The realization problem for finitely generated refinement monoids

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 QK (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)).