A TORSION PROJECTIVE CLASS FOR A GROUP ALGEBRA

IAN J. LEARY

1. Introduction

Let G be the group given by the following presentation:

G = ⟨a, b, c : a^{2} = b^{2} = c^{3} = 1, c^{a} = c^{b} = c^{2}⟩.  (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 = ⟨a, c⟩ and K = ⟨b, c⟩ are both dihedral of order 6, and G is isomorphic to the free product of H and K amalgamating L = H ∩ K. We study K_{0}(kG), the Grothendieck group of isomorphism classes of finitely generated projective kG-modules, and in particular the dependence of K_{0}(kG) on the choice of field k. As usual, let ℚ, ℝ and ℂ stand for the rationals, reals and complex numbers respectively. We prove

Theorem 1. There is an element of order two in K_{0}(ℚG), whose image in K_{0}(ℝG) is non-zero, but whose image in K_{0}(ℂG) is zero.

There are two published accounts of groups G and fields k for which K_{0}(kG) contains torsion. P. H. Kropholler and B. Moselle exhibited crystallographic groups G for which K_{0}(kG) contains elements of order two, for any field k of characteristic 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 crystallographic groups for which K_{0}(kG) contains 3- and 4-torsion for any field k of characteristic zero [5]. Lorenz’s techniques appear to be unable to detect torsion that is annihilated by field extensions. Moselle’s Ph. D. thesis contained examples of crystallographic groups with torsion in K_{0}(ℚG) but not in K_{0}(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 K_{0}(kG), and torsion in K_{0}(ℚG) that dies in K_{0}(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. Swan:

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 vectorbundles over X to R-modules taking a bundle to its sections induces an isomorphism from the real topological K-group KO^{0}(X) to the algebraic K-group K_{0}(R).

Work supported in part by EPSRC grant GR/L69398 and by a grant from the Nuffield Foundation

1991 Mathematics Subject Classification 19A31, 16S34, 55N15.

2 ian j. leary

We also use the Morita invariance of K_{0}: for any R, the non-unital ring homomorphism including R in the top left corner of the matrix ring M_{n}(R) induces an isomorphism from K_{0}(R) to K_{0}(M_{n}(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, R^{n}. 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, and 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 α (resp. α′) denote the faithful irreducible 2-dimensional representation of H (resp. of K). The element of K_{0}(ℚG) whose existence is claimed in Theorem 1 is ξ defined by

ξ = Ind_{H}^{G}(α) − Ind_{K}^{G}(α′).  (2.1)

In terms of idempotents, α may be represented by \frac{1}{6}(1 + a)(2 − c − c^{2}) in ℚH, and Ind_{H}^{G}(α) by the same element in ℚG. Similarly, α′ and Ind_{K}^{G}(α′) are represented by the element \frac{1}{6}(1 + b)(2 − c − c^{2}).

Over ℂ, there is a 1-dimensional faithful representation γ of L = H ∩ K = ⟨c⟩ such that α = Ind_{L}^{H}(γ) and α′ = Ind_{L}^{K}(γ). It follows that in K_{0}(ℂG),

ξ = Ind_{H}^{G}(α) − Ind_{K}^{G}(α′) = Ind_{L}^{G}(γ) − Ind_{L}^{G}(γ) = 0.  (2.2)

The same argument applies over any field of characteristic zero containing ω, a primitive cube root of 1. In terms of idempotents, the idempotents representing α and α′ are both conjugate to \frac{1}{3}(1 + ωc + ω^{2}c^{2}) in ℂG.

The representation γ is not defined over every field of characteristic zero, but there is a 2-dimensional representation β of L defined over ℚ such that 2α = Ind_{L}^{H}(β) and similarly for α′. It follows that in K_{0}(ℚG),

2ξ = 2Ind_{H}^{G}(α) − 2Ind_{K}^{G}(α′) = Ind_{L}^{G}(β) − Ind_{L}^{G}(β) = 0.  (2.3)

It remains to show that the image of ξ in K_{0}(ℝG) is non-zero. Let R denote the ring of continuous functions from the circle ℝ/ℤ to ℝ. The matrix ring M_{2}(R) may be identified with the ring of continuous functions from the circle to M_{2}(ℝ). Define a function ϕ from the generators of G to M_{2}(R) by

ϕ(a) = (t ↦ ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ))
ϕ(b) = (t ↦ ( \begin{matrix}cos 2πt & sin 2πt \\ sin 2πt & − cos 2πt \end{matrix} ))
ϕ(c) = (t ↦ ( \begin{matrix}cos 2π/3 & − sin 2π/3 \\ sin 2π/3 & cos 2π/3 \end{matrix} ))

(2.4)

The group relations are satisfied by the images, and ϕ extends ℝ-linearly to a homomorphism ϕ : ℝG → M_{2}(R). View M_{2}(R) as the endomorphism ring of R^{2}, and view R^{2} as the sections of a 2-dimensional trivial vector bundle over the circle. The idempotents representing α and α′ are sent by ϕ to a projection onto a 1dimensional trivial sub-bundle and to a projection onto a Möbius band sub-bundle

a torsion projective class for a group algebra 3

respectively. It follows that α and α′ represent distinct elements of K_{0}(ℝG), since the Möbius bundle over the circle is not stably trivial [6].

The above argument shows that ξ ∈ K_{0}(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 ξ ∈ K_{0}(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 thanks 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), 434–487.

3. P. H. Kropholler and B. Moselle, ‘A family of crystallographic groups with 2-torsion in K_{0} of the rational group algebra’, Proc. Edinburgh Math. Soc. 34 (1991), 325–331.

4. I. J. Leary, ‘The Euler class of a Poincaré duality group’, in preparation.

5. M. Lorenz, ‘Picard groups of multiplicative invariants’, Comment. Math. Helvetici 72 (1997) 389–399.

6. J. W. Milnor and J. D. Stasheff, Characteristic Classes Annals of Math. Studies 76 (Princeton Univ. Press 1974).

7. B. Moselle, ‘Homological properties of polycyclic group algebras’, Ph.D. thesis, Queen Mary College, London (1990).

8. F. Quinn, ‘Algebraic K-theory of poly-(finite or cyclic) groups’, Bull. Amer. Math. Soc. ser. (1) 12 (1985), 221–226.

9. J. Rosenberg, ‘Algebraic K-theory and its applications’, Graduate Texts in Mathematics 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 and II, Annals of Math. 108 (1978), 135–204 and 205–256.

Faculty of Mathematical Studies University of Southampton Southampton SO17 1BJ