维数群图对半格图的提升
Liftings of diagrams of semilattices by diagrams of dimension groups
摘要：
本文研究了赋给偏序阿贝尔群G的函子Idc的范畴性质和归并性质，给出了赋给偏序阿贝尔群G的紧理想半格Idc G的函子Idc的性质.我们的主要结果如下。定理1.有限布尔半格的每一个由有限可分偏序集索引的图都可以通过伪单纯向量空间的图相对于Idc函子提升。伪单纯向量空间是一类特殊的有限维偏序向量空间（有理数上）。所介绍的方法也使证明下列环理论结果成为可能。定理2.对于任意可数分配并半格S和T以及任意域K，对于K上的可数维局部矩阵代数A和B，任意（v，0）-同态$f: S\to T$都可以被K-代数的同态$f: A\to B$关于环上的Idc函子提升.我们还陈述了定理2（关于Conc函子）的一个格论类似物，并且我们提供了各种相关陈述的反例。特别地，我们证明了定理1的结果不能单独用单纯向量空间来实现。
英文摘要：
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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