Simplicial and dimension groups with group action and their realization

TL;DR

This paper introduces and studies simplicial and dimension groups with group actions, showing their structure and realization via graded rings, extending previous theories to non-abelian groups and graded ring contexts.

Contribution

It generalizes simplicial and dimension groups with group actions, proves their structure as direct limits, and demonstrates their realization through graded rings, including non-abelian groups.

Findings

Dimension $ ext{Γ}$-groups are direct limits of simplicial $ ext{Γ}$-groups.

Every simplicial $ ext{Γ}$-group with an order-unit can be realized by a graded matricial ring.

Countable dimension $ ext{Γ}$-groups can be realized by $ ext{Γ}$-graded ultramatricial rings.

Abstract

We define simplicial and dimension $\Gamma$-groups, the generalizations of simplicial and dimension groups to the case when these groups have an action of an arbitrary group $\Gamma.$ Assuming that the integral group ring of $\Gamma$ is Noetherian, we show that every dimension $\Gamma$-group is isomorphic to a direct limit of a directed system of simplicial $\Gamma$-groups and that the limit can be taken in the category of ordered groups with order-units or generating intervals. We adapt Hazrat's definition of the Grothendieck $\Gamma$-group $K_0^{\Gamma}(R)$ for a $\Gamma$-graded ring $R$ to the case when $\Gamma$ is not necessarily abelian. If $G$ is a pre-ordered abelian group with an action of $\Gamma$ which agrees with the pre-ordered structure, we say that $G$ is {\em realized} by a $\Gamma$-graded ring $R$ if $K_0^{\Gamma}(R)$ and $G$ are isomorphic as pre-ordered $\Gamma$-groups with an isomorphism which preserves order-units or generating intervals. We show that every simplicial $\Gamma$-group with an order-unit can be realized by a graded matricial ring over a $\Gamma$-graded division ring. If the integral group ring of $\Gamma$ is Noetherian, we realize a countable dimension $\Gamma$-group with an order-unit or a generating interval by a $\Gamma$-graded ultramatricial ring over a $\Gamma$-graded division ring. We also relate our results to graded rings with involution which give rise to Grothendieck $\Gamma$-groups with actions of both $\Gamma$ and $\mathbb Z_2$. We adapt the Realization Problem for von Neumann regular rings to graded rings and concepts from this work and discuss some other questions.