Liftings of Diagrams of Semilattices by Diagrams of Dimension Groups
Copyright policy )Liftings of Diagrams of Semilattices by Diagrams of Dimension Groups
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.
flat, 15A48, dimension group, sectionally complemented modular lattice, 16E50, dimension vector space, 15A03, 19A49, 15A24, [MATH.MATH-KT] Mathematics [math]/K-Theory and Homology [math.KT], [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], direct limit, Complemented modular lattices, continuous geometries, General Mathematics (math.GM), \(K_0\) of other rings, [MATH.MATH-RA] Mathematics [math]/Rings and Algebras [math.RA], FOS: Mathematics, Grothendieck groups, \(K\)-theory, etc., congruence lattice, Mathematics - General Mathematics, 06A12, locally matricial algebra, generic, 19K14, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Distributive semilattice, Semilattices, [MATH.MATH-GM] Mathematics [math]/General Mathematics [math.GM], K-Theory and Homology (math.KT), Mathematics - Rings and Algebras, Ordered abelian groups, Riesz groups, ordered linear spaces, 16E20, dismantlable poset, ideal lattice, 06A12, 06C20, 06F20, 15A03, 15A24, 15A48, 16E20, 16E50, 19A49, 19K14, \(K_0\) as an ordered group, traces, Rings and Algebras (math.RA), semilattice, ultramatricial algebra, compact ideal, Mathematics - K-Theory and Homology, [MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT], von Neumann regular rings and generalizations (associative algebraic aspects), 06F20, 06C20
flat, 15A48, dimension group, sectionally complemented modular lattice, 16E50, dimension vector space, 15A03, 19A49, 15A24, [MATH.MATH-KT] Mathematics [math]/K-Theory and Homology [math.KT], [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], direct limit, Complemented modular lattices, continuous geometries, General Mathematics (math.GM), \(K_0\) of other rings, [MATH.MATH-RA] Mathematics [math]/Rings and Algebras [math.RA], FOS: Mathematics, Grothendieck groups, \(K\)-theory, etc., congruence lattice, Mathematics - General Mathematics, 06A12, locally matricial algebra, generic, 19K14, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], Distributive semilattice, Semilattices, [MATH.MATH-GM] Mathematics [math]/General Mathematics [math.GM], K-Theory and Homology (math.KT), Mathematics - Rings and Algebras, Ordered abelian groups, Riesz groups, ordered linear spaces, 16E20, dismantlable poset, ideal lattice, 06A12, 06C20, 06F20, 15A03, 15A24, 15A48, 16E20, 16E50, 19A49, 19K14, \(K_0\) as an ordered group, traces, Rings and Algebras (math.RA), semilattice, ultramatricial algebra, compact ideal, Mathematics - K-Theory and Homology, [MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT], von Neumann regular rings and generalizations (associative algebraic aspects), 06F20, 06C20
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