Controlling the Dimensions of Formal Fibers of a Unique Factorization Domain at the Height One Prime Ideals

Completely Controlling the Dimensions of Formal Fiber Rings at Prime Ideals of Small Height

Let $T$ be a complete equicharacteristic local (Noetherian) UFD of dimension $3$ or greater. Assuming that $|T| = |T/m|$, where $m$ is the maximal ideal of $T$, we construct a local UFD $A$ whose completion is $T$ and whose formal fibers at height one prime ideals have prescribed dimension between zero and the dimension of the generic formal fiber. If, in addition, $T$ is regular and has characteristic zero, we can construct $A$ to be excellent.

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。

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。

Examples of non-Noetherian domains inside power series rings

Let R* be an ideal-adic completion of a Noetherian integral domain R and let L be a subfield of the total quotient ring of R* such that L contains R. Let A denote the intersection of L with R*. The integral domain A sometimes inherits nice properties from R* such as the Noetherian property. For certain fields L it is possible to approximate A using a localzation B of a nested union of polynomial rings over R associated to A; if B is Noetherian, then B = A. If B is not Noetherian, we can sometimes identify the prime ideals of B that are not finitely generated. We have obtained in this way, for …

簇代数的阶式性与类群

Factoriality and class groups of cluster algebras

本文研究了簇代数的类群结构，证明了克鲁尔域簇代数的类群有限生成，并通过初始交换矩阵描述了非奇异种子簇代数的类群，扩展了克鲁尔型簇代数的阶式性分类。

Presentations and module bases of integer-valued polynomial rings

Let D be an integral domain with quotient field K. For any set X, the ring Int(D^X) of integer-valued polynomials on D^X is the set of all polynomials f in K[X] such that f(D^X) is a subset of D. Using the t-closure operation on fractional ideals, we find for any set X a D-algebra presentation of Int(D^X)$ by generators and relations for a large class of domains D, including any unique factorization domain D, and more generally any Krull domain D such that Int(D) has a regular basis, that is, a D-module basis consisting of exactly one polynomial of each degree. As a corollary we find for all s…