Refinement Monoid - Nonstable K-theory of Von Neumann Regular Rings

Nonstable K-theory of Von Neumann Regular Rings

For a ring (with unit) R, denote by FP(R) the class of finitely generated projective right R-modules. Equivalently, the objects of FP(R) are the direct summands of all modules of the form Rn, with n a positive integer, viewed as a right module over itself. Denote by the isomorphism type of an object X in FP(R). Then the set V(R) of all isomorphism types of members of FP(R), endowed with the addition defined by, is a conical commutative monoid. In addition, if R is von Neumann regular, then V(R) is a refinement monoid. It has the order-unit . We say that V(R) encodes the nonstable K-theory of R.

For example, if R is a division ring, then the members of FP(R) are exactly the finite-dimensional right vector spaces over R, and two vector spaces are isomorphic if and only if they have the same dimension. Hence V(R) is isomorphic to the monoid of all natural numbers, endowed with its usual addition.

A slightly more complicated example can be obtained as follows. A matricial algebra over a field F is a finite product of rings of the form =ring of all square matrices with n rows and entries in F, for variable positive integers n. A direct limit of matricial algebras over F is a locally matricial algebra over F. Every locally matricial algebra is von Neumann regular. For any locally matricial algebra R, V(R) is the positive cone of a so-called dimension group. By definition, a dimension group is a partially ordered abelian group whose underlying order is directed, whose positive cone is a refinement monoid, and which is unperforated, the letter meaning that mx≥0 implies that x≥0, for any element x of G and any positive integer m. Any simplicial group, that is, a partially ordered abelian group of the form, is a dimension group. Effros, Handelman, and Shen proved in 1980 that dimension groups are exactly the direct limits of simplicial groups, where the transition maps are positive homomorphisms. This result had already been proved in 1976, in a slightly different form, by P.A. Grillet. Elliott proved in 1976 that the positive cone of any countable direct limit of simplicial groups is isomorphic to V(R), for some locally matricial ring R. Finally, Goodearl and Handelman proved in 1986 that the positive cone of any dimension group with at most ℵ₁ elements is isomorphic to V(R), for some locally matricial ring R (over any given field).

Wehrung proved in 1998 that there are dimension groups with order-unit whose positive cone cannot be represented as V(R), for a von Neumann regular ring R. The given examples can have any cardinality greater than or equal to ℵ₂. Whether any conical refinement monoid with at most ℵ₁ (or even ℵ₀) elements can be represented as V(R) for R von Neumann regular is an open problem.