Congruence Lattice Problem - Congruence Lattices of Lattices and Nonstable K-theory of Von Neumann Regular Rings

Congruence Lattices of Lattices and Nonstable K-theory of Von Neumann Regular Rings

We recall that for a (unital, associative) ring R, we denote by V(R) the (conical, commutative) monoid of isomorphism classes of finitely generated projective right R-modules, see here for more details. Recall that if R is von Neumann regular, then V(R) is a refinement monoid. Denote by Id_c R the (∨,0)-semilattice of finitely generated two-sided ideals of R. We denote by L(R) the lattice of all principal right ideals of a von Neumann regular ring R. It is well known that L(R) is a complemented modular lattice.

The following result was observed by Wehrung, building on earlier works mainly by Jónsson and Goodearl.

Theorem (Wehrung 1999). Let R be a von Neumann regular ring. Then the (∨,0)-semilattices Id_c R and Con_c L(R) are both isomorphic to the maximal semilattice quotient of V(R).

Bergman proves in a well-known unpublished note from 1986 that any at most countable distributive (∨,0)-semilattice is isomorphic to Id_c R, for some locally matricial ring R (over any given field). This result is extended to semilattices of cardinality at most ℵ₁ in 2000 by Wehrung, by keeping only the regularity of R (the ring constructed by the proof is not locally matricial). The question whether R could be taken locally matricial in the ℵ₁ case remained open for a while, until it was disproved by Wehrung in 2004. Translating back to the lattice world by using the theorem above and using a lattice-theoretical analogue of the V(R) construction, called the dimension monoid, introduced by Wehrung in 1998, yields the following result.

Theorem (Wehrung 2004). There exists a distributive (∨,0,1)-semilattice of cardinality ℵ₁ that is not isomorphic to Con_c L, for any modular lattice L every finitely generated sublattice of which has finite length.