格序阿贝尔群的维数与Prüfer整环上模的模型论
Dimensions on Lattice Ordered Abelian Groups and Model Theory of Modules over Prüfer Domains
摘要：
我们证明了一个转移定理，当结合Jaffard-Kaplansky-Ohm定理，允许在Bézout域上的模的模型论中的结果通过它们的值群被转化为Prüfer域上的结果。推广了Puninski和Toffalori的工作，证明了Prüfer整环的值群的扩张正锥有m维当且仅当其pp- $1$ -公式格有宽度（等价宽度）且这些维数相等.进一步，我们证明了这些维数的存在性等价于pp- $1$ -公式的格具有m维（从而等价于它的Ziegler谱具有Cantor-Bendixson秩）和不存在可叠加的纯内射模.最后，我们根据Prüfer整环的值群的扩张正锥的m维给出了其pp-PDMATHAMM-格的m维的一个最佳可能上界，并证明了对于PDMATHAMO为极限序数的情形，所有不具有$λ+1$形式的序数都作为Bézout整环的pp- $1$ -格的m维出现.
英文摘要：
We prove a transfer theorem which, when combined with the Jaffard-Kaplansky-Ohm Theorem, allows results in model theory of modules over Bézout domains to be translated into results over Prüfer domains via their value groups. Extending work of Puninski and Toffalori, we show that the extended positive cone of the value group of a Prüfer domain has m-dimension if and only if its lattice of pp-$1$-formulae has breadth (equivalently width) and that these dimensions are equal. Further, we show that the existence of these dimensions is equivalent to the lattice of pp-$1$-formulae having m-dimension (and hence to its Ziegler spectrum having Cantor-Bendixson rank) and the non-existence of superdecomposable pure-injective modules. Finally, we give a best possible upper bound for the m-dimension of the pp-$1$-lattice of a Prüfer domain in terms of the m-dimension of the extended positive cone of its value group and show that all ordinals which are not of the form $λ+1$ for $λ$ a limit ordinal occur as the m-dimension of the pp-$1$-lattice of a Bézout domain.
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