Simple dimension groups that are isomorphic to stationary inductive limits

TL;DR

This paper provides a concrete structural description of stationary simple dimension groups, showing how they are built from the kernel and image of their trace, and explores related algebraic properties.

Contribution

It characterizes stationary simple dimension groups explicitly in terms of their trace kernel and image, extending previous results to a broader class of groups.

Findings

Every stationary simple dimension group contains the direct sum of the kernel and image of its trace.

Such groups are generated by this direct sum plus finitely many additional elements.

For any positive integer m and integer matrix B, there exist two distinct powers of B whose difference is divisible by m.

Abstract

A dimension group is an ordered abelian group that is an inductive limit of a sequence of simplicial groups, and a stationary dimension group is such an inductive limit in which the homomorphism is the same at every stage. If a simple dimension group is stationary then up to scalar multiplication it admits a unique trace (positive real-valued homomorphism), but the short exact sequence associated to this trace need not split. In an earlier paper, Handelman described these ordered groups concretely in the case when the trace has trivial kernel---i.e., the group is totally ordered---and in the case when the group is free. The main result here is a concrete description of how a stationary simple dimension group is built from the kernel and image of its trace. Specifically, every stationary simple dimension group contains the direct sum of the kernel of its trace with a copy of the image, and is generated by that direct sum and finitely many extra elements. Moreover, any ordered abelian group of this description is stationary. The following interesting fact is proved along the way to the main result: given any positive integer m and any square integer matrix B, there are two distinct integer powers of B, the difference of which has all entries divisible by m.