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A note on the isomorphism problem for SK[G]
Published online by Cambridge University Press: 12 March 2014
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Let G be an infinite abelian p-group and let K be a field of characteristic ≠ p. Let K[G] be the set of all sums Σg∈Gagg where the ag are in K, and all but finitely many ag are 0. Then K[G] is a K-algebra, with multiplication induced by the group multiplication on G.
If G is countable, then the isomorphism type of K[G] has been completely described by S. D. Berman [1]. If G is a direct sum of countable groups, one can also describe K[G], as K[⊕iGi] ≃ ⊗iK[Gi]. If K contains all pn-th roots of unity, then K[G] is isomorphic to the ring of continuous functions from a Boolean space X to the field K with the discrete topology. In that case, the group UK[G] of invertible elements of K[G] is isomorphic to the direct sum of ∣G∣ copies of K×. More generally, if K is of the second kind with respect to p (see below for the definition), the group UK[G] has a simple description.
Consider the subgroup SK[G] of elements Σgagg which have order a power of p and such that Σg ag = 1. This group is of course much simpler than UK[G]. Classifying SK[G] up to isomorphism reduces to the case where G has no element of infinite height, see [7]. If G is a direct sum of cyclic groups then the isomorphism type of SK[G] has been completely determined, in [2, 3, 7, 8]. The aim of this note is to show that a similar result is in general not possible for uncountable G. We use an invariant Γ associated to abelian groups, and for any regular uncountable cardinal κ, exhibit 2κ groups G for which Γ(G) = Γ(SK[G]) are pairwise distinct. Our work is based on a construction of Shelah [9], who constructed 2κ non-isomorphic abelian p-groups of cardinality κ for κ an uncountable regular cardinal.
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