The Realization Problem for finitely generated refinement monoids
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We show that every finitely generated conical refinement monoid can be represented as the monoid V(R) of isomorphism classes of finitely generated projective modules over a von Neumann regular ring R. To this end, we use the representation of these monoids provided by adaptable separated graphs. Given an adaptable separated graph (E, C) and a field K, we build a von Neumann regular K-algebra Q_K(E,C) and show that there is a natural isomorphism between the separated graph monoid M(E,C) and the monoid V(Q_K(E,C)).
Subjectsvon Neumann regular ring; refinement monoid; realization problem; universal localization
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