A TORSION PROJECTIVE CLASS FOR A GROUP ALGEBRA
IAN J. LEARY
1. Introduction
Let G be the group given by the following presentation:
G = : (1.1)
The subgroup generated by ab is infinite-cyclic and is normal, with quotient the
dihedral group of order 6, and so G is cyclic-by-finite. The subgroups H = and
K = ** are both dihedral of order 6, and G is isomorphic to the free produc*
*t of
H and K amalgamating L = H \ K. We study K0(kG), the Grothendieck group of
isomorphism classes of finitely generated projective kG-modules, and in particu*
*lar
the dependence of K0(kG) on the choice of field k. As usual, let Q, R and C sta*
*nd
for the rationals, reals and complex numbers respectively. We prove
Theorem 1. There is an element of order two in K0(QG), whose image in
K0(RG) is non-zero, but whose image in K0(CG) is zero.
There are two published accounts of groups G and fields k for which K0(kG)
contains torsion. P. H. Kropholler and B. Moselle exhibited crystallographic gr*
*oups
G for which K0(kG) contains elements of order two, for any field k of character-
istic zero [3]. They used F. Waldhausen's work on the algebraic K-theory of free
products with amalgamation [11 ]. More recently, M. Lorenz has exhibited crys-
tallographic groups for which K0(kG) contains 3- and 4-torsion for any field k *
*of
characteristic zero [5]. Lorenz's techniques appear to be unable to detect tors*
*ion
that is annihilated by field extensions. Moselle's Ph. D. thesis contained exam*
*ples of
crystallographic groups with torsion in K0(QG) but not in K0(CG) [7], but relied
on a theorem of F. Quinn [8] for which no full proof has appeared.
It seems to have gone unnoticed, or at least unremarked, that results due to
G. M. Bergman [1], W. Dicks [2], and F. Waldhausen [11 ] can be used to exhibit
groups G having torsion of any order in K0(kG), and torsion in K0(QG) that dies
in K0(CG). Examples of this kind are to be given in [4]. Our purpose here is to
give a topological proof of Theorem 1, using the following theorem of R. G. Swa*
*n:
Theorem 2 (Swan, [10 ]). Let X be a compact Hausdorff space, and let R be
the ring of real-valued continuous functions on X. The functor from real vector-
bundles over X to R-modules taking a bundle to its sections induces an isomorph*
*ism
from the real topological K-group KO0(X) to the algebraic K-group K0(R).
____________________________________________________________________________
Work supported in part by EPSRC grant GR/L69398 and by a grant from the Nuf*
*field
Foundation
1991 Mathematics Subject Classification 19A31, 16S34, 55N15.
2 ian j. leary
We also use the Morita invariance of K0: for any R, the non-unital ring homo-
morphism including R in the top left corner of the matrix ring Mn(R) induces an
isomorphism from K0(R) to K0(Mn(R)). The class of the free R-module of rank
one is mapped by this isomorphism to the class of the module of column vectors,
Rn. As a general reference for K-theory, including proofs of Swan's theorem and
Morita invariance, we recommend [9].
2. The proof
Throughout this section, G stands for the group presented in equation 1.1, a*
*nd
H, K are the subgroups generated by {a; c} and {b; c} respectively.
Proof. The groups H and K are both isomorphic to the dihedral group of order
6, so have 3 isomorphism classes of irreducible representations over Q. Let ff *
*(resp.
ff0) denote the faithful irreducible 2-dimensional representation of H (resp. o*
*f K).
The element of K0(QG) whose existence is claimed in Theorem 1 is defined by
= IndGH(ff) - IndGK(ff0): (2.1)
In terms of idempotents, ff may be represented by 1_6(1 + a)(2 - c - c2) in QH,*
* and
IndGH(ff) by the same element in QG. Similarly, ff0 and IndGK(ff0) are represen*
*ted
by the element 1_6(1 + b)(2 - c - c2).
Over C, there is a 1-dimensional faithful representation fl of L = H \ K = <*
*c>
such that ff = IndHL(fl) and ff0= IndKL(fl). It follows that in K0(CG),
= IndGH(ff) - IndGK(ff0) = IndGL(fl) - IndGL(fl) = 0: (2.2)
The same argument applies over any field of characteristic zero containing !, a
primitive cube root of 1. In terms of idempotents, the idempotents representing*
* ff
and ff0 are both conjugate to 1_3(1 + !c + !2c2) in CG.
The representation fl is not defined over every field of characteristic zero*
*, but
there is a 2-dimensional representation fi of L defined over Q such that 2ff =
IndHL(fi) and similarly for ff0. It follows that in K0(QG),
2 = 2IndGH(ff) - 2IndGK(ff0) = IndGL(fi) - IndGL(fi) = 0: (2.3)
It remains to show that the image of in K0(RG) is non-zero. Let R denote
the ring of continuous functions from the circle R=Z to R. The matrix ring M2(R)
may be identified with the ring of continuous functions from the circle to M2(R*
*).
Define a function OE from the generators of G to M2(R) by
OE(a)= t 7! 01 10
OE(b)= t 7! cos2sstsisin2sstn2sst- cos2sst (2.4)
OE(c)= t 7! cos2ss=3s-isin2ss=3n2ss=3cos2ss=3
The group relations are satisfied by the images, and OE extends R-linearly to a
homomorphism OE : RG ! M2(R). View M2(R) as the endomorphism ring of R2,
and view R2 as the sections of a 2-dimensional trivial vector bundle over the c*
*ircle.
The idempotents representing ff and ff0 are sent by OE to a projection onto a 1-
dimensional trivial sub-bundle and to a projection onto a M"obius band sub-bund*
*le
a torsion projective class for a group algebra 3
respectively. It follows that ff and ff0 represent distinct elements of K0(RG),*
* since
the M"obius bundle over the circle is not stably trivial [6].
The above argument shows that 2 K0(kG) is non-zero when k embeds in R,
and is zero when k contains a primitive cube root of 1. There are subfields of C
that satisfy neither of these conditions. An argument using either Bergman's or
Waldhausen's exact sequence for the K-theory of free products [1, 11] shows that
2 K0(kG) is zero if and only if k contains a primitive cube root of 1 [4].
Acknowledgements. The author thanks Brian Bowditch who gave a series
of lectures on K-theory while the author was considering this problem, and than*
*ks
Peter Kropholler for encouragement with this work. The author also thanks Warren
Dicks for helpful comments on an earlier version of this paper.
References
1. G. M. Bergman, `Modules over coproducts of rings', Trans. Amer. Math. Soc. *
*200
(1974), 1-32.
2. W. Dicks, `The HNN construction for rings', J. of Algebra 81 (1983), 434-48*
*7.
3. P. H. Kropholler and B. Moselle, `A family of crystallographic groups with *
*2-torsion in
K0 of the rational group algebra', Proc. Edinburgh Math. Soc. 34 (1991),*
* 325-331.
4. I. J. Leary, `The Euler class of a Poincare duality group', in preparation.
5. M. Lorenz, `Picard groups of multiplicative invariants', Comment. Math. Hel*
*vetici 72
(1997) 389-399.
6. J. W. Milnor and J. D. Stasheff, Characteristic Classes Annals of Math. Stu*
*dies 76
(Princeton Univ. Press 1974).
7. B. Moselle, `Homological properties of polycyclic group algebras', Ph.D. th*
*esis, Queen
Mary College, London (1990).
8. F. Quinn, `Algebraic K-theory of poly-(finite or cyclic) groups', Bull. Ame*
*r. Math.
Soc. ser. (1) 12 (1985), 221-226.
9. J. Rosenberg, `Algebraic K-theory and its applications', Graduate Texts in *
*Mathemat-
ics 147 (Springer 1994).
10. R. G. Swan, `Vector bundles and projective modules', Trans. Amer. Math. Soc*
*. 105
(1962), 264-277.
11. F. Waldhausen, `Algebraic K-theory of generalized free products', Parts I a*
*nd II,
Annals of Math. 108 (1978), 135-204 and 205-256.
Faculty of Mathematical Studies
University of Southampton
Southampton
SO17 1BJ
**