Liftings of diagrams of semilattices by diagrams of dimension groups

Details Liftings of diagrams of semilattices by diagrams of dimension groups Liftings of diagrams of semilattices by diagrams of dimension groups

2005-01-22

arxiv.org/abs/math/0501376v1

Proceedings of the London Mathematical Society 87, no. 3 (2003) 1--28

Mathematics

General Mathematics

Scientific paper

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 $f: S\to T$ can be lifted, with respect to the Idc functor on rings, by a homomorphism $f: A\to B$ 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 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.

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