Ranks of Kn and Gn of orders and group rings of finite groups over integers in number fields

JOURNAL OF PURE AND APPLIED ALGEBRA ELSEVIER

Journal of Pure and Applied Algebra

138 (1999)

39-44

Ranks of K, and G, of orders and group rings of finite groups over integers in number fields Aderemi 0. Kuku International Centre for Theoretical Physics, I-34100 Trieste. Italy Communicated

by CA.

Weibel; received

10 June 1997; received in revised form 28 October

1997

Abstract Let R be the ring of integers in a number field F, A any R-order in a semi-simple Falgebra C, r any maximal R-order containing A. We show in this paper that for all n 2 2 rank&(n) = rank G,(n) = rank K,(T) = rank K,(C). Hence if G is a finite group, rank K,(RG) = rank Gn(RG) = rankK,,(FG). @ 1999 Elsevier Science B.V. All rights reserved. AMS

clussifiicatiun:

19D50; 19F27

Introduction In this paper, we continue our study of higher K-theory of finite groups over integers in number fields.

of orders and group rings

In earlier papers, [7, 10-121, the author had proved that if R is the ring of integers in a number field F, and LI any R-order in a semi-simple F-algebra C, then &(A), G,(n) are finitely generated Abelian groups for all n 2 1. We now obtain information about the ranks of these groups and show precisely that if r is a maximal order containing A, then rank&(n) = rank G,(n) = rank K,,(T) = rankK,(C) for all II 22. It then follows that if /I = ZG, then rankK,,(ZG) = rankG,(ZG)=rankK,(T)=rankK,(QG). Notation. For any ring A, we write K,,(A) for the Quillen K-groups n,+l(BQP(A)) = z,(BGL(A)+) where P(A) is the category of finitely generated projective A-modules. If A is Noetherian, we write G,,(A) for rcn,+l(BQM_(A)) where M_(A) is the category of finitely generated A-modules. If R is the ring of integers in a number field F, and n is an R-order in a semi-simple F-algebra C, we write SK,(,4)=Ker(ZC,,(ci)--+K,(C)).

0022-4049/99/$ - see front matter @ 1999 Elsevier Science B.V. All rights reserved PII: SOO22-4049(98)00010-3

A. 0. Kuku I Journul of Pure and Applied Algebra 138 (I 999) 39-44

40

Theorem 1.1. Let R be the ring of integers in a number field F, A any R-order in a semi-simple F-algebra

C, r a maximal R-order containing A. Then for all n 2 2

rankK,(~)=rankK,(T)=rankG,(/i)=rankK,(C). The proof of Theorem 1.l will be in several steps 1.2-1.5 below. Theorem 1.2. Let R be the ring of integers in a number jield F, r a maximal Rorder in a semi-simple F-algebra C. Then the canonical map K,,(T)+ kernel and torsion cokernel for all n 2 2. Hence, rank K,,(f)

K,(C) hasJinite

= rank K,,(C).

Proof. Since r, .Z are regular, we have K,(T)? G,(T) and K,(C)? G,(C). So, we show that G,(T) t G,(C) has finite kernel and torsion cokemel. Now, SG,(T) = SK,(T) is finite for all n > 1 (being finitely generated and torsion see [6,9, lo]). Also the localisation sequence of Quillen yields . .

-Gn+1(r)--tGn+,(c)-t~Gn(r/pr)--t ..‘2

where p runs through the prime ideals of R. Now, for n > 1, each G,,(T/pT) is finite since F/pT) is finite (see [7]). So, $,G,(T/pT) is torsion. Hence, G,+t(Z)/lm (G,+t (f) is torsion, as required.

0

Lemma 1.3 (Serre). Let A +B EEK --) C @ L+ D be an exact sequence of Abelian groups. If A, B, C and D are finite (resp. torsion), then the kernel and cokernel of K -+ L are both finite (resp. torsion).

Theorem 1.4. Let R be the ring of integers in a number field F, A any R-order in a semi-simple F-algebra, r a maximal order containing A. Then, for all n > 1, the map G,,(T)-+G,,(A)

induced by the functor M_(T)-+M_(A) given by restriction of scalars

has finite kernel and cokernel. Hence,

rank G,(T) = rank K,,(T) = rank G,(A).

Proof. There exists a non-zero element s E R such that A c r c A( l/s). Let S = {s’}, i 2 0. Then As = A @‘R Rs ‘v r aR Rs = Ts. We show that for all n > 1, c(, : G,(T) + G,(A) has finite kernel and cokemel. Consider the following commutative diagram of exact sequences: . . . -+

Gn(rjsr)-3 G,(r)5 G,(rs)-

cn-l(r/sr)

-

..

(1) -

G,,(A/sA) A

G,(A) L

where 6 is an isomorphism.

G,(As) i

G,_l(A/sA)

-.

..

41

A. 0. Kuku/ Journal of Pure and Applied Algebru 138 11999) 39-44

From (I) we extract the Mayer-Vietoris

Now, since T/ST and A/s/i finite [7] except Gn(T)

sequence

are finite and it 2 1 all the groups in (II) above are

and G,,(A). The result now follows from Lemma

q

1.3.

Theorem 1.5. Let R be the ring of integers in u number field F, A any R-order in u semisimple F-algebra, K,,(A)--+Kn(T)

T a maximal order containing A. Then for all n 2 1 the map

has finite kernel and cokernel. Hence rank K,,(A)=rankK,,(T).

To be able to prove Theorem

1.5, we first prove the following:

Theorem 1.6, Let R be the ring of integers in a number field F, A any R-order in a semisimple F-algebra

C, T a maximal order containing A. Then for all n 2 1 the

map K,,(A) --) K,,(T) (induced by the inclusion map A H r),

is un isomorphism mod

torsion.

Proof. First note that since every R-order is a B-order, there exists a non-zero s such that A c r c A(l/s). A

Put q - = sr. Then we have a Cartesian

-+r

I A/q --

(1)

1 r/q -

Now, by tensoring

(I) with Z(l/s),

if we write A(l/s)

group A, we have long exact Mayer-Vietoris

.

integer

square

sequence

for A@ Z(l/s)

for any Abelian

(see [2] or [18])

..K~+.(,,)(~)-K.o(j)‘K,(r)(~)@K&/g)($) :K,(r,q)

(-!)

-K,-,(A)

(-!)

+ ....

(II)

Now A/q and r/q are finite rings and so Kn(A/q) and K,(T/q) (see [l 1, 121). The Result is now immediate from Lemma 1.3. 0

Proof of 1.5. Let CL,: K,,(A)+Kn(T)

denote the map. By Theorem

are finite groups

1.6, the kernel and

cokemel of c(, are torsion. Also, for all IZ > 1, Kn(A) and K,(T) are finitely generated (see [ 11, 2.11). Hence, the kernel and cokemel of c(,, are finitely generated, hence finite. So rank K,(T) = rank Kn(A). As a fall out from the above, we now have the following that SK,(A)

is finite for any R-order A.

Cl

result which also proves

42

A.O. Kukul Journal of Pure and Applied Algebra

138 (1999)

39-44

Theorem 1.7. Let R be the ring of integers in a number field F, A any R-order in a semi-simple F-algebra C. Then the canonical map K,,(A) -+K,(.Z)

has jnite kernel

and torsion cokernel.

Proof. From the commutative diagram

we have an exact sequence K,(r) InWA)

0 -+kercc-+SK,(/i)-+SK,(T)-+ K,(C) K,(z) +o * ImK,(/1) 3 Im(K,(T) ’

Now, by Theorem 1.5 kercr is finite and by [S], SK,(T) =SG,,(T) is finite for all 12> 1. Hence, from the exact sequence (I) above, SK,(A) is finite. Also, by Theorem 1.5, K,,(r)/ImK,,(A) is finite, and by Theorem 1.2, K,(C)/Im(KJ) is torsion. Hence the result. 0 Remarks 1.8. (i) The above results hold for ,4 = RG where G is any finite group (ii) The ranks of K,(R) and K,(F) are well known and are due to Bore1 (see [I] or [41X More precisely, let ri be the number of embeddings of F in Iw and r2 the number of distinct conjugate pairs of embeddings of F in @ with image not contained in Iw. Then 1 CC

rank K,(F) =

0 rl +rz r2

1 q+r2--1

rank K,(R) =

rl + r2 r2

0

if if if if if

n =0, n= 1, n=2k

k > 0,

n=4k+

1,

n=4k+3,

if if if if if

n = 0, n=l, n=4k+ 1, n=4k+3, n=2k k > 0.

It then means that if Z is a direct product of matrix algebras over fields and f is a maximal order in C, then m&K,(T) = rank K,(C) is completely determined since C=17M,,(Fi) and f = nM,,(Ri) where Ri is the ring of integers in Fi. Also, by Theorem 1.2, this is equal to rank G,(n) as well as rank K,(A) if /1 is any R-order contained in r.

43

A. 0. Kuku I Journal of Pure und Applied Algebra 138 (I 999) 39-44

However,

E of F which splits

if C does not split, there exists a Galois extension

C, in which case we can reduce the problem

of ranks of K, of

to that of computation

fields. (iii) In [5], Jahren proves that if G is a finite group with r irreducible sentations,

c of them of complex r

if

n-_ l(4),

if

n~3(4),

{ 0

if

n is even.

(iv) Even though p-adic

orders

cokemel

type, then for all n > 1, we have

c

rankK,(ZG)=

we do not have finite generation

A, the following

real repre-

results,

Theorem

results

(1.8),

for K,(A),

concerning

G,(A),

finite kernel

of and

are quite interesting.

Theorem 1.8. Let R be the ring of integers in a p-adic field F, A any R-order in a semi-simple F-algebra

C, r a maximal R-order containing A. Then, for all n > 2,

(i) the canonical map K,,(T)+

K,,(C) has jinite kernel and cokernel,

(ii) the canonical map G,(A)-+ G,(C) has jnite kernel and cokernel, (iii) a, : G,,(T) + G,,(A) has jinite kernel and cokernel where c(,, is the map induced by the functor M_(T)-+M_(A) given by restriction of scalars. Proof.

Since r and C are regular,

(i) is a special case of (ii) and (iii) follows from

both (i) and (ii) as the map in (i) factors as K,(T)=G,(T)*G,(A)+G,(C)=K,(C). So, it suffices to prove (ii). Now, C = A[ l/p] and so, there is an exact sequence G,(A/PA>-G,(/~)~G,(C)-,G,-I(AIPA)

Since n > 2, and A/pA kernel and cokemel

is finite, the end terms are finite. Now by Lemma

of G,,(A) + G,(Z)

are finite.

1.3, the

0

Acknowledgements I would like to thank Chuck Weibel for useful conversations Science Foundation

and the Swiss National

for support during a short visit to the University

part of this work was done. I also thank the Institute

of Lausamre where

of Mathematics

at Lausanne

for

hospitality. References [I] A. Borel, Stable real cohomology of arithmetic groups, Ann. Sci. Ecole. Norm. Sup. (4), Serie 7 (1974) 235-272. [2] R. Charney, A note on excision in K-theory, Proc. Bielefeld Conf., Springer Lecture Notes, vol. 1046, 1984, 49-54.

A.O. Kukul Journal of Pure and Applied Algebra 138 (1999)

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[3] A. Dress, A.O. Kuku, The Cartan map for equivariant higher K-groups, Commun. Algebra 9 (7) (1981) 727-746. [4] D. Grayson, On the K-theory of fields, in: Algebraic K-Theory and Algebraic Number Theory, Contemp. Math. 83 (1989) 31-55. [5] B. Jahren, On the rational K-theory of group-rings of finite groups, preprint. [6] A.O. Kuku, Some finiteness theorems in the K-theory of orders in p-adic algebras, J. London Math. Sot. 2 (13) (1976) 122-128. [7] A.O. Kuku, SK, of Orders and G, of Finite Rings, Springer Lecture Notes, vol. 551, Springer, Berlin, 1976, pp. 60-68. [8] A.O. Kuku, SG, of orders and group-rings, Math. Zeit. 165 (1979) 291-295. [9] A.O. Kuku, Higher algebraic K-theory of group-rings and orders in algebras over number fields, Commun. Algebra 10 (8) (1982) 805-816. [IO] A.O. Kuku, K-theory of group-rings of finite groups over maximal orders in division algebras, J. Algebra 91 (1) (1984) 18-31. [l l] A.O. Kuku, K,, SK, of integral group-rings and orders, Contemp. Math. 55 (1986) 333-338. [12] A.O. Kuku, Some finiteness results in the higher K-theory of orders and group-rings, Topology Appl. 25 (1987) 185-191. [13] R. Oliver, Whitehead Groups of Finite Groups, Cambridge University Press, Cambridge 1988. [14] D.G. Quillen, Higher Algebraic Theory I, Springer Lecture Notes, vol. 341, Springer, Berlin, 1973, pp. 77-139.

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Integral Representations,

1979. [ 161 E. Spanier, Algebraic Topology, [17] A. Suslin, Stability in Algebraic pp. 304-333. [18] C. Weibel, Mayer-Vieloris

Springer Lecture Notes, vol. 744, Springer,

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and module structures

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Berlin (1981) pp. 466-493.

Berlin,

K-theory

Berlin,

1982, Conf.,