basis for Fpp-1over Fp. In other words, in view o*
*f the the
formula aeT(a!) = aeT(a)oeabove, aeT(a) and its conjugates should form a normal*
* basis
of Fpp-1=Fp.
Lemma 4.1. The image under aeT : Sl ! Fpp-1of the p-th roots of unity consist*
*s of
the elements of the form up-1. (These are the elements of norm 1.)
Proof.As mentioned in Section 2, Qp(T ) contains a pth root of unity z. Now aeT*
*(z) 2
Fp, and since aeT(zr) = raeT(z), we may assume that aeT(z) = 1. It is a consequ*
*ence of
the Skolem-Noether Theorem that any p-th root of unity in D is of the form zt; *
*t 2 Dx .
We may write t = T ru(1 + vT ); u 2 W \ O, v 2 O. Then aeT(zt) = up-1 mod p. _*
*_|_ |
Lemma 4.2. The extension Fpp-1=Fp has a normal basis of the the form up-1.
Proof.The characteristic polynomial of oe factorizes over Fp, so Fpp-1decomposes
over Fp into eigenspaces for oe. If r is a generator of the multiplicative grou*
*p Fxp, then
the eigenvectors are er; e2r; ::; ep-1rof eigenvalues r; r2; ::; rp-1 respectiv*
*ely, and they
form a basis. Note that oeer = ep-1rby definition of oe, and oeer = rer by defi*
*nition
of er, so ep-1r= r. The group algebra Fp contains idempotents which project
p-1
ontoPeach eigenspace, thus x; xoe; ::; xoe form a basis if and only if, when w*
*e write
x = p-1i=1ieir, no i is 0.
Now ! !
p - 1 p - 1 2 p-1
(1 + er)p-1 = 1 + er + er + :: + er :
1 2
Since ep-1r= r, this shows that (1 + er)p-1 and its conjugates form a normal ba*
*sis
provided that 1 + r 6= 0 ie. p 6= 3. If p = 3 there are four elements of norm 1*
*, so they
are not all contained in F3. __|_ |
Remark 4.3. If we had used S instead of T , then the pth roots of unity would *
*have
had image under ae of norm -1.
By Corollary 3.8, there is anpx-21Sl such that ff = x-1ax gives rise to a fre*
*e product
G = * * : :*: and ae(ff) = ae(a). The inclusion of G in Sl *
*induces an
8 V. GORBOUNOV, M. MAHOWALD, AND P. SYMONDS
isomorphism G=G* ~=Sl=Sl*. By proposition 2.1 this implies that G is dense in S*
*l.
This proves 2.2 completely.
Also, since the cohomology of G is generated by H 1and its Bocksteins, we have
the following easy proposition about the cohomology on continuous cochains of S*
*l.
Proposition 4.4. There is an epimorphism
H *c(Sl; Fp) ! H*(G; Fp):
5. The proof of the main theorem for p=3
We give now an elementary proof of the main theorem for the primepp_= 3. We
will be able to give precise formulas for the embedding of Q3 [ 31] into Sl whi*
*ch
we believe will be useful for computing the cohomology of Sl. In this case S2 *
*is a
subgroup of units of the maximal order of the division algebra (W=Q3 ; oe; 3), *
*where
oe, the Frobenius, is the generator of Gal(W=Q3 ).
Since W is an unramified extension of degree 2 of Q3 it can be constructed by
adjoining to Q3 a fourth root of unity, i. Since Q3 contains a square root of -*
*2, we
can set
p ___(i + 1)
i = -2 ______;
2
and it is easy to check that i is an eighth root of unity such that i2 = -i and
p ___(1 - i)
iS = i3 = -2 ______:
2
Proposition 5.1. In the above notation a cube root of unity z 2 D is given by t*
*he
formula
1 p___(i + 1) 1 i
z = -__+ -2 ______ S = -__+ __S:
2 4 2 2
Proof. This is a simple computation in number theory.pAs_it was mentioned abo*
*ve
S2 contains a square root of -3, e.g. ! = iS and Q3[ -3 ] = Q3[z]. Therefore
(5.2) z = a + b!
where a; b 2 Q3. Using the fact that z3 = 1 we immediately obtain that
a = -1_2, b = +_1_2
p __
Remark 5.3. The above method of constructing an embedding of Qp [ p1] into Sp-1
theoretically works for any prime p, but the coefficients in an expansion simil*
*ar to
5.2 are not rational numbers.
Since Q3 does not contain a primitive cube root of unity, the norm of z is eq*
*ual to
1 and we conclude that z 2 Sl.
Theorem 5.4. The subgroup G of Sl generated by z and zS is isomorphic to a free
product of two copies of Z=3. Moreover G is dense in Sl.
9
Proof.We will use the matrix representation of this cyclic algebra which was d*
*e-
scribed above. z and zS are two matrices of order three with entries in W:
0 1 p ___(i-1)1 0 1 p ___(i+1)1
-_2 -3 -2 ____4 -_2 -3 -2 ____4
z = B@p___ CA; zS = B@p___ CA:
-2 (i+1)_4 -1_2 -2 (i-1)_4 -1_2
Because of the form of the entries, we see that we can also consider these ma*
*trices
as elements of SL2(C). The subgroup they generate in SL2(C) is isomorphic to the
subgroup they generate in SL2(W). Conjugating the two generators by the matrix
!
1 p0__
0 3
we can get them to lie in SU(2). Using now the homomorphism SU(2) ! SO(3)
we reduce ourselves to two matrices in SO(3). Using the formulas in [3] one se*
*es
that these are two rotations by 120O about two perpendicular axes. They generate
a subgroup of SO(3) which is isomorphic to Z=3 * Z=3 due to the following theor*
*em
proved by F. Hausdorff [4].
Theorem 5.5. Consider a half turn rotation g (so g2 = 1) and a one third rotat*
*ion
h (so h3 = 1) of R3, the angle between axes being ss=4. Then g and h generate *
*in
SO(3) a group isomorphic to Z=2 * Z=3.
The subgroup we are interested in is the one generated by h and hg. So the
statement of 2.2 is an easy corollary of the above theorem.
To prove that G is dense in Sl we will prove that z and zS are topological ge*
*nerators
of Sl. According to 2.1 we need to show that ae(z) and ae(zS) are generators of
Sl=Sl* ~=F9+:
Direct computation using the formula from 5.1 shows that
ae(z) = i
ae(zS) = i3
Since i and i3 generate the additive group of F9 the theorem is proven. __|_ |
10 V. GORBOUNOV, M. MAHOWALD, AND P. SYMONDS
6.Applications
We start by describing a certain E1 spectrum En. The ring of its coefficient*
*s is
isomorphic to
WFpn[[u1; : :u:n-1]][u; u-1]:
One can obtain such a spectrum by completing the Ravenel - Wilson spectrum E(n)
in the appropriate way. M. Hopkins and H. Miller proved that the Morava stabili*
*zer
group Sn acts by E1 maps on En. This implies that for any discrete subgroup H
of Sn one can form a spectrum EHn of homotopy fixed points of the action of H on
En. This spectrum gives an approximation of the Bousfield localization of the s*
*phere
spectrum with respect to Morava K-theory K(n). The latter is denoted by LK(n)S0.
The computation of the homotopy groups of the spectra LK(n)S0 can be done usi*
*ng
the homotopy fixed point spectral sequence which starts with the continuous Gal*
*ois
cohomology of Sn, H*c(Sn; En;*)Gal. The spectra EHn for different subgroups H o*
*f Sn
approximate LK(n)S0 and therefore can be useful as a substitution for LK(n)S0. *
*The
homotopy groups of EHn can be computed in a similar manner to those of LK(n)S0
using the homotopy fixed point spectral sequence which starts with H*(H; En;*).
Now let p be an odd prime and n = p - 1. The authors of [7] consider the maxi*
*mal
finite subgroup H of Sp-1, which is unique up to isomorphism, and is isomorphic
in this case to Z=pZ=(p - 1)2 [5] . They call the spectrum EHp-1the "higher re*
*al
K-theory" and denote it by EOp-1. The reason for this is that, at the prime 2, *
*S1
is just the group of units in Z2 and so its maximal finite subgroup is Z=2. The*
* KO
spectrum is the homotopy fixed point spectrum of this Z=2 action on the spectrum
of complex K-theory completed at 2, which is E1.
Now we are ready to outline the applications of the main theorem.
7. The spectrum EL for p = 3
We deal here with the case p = 3 and n = 2. The maximal finite subgroup H
of S2 in this case is isomorphic to Z=3Z=4. It is not hard to see that this gr*
*oup is
generated by z and i2, for example. The infinite subgroup G of Sl defines an in*
*finite
subgroup L of S2 which we will describe now. Note that S conjugates z in the sa*
*me
way as the 8-th root of unity i. Indeed
1 i S 1 i3
zS = (-__+ __S) = -__+ __S
2 2 2 2
On the other hand
1 i 1 i11 1 i3
zi = i7(-__+ __S)i = -__+ ___S = -__+ __S
2 2 2 2 2 2
2 i2 2
However zS = z and z = z .
11
We define the group L to be the semi-direct product
(Z=3 * Z=3)Z=8
where Z=8 is the group generated by i and the free product is generated by z and
zi. So we have the following sequence of subgroups of S2:
S2 > (Z=3 * Z=3)Z=8 > Z=3Z=4;
and therefore the map of spectra EL ! EH . As we pointed out in the introduction
the spectrum EL should be a better approximation to the spectrum LK(2)S0.
8.Cohomology
Let us describe the computation of H*(S02; F3) from [13]. First we need some
background. There is a profinite Hopf algebra S(n) such that
S(n)* Fpn~= Fpn[S0n]:
It was shown in [13] that
Ext*S(n)(Fp) Fpn~= H*c(S0n; Fpn):
Let p = 3, n = 2 and consider
R = E(h10; h11) P (b10; b11)=I
where
(8.1) I = (h10h11; b210+ b211; h10b10- h11b11; h11b10+ h10b11):
Theorem 8.2. [13] H*(S(2); F3) is isomorphic as an algebra to E(i1; i2) R, wh*
*ere
R is as defined above, i1 is an element of H1(S(2); F3) determined by the deter*
*minant
map, and i2 is a certain four-fold Massey product which belongs to H2(S(2); F3).
Here is the interpretation of a part of this computation in terms of the infi*
*nite
subgroups described above.
Proposition 8.3. After extending the coefficients to F9, R becomes isomorphic to
H*(Z=3 * Z=3; F9).
Proof.Let i 2 F9 be an element such that i2 = -1. Consider the following basis *
*of
R over F9:
h1 = h10+ ih11; h2 = h10- ih11; b1 = b10- ib11; b2 = b10+ ib11:
Direct computation using 8.1 shows that
R ~=E(h1) P (b1) E(h2) P (b2) ~=H*(Z=3 * Z=3; F9);
except in degree 0. __|_ |
12 V. GORBOUNOV, M. MAHOWALD, AND P. SYMONDS
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Mathematics Department, Northwestern University, Evanston IL 60208
E-mail address: vgorb@math.nwu.edu, mark@math.nwu.edu, symonds@math.nwu.edu