Swan , Projective modules over binary polyhedral groups 87

Proof . (1) is clear by Lemma 6.2 and the following remark. Note r\n impUes H^.czlH^^ by Corollary 5.7.

( 4 ) If 4|«, ÖczH*c:H*„. If OczH*^ then Qte'^OczH*^ so 4|n by (1).

( 5 ) If 5\n, IœH*^œH^„. If IczH*^ then Q^oCzIczH*^ so 5|« by (1).

( 2 ) If 3|«, 012^^6 ^^2и- Suppose 3J^n. If ß^^^ln^ Corollary 5.7 gives ^2и®о^б~^^2п- Therefore these algebras have the same invariants. This is clear at 00. Since IH^ ramifies only at 3 and oo, «Ф/?*" (/7 = 3 is excluded) so n is composite. Therefore the invariant ^ of H^ at 3 must go to 0 so ^^2и/0 "itist have even local degree at 3 but this degree is the order of 3 in (Z/«Z)*/{±1}.

( 3 ) If Qs^l^n then tt^czH^^ so fciHlczH*^ and conversely since Q^<=:f. The last part follows by the argument of (2) with 3 replaced by 2.

In the remainder of this section I will discuss the cases of Л205 ^24» ^42 ^^^ Л120 more fully. In §6 we saw that apsc>\ for these orders but left open the question of whether mpsc= 1. I will show that in fact mpsc= 1 for the first three of these orders. Earlier examples of maximal orders with mpsc= 1 <apsc were given by Vignéras [VC]. The statement in [V] that it is sufficient to check cancellation for stably free modules should, of course, be inteфreted as applying to the totality of maximal orders rather than to a single such order.

Case A2Q. If Г is a maximal order in H^q, the above theorems show that Г^ must be one of the following groups: C„ for « = 2,4,6,10,20; Ö„ for « = 8,12,20,40; for /. We can omit the groups of odd order since -1 6 Г^.

The Eichler-Vignéras formula gives E ^Г^ =:^ so W|^24. Therefore F^q can only be one of 640^ fotl ^

1 For the stably free class we have ^^=40 so T. ^i^-TR- ^^^^ ^^e only possi-

1^2 OÜ

bility is r^Q = 7. Since only one such Г^ will occur by Corollary 5.11 we see that ^2 = 60.

Therefore the stably free class has two elements with Г^ and w being Q^, w = 40

and /, w = 60.

In the other stable class, fj = Q^ and Î will not occur by Corollary 5.11 so r^-f and vv= 12 or 24. Therefore there is only one class and Г^-f, w = 24.

CaseA2^^ By the above theorems, Г^ must be one of C„, « = 2,4,6,8,12,24; e„, « = 8,12,16,24,48; for (5.

Here E wT^ =— so only C24, ßi6. Ô24. 048» ^ and Ö are possible. 10

1 ^ , 1 In the stably free class, h^i =-^^ so L ^1 =:^7-

^ I2t2 ^^

12 *