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Article Dans Une Revue Proceedings of the London Mathematical Society Année : 2003

Liftings of diagrams of semilattices by diagrams of dimension groups

Jiri Tuma
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Résumé

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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Dates et versions

hal-00004021 , version 1 (21-01-2005)

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Jiri Tuma, Friedrich Wehrung. Liftings of diagrams of semilattices by diagrams of dimension groups. Proceedings of the London Mathematical Society, 2003, 87 (3), pp.1-28. ⟨10.1112/S0024611502013941⟩. ⟨hal-00004021⟩
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