Fundamental Results on $s$-Closures

Topics on strict closure of rings

In 1971, J. Lipman introduced the notion of strict closure of a ring in another, and established the underlying theory in connection with a conjecture of O. Zariski. In this paper, for further developments of the theory, we investigate three different topics related to strict closure of rings. The first one concerns construction of the closure, and the second one is the study regarding the question of whether the strict closedness is inherited under flat homomorphisms. We finally handle the question of when the Arf closure coincides with the strict closure. Examples are explored to illustrate …

极小闭包与F-理性质

Tight closure of products and F-rational singularities

本文通过极小闭包理论研究F-理性质，并在高维情况下扩展了曲面奇点的特征化结果。

Nilpotence of Frobenius actions on local cohomology and Frobenius closure of ideals

The study of Frobenius actions on local cohomology modules over a local ring of prime characteristic has interesting connections with the theory of tight closure. This paper establishes new connections by developing the notion of relative Frobenius actions on local cohomology. As an application, we show that a ring has $F$-nilpotent singularities if and only if the tight closure of every parameter ideal is equal to its Frobenius closure.

关于理想的整闭包

On the integral closure of ideals

在过去的几十年中，人们定义和研究了理想$I$的几种闭包类型，其中整闭包$\bar{I}$具有中心地位，是最早和最相关的闭包之一。尽管有这种作用，但一旦$I$的生成器已知，要具体描述它往往是一个困难的挑战。我们在这篇文章中的目的是表明，在一个广泛的理想类，他们的根发挥了根本性的作用，在测试积分封闭性，并在$I\neq \bar{I}$的情况下，$\surd{I}$仍然是有助于在$\bar{I}\setminus I$中找到一些新鲜的新元素。其中类理想正在考虑的是：完全相交理想的余维2，一般完全相交理想，一般Gorenstein理想。

具有非Prüfer整闭包的Gorenstein Prüfer整环

Gorenstein Prüfer domains with non-Prüfer integral closure

该研究证明存在一维局部Gorenstein Prüfer整环，其整闭包非Prüfer，还确立了有限生成理想除子性在超积与可分素理想商下的不变性，提供一致矩阵分解并区分相关失败情形。

Galois closure data for extensions of rings

To generalize the notion of Galois closure for separable field extensions, we devise a notion of $G$-closure for algebras of commutative rings $R\to A$, where $A$ is locally free of rank $n$ as an $R$-module and $G$ is a subgroup of $\mathrm{S}_n$. A $G$-closure datum for $A$ over $R$ is an $R$-algebra homomorphism $φ: (A^{\otimes n})^{G}\to R$ satisfying certain properties, and we associate to a closure datum $φ$ a closure algebra $A^{\otimes n}\otimes_{(A^{\otimes n})^G} R$. This construction reproduces the normal closure of a finite separable field extension if $G$ is the corresponding Galo…