The Realization Problem for Finitely Generated Refinement Monoids

We show that every finitely generated conical refinement monoid can be represented as the monoid $\mathcal{V}(R)$ 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 $\mathcal{V}(Q_K(E,C))$.