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AbstractWe 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 K0Γ(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 K0Γ(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 ℤ2{\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.
Forum Mathematicum – de Gruyter
Published: May 1, 2022
Keywords: Dimension group; simplicial group; group action; Grothendieck group; ordered abelian group; realization; graded ring; 19A49; 06F20; 16W50; 16E20; 19K14
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