2005/01/22 by Jiri Tuma, Friedrich Wehrung
Mathematics · #math.GM #math.KT #math.RA #msc:06A12 #msc:06C20 #msc:06F20 #msc:15A03 #msc:15A24 #msc:15A48 #msc:16E20 #msc:16E50 #msc:19A49 #msc:19K14
published as Proceedings of the London Mathematical Society 87, no. 3 (2003) 1--28
arxiv created 2005/01/22 · arxiv updated 2009/12/01
We investigate categorical and amalgamation properties of the functor Idc assigning to every partially ordered abelian group G its semilattice of compact ideals Idc 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 Idc functor, by a diagram of pseudo-simplicial vector spaces. Pseudo-simplicial vector spaces are a special kind of finite-dimensional partially ordered vector spaces (over the rationals) with interpolation. The methods introduced make it also possible to prove the following ring-theoretical result. Theorem 2. For any countable distributive join-semilattices S and T and any field K, any (v,0)-homomorphism f: S→ T can be lifted, with respect to the Idc functor on rings, by a homomorphism f: A→ 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 Conc 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.