Completions of Countable Excellent Domains and Countable Noncatenary Domains

Characterization of Completions of Noncatenary Local Domains and Noncatenary Local UFDs

We find necessary and sufficient conditions for a complete local ring to be the completion of a noncatenary local (Noetherian) domain, as well as necessary and sufficient conditions for it to be the completion of a noncatenary local (Noetherian) unique factorization domain. We use our first result to demonstrate a large class of quasi-excellent domains that are not excellent, as well as a large class of catenary domains that are not universally catenary. We use our second result to find a larger class of noncatenary local UFDs than was previously known, and we show that there is no bound on ho…

Completions of Uncountable Local Rings with Countable Spectra

We find necessary and sufficient conditions for a complete local (Noetherian) ring to be the completion of an uncountable local (Noetherian) domain with a countable spectrum. Our results suggest that uncountable local domains with countable spectra are more common than previously thought. We also characterize completions of uncountable excellent local domains with countable spectra assuming the completion contains the rationals, completions of uncountable local unique factorization domains with countable spectra, completions of uncountable noncatenary local domains with countable spectra, and …

非悬链唯一分解整环

Noncatenary Unique Factorization Domains

本文构造了一类在无穷多个位置非悬链的局部诺特唯一分解整环，并证明其存在无穷多个高度为一的素理想使商环非悬链，还构造了所有高度为一素理想商环均非悬链的可数局部UFD。

素特征中的非链状裂体

Noncatenary splinters in prime characteristic

本文在素特征中构造了非链状的局部Noetherian裂体及非Cohen-Macaulay的弱$F$-正则局部UFD，核心方法是通过寻找充分条件确保完备局部环成为弱$F$-正则局部整环的完备化。

构造非链状拟优局部预完备化

Constructing Noncatenary Quasi-Excellent Precompletions

本文研究局部环中两个素理想的粘合问题，给出存在拟优局部子环同时保持完备化与素谱性质的充分条件，并据此构造出素谱含非链状有限子集的拟优局部环。

极非链状诺特唯一分解整环的完备化

Completions of Extremely Noncatenary Noetherian UFDs

本文给出完备局部环$T$成为满足特定素理想条件的局部诺特唯一分解整环$A$的完备化的充要条件，以及$T$成为对所有高度1素理想$J$有$B/J$非链状的局部诺特唯一分解整环$B$的完备化的充分条件。