Simplicial and dimension groups with group action and their realization

Abstract

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 ) {K_{0}^{\Gamma}(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 realized by a Γ-graded ring R if K 0 Γ ⁢ ( R ) {K_{0}^{\Gamma}(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 ℤ 2 {\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.