Simplicial and dimension groups with group action and their realization

Abstract

We define simplicial and dimension $Γ$-groups, the generalizations of simplicial and dimension groups to the case when these groups have an action of an arbitrary group $Γ.$ Assuming that the integral group ring of $Γ$ is Noetherian, we show that every dimension $Γ$-group is isomorphic to a direct limit of a directed system of simplicial $Γ$-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 $Γ$-group $K_0^Γ(R)$ for a $Γ$-graded ring $R$ to the case when $Γ$ is not necessarily abelian. If $G$ is a pre-ordered abelian group with an action of $Γ$ which agrees with the pre-ordered structure, we say that $G$ is {\em realized} by a $Γ$-graded ring $R$ if $K_0^Γ(R)$ and $G$ are isomorphic as pre-ordered $Γ$-groups with an isomorphism which preserves order-units or generating intervals. We show that every simplicial $Γ$-group with an order-unit can be realized by a graded matricial ring over a $Γ$-graded division ring. If the integral group ring of $Γ$ is Noetherian, we realize a countable dimension $Γ$-group with an order-unit or a generating interval by a $Γ$-graded ultramatricial ring over a $Γ$-graded division ring. We also relate our results to graded rings with involution which give rise to Grothendieck $Γ$-groups with actions of both $Γ$ 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.