Comment. Math. Helvetici $8 (1983) 291-311 0010-2571/83/002291-21501.50 + 0.20/0 �9 1983 Birkh~iuser Verlag, Basel

D ( Z ' n ' ) + and the Artin cokernel

R O B E R T O L I V E R

Let p be an odd prime. If -rr is a p-group and ~ Q 1 r is a maximal order containing 2~r, then we set

D(ZIr) = Ker [K0(2~1r) ~ K0(~R)]

as usual. Since D(7/Tr) is a p-group [2], the involution g ~-* g-1 induces a natural splitting

D(7/Tr) = D(2~Tr) + ~ D(gr r ) - ;

where D(7/1r) § and D(7/1r)- are the +1 and - 1 eigenspaces, respectively. The involution can be extended to a linear action of ~:* on D(2~1r); and this induces further eigenspace decompositions

(p--3)/2 (p--3)/2

D(77~r) += ~'. 2iD(7/-tr) and D(7/~r)-= ~ 2~+ZD(7/'tr). i =O i =0

The groups D(7/1r)- and 2~+ID(ZIr) have been studied extensively in [3] and [14]; and the order of D(2~r)- for abelian ~r is computed in [3]. In this paper, attention is focused on ~ This is the part of D(2~Tr) § which is independent of number theoretic properties of p; and in fact, ~ D(Z~r) + whenever p is

regular. For any finite group zr, we define the "Art in cokernel" Ao(cr) to be the group

Ao(1r) -- Coker [Ind: ~ {Ro(tr) :or c_ -tr, tr cyclic}--~ Ro(~r)],

where Ro(~r) denotes the rational representation ring. The main result here is that ~ for any p-group ~r (p odd). Among other consequences, this gives new insight into Martin Taylor 's result [13] that the image T(771r) of the Swan homomorphism has order equal to the Artin exponent of "rr : T(Z~r) corres- ponds under this isomorphism to multiples of the identity in Ro(1r).

291

292 R O B E R T O L I V E R

We end by deriving a formula for I~ fact, a formula for the order of Ac~(cr) for arbitrary finite 7r. Also, for the sake of completeness, the formula for ID(Z~')-I in [3] is generalized to cover arbitrary p-groups; thus giving a complete calculation of ID(7/Tr)l when -r: is a p-group and p any odd regular prime.

For n = 1, Qr always denotes the field generated by the n-th roots of unity (and similarly for ;7/~,, Z'o~, etc.) Also, q~(n) will always mean the Euler q~- function.

Throughout the paper, unless otherwise stated, p will be a fixed odd prime. We start with the following well known description of Q~- when 7r is a p-group.

P R O P O S I T I O N 1. Let 1r be a p-group. Then Qrr is a product of matrix rings over [ields Q~p, for various s >- O. Furthermore, for each s >= O, the number of simple summands isomorphic to matrix rings over Q~p, is equal to the number o[ conjugacy classes of cyclic subgroups or ~_ 7r such that

Io'/[or, N(or)]l = Ior[" Iz(or)lllN(or)l = p'.

Proof. That Q1r is a product of matrix algebras over the QCp, is shown (though not explicitly stated) by Roquet te in [11]. In Section 2 of [11] he shows that the division algebra for any irreducible representat ion M of zr is isomorphic to the division algebra of a primitive faithful representation of some subquotient of -a-; and in Section 3 he shows that the only p-groups with primitive faithful represen- tations are the cyclic groups.

Now, for all s >= 0, let w~ be the number of simple summands of Q~- which are matrix algebras over Q~p, ; and let v~ be the number of conjugacy classes of cyclic subgroups or_ 7r with Io-/[o -, N(or)]l = pL Let p " = I~1. We say that two elements g, h ~ ~r are Q~p--conjugate (any m >_- 0) if g is conjugate to h" for some

a ~ Gal (OCpdQCpm) ~_ Gal (QGdT/) = (Z/p")*.

In other words, a = 1 ( m o d p " ) if m >- 1, or p X a if m = 0 . For each cyclic or ~ 7r, the number of conjugacy classes of generators of or is

just q~(lor/[or, N(or)][). So for any m >= 0, the number of Q~pm-conjugacy classes in 7r is

vs" min {q~(p"~), q~(p~)}. (1) s ~ - 0

On the other hand

rk Ko(Qs = ~ w~" min {~(p~), ~o(p~)}; (2) s ~ -O

D(7/w) + and the Artin cokernel 293

and by Theorem 21.5 in [1], the numbers in (1) and (2) are equal for all m. So v~=w~ for alls>=0. []

Thus, if ~r is a p-group and L = Z(Qw) is the center, then the reduced norm

~' : Kl(~pTr) ~ , (/~p)*

is just the product of the determinant maps for the simple summands of QpTr. If ~ff~Q~r and 0~_L are maximal orders, then v induces an isomorphism of Kl(ff)~o) with (alp)*: by [9, Theorem 21.6], ~)2p is a product of matrix algebras over the components of ~p. In particular, K l (~p ) can be regarded as a subgroup of Kl(Qp~r). We let K~(~9~) and K~(~_oTr) denote the images of Kl(~l~) and Kl(2~pTr) in Kl(QpTr); and let K~(~0~) ̂ denote the p-adic closure of K~(~0~).

We will use the following description of D(;Y'rr), based on the formulas in [4].

PROPOSITION 2. Let 7r be a p-group, and let ~fRc_~_w be a max imal order in

QTr. Then there is an isomorphism

p ^

D(2~Tr) --~ Cokcr [Kl(7]pTr) ~ Kl(ff~p)/K'~(~,~) ^]

which is natural in or.

Proof. Let L ~_Qqr be its center, and let ~7 ~ L be its maximal order. Since 77q[Tr]=~]~2, for all primes q ~ p [9, Theorem 41.1], Theorems 1 and 2 in [4] reduce to the formula

D(7].tr) ~ K,(~o)/K'~(~)'O ' ^ �9 KI(Y-~r).

p ^

Furthermore, Kl(ff)~p)/Kl(7/o~r) is finite [12, Proposition 8.15]; and so K~(~I~) can be replaced by its p-adic closure. []

Now let

K : ~:* ~ ( 2 p ) *

denote the inclusion into the group of ( p - D - s t roots of unity (with K(a)= a (mod p) for a ~ ~:*). If ~r is an abelian p-group, then the group homomorphisms g--* g'(") for a ~ : * and g~-rr induce actions of F* on Ki(2~Tr), Ki(2p*r), D(?Trr), etc. The next problem is to find a natural way to do this when ~r is non-abelian.

294 ROBERT OLIVER

Let ,r be an arbi t rary p-group. Then by Proposi t ion 1, the center Z(Qr is a product of fields Q~p, for various s > = 0. T h e natural embedding g:*c__ Gal (Q~p./Q) (where a ~ F* sends ~p. to r induces actions of I:* on the centers Z(QIr ) and Z(Qprr); and hence on Kl(Qo~r). If ot : 1r ~ It' is a homomorph i sm of p-groups, then

or. : Kl(~p,i'r ) ~ Kx(~p,tr' )

is a p roduc t of norm, includion, and diagonal maps be tween the groups of units of the field componen ts of the centers; and is hence B:*-linear.

P R O P O S I T I O N 3. Le t ~r be a p-group. Then

(i) K~(~p-tr) is an ~:*-invariant subgroup of Kl(~p'tl'). (ii) For any 0 <-_ t <= p - 2, set

l t ^ t ^ . KI(Zolr) ={x E Kl(Zp~r)~p~. z , ( x ) = x (a ) ' �9 x for all a ~F*}

(here ra denotes the action of a ~D:*o). Then for t ~ 1, ' ' ^ Kl(7/p'rr) is generated by

induction from cyclic subgroups.

Proof. By [8, T h e o r e m 2], there is an exact sequence

^ ~ i ^ , . i T a b 0 ~ (Xg : X ~ 7/p, g ~ It) c_~ KI(Zp~') _~r I(~plr) --~ ~ 0,

natural in ~r, where

I(~_pcr) = Ker [e : 2~01r --~ Zp]/(gxg -1 - x : g e ~r, x ~ Zp~')

and

e(~. Xigi) = ~ X,; t o ( ~ X,g,) = l'I gt'.

For a ~F*, let ~ be the action of a on I(Zplr) given by: § X~g~) = ~ A~g~ ~ . This clearly leaves Ker ( to)= Im (F) invariant. By the definit ion of F in [8], F is F*-l inear when ~ is abelian.

Now set

X(~r) = Im find: )-'. {K~(Zptr) : tr ~ m tr cyclic} ~ K~(2olr)]

and

Y(~r) = F-l(lI(~_p~r)) t"l Ker [ e , : K~(Z~,n') --~ Z*]. (1)

D(77r § and the Artin cokernel 295

Since I(~plr) is generated by cyclic induction, Ker ( F ) c X(Tr), and

'I(~p~r) = 'Ker (oJ) = ' Im (F) (2)

for t ~ 1 (0_-< t -< p - 2); it follows that

K~(~prr) = X(er) + Y(w). (3)

By naturality, X(1r) is an ~:*-invariant subgroup of Kl(~plr), and I'lX(~r) is ~:*-linear. It follows that for n large,

Y ( q l ' ) p" ~--- 1X("JT) . (4)

Furthermore,

tots (Kl(~p~r))(o~__ 1Kl(~o~r):

the torsion in Kl(~p~r) comes from roots of unity in the center. It follows by (4) that Y(-tr)c_lKl(~pcr); and hence by (3) that K~(~pTr) is ~:*p-invariant. Further- more, by (1), this shows that F is ~:*o-linear. Since I(2~pZr) is generated by cyclic induction, 'K~(2~pTr) is generated by cyclic induction for tq: 1 by (2). []

Propositions 2 and 3 now imply the existence of natural actions of D:~ on D(7]1r):

PROPOSITION 4. For any p-group 7r, there is a natural linear action of ~:* on

D(~_Tr) such that the isomorphism of Proposition 2 is g:*-linear. []

In particular, for any p-group 1r and 0-< t _-< p - 2, set

'D(7] Ir) = {x ~ D(7/~r) : ~'a(x) = K(a)' �9 x for all a ~ I:*}.

Since D(7/Tr) is a p-group [2], and p A" ID:*~I,

p--2 (p--3)/2 D(7]~r)= ~ 'D(7/1r) and D(7/~r) += ~ 2~D(7/zr).

~=0 i = 0

Here, D(7/~r) § is the group of elements invariant under the involution "r-l; induced by complex conjugation on Z(QTr).

If p is regular, then results of Iwasawa (see, fx, [7, Theorem 7.5.2]) show that

296 ROBERT OLIVER

for any s ->__ 0 and any even 0 < t <_-- p - 3,

' [ (Z~, ) ^] = ' (2~ . ) * .

So by Proposi t ion 1, if ~ I ~ 7/r is a maximal order in Qrr, then tK~(RR) ̂ = ' K l ( ~ p ) ; and hence 'D(~rr) - - -0 for such t. In o ther words:

P R O P O S I T I O N 5. I f p is an odd regular prime and rr is a p-group, then

D(-Zrr) + = ~ []

In order to study the groups ~ we must first describe

~ ^]

for any s -----_ 0. The following result must be well known, but we have been unable

to find a reference.

P R O P O S I T I O N 6. For any s >- O,

~ - 2~.

Proof. This is clear if s = 0. So fix s > 0, let K c Qs be the fixed subfield of F*, and let R c K be the ring of integers. Then F = Gal (K/Q) is cyclic of order

p,-1. Set s = s and let y ~ F be the genera tor : y(s = ~p+l.

Le t

= z = ( I - ~ ' " < ~ _oR

be the pr ime ideal over p. Set

U ' = Ker I N : ( /~)* --+ (2p)*],

where N is the no rm o f / ~ p / ~ 0. Note tha t U ' c l+ta/~p. Fix u e U' . By Hi lber t ' s T h e o r e m 90, there is x ~ K* such that u = y(x ) / (x ) .

Write x = z~v, where v e (/~p)* and z is the e lement defined above. Then

~'7T 1 / 1 - - y(p+l)K(a) '~i

= o=lli t )

D(2~w) + and the Artin eokernel 297

Furthermore, N ( v ) ~ Kff*)x (1+p~2o) (the norm group has index p~-1 by local class field theory). So there exists w ~ (20)* such that N ( w ) = w p'-' = N(v) . Then

N ( v w -1) = 1 and " ~ ( 1 ) W - - 1 ) / ( ' I ) W - 1 ) = " y ( l ) ) / V ~- X " ('~/(zi)/zi) -1.

In other words,

U'={ 'y(v) /v :v ~ U'}- (R* N (1 +p)). (1)

But U' is a 2o[F]-module, and the closure of R * N ( I + p ) is a 2o[F ]- submodule. Since 2[F] is a local ring with maximal ideal generated by p and ~ - 1, no proper submodule of U' can generate

U 7 ( ' y ( v ) / v , v ~ : v e U').

So by (1), R * N ( I + p ) is dense in U ' = K e r ( N ) ; and

~ = [(-~,)*l(R*)^](p)~ (Im (N))(p) = 2 o. []

By Proposition 6, if 7r is a p-group and ~ Q ~ " is a maximal order, then ~ is a sum of one copy of 2 o for each irreducible Q1r-module; and is thus (abstractly, at least) isomorphic to 2 0 | The key remaining step is to construct a natural isomorphism between these groups; once this is done the isomorphism between ~ and the Artin cokernel will follow easily.

We temporarily allow p to be an arbitrary prime (possibly p = 2). If A is a ~o-algebra, and V is an A-modu le with d in~ , (V)<o% let det (u, V), for u ~A, denote the determinant over ~o of u : V ~ V. Define

L : (20)* ~ 2 0

by setting L ( u ) = 1/plog(u/K(fO) for u~(;~p)* and f ieF* its reduction m o d p

(note that u/K(fi)e 1+02o). Now assume A is a finite dimensional semisimple ~p-algebra, and let 91 c A

be any order. Let V1 . . . . Vk be the distinct irreducible A-modules , and set

r~ = [Endh (Vi):Qp].

Define a homomorphism

= &a : K1(91) --> ~p ~ z K o ( A )

2 9 8 ROBERT OLIVER

by setting, for any matrix u ~ GL,(92),

~([u]) = ~. 1 L(det (u, V~i)) �9 [Vi]. i = 1 rli

PROPOSITION 7. For any prime p, the maps ~ are natural with respect to homomorphisms between orders in semisimple Qp-algebras.

Proof. We must show, for any homomorphism a:A- -~ B, orders 92_ A and _c B such that a (92) __ ~ , and u e 92*, that

a.(&~(u)) - 8~(ot (u)) e O,|

Let V1 . . . . . V, be the irreducible A-modules, and Wt . . . . . Wt the irreducible B-modules. Define aij, b~/~ 7/by setting

i = 1 i = l

(where tx.(V~)= B | Vi, and ol*(W~) is W i regarded as an A-module). We also set

m, = [EndA (Vi):0p], nj = [Enda (Wi):~o],

and write Li. = L(det (u, Vi)) for short. Then

(' ) a . ( ~ ( u ) ) -- ~ . ~ (~/m,)[Vi] = ~ (aJ_Jm,)[Wj] i = 1 i , j

and

~ ( ~ ( u ) ) = ~ n~-~L(det (u, a*(Wj)))[Wj] = ~. (b~i~/r~)[Wj]. i = 1 id

It remains to check that (bjn~)= (a~/m~) for all i, j. But

dim HomA (V. Wi)= mibij, dim HomB ( a . V , W i) = nja~;

and these two dimensions are equal by [1, Theorem 2.19]. []

We now again restrict to the case where p is odd.

D(Z~r) + and the Artin cokernel 299

P R O P O S I T I O N 8. Let ~r be a p-group, let ~Rc_Q~r be any maximal order, and set 87 = 8r Then

Im [8,~ : K , ( ~ p ) ~ ~p OKo(~pw)] = ~_p OKo(~pw) ;

and 8~ induces an isomorphism

8' = 8~ : ~ ~ , ~_p 0 Ko(Op'tr) ~ ~_p 0 Ro( ~).

Proof. Using 'Proposi t ion 1, it will suffice to show that whenever A --M,(Q~p.) and ~R_ A is a maximal order , then 8 = 8ao induces an isomorphism

8' : ~ ~- , ~_p t~ Ko(A ).

By [9, T h e o r e m 21.6], we may assume that ~R = Mr(T]s Le t V ~ ( ~ o ~ r r be the irreducible fi~o-representation. For any u ~ l + J ( ~ 2 v )

(where J(~I~,) is the Jacobson radical),

8 ( u ) = ( ~ L ( d e t ( u , V ) ) - [ V ]

1 - p~(pS--~) log (Nopc;/~v(det~c~ (u))- [V].

Fur the rmore , by local class field theory,

N o det (1 + J ( ~ p ) ) = 1 + p~2~ o

(or 1 + p2p if s = 0). Since log (1 + p~2~p) = p~2~p for s - 1, we have

8(K~(?fJ~r,)(p ) = 8(1 + J ( ~ p ) ) = ~p- [V] = 2p |

If u is a global unit, then N(de t (u)) = • and so 8(u) = 0. Fur thermore , 8 is F~-linear when Ko(A) is given the trivial action; and so 8 induces a surjection

8' : ~ K ~ ( ~ p ) / K'~ (~IR)^ ](p ) ~ ~_p ~ Ko( A ) ~- ~-p.

But the two groups are isomorphic by Proposi t ion 6, and so 8' is an

isomorphism. [ ]

300 R O B E R T O L I V E R

We can now prove the main result. Recall that the Artin cokernel Ao('rr) is defined by

Ao(~) = Coker l ind: ~. {/~a(tr) : tr _ It, tr cyclic} --~ Ro(~r)].

T H E O R E M 9. For any p-group 1r (p-odd), 8' induces an isomorphism

8~':~ -~ ~Ao('n').

Proof. Let C be the set of cyclic subgroups of ~r. Propositions 2, 7, and 8 combine to give the following commutative diagram with exact rows:

Kl(Zpo') , ~ ~p~Ro(c r ) z0 , ~ OD(Zcr ) ~0

0K[(2~p~r) . . , ~p@Ra(~r ) o , OD(Z~r ) , 0

Here 11, /2, and 13 are the induction maps; and 0r and ~1,~ are the composites (~IR ~_ 07r a maximal order):

0~, :)_p @tLa(~r) c~')-I , ~ , ~

(the second map being the map of Proposition 2), and

By Proposition 3(ii), 11 is onto. Assume that ~ = 0 for any cyclic p-group tr. It then follows by diagram chasing that

~ --- Coker (/2) ~ Ao(rt).

(AaOr) is a p-group by the Artin induction theorem: see, for example, [1, Theorem 15.4]).

It remains to check that ~ = 0 for cyclic or: This is implicit in [6], [5], and [15]; but doesn' t seem to be stated explicitly. If Itrl-<__p, then D(2~tr)= 0 by [10, Theorem 6.24].

D(?71r) + and the Artin cokernel 301

So assume lal = p~ for n >_- 2. Let p __q ~r be the order p subgroup, and assume inductively that ~ 0. There is a commuta t ive diagram

2o| ~" , 2 o | J" ,2o|

0 = ~ ,~ , ~ = O,

where i , and j , are induced by inclusion and project ion. Since Kl(2p~r) maps

on to Kt(2~p[tr'0]),

j . ( K e r (0,)) = Ker (0,~/o) = 2~ o @RQ(tr/o).

In o ther words, 0~ ] Ker ( j , ) is onto. Fur thermore , Ker ( j , )~_Im (i ,) : if VmQffp. and W~OCp are the faithful irreducible •cr- and O0-representa t ions , then [V] generates Ker (j ,) , and V = Ind~ (IV). We thus get that 0 r (i ,) is onto, but 0r o i , = 0, and so ~ = 0.

One easy consequence of T h e o r e m 9 is an al ternate proof, for odd p-groups, of Mart in Taylor ' s theorem [13] involving the image T(7/rr) of the Swan homomorphism. T(7/~r) is the group of all e lements

[Z, n] - [7/'rr] e D(7/rr),

for (n, I~rl) = 1, where [~, n] is the project ive module

So if 7r is a p-group and ~/R___7/1r is a maximal order in QTr, then [,~, n]-[771r] corresponds, under the identification in Proposi t ion 2, to the e lement of Kl(~2p) which is n ~ (2~0)* at the identi ty componen t and 1 at all o ther components (in particular, T(7/Tr) _ ~ The isomorphism of Theo rem 9 thus sends T(PT~-) to the group of multiples of the identi ty in

Ro(Tr)/~. {Ind:(Ro(o')) : o" __q ~" cyclic}.

In o ther words:

T H E O R E M 10. (M. Taylor [13]) For any p-group w, T(7/w) is cyclic of order equal to the Artin exponent of ~r. []

3 0 2 ROBERT OLIVER

The computation of I~ can now be carried out, using the same idea as for the calculation in [3]: that of comparing discriminants. We first consider the Artin cokernel of an arbitrary finite group.

T H E O R E M 11. Let ~r be any finite group, and write

k

o ~ I ] M,,(D,), i = 1

where the Di are division algebras. Let X be a set of conjugacy class representatives for cyclic subgroups o" c_ It. Then

IAo( )l --- [QOx (I'll) :Q])]l/2 Proof. For convenience, set

G = Y. Ind2 (Ro(o')) _ Ro(~r). o-EX

Then

IRoOr)/GI = [d(G)/d(Ro('rr))]l/2; (1)

where d ( - ) denotes discriminant with respect to the usual inner product

1 (IV], [W])=r-~i ~.. Xv(g)xw(g).

gE'a"

For each i, let Vi denote the irreducible representation of M,,(D~); then

So

([V~]. [Vj]) = dirno (Homo.. (V, V/)) = 0

= [I), :o ]

k

d(ro(,.,.,-)) = l - I [D, :0]. i = 1

To compute d(G). consider first the set

S ='{[Q(Tr/cr)] : o, e X}___ Ro(w);

if i=#j

if i=j.

(2)

D(;~r) + and the Artin cokernel 303

where Q(Tr/o-) denotes the permuta t ion representat ion with Q-basis 1r/o-. These elements genera te G : since

Q(~r/cr) = Ind~ (Q) c G,

and Ro(o-) (o-e X) is genera ted by the elements

{[Q(o-/1-)] = Ind , ~ ([Q]) : ~- c ~r}.

Also, rk (Ro(Tr))= Ixl (see [1, T h e o r e m 21.5]); and so S is a basis for G. It follows that

d(G) = det (M) (3)

where M = (M~) . . . . x is the matrix defined by

M ~ = <[Q(rr/c,)], [O(~r/T)]>.

For o- ~ X, let X~, deno te the character of Q(~-/~r). For any x e ~r,

X,,(x) = ~-~" #{g e "n" : xgo" = go'} = �9 #{g e .tr : x e gcrg-1}.

Hence , for o', ~- e X,

M~ = 1 ~, X,,(x)x,(x) = 1 . 1 ~ Igo'g_ ! Cl h~'h-l[ I~l x~,, Io'l " b'l I~1 . . ,~ , ,

1 - Io'I 7 I,I y~ Io" n g~-g-11. ( 4 )

gE' l r

To simplify what follows, define, for n---1 and m >= 1,

,Ore(n) = n - Y. ~o(d). d i n

d < rrt

N o t e in part icular that q~l(n)=n, ,.(n)=q~(n), and q~,.(n)=O for m>n. Let N = m a x {Io'I : ~ s x } ; and define, for 1 <_- m <_- N :

Xm={o 'eX: lo ' l=m} Y~={o-ex:lo-l>=m} = U Xi. i > r a

304 ROBERT OLIVER

= r a ) For all 0 < m =< N, define a matrix M ~ = ( M ~ ) . . . . v~, by setting

1 ~ ) - M - b l y: q~m(l~ g'rg-l})"

g ~ ' n "

In particular, M ~ ) = M. Fix l < - - m < - N . For or, " reX, , (i.e., Icrl=[vl = m),

M(~ )= m -2 ~ w,,,(Iong'rg-'l) g E "i'r

= ( q ~ ( m ) / m 2 ) �9 # { g ~ "rr : o" = g1"g-'} = 0

_ ,p (k,I) Iot

- - - " I N ( o ' ) / o ' l

if ~ r~ ' r

if t r = ?

In particular,

o'~X N (5)

If 1 <_- m < N and o-, -r e Y,,+I (i.e., Icrl, I~[ > m), consider the entries /W~ ) for p e X , , ([Pl = m). By definition, M ~ ) = 0 unless [cr fq gpg-1]->_ m for some g; i.e., unless g P g - ' ~ or. If m 4" I~r[, then these /W~ ) all vanish; and also M ~ )= M ~ +') (q~,,(n) = , , ,+ l (n ) if m ~c n). ff m 110"l, let p ~ X,, be the unique e lement conjugate to a subgroup of o'; then

M(:~) m) m) 1 [g~.n- (~rrl (I 0" 0 gTg--ll) - X (~Orrl ([P N gTg--l[)]

1 - Io-I -b-I Z w,,+l(IO- n gvg-ll) = M ~ ). g~Ir

In o ther words, for all o', ,r ~ Ym+l,

M~._~+" = M ~ ) - y. (M~VM(T) �9 M(o~+'; pEXm

a n d / W "+1) is obta ined f rom M ~") by e lementary opera t ions which eliminate all

entries MC~ ) for p e X m , r e Y , , + , . It follows that

det (M ~")) = det ( M ~ + ' ) ) �9 I ' l M(~ )

�9 r I p ( M ) IN(, , ) / , , I ] = det (M~"+t') �9 o~x. [ - - ~ "

D(2~.n-) + and the Artin cokernel 305

Combining this with (5) gives

d(G) = det (M) ~ det (M ('~) = I-I [~(l(rl)" IN(o-)/orl].

Finally, combined with (1) and (2), this gives the desired formula for [Ro(~r)/G[. []

When 7r is a p-group (recall that p is always odd), the above formula can be reformulated solely in terms of cyclic subgroups:

T H E O R E M 12. Let ~r be a p-group, and let X be a set of conjugacy class representatives for cyclic subgroups cr c It. Then

I- n I ~ = lff•x ~ J "

QTr =rl i=l Mr (Di), when the Di are fields, and Proof. By Proposition 1, _ k ,

r I [oi :Q] = r I ~(1~1. Iz(,~)IIIN(<~)I) i - - 1 o - ~ X

-- I-I [<p(Io-I)- IZ(,~)IIIN(,~)I]. o ' E X

The result now follows by substitution into the formula of Theorem 11. []

Finally, for the sake of completeness, we extend FrShlich's formula for ID(2~r)-[ in [3] to arbitrary (not necessarily abelian) p-groups 7r. For any such 7r,

will denote the group ring modulo conjugation:

(~.~ -- ( ~ o ~ l ( x - g x g - l : x ~ ~ r , g ~ ~ ) = ( ~ l ( x y - y x : x, y ~ ~0~).

This can be regarded as the ~,-vector space with basis the set of conjugacy classes in ~r. Let 7/orrc_Qo~r be the image of 2o~r; and let ~___QoTr denote the image of any maximal order ~[E___ ~o~r.

If F is any field and r >= 1, it is easy to check that

(xy - yx : x, y e M~(F)) = Ker [tr: M,(F) ~ F].

Thus, if ~ o ~ " - r [ M,,(F~), then ~ 1 r = r l Fi, and the projection Qo~r ~ ~p~" is the product of the trace maps. If R~ E F~ is the ring of integers, then any maximal

3 0 6 ROBERT OLIVER

order ~ i ~_ M,,(Fi) is conjugate to Mr,(Ri) [9, Theorem 21.6]; and so tr (~Ri) = R~. In particular, ~ = l-I Ri under the above identification (and is thus independent of the choice of maximal order).

PROPOSITION 13. Let zr be a p-group, and let ~_plr, ~c_(~plr be as above. Then, for any odd 1 <-_ t <- p - 2,

ttD(727r)l = tt(~/?_p~r)l if t # 1

--II(~/~PTr)l'ltors~ (Q~r)*l i[ t = 1.

Proof. Let ~IR_ ZTr be a maximal order, and write

k k

i = 1 i = l

where F~ are fields and ~IR~ ~_ A~ is a maximal order for all i. Given any x e ~0~

which is topologically nilpotent (i.e., p Ix" for some n), the series

X 2 X 3

log (1+ x) = x - ~ - + ~ - . . . .

converges in A~. We claim that for such x,

tr (log (1 + x)) = log (det (1 + x)) ~ F~. (1)

To see (1), choose n such that p l x p". Then for any m > 0 , ( l + x ) " ' ~ = l + p ' + l y for some y e~02, and

log (det (1 + x) p'+') = log (det (1 + p '+ ly ) ) = p'+~ �9 tr (y)

= tr (log (1 + p'+Xy)) = tr (log (1 + x) p~§ (rood p2m+2).

So for all m ~ 0,

log (det (1 + x)) = tr (log (1 + x)) (mod p " - " +2);

and (1) holds. In particular, log (1 + x) for x E J(~p~r) or x e J(~)~p) induces homomorphisms

:" ' ' ~ ' ~ ; L2 K~(~.)~)--*~ Lt . Kl(Zpcr)~p~

D(Zw) § and the Artin cokernel 307

# ^

such that Ll=L21Kl(~_pTr)~p). (Here J means Jacobson radical; note that J(~plr) ~ J (~p) in general.) Furthermore,

Ker (L2) = torsp ( ~ = torso (O~r)*; Ker (L0 = 1r ab ;

and so by Proposition 2, for all odd t:

I'D(7]rr)l = [ 'Im (L2) : ' Im (L0]. (2)

For any ;~p-lattices M1, M E - ~prr, we write for short

[ M, : M~] = [ M , :M , n M~]I[M~ :M, n M~].

By Theorem 2 in [8], for any 1-<-t <--p- 2,

1 ^ ( 1 - p ~ ) ('Im (L0) = 'Zp~r if i ~ l

=Ker[l~p~r-->Ir ab] if t = l

Here, ~(z~ Aigi) = ~ )hg~'; and �9 is nilpotent ('~prr lies in the augmentation ideal, since tO: 0). So

1 4~)= 1, det (1 -p -

and hence

['Zp~r : ' Im ~ 0 ] = 1 if t ~ l (3)

=l#<*l if t = l .

Finally, note that for s >_-0,

['2p~p.:log('(~p~p,)*)]=l if t=/=l (4)

= p" if t= 1:

this follows by noting that log(l+p~_p~p,)=p~_p~p., and then counting orders of the quotients. Since by (1),

k I t

Im (L2) = 11 log (R*)(p)c r l F~ ----(Ip,tr i = 1 i = 1

3 0 8 ROBERT OLIVER

(R~ _ F~ the ring of integers), (4) implies that

[ ' ~ : ' I m (L2) ] = 1 if tv ~ 1 (5) = 11 ttor% (R*)t = ttOrSp (Qlr)*/ if t = 1.

So (2), (3), and (5) combine to prove the proposition. []

Generalizing Frrhlich's formula for [D(ZIr)-] is now straightforward:

T H E O R E M 14. Let 7r be a p-group (p odd). Let S ~_ 7r be a set of conjugacy class representatives for all 1 v ~ g ~ ~r. Set

p"=lTr"bl and pk= 11 [Z(g)l. g c S

For s >= 1, let ws be the number of simple summands of Orr which are matrix algebras over Q(,p,. Then ]D(7/,rr)-] = pN, where

1[ ] N=-~ k + 4 n - ~, w ~ ( s p ~ - ( s + l ) p S - l + 4 s + l ) .

s ~ l

Proof. Let 2 p ' ~ ' _ ~ Q p T r be as above. By Proposition 13,

I D ( ~ ) - I = l ( ~ / ~ , r ) - I " p"" p~*" �9 (1)

Write Qp,rr = I-[~= ~ At, where Ai ~-M,,(Ft) and the ~ are fields. As before, the trace maps trt : At ~ F~ induce an identification of Q~w wit____h 11 F~. Let pr, : (~pTr --, At be the projection; and define an inner product on Qp-rr by setting

k

(x, y)--- Y. trvj~o (trt o Prt (x). tri o prt (y)) (x, y ~ QpTr). i = l

Since ~2~_I-I F~ is the product of the rings of integers, we have by definition discriminants

aWt) and

Here A.(F0, A(Ft t')R) denote the discriminants over Q; and the power of 2 arises due to' using the trace over Ft instead of Ft fqR.

D(Zw) + and the Artin cokernel 309

By [16, Proposition 7-5-7], for s ~ 1,

A ({l~p~) = t? I~(ps -s -1) .

By the same proof, or by the composition formula applied to the fields OG,/~ n OG~ [16, Corollary 3-7-20]:

Hence, d(~[l~-)= pNo, where

N o = ~ ~ (sp ~-(s + 1)p s - l+ 1). (2) s ~ l

Now fix g, h c w. For any given 1 =< i _--< k,

trv,/o~ (g) -tr~ opr~ (h))= ~ xi(g)xj(h), j = l

where Xi . . . . . X, are the distinct irreducible (complex) characters contained in the summand A~. Let w* denote the set of all irreducible complex characters. Then

<g, h>= ~ x(g)x(h)=O if g not conjugate to h -1 X E-n-*

=lZ(g)[ if g is conjugate to h -1

by the second orthogonality relation [1, Proposition 9.26]. Hence, eliminating factors prime to p,

r -11/2 a(L ) = / H LZ(g)L] = (3)

Lg~S

By (2) and (3), l(~7~'2prr)-t = pN,, where

N i = l [ k - ~, (sp~-(s+l)p~-l+l)]. s > l

Together with (1), this proves the theorem. []

This can also be reformulated solely in terms of cyclic subgroups of w:

310 ROBERT OLIVER

T H E O R E M 15. Let rt be any p-group, and let Xo be a set o f conjugacy class

representatives for cyclic subgroups 14= cr c_ ~r. For each cr~ Xo, set

a~ = ordp IN(cr)/al; b~ = ordp ( lal" IZ(o')l /IN(a)l) .

Then

ord o ID(~r)-I = ordp Icr~b I +�88 ~. [(a~ - 1)q~(p b-) + p b - 4 b ~ - 1]. o'EXo

Proof. Let w~(s=> 1) and k be as in T h e o r e m 14. Then each a ~ X o has r conjugacy classes of genera tors , and so

k = ~ ~0(pb~) �9 (a~ + b,,). o'EXo

By Propos i t ion 1, w~ is the n u m b e r of o, e X0 such that b,, = s. So T h e o r e m 14 takes the fo rm

ordp I D(7/~r)-[ = ordp ler~b I

+�88 ~. [ (a , ,+b, , )q~(p%)-b, ,pb~+(b, ,+ l ) p b - - ~ - 4 b , , - 1 ] creXo

= o r d o l~'abl+�88 ~ [a, ,~(pb~)+pb~-'--4b, ,--1] crEXo

= o r d p l ~ r ~ [ + l ~ [ ( a , , - l l ~ ( P ~ ) + P b ' - 4 b ~ , - 1 ] . [] o'eXo

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D(Zw) + and the Artin cokernel 311

[8] R. OLIVER, S K l for finite grouprings: II, Math. Scand. 47 (1980), 195-231. [9] I. REINER, Maximal Orders, Academic Press (1975).

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(1958), 241-250. [12] R. SWAN and F. EvANs, K-theory of finite groups and orders, Lecture notes in mathematics no.

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231-239. [16] E. WEISS, Algebraic Number Theory, McGraw Hill (1963).

Dept of Mathematics Aarhus University Ny Munkegade 8000 Aarhus Denmark

Received August 5, 1982