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