Liftings of Diagrams of Semilattices by Diagrams of Dimension Groups

Abstract

We investigate categorical and amalgamation properties of the functor $\operatorname{Id_c}$ assigning to every partially ordered abelian group $G$ its $\langle \vee, 0 \rangle$-semilattice of compact ideals $\operatorname{Id_c} G$. Our main result is the following.

Theorem 1. Every diagram of finite Boolean semilattices indexed by a finite dismantlable partially ordered set can be lifted, with respect to the$\operatorname{Id_c}$functor, by a diagram of pseudo-simplicial vector spaces.

Pseudo-simplicial vector spaces are a special kind of finite-dimensional partially ordered vector spaces with interpolation over the field of rational numbers. The methods introduced also make it possible to prove the following ring-theoretical result.

Theorem 2. For any countable distributive$\langle \vee, 0 \rangle$-semilattices$S$and$T$and any field$K$, any$\langle \vee, 0 \rangle$-homomorphism$f \colon S \to T$can be lifted, with respect to the$\operatorname{Id_c}$functor on rings, by a homomorphism $f \colon A \to B$of$K$-algebras, for countably dimensional locally matricial algebras$A$and$B$over$K$.

We also state a lattice-theoretical analogue of Theorem 2 (with respect to the $\operatorname{Con_c}$ functor), and we provide counterexamples to various related statements. In particular, we prove that the result of Theorem 1 cannot be achieved with simplicial vector spaces alone.