Colimits of Functors, and Grothendieck Groups of Infinite ...lorenz/papers/40.pdffinite, but...

Contemporary Mathematics Volume 130, 1992

Colimits of Functors, and Grothendieck Groups of Infinite Group Algebras

K.A. BROWN AND M. LORENZ

Dedicated to our friend and colleague Bob Warfield

ABSTRACT. Let f be a group and let X be a collection of finite subgroups of r which is closed under taking intersections and under r-conjugation. Viewing X as a category with morphisms given by inclusion and f-con-jugation, we study the colimits lim Go(kX) and lim Ko(kX) for

~xex ~xex

any field k. In particular, we determine the ranks of these colimits and investigate their torsion subgroups. For example, we show under mild as-sumptions on X and k that the presence of q-torsion (q prime) in either colimit forces q to divide the order of some X E X.

Particular emphasis is given to the special case where the group r is polycyclic-by-finite and X consists of all finite subgroups of r. The above results in conjunction with Moody's induction theorem [Mo] have the fol-lowing consequence. Let R ~ Q be obtained from Z by adjoining q-1 for each prime q such that r has q-torsion and let T(f) denote the set of con-jugacy classes in r of elements of finite order not divisible by chark. Then Ga(kf) ®z R ~ RT(r), provided k is large enough.

Introduction

A special case of the theorem of J .A. Moody [Mo] asserts that, if R is a Noetherian ring and r is a polycyclic-by-finite group, then the Grothendieck group Go(Rr) is generated by the images under induction of the groups Go(RH), for H a finite subgroup of r. This result opens the way for the explicit calculation of these Grothendieck groups. For example, when R is a field or the integers and r is torsion free, one immediately deduces that G0 (Rf) ~ ~, a result obtained (for certain such R) by other methods in [FH].

In this paper we study the case where r has torsion, assuming for the most part that R is a field, denoted k, (so that [Mo] implies at once that G0(kf) is

1991 Mathematics Subject Classification. Primary 16E20, 16S34; Secondary 19A31, 20C07. This paper is in final form, and no version of it will be submitted for publication elsewhere.

89

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90 K.A. BROWN AND M. LORENZ

finitely generated abelian group). In the first part of the paper (§§0 and 1), we present a series of results culminating in the calculation of the torsion free rank of G0 (kf), (Corollary (1.8)). Let k have characteristic p 2: 0, and let T(f, k) denote the set of orbits of elements of r of finite order prime to p, under the combined action of conjugation by r and by the Galois group of a splitting field extension of k for the finite subgroups of r. (The precise definition is in (0.3); if k is large enough, then of course only f-conjugation features here.) Generalizing the result for finite groups, we show that

rankG0(kf) = IT(f,k)j. This formula was the main result of [L]. The proof is given here in a more

categorical form, as a result in fact about the rank of certain colimits, which we hope may have applications in the case where r is not polycyclic-by-finite. Whereas the proof [L] made use of ordinary or Brauer characters, here we employ Hattori-Stallings ranks. While Hattori's Lemma [H] shows that the difference is purely formal, ranks are better behaved under induction (see (1.2)), so yielding simpler arguments. Furthermore, some of the lemmas proved in §§0 and 1 are used in §§2 and 3, where the torsion subgroups of G0(kf) and G0 (kX) are studied. Here, X denotes a collection of finite subgroups of the group f, closed under taking intersections and f-conjugates, and G0 (kX) denotes the colimit lim Go(kX), the quotient of L~EX Go(kX) by the subgroup generated by ---+XEX all relations given by f-conjugation and inclusion, (0.2). Denote the torsion subgroup of G 0 (kr) by To(kf); when r is polycyclic-by-finite,

T0 (kf) is an image of the torsion subgroup of G0 (k:F(f)),

where :F(r) is the collection of all finite subgroups of r, (Corollary (1.8)). First, torsion in G0(kX) and K 0 (kX) is examined, in §2. The main results are

listed in (2.1). They are obtained without the hypothesis that f is polycyclic-by-finite, but assuming instead that X consists of only finitely many r -conjugacy classes, (as is always the case for f polycyclic-by-finite). The central finding is that

G0 (kX) contains an element of order q, for q prime, only if q is the order of an element off (at least when k is large enough for r) {2.5).

In view of the aforementioned map of torsion groups onto T0 (kf), when r is polycyclic-by-finite, the results of §2 yield immediate consequences for To(kf). Further information about T0 (kr) is contained in §3. When the characteristic p of k is positive and r involves only p-torsion, it is not hard to see that To(kr) is a p-group (independently of the size of k); and [LP] discussed certain classes of groups r for which this p-group can be explicitly calculated, (and shown to be isomorphic to the torsion subgroup of Go(k:F(f))). However, a recent result of [KM] shows that T0 (kf) can be non-zero even when r has no p-torsion, in fact even when k = Q, and accordingly our effort in §3 is directed towards bounding



GROTHENDIECK GROUPS OF GROUP ALGEBRAS 91

the order of To(kf) independently of the characteristic of k. The key point is that

there is a finite image G of r such that the canonical homomorphism

from r onto G induces an isomorphism of colimits (1.6}, (1. 7).

Thus methods from the representation theory of finite groups can be applied. We state a result, (3.3), only for the case where r has a (torsion free) abelian normal subgroup A of finite index, (although the method can clearly be applied in general). It yields in particular for this case that, letting h(f) denote the rank of A,

the exponent ofTo(kG) is a divisor of If: Alh(r)+l.

Acknowledgements. Part of this work was done while the authors were at-tending the MSRI Microprogram on Noncommutative Rings at Berkeley, Ca. in July 1989, and part while the first author was visiting the University of Wash-ington (Seattle), immediately thereafter. These visits were supported by the N.S.F. and the Carnegie Trust for the Universities of Scotland. The second author's research was supported in part by an N.S.F. grant.

0. Notations and basic facts

0.1. In this section, we establish notations and collect some background ma-terial that will be used later on. Throughout,

r will be a group, X is a collection of finite subgroups of r which is closed under taking

intersections and under r-conjugation, lXI is the least common multiple of {lXI :X EX} whenever this integer is

defined. F(f) is the set of all finite subgroups of r,

k is a commutative field, and p > 0 is the characteristic of k.

Modules are understood to be right modules.

0.2. Colimits. We will view X as a category, with morphisms given by inclusion off-conjugates: If Xf ~ X2 (X; EX, g E f) then we have a morphism Xt ---> x2 in X which is given by X---> x9 (x EXt)· For any covariant functor M :X --->mod-S of X into the category of modules over some ringS, we let

M(X) = lim M(X) ---+

XEX

denote the colimit of M (cf. [MacL], p. 67). Explicitly, M(X) is the quotient of ffixEX M(X) by the submodule that is generated by all elements of the form /-LX 2 (M(f)(x))- /-LX 1 (x), where f: X1---> X2 is a morphism in X, x E X 1, and




t-tx, : M(X;) --+ ffixEX M(X) are the canonical injections. In particular, we have the following abelian groups:

Ko(kX), Go(kX), and Go(kX),

corresponding to the functors X --+ mod- fZ that are given by X ~-+ K0 (kX), X~-+ G0(kX), and X~---+ G0(kX). Here K0 (-) and G0 (-) are the usual Grothen-dieck groups and

G0 (kX) is the cokernel of the Cartan map ex : Ko(kX) --+ Go(kX).

All this of course applies to the special case where X = F(f).

0.3. Conjugacy classes and Galois actions. An element of f will be called p-regular if it has finite order not divisible by p. (So 0-regular just means torsion.) We let

fp,, Xp'' and Xp'

denote the sets of p-regular elements of r, X, and UxEX X. Fix a Galois ex-tension K 2 k in some algebraic closure k of k and assume that K contains all subgroups

f.lX = {( Ek I (e(X)p' = 1} (X EX), where e(X)p' denotes the p'-part of the exponent of X. (We could take K to be the field generated by all t-tx over k, but it will be advantageous to keep K variable.) For each X EX, we have homomorphisms

restriction Gx,k := Gal(K/k) -----+ Gal(k(t-tx )/k) <-+ U(/Zfe(X)p'.IZ),

and U(.IZ/e(X)p'IZ), the group of units of /Zfe(X)p'IZ, acts as a permutation group on Xp' by setting xt+e(X)p,z = xt. In this way, Gx,k acts by permutations on each Xp' and these actions combine to give a permutation action on Xp'. Note that this action commutes with the conjugation action of r on Xp'· Thus r x Gx,k acts on Xp' and X x Gx,k acts on Xp' (X EX). The orbit sets will be denoted by T(X,k) and T(X,k), so

T(X,k) = Xp'/f x Gx,k, T(X,k) = Xp'/X x Gx,k·

A typical orbit in T( X, k) will be written as

xXxGx,k E T(X, k) (x E Xp' ),

and similarly for T( X, k). In the special case when X = F(r) we put

T(f,k) = T(F(f),k) = fp,/f x G:F(r),k.

If k is large enough for X, that is t-tx <;:;: k holds for all X E X, then G x,k acts trivially on Xp'; and so T( X, k) = Xp' /f is the set of f -conjugacy classes in Xp' and T( X, k) = X p' /X is the set of all p-regular conjugacy classes of X.




0.4. p-modular systems and the Cartan-Brauer diagram. Assume that k is given with char k = p > 0. Then there exists a discrete valuation ring A with maximal ideal mA = pA such that A is complete in its mA-adic topology, the residue class field A/mA is isomorphic to k, and the field offractions F = Fract(A) has characteristic 0 ([Bou], Propositions 1 and 5 on p. 17 and p. 23). The triple (F, A, k) is called a p-modular system.

For any finite group X, there exists a commutative diagram

Ka(FX)

II Ga(FX)

dx ----->

ex -Go(kX)

lex Ko(kX)

the Cartan-Brauer diagram. The notation is as follows. ex is the Carlan map. It is injective and has cokernel annihilated by IXIp,

the p-part of lXI. dx is the decomposition map and is surjective. ex is a split injection which is defined to be the inverse of the reduction

C><

isomorphism(-) 0A k : Ka(AX) -=.. Ko(kX) followed by the localization map(·) 0A F: Ka(AX)---+ Ka(FX).

The maps c, d and e commute with the induction maps: If f : X ---+ Y is a monomorphism of finite groups and Ind:k denotes all three induction maps, Ko(kX) ---+ Ko(kY), Go(kX) ---+ Go(kY), and Ka(FX) ---+ Ka(FY), then Ind:k o ex = cy o Ind:k, and similarly for d and e. For all this, see [CR], in particular Theorems (21.16), (21.20) and (21.22).

Using the universal property of colimits, we get a corresponding commutative diagram:

Ka(FX)

II ex

Ga(FX) - Ko(kX) Again, dx is surjective. This follows from the fact that colimits are right exact (which is clear from the explicit description of colimits given above, or from their universal property). We will also see later (2.3) that ex is injective, and hence SOlS ex.

1. Ranks

1.1. Our main goal in this section is the computation of the ranks of K0 (kX) and G0 (kX). We will show in (1.5) that

ranki<o(kX) = rankG0 (kX) = IT(X,k)i,

where T(X, k) is as in (0.3). In the special case where r is a polycyclic-by-finite group, this result combined with Moody's induction theorem [Mo] yields the




following formula from [L]:

rank G0 (kf) = JT(r, k)J (see (1.7)).

1.2. Hattori-Stallings ranks. Let P denote a finitely generated projective module over the group ring Rr, where R is some commutative ring. Then P <:::: e(Rr)n for some n and some idempotent matrix e = e2 = (e;i) E Mn(Rr). The H attori-Stallings rank of P is defined to be the function

n

XRr(P): r-+ R, g ~---+ 'L:tr9 (e;;), i=l

where tr9 : Rr-+ R is the R-linear map that is defined by

tr9 (h) = { ~ if h E r is conjugate to g

otherwise.

Clearly, the function XRr(P) is constant on conjugacy classes of r, and vanishes on all but finitely many conjugacy classes. It can thus be viewed as an element of R(T(r)), the free R-module on the set T(r) of conjugacy classes of r. One checks that XRr yields a well-defined group homomorphism

XRr : Ka(Rf) -+ R(T(r)), [P] ~---+ XRr(P).

A useful property of Hattori-Stallings ranks is the fact that they commute with induction maps. Indeed, if f : r -+ ~ is a group homomorphism and if the idempotent e E Mn(Rr) corresponds to [P] E Ko(Rf) then f(e) E Mn(R~) corresponds to lnd~[P] = [P 0Rr R~] E K 0 (R~). Thus the following diagram commutes:

Ka(Rr) XRI' R(T(r)) --+

lnd~ 1 1 R(T(J))

Ka(R~) XR.O. R(T(L1)) --+

Here T(f) : T(r) -+ T(~) sends r-conjugacy class of g E r to the ~-conjugacy class of f(g) E ~,and R(T(f)) is the R-module extension ofT(!). Because of this fact, it will be more convenient to work with Hattori-Stallings ranks rather than with ordinary characters, although there are close connections as the following Lemma and its proof show.

LEMMA. Let X be a finite group and let k be a field. Then Xkx yields a homomorphism

Xkx : Ko(kX) -+ ecx,k)

such that the corresponding k-linear map

XkX = XkX 0z idk : Ko(kX) 0z k-+ ecx,k)




is an isomorphism.

PROOF. The first assertion says, more explicitly, that for each finitely gen-erated projective kX-module P, nx(P) vanishes on X\Xp' and is constant on orbits of the action of Gal(k(Jlx )/k) on X (as in 0.3). For the former assume p > 0 and choose q = pr so that xq E Xp' holds for all x EX; (q = IXIp would do). Choose a generating idempotent e = (eij) E Mn(kX) for P and, for each x EX, define a k-linear map trx: Mn(kX)---> k by

Then trx vanishes on [Mn(kX), Mn(kX)]. Using [P], Lemma 2.3.1, we see that e = eq = diag(ei1 , ... , e~n) + c with c E [Mn(kX), Mn(kX)]. Furthermore, if L~1 eii = LyEX eyy E kX, then 2:7=1 e{i = LyEX ezyq +d with dE [kX, kX]. Therefore

trx(e) = trx(eq) = trx (I>~yq). By choice of q, all yq belong to Xp'· So, if x E X\Xp' then no yq is conjugate to x, whence nx(P)(x) = trx(e) = 0, as required.

To deal with the action of Gal(k(Jlx)/k) on X, choose a p-modular system (F, A, k(Jlx)) and let - : AX ---. k(Jlx )X be the reduction map that extends A ---. k(Jlx ). Using the reduction isomorphism Ko(AX) ~ Ko(k(Jlx )X) lift [P] E K0 (kX) to [Q] E Ko(AX) with Q = Q ®A k(Jlx) = P ®k k(Jlx ). Then for all x E Xp''

XAx(Q)(x) = nx(P)(x).

Furthermore, by Hattori's Lemma {[H] or [B], 5.8),

XAx(Q)(x- 1 ) = ICx{x)l- 1 ch{Q)(x),

where Cx(x) denotes the centralizer of x in X and ch(Q)(x) is the ordinary character of Q evaluated at x, that is the trace of the A-linear map on Q that is given by the action of x. Now, if r E Gal{k(Jlx )/ k), then T acts on x via the

restriction map Gal(k(Jlx )/k) Gal(k(Jlm)/k) ---. U(7l/m7l), where m denotes the order of x and Jlm = {( E k(Jlx) I (m = 1}. So xr = xt for some t E 7l with (xt) = (x), and hence Cx(xr) = Cx(x). Moreover, A contains all m-th roots of unity in some algebraic closure ofF = Fract(A). For, if M ~ U(A) denotes the preimage of J-tm under - then the extension

1 ---> 1 + ffiA ---> M ---> Jlm ---> 1

is split, because IJlm I = m is not divisible by p and so the endomorphism a 1--+ am

of 1 + mA is invertible ([Bou], Exercise 12a, p. 72). Fix a splitting s : Jlm ---. M. Then

ch(Q)(x) = L s((), (EE(x)




where E(x) is the subset of J.lm consisting of the eigenvalues of the operation of x on P ®k k(J.Lx ). Furthermore,

where the last equality follows from the fact that the matrix describing the operation of x on P ® k(J.Lx) has entries in k. Therefore,

ch(Q)(x 7 ) = ch(Q)(x).

This implies XAx(Q)(x 7 ) = XAx(Q)(x) for all x E Xp', and hence nx(P)(x7 ) = Xkx(P)(x), as we have claimed. Therefore, the map XkX defines a homomor-phism Ko(kX)---+ kT(X,k).

Finally, as to the assertion that Xkx : K 0 (kX) ®z k ---+ kT(X,k) is a k-linear isomorphism, we recall that Ko(kX) is a free abelian of rank r = IT(X, k)l, with generators the images of the principal indecomposable kX-modules ([CR], Theorems (21.5) and (21.25)). Let's say these are P; = e;kX, withe;= e[ E kX (i = 1, 2, ... , r), and put Xi= XP, E kT(X,k). It suffices to show that the Xi are linearly independent over k. So suppose that I:~=l a;x; = 0 for suitable a; E k. Then putting a= I:~=l a;e; E kX we have, for all x EX,

r r

0 = L a;x;(x) =La; trx(e;) = trx(a). i=l

Therefore, a E [kX,kX]. Writing kX/rad(kX) = EBi= 1 Mn,(Di) and letting 7r;: kX---+ Mn,(D;) be the projection map, we conclude that the element 7r;(a) = diag(a;, 0, ... , 0) belongs to [Mn.(Di), Mn,(D;)]. Hence, taking matrix traces in D;j[ D;, D;], we get

a;+ [D;,D;] = trace(7r;(a)) = 0.

Therefore, a; E k n [D;, D;] = 0, where this last equality is clear for char k = 0, for example via the reduced trace of D;. For char k = p > 0 it follows from the fact that, in this case, all D; are commutative ([Se], p. 136). This completes the proof of the lemma. 0

1.3. Lifting ranks to characteristic 0. Let X be a given finite group of exponent e and let k be a field of characteristic p > 0. Choose a p-modular system ( F, A, k). Our goal here is to construct a homomorphism

chkX : Go(kX) ---+ FT(X,k)

by imitating the usual construction of Brauer characters but usmg Hattori-Stallings ranks instead of ordinary characters. First, we claim that there is a canonical inclusion of sets

T(X, k) <;;; T(X, F).

For this, consider the following groups:

Ilk= {( E k I c = 1} = {( E k I C•' = 1},




Jl-F = {( E F' 1 ce = 1} and

Jl-F,p' = {( E F I (eP' = 1}, where ep' is the p'-part of e, and k and F denote algebraic closures of k and F. Then the Galois groups Gal(k(pk)/k) and Gal(F(P,F,p' )/F) are canonically isomorphic to the same subgroup of U(7!..fep,7J..) (see ([CR], Exercise 5 on p. 512)), and hence they give rise to equivalent permutation actions on Xp' as described in (0.3). Similarly, Gal(F(pF )/F) acts on X and this action, when restricted to Xp', stabilizes Xp' and factors through the restriction map Gal(F(pF )/F) --+

Gal(F(PF,p' )/F). Thus we have the following inclusions of orbit sets:

T(X,k) = Xp'/X x Gal(k(pk)/k) = Xp'/X x Gal(F(P,F,p')/F) ~ X/ X x Gal(F(JLF )/F) = T(X, F).

Using this inclusion we can define a homomorphism, (where 1r denotes projection)

Ka(FX) ~ FT(X,F) ~ FT(X,k)_

This homomorphism factors through the decomposition map dx : Ko(F X) --+

G0(kX), since by [CR], Exercise 2 on p. 427, and Hattori's Lemma (to pass from ordinary characters to Hattori-Stalling ranks) one has

kerdx = {[V] E I<a(FX): XFx(V)Ix, = 0}. p

Thus we obtain the desired homomorphism chkx : G0 (kX) --+ FT(X,k). Of course, this map could also have been defined by using Brauer characters, suit-ably renormalized according to Hattori's Lemma.

LEMMA. Let chkx : G0 (kX)--+ FT(X,k) be the above homomorphism. Then

(i) chkx = chkX ®z idF : G0(kX) ®z F --+ FT(X,k) is an F -linear isomor-phism.

(ii) The maps ch commute with induction: Iff :X --+ Y is a monomorphism of finite groups then the following diagram commutes.

Ga(kX) chkx FT(X,k) ---+

Indk 1 1 FT(J,k)

Ga(kY) ---+ chkY

FT(Y,k)

Here, T(f, k) sends the orbit of x E Xp' to the orbit off( x) E Yp'.

PROOF. (i) By Lemma 1.2, XFX = XFX ®z idF gives an F-linear isomor-phism I<a(F X) ®z F ~ FT(X,F), and the above description of ker dx shows that ker(dx 0 idF) = ker(7roXFx). This implies (i).

(ii) This follows from the fact that the decomposition maps commute with induction ([CR], p. 502) and so do Hattori-Stallings ranks (1.2). 0




1.4. Colimits and conjugacy classes. It has been implicit in the foregoing that the assignment

X~----+ T(X, k) defines a functor T(-, k): X --+Sets. For each morphism f: X1 --+ X2 in X one has a map T(f, k): T(X1, k)--+ T(X2, k) sending X1 x Gx,k-orbit of x E (Xl)p' to the x2 X Gx,k-orbit of f(x) E (X2)p' (notations as in (0.3)). Furthermore, if one defines maps fx : T(X, k) --+ T(X, k) by sending the X x Gx,k-orbit of X E Xp' to the r X Gx,k-orbit of x, then, for each morphism f: xl--+ x2 in X, one has a commutative diagram

T(X1,k) ~ T(X2,k) T(j,k)

which is universal in the following sense. Suppose that gx : T(X, k) --+ S (X EX) is any collection of maps into some set S such that gx, = gx2 o T(f, k) holds for all morphisrns f : X1 --+ X2 in X. Then one obtains a unique map g: T(X, k)--> S with gx =go fx for all X EX: To define the image of a given orbit xrxGx,k E T(X, k) (x E Xp' ), choose any X in X with x E X and put g(xrxGx,k) = gx(xXxGx,k) E S. This definition is independent of the choice of X EX. For, if xl and x2 both contain X then so does xl n x2 =X EX, and the inclusions fi :X--+ Xi (i = 1, 2) are morphisms in X. Therefore,

gx,(xX;xGx,k) = (gx, 0 T(/i, k))(xXxGx,k) = gx(xXxGx,k) (i = 1, 2),

which shows that g is well-defined. Since gx =go fx is clear, by definition, we see that

T(X,k) ~ lim T(X,k). ___. XEX

For a given field F, the functor Sets --> F - vector spaces, S ~----+ F( S), has a right adjoint (the "forgetful functor"), and hence it preserves colimits ([MacL], p. 115). Thus we obtain the following lemma.

LEMMA. For any field F, we have an F-linear isomorphism

lim FT(X,k) ~ F(T(X,k)) _ ___. XEX

1.5. PROPOSITION. rankG0(kX) =rankKo(kX) = IT(X,k)l.

PROOF. First, for each X E X, the Cartan map yields an isomorphism Ko(kX)®;z. Q ~ G0 (kX)®;z.Q which commutes with induction maps ([CR], The-orem (21.22)). Since colimits commute with tensor products ([MacL], p. 115), we conclude that

Ko(kX) ®;z. Q ~ Go(kX) ®z Q, and so we can concentrate on Go(kX) in the following.




In the case where char k = p > 0 fix a p-modular system ( F, A, k) and put xx = chA:x, as in Lemma (1.3)(i). If chark = 0 put F = k and xx = XkX as in Lemma (1.2). In both cases char F = 0 and the maps

Xx : Go(kX) 0z F --> FT(X,k) (X E X)

are F-linear isomorphisms which commute with induction maps. Hence, by Lemma (1.4), they yield an F-linear isomorphism

Go(kX) 0z F ~ lim FT(X,k) ~ F(T(X,k))_ --+

XEX

The proposition follows from this isomorphism. 0

1.6. Changing the group. Recall that, for any group f, F(f) = {all finite subgroups of r} is a category, as in (0.2). Any group homomorphism r.p: r- A. induces a functor F(r.p) : F(f) --> F(A.) by F(r.p)(X) = r.p(X) (X E F(f)) and F(r.p)((-)9) = (·)'P(o) (g E f). We are especially interested in the case when F(r.p) is a full functor: that is, for each pair xl' x2 of finite subgroups of r and each morphism g : r.p(Xl) --> r.p(X2) in F(A.), there exists a morphism f : X1 --> X2 in F(r) with g = F(r.p)(f). Explicitly, this means that if r.p(Xl)d ~ r.p(X2 ) for some dE A. then there exists g E f with Xf ~ X2 and r.p(g)d- 1 E C~(r.p(Xl)). An immediate consequence of fullness ofF( r.p) is that ker r.p must be torsion-free. For otherwise there exists {1) #- X E F(f) with r.p(X) = r.p({1)) and fullness implies xu ~ {1) for some g E r, a contradiction. The following lemma says, roughly speaking, that Go(kF(r)) and T(f, k) = T(F(r), k) can be computed using a suitable X inside A. instead of F(f) ifF( r.p) is known to be full.

LEMMA. Let r.p : r - A. be a group homomorphism such that F( r.p) is full and put X = { r.p( X) I X E F(f)}, viewed as a full subcategory ofF( A.). Then:

(i) Go(kF(f)) ~ Go(kX), and similarly for Ko and G0 .

(ii) The map T(r.p, k) : T(f, k) - T(A., k), grxG:F(r).k - r.p(g)~xG:F(aJ,k, is injective and has image T(X, k).

PROOF. (i) The functor F : F(f) --> X, F(X) = r.p(X) = F(r.p)(X) and F(f) = F(r.p(f)), is surjective on objects and morphisms, the latter by fullness of F(r.p). Hence F is clearly a final functor in the sense of [MacL], p. 213. Thus, by [MacL], Theorem 1 on p. 213, we have

lim Go(kr.p(X)) ~ Go(kX). --+

XEF(r)

Furthermore, since r.p(X) ~ X holds for all X E F(f), one has G0(kX) e:<

Go(kr.p(X)). Hence X~----+ Ga(kX) and X~----+ G0 (kr.p(X)) are naturally equiva-lent functors F(r) -->mod- Z. Therefore,

lim Go(kr.p(X)) ~ lim Ga(kX) = Go(kF(f)). --+ --+

XEF(r) XEF(r)




The same argument covers the cases of I<0 and Go as well. (ii) Suppose that <p(g!) and <p(g2) have the same orbit under Ll x G:F(A),k

(gl, Y2 E f p'). In other words,

holds for some d E Ll and some t E IZ. which is coprime to the order of <p(gi), which equals the order of Yl· Thus (<p(g2)) = (<p(gl))d, and so (g2) = (g1)g for some g E f with <p(g )d-1 E CA ( <p(gl)), by fullness of :F( <p). Therefore,

<p(g2) = <p(gi)'l'(g l = <p( (gDg),

and so g2 = (gl)g, since <p is injective on (g2) = (gf). This shows that Yl and Y2 have the same orbit under r x G:F(r),b and hence proves injectivity of T(<p, k). The assertion about the image ofT( <p, k) is clear. 0

1. 7. Lemma (1.6) applies in particular when f is polycyclic-by-finite, as is shown by the following result. Part (i) is [L], Lemma 1, and it is easy to check that the proof given there also proves (ii).

LEMMA. Suppose that f is polycyclic-by-finite.

(i) There is an epimorphism <p : r -+ G, with G finite, such that :F( <p) is full.

(ii) Suppose that r is abelian-by-finite, with Hirsch number h(f). Let A be a maximal torsion free abelian normal subgroup of r. Then the group G in ( i) can be chosen to be r /(AI:F(r)l), and so to have order

If : AII:F(rWank(A).

An analogue of (ii) for any polycyclic-by-finite group r can be obtained in a similar fashion. But we leave the derivation of the exact statement to the interested reader, because we hope in the future to obtain better bounds for the exponent of the torsion subgroup of Go(kf) than it would yield in conjunction with the argument of (3.3).

1.8. Application to polycyclic-by-finite groups. In this subsection, f denotes a polycyclic-by-finite group and X is a collection of finite subgroups of r that is closed under r -conjugation and under taking intersections, as usual. The induction maps Ind~ : G0 (kX) -+ G0(kf) (X E X) combine to give a homomorphism

Ind~ : G0(kX)-+ Go(kf), and similarly for I<o instead of Go.

PROPOSITION. The homomorphism Ind~ has finite kernel, for both Go and I<o.

PROOF. By Lemma (1.7), we can choose a finite image G of r such that :F( <p) is full, where <p : r -+ G. It follows at once that X has only finitely many




f-orbits (at most the number of G-orbits in F(G)). So Go(kX) and I<o(kX) are finitely generated abelian groups, and it suffices to show that ker(Ind~) is torsion. Since I< a( kX) @z Q = Go( kX) @z Q, via Cartan maps ( cf. the proof of Proposition (1.5)), we can concentrate on G0 . Furthermore, exactly as in the proof of Proposition (1.5), there is a field F with char F = 0 and F-linear isomorphisms

Xx : Go(kX) ®z F ~ pT(X,k) (X E X or X = G)

which commute with induction maps, and hence give an F-linear isomorphism

XX : Go(kX) @z F _=-. pT(X,k)_

Using Lemma (1.6)(ii) for the above epimorphism cp, we see that T(cp, k) injects T(X, k) ~ T(f, k) into T(G, k). Since N = kercp is torsion-free, the maps cplx are all injective, and hence we have induction maps

Ind~ : G0(kX) ___. Go(kG) and Ind~ : Go(kX) ___.Go( kG).

The equalities pT(cp,k) o xx = XG o Ind~ (X E X) imply the commutativity of the following diagram:

pT(X,k) pT(v>,k)

pT(G,k)

xx r ~ ~ r Xa ((-) = (-) ®z idp ).

Go(kX) ®z F ---+ Ind~

Go(kG) ®z F

Thus, since pT(cp,k) is injective, so is Ind~. This shows that the kernel oflnd~ is torsion. To finish the proof, it suffices to show that Ind~ = ~ o Ind~ for a suitable homomorphism~ : Go(kf) ---> Go(kG). For this, note that the kernel N of cp is torsion free and hence of finite homological dimension ([P), 10.3.13); so we can define, for each finitely generated kf -module V,

~([V]) = L(-1)i[Hi(N, V)] E Go(kf/N) = G0(kG), i~O

as in [B2), p. 454. To check the claimed factorization of Ind~ we show that Ind~ = ~ o Ind~ holds for all X E X. Indeed, if W is a finitely generated kX-module then W®kX kfikN is free over kN, since NnX = (1). So H;(N, W®kX kf) = 0 fori"# 0 and Ho(N, W ®kX kf):::! W ®kX kf ®kN k ~ W ®kX kG. This proves the required equality and thus completes the proof of the proposition. D

In the special case when X = F = F(f) one knows that lnd~ is onto, by Moody's induction theorem [Mo]. Therefore, the above proposition combined with Proposition (1.5) imply the following result from [L].




CoROLLARY. If r is polycyclic-by-finite then rank Go(kf) = jT(f, k)J, and

the torsion subgroup of Go(kr) is a homomorphic image of the torsion subgroup

of Go(kF(f)).

1.9. According to [Q], Corollary (1.5)(c), G0(Qr) ~ G0 (QF(f)) when r is polycyclic-by-finite. It would be interesting to have a purely algebraic proof of this fact, and to know whether the analogue for other coefficient fields is valid. The fact that this is the case for a field k of characteristic p > 0, for certain polycyclic-by-finite groups r with only p-torsion, is the main result of [LP].

2. Torsion in the colimits

2.1. In this section, we study torsion in the colimits G0(kX) and K 0 (kX). We put

1r(X) =the set of all primes q with qllXI for some X EX, IXlp = sup{jXjp: X EX}; (if p = 0 we set IXlp = 1).

For simplicity, we assume throughout that X consists of finitely many r -conjug-acy classes. This is always true if r is polycyclic-by-finite (1. 7) or if r is arith-metic (cf. [S], Theorem 3 on p. 172). In particular, IXIP < oo, 1r(X) is a finite set, and Go(kX) and Ko(kX) are finitely generated abelian groups. Our main results are as follows:

• If p = char k > 0 then Ko(kX) and G0 (kX) ·IXlp have no p-torsion and the p'-torsion subgroups of Ko(kX) and Go(kX) are isomorphic (2.2 and 2.3).

• Go(kX) and Ko(kX) have at most X-torsion, at least when k is large enough for X: the group Ga(kX)0zZ[7r(X)- 1] = Ko(kX)0zZ[7r(X)- 1]

is torsion-free (2.5). • Reduction to characteristic 0: If p = chark > 0 and (F,A,k) is a p-

modular system then the torsion subgroup of K 0 (kX) embeds as a direct summand into the torsion subgroup of G0(F X)= Ka(F X) (2.4).

2.2. LEMMA. If char= p > 0 then K 0 (kX) has no p-torsion.

PROOF. We use the k-linear isomorphisms

XkX = XkX ®z idk : Ka(kX) ®z k-+ e<X,k)

of Lemma (1.2). Since the Hattori-Stallings ranks XkX commute with induction, we obtain an isomorphism

XH' : Ko(kX) ®z k ~ lim (Ko(kX) ®z k) ~ lim kT(X,k) ~ e(x,k) ---+ ---+

XEX XEX

where the last isomorphism follows from Lemma (1.4). By Proposition (1.5), we also know that K 0 (kX) has rank jT(X, k)j. Thus all torsion in Ko(kX) must be annihilated by(-) ®z k, and hence there can be no p-torsion. 0




2.3. The Cartan map. Recall that Go(kX) denotes the cokernel of the Cartan map ex : K 0 (kX) -+ G0(kX). Thus, for each finite group X, one has a short exact sequence

ex -0 -+ Ko(kX) --+ Go(kX) -+ Go(kX) -+ 0,

and G0(kX) ·IXIv = 0, ([CR], Theorem 21.22). The following lemma establishes analogous facts for colimits.

LEMMA. There is a short exact sequence

cx=limcx ---+

0-+ K 0 (kX) ----+ Go(kX) -+ Go(kX) -+ 0,

and Go(kX) ·IXIv = 0.

PROOF. We may assume that p = char k > 0, for otherwise ex is an isomor-phism. Since G0 (kX) is generated by the images of all Go(kX) (X E X), it is clearly annihilated by jXjp and the canonical map Go(kX) -+ Go(kX) is onto. Thus only injectivity of ex = lim ex must be proved. For this note that the

--+X ex map

K 0(kX) ®z IZ [t] ~ Go(kX) ®z IZ [t] is an isomorphism, because this is true in each component. Since K0 (kX) embeds into K 0 (kX) ®z IZ [*], by Lemma (2.2), we deduce that Ko(kX) -+ Ko(kX) ®z

IZ [*] -+ G0(kX) ®z IZ [*] is a monomorphism, and hence so is ex. D

CoROLLARY. If char k = p > 0, then Go( kX) · IX lv has no p-torsion and the p'-torsion subgroup ofGo(kX) is isomorphic to the torsion subgroup of Ko(kX).

PROOF. In view of Lemma (2.2), all p-torsion of Go(kX) injects into G0 (kX) and hence is annihilated by IXIv· The assertion about p'-torsion is clear, since Go(kX) is a p-group. D

2.4. Reduction to characteristic 0.

LEMMA. Let (F, A, k) be a p-modular system, with p = char k > 0. Then the torsion subgroup of Ko( kX) embeds as a direct summand into the p'-torsion subgroup of Ko(F X) = Go(F X).

PROOF. We use the Cartan-Brauer diagram (0.4 and 2.3): dx

Go(FX) - Go(kX)

Ko(kX) = Ko(kX)

By Corollary (2.3), ex yields an isomorphism of the torsion subgroup of K 0 (kX) with the p'-torsion subgroup of Go(kX). Thus the lemma follows by restricting the above diagram to p'-torsion subgroups. D




2.5. THEOREM. Put R := .:l [11"(X)- 1 , 8x\) where 8x,k is the degree of the

field extension of k that is obtained by adjoining all J.tx (X EX) to k. Then

I<o(kX) @z R ~ Go(kX) @z R ~ R(T(X,k))_

PROOF. The first isomorphism being clear, we can concentrate on K 0 . More-over, in view of Proposition (1.5), the proof of the theorem amounts to showing that Ko(kX) @z R is torsion-free. Put kx k(J.tx IX E X) ~ k. Then the composite map

scalar extension restriction Ko(kX) -----+ Ko(kxX) ---> Ko(kX)

is just multiplication by 8x ,k, because this is true for each X E X. Thus I<0 (kX) Q9 R embeds into I<0(kxX) Q9 R and so, after replacing k by kx, we may assume that k is large enough for X. If char k = p > 0 then choose a p-modular system ( F, A, k) so that F is large enough for X. (The required roots of unity of p'-order are automatically in F, and the p-power roots of unity can be adjoined if necessary.) Lemma (2.4) allows us to replace k by F, and so we can assume that chark = 0 in the following. Put Qx = Q(J.tx: X EX)~ k,

as above. Then scalar extension gives isomorphisms from K0 (QxX) to K0 (kX) for all X EX, and these isomorphisms yield K0 (QxX) ~ I<0 (kX). Thus, upon replacing k by Qx, we may assume that [k: Q] < oo.

Now let q be a given prime with q tf_ 11"(X). Extend the q-adic valuation of Q to k and complete k with respect to this valuation to obtain an extension E of k that is complete with respect to a discrete valuation and with residue field, L say, of characteristic q. Note that L is large enough for X (e.g., [CR], Cor. 17.2) and that I<0 (kX) ~ K 0(LX) via extension of scalars, as above. The decomposition map

dx : Ko(EX) -+ I<o(LX)

is an isomorphism because, for each X EX, dx : K0 (EX)-+ K0 (LX) is surjec-tive and both groups are free abelian of rank IT(X)I, the number of conjugacy classes of X. Thus all dx are isomorphisms, and hence so is dx. But K0 (LX) has no q-torsion, by Lemma (2.2), and hence neither does K 0 (EX). This completes the proof. D

Of course, it would be useful to dispense with adjoining 8x \ in the above theorem. For this it would be enough to show that the scalar extension map K0 (kX)-+ K 0 (KX) for a finite field extension K/k is injective, at least when chark = 0.

2.6. Application to polycyclic-by-finite groups. Let r be a polycyclic-by-finite group and put

:F = :F(r), 71" = 11"(F), the set of all primes q for which r has non-trivial q-torsion, 8 = 8:F,k, the field degree of Theorem (2.5).



GROTHENDIECK GROUPS OF GROUP ALGEBRAS

Then, by Moody's theorem [Mo], the induction map

Ind~: Go(k:F)-+ Go(kf)

105

is surjective and, by Proposition (1.7), it has finite kernel. Thus Theorem (2.5) implies the following result.

THEOREM. Put R = ~[6- 1 , 11"- 1). Then

Go(kr) ®z R ~ Go(k:F) ®z R = R(T(r,k))_

We now briefly consider the Cartan map

cr : Ko(kf)-+ Go(kr)

in the case when char k = p > 0. Define: Ind~ cp

A= kernel of a: Ko(kr)-- Ka(H)-+ crKo(kf); B = kernel of f3 = Ind~ : Go(k:F) -+ Go(kr); C = kernel of 1 = Ind~ : Go(k:F) -+ Go(kf); and D = co kernel of a.

By Moody's theorem the maps f3 and 1 are surjective.

LEMMA. The groups A, B, C and D are finite abelian groups and C and D are p-groups. Moreover, there is an exact sequence

0 -+ A -+ B -+ C -+ D -+ 0.

PROOF. By Lemma (2.3), we know that Go(kF) is a finite p-group, and hence so is C. Furthermore, by Proposition (1.8), B is finite (in fact, a finite 1r U {6}-group). Finally, the exact sequence follows from the Snake Lemma applied to the diagram:

0---+ Ko(k:F) CF

---+ Go(k:F) ---+ Go(k:F) - 0

1a 1{3 1~ 0 ---+ crKo(kr)

incl. Go(kr) ---+ Go(kr) -o. o

We conclude this section by mentioning the following two facts: ( 1) C = 0 if all finitely generated k r -modules are almost injective for all kX

(X EX), in the sense of[LP]. This happens, for example, if r has a torsion-free normal subgroup N having

finite index in r such that, for all X E :F, the group algebra kN is Noetherian as a module over the fixed ring (kN)x (which is certainly the case if N is abelian and might in fact be always true) and {n EN I Cx(n) =F (1)} is finite (see [LP]). To prove (1), one can proceed as in Dress' induction theorem for finite groups, [D], or [0], or see Lemma (3.2) below, using the modified restriction map from [LP] instead of ordinary restriction.

(2) A = 0 if Ko(k:F) is torsion-free.




This is of course trivial. The remark applies, however, when r has only p-torsion (p = chark > 0), for then K 0 (kX) = Ind~) K0 (k) holds for all X E :F and so K 0 (k:F) ~ ~- K 0 (k:F) is also torsion-free if any two distinct maximal finite subgroups of r have trivial intersection. For, in this case one has

Ko(k:F) ~ Ko(k) EB Lel Ho(Nr(X), Ko(kX)), X

where X runs through a complete set of non-conjugate maximal finite subgroups of rand Ko(kX) = Ko(kX)/([kX]) is the projective class group of kX.

In view of the above lemma, (1) and (2) together imply that B = D = 0 as well, and hence

Go(kf) ~ Go(k:F), Go{kf) ~ Go(k:F), crKo(kf) ~ Ko(k:F).

These remarks apply to the explicit examples of polycyclic group algebras that were considered in [LP], for example, but the description of A, B, C and Din general needs further work.

3. The torsion subgroup of G0{kf)

3.1. Throughout this section we assume that r is polycyclic-by-finite. By Moody's theorem [Mo], G0(kf) is an epimorphic image of G0(k:F(f)) under Ind~, and by Corollary (1.7) this yields an epimorphism on torsion subgroups. Let T0 (kf) denote the torsion subgroup of G0(kf), so T0 (kf) is finite by the above. The results of §2 yield the following:

THEOREM.

(i) When k is large enough for :F(r), T0 (kf) is an :F(f)-group. (i) Suppose char k = p > 0. The p-primary component of T0(kf) has expo-

nent dividing I:F(f)lp·

In view of the remarks preceding this theorem, (i) is a consequence of (2.5), and (ii) follows from Corollary (2.3). Cases where the maximum exponent per-mitted by (ii) is actually attained are described in [LP]. On the other hand, one cannot replace "dividing" by "equal to" in (ii). For example, let p be any prime, C the cyclic group of order p, f = ~I C, and let k be a field of charac-teristic p. The base group A of r is a free ~C-module, so that H 1(C,A) = 0 and r therefore has only one conjugacy class of maximal finite subgroups. Since G0 (kC) = ([k]) ~ ~. [Mo] yields Go(kC) =~;so To(kf) = {0} in this case.

3.2. Let G be a finite group. Recall that G0(kG) is a commutative ring with identity element [k], the multiplication being defined by [M][N] = [M QSlk N], for finitely generated kG-modules M and N ([CR], §16B). Let X be a collection of finite subgroups of G closed under taking intersections and G-conjugates. For H E X, L a finitely generated kH -module and M a finitely generated kG-module, define

[M][L] = [resj}(M) @k L] E Go(kH).




In this way Go(kH) is a Go(kG)-module. Thus so too is L:tex Go(kH), and the Frobenius identities show that the subgroup factored out to construct G0 (kX) is a G0 (kG)-submodule. Hence, Go(kX) is a Go(kG)-module, and Frobenius Reciprocity ([CR], Theorem (80.29)) ensures that Ind~ is a homomorphism of Go(kG)-modules.

Set G0 (kG)x = {y E Go(kG) : res~(y) = 0 for all HE X},

an ideal of Go( kG) by the Frobenius identities. By [CR], Corollary (80.33),

(1) IGIGo(kG) ~ (imind~) EB Ga(kG)x ~ Go(kG).

Let K<j denote kerlnd~. In view of Proposition (1.7) and the fact that Go(kG) is torsion free, K<j is the torsion subgroup of Ga(kX).

We claim that the annihilator of the G0 (kG)-module K<j contains the ideal (imind~) EB G0 (kG)x of G0(kG). Since it is clear that

(2) Go(kG)x · Go(kX) = 0,

our claim follows from:

LEMMA. (imind~) · K<j = 0.

PROOF. Let x E Kf, and let L:tEX L; ny,;XH,i E L~EX Ga(kH) be a preimage for x. Here nH,i E IZ for all H and i and, for given H E X, the elements XH,i E G0 (kH) denote the elements of the Grothendieck group representing the distinct irreducible kH-modules. Thus

(3) 0 = Ind~(x) = L L ny,;(xH,i ®kH kG). HEX i

(Here we are abusing notation by writing XH,i for a module as well as for its representative in the appropriate Grothendieck group.)

Let y be a generator of imind~, say y = [M ®kK kG], where K EX and M is a finitely generated kK-module. Calculating in G0(kX), and using Mackey's Subgroup Theorem ([CR], (10.13)) at (i) and (v), (with standard notation g H = gHg- 1 , etc.),

x · y = L L ny,i{xH,i · res~(y)} HEX i

(i) = L L nH,i L {xH,i ®k (9 M ®k(Hn•K) kH)} HEX i gEH\G/K

= L LnH,i 2: {(xH,i ®k 9 M) ®k(Hn•K) kH)} HEX i gEH\G/K

(ii) = L LnH,i E {(xH,i ®k 9 M) ®k(Hn•K) k9 K)} HEX i gEH\G/K

(iii) = L LnH,i E {(xH,i ®k(Hn•K) k9 K) ®k 9 MloK} HEX i gEH\G/K




(iv) L L nH,i L {(xk,; ®k( 9 -1 HnK) kK) ®k MIK} HEX i gEH\G/K

(v) = L LnH,d(xH,i ®kH kG)IK ®k MIK} HEX i

In the above, (ii) and (iv) are consequences of the defining relations for G0(kX), {iii) is a consequence of Frobenius Reciprocity ([CR], {10.29)), and (vi) follows from {3). 0

3.3. From {1) and {2) and Lemma {3.2) we deduce that

{4) IGI· KCj = 0.

Applying this conclusion in conjunction with Lemmas (1.7) and {1.6), one can obtain bounds on the exponent of the torsion subgroup G0(kf), where r is polycyclic-by-finite. In the expectation that sharper results will be obtained in the future, we state an explicit bound here only for the case where r is abelian-by-finite. For a non-zero integer n and a prime p dividing n, we write np for the p-part of n.

THEOREM 0 Let r be a finitely generated group with torsion free abelian normal

subgroup A of finite index. Let k be a field.

(i) The exponent ofT0 (kr) is a divisor of

If : AIIF{f) I rank( A).

(ii) Suppose that k is large enough for r. Then the exponent ofT0 (kf) is a

divisor of

PROOF. (i) By Lemmas (1.6) and {1.7) there is an epimorphism <p: r--+ G, with IGI = If : AIIF(r)l"ank(A), such that

Go{kF(f)) ~ Go(k<p(F{f))).

Hence, by ( 4), the exponent of the torsion subgroup of G0 (k.F(f)) has exponent dividing IGI, so the result follows from Corollary (1.8).

Now (ii) can be deduced from (i) and {3.1){i). 0

3.4. Theorem (3.3){i) shows that, when r is a split extension of a torsion free abelian group, the restriction to large enough fields can be removed from Theorem (3.1 )(i).



[B)

[B2) [Bou) [CR)

[D)

[FH)

[H)


REFERENCES

H. Bass, Euler characteristics and characters of discrete groups, Inventiones Math. 35 (1976), 155-196. ___ ,Algebraic K-theory, Benjamin, New York, 1968. N. Bourbaki, Algebre commutative, chap. 9, Masson, Paris, 1983. C.W. Curtis and I. Reiner, Methods of representation theory, 2 volumes, Wiley and Sons, New York, 1981 and 1987. A. Dress, Induction and structure theorems for orthogonal representations of finite groups, Annals of Math. 102 (1975), 391-325. F.T. Farrell and W.C. Hsiang, The Whitehead group of poly-(finite or cyclic) groups, J. London Math. Soc. 14 (1981), 308-324. A. Hattori, The rank element of a projective module, Nagoya J. Math. 25 (1965), 113-120.

[KM) P.H. Kropholler and B. Moselle, A family of crystallographic groups with 2-torsion in K 0 of the rational group algebra, to appear, Proc. Edin. Math. Soc. (1989).

[L) M. Lorenz, The rank of Go for polycyclic group algebras, In: Banach Center Publica-tions, Vol. 26, PWN, Warsaw, 1990, pp. 45-53.

[LP) ___ and D.S. Passman, The structure of Go for certain polycyclic group algebras and related algebras, Contemporary Math. 93 (1989), 283-302.

[MacL) S. MacLane, Categories for the working mathematician, Springer, New York, 1971. [Mo) J.A. Moody, Brauer induction for Go of certain infinite groups, J. Algebra 122 (1989),

1-14. [OJ R. Oliver, The Whitehead groups of finite groups, Cambridge University Press, Cam-

bridge, 1988. [P) D.S. Passman, The algebraic structure of group rings, Wiley and Sons, New York, 1977. [Q) F. Quinn, Algebraic K -theory of poly-(finite or cyclic) groups, Bull. Amer. Math. Soc.

12 (1985), 221-226. [SJ D. Segal, Polycyclic groups, Cambridge University Press, Cambridge, 1983. [Se) J.P. Serre, Representations lineaires des groupes finis, 2nd ed., Hermann, Paris, 1971.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF GLASGOW, GLASGOW G12 BQW, U.K.

DEPARTMENT OF MATHEMATICS, TEMPLE UNIVERSITY, PHILADELPHIA, PA 19122, USA



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