Analytic spread and integral closure of integrally decomposable modules

Ratliff-Rush Closure of Ideals in Integral Domains

This paper studies the Ratliff-Rush closure of ideals in integral domains. By definition, the Ratliff-Rush closure of an ideal $I$ of a domain $R$ is the ideal given by $\tilde{I}:=\bigcup(I^{n+1}:_{R}I^{n})$ and an ideal $I$ is said to be a Ratliff-Rush ideal if $\tilde{I}=I$. We completely characterize integrally closed domains in which every ideal is a Ratliff-Rush ideal and we give a complete description of the Ratliff-Rush closure of an ideal in a valuation domain.

强拟n-部图边理想的整闭包组合学

Combinatorics of the integral closure of edge ideals of strong quasi-n-partite graphs

本文研究了强拟n-部图边理想的组合性质，证明了其非整闭性并给出整闭包表达式，同时确定了相关顶点覆盖理想的结构。

非整闭的 Kronecker 函数环与具有唯一极小扩环的整环

Non-integrally closed Kronecker function rings and integral domains with a unique minimal overring

本文将 Kronecker 函数环的构造推广到非整闭整环，通过研究具有唯一极小扩环的整环及其 Nagata 环的交集，并构造整闭包远离赋值整环的新例子。

On asymptotic depth of integral closure filtration and an application

Let $(A,\mathfrak{m})$ be an analytically unramified formally equidimensional Noetherian local ring with $\ depth \ A \geq 2$. Let $I$ be an $\mathfrak{m}$-primary ideal and set $I^*$ to be the integral closure of $I$. Set $G^*(I) = \bigoplus_{n\geq 0} (I^n)^*/(I^{n+1})^*$ be the associated graded ring of the integral closure filtration of $I$. We prove that $\ depth \ G^*(I^n) \geq 2$ for all $n \gg 0$. As an application we prove that if $A$ is also an excellent normal domain containing an algebraically closed field isomorphic to $A/\m$ then there exists $s_0$ such that for all $s \geq s_0$ a…

Closures and co-closures attached to FCP ring extensions

The paper deals with ring extensions $R\subseteq S$ and the poset $[R,S]$ of their subextensions, with a special look at FCP extensions (extensions such that $[R,S]$ is Artinian and Noetherian). When the extension has FCP, we show that there exists a co-integral closure, that is a least element $\underline R$ in $[R,S]$ such that $\underline R \subseteq S$ is integral. Replacing the integral property by the integrally closed property, we are able to prove a similar result for an FCP extension. The radicial closure of $R$ in $S$ is well known. We are able to exhibit a suitable separable closure…

积分闭合诺特环的几何配置

Geometric configuration of integrally closed Noetherian domains

本文通过研究克鲁尔环和超度量球体，揭示了位于Z[X]和Q[X]之间的积分闭合诺特环的几何结构，并探讨了其唯一分解环的性质。

Asymptotic resurgence via integral closures

Given an ideal in a polynomial ring, we show that the asymptotic resurgence studied by Guardo, Harbourne, and Van Tuyl can be computed using integral closures. As a consequence, the asymptotic resurgence of an ideal is the maximum of finitely many ratios involving Waldschmidt-like constants (which we call skew Waldschmidt constants) defined in terms of Rees valuations. We use this to prove that the asymptotic resurgence coincides with the resurgence if the ideal is normal (that is, all its powers are integrally closed). For a monomial ideal the skew Waldschmidt constants have an interpretation…

Integral closures of ideals in completions of regular local domains

In this paper we exhibit an example of a three-dimensional regular local domain (A, n) having a height-two prime ideal P with the property that the extension PA^ of P to the n-adic completion A^ of A is not integrally closed. We use a construction we have studied in earlier papers: For R=k[x,y,z], where k is a field of characteristic zero and x,y,z are indeterminates over k, the example A is an intersection of the localization of the power series ring k[y,z][[x]] at the maximal ideal (x,y,z) with the field k(x,y,z,f, g) where f, g are elements of (x,y,z)k[y,z][[x]] that are algebraically indep…

On the depth and Stanley depth of integral closure of powers of monomial ideals

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Assume that $G$ is a graph with edge ideal $I(G)$. We prove that the modules $S/\overline{I(G)^k}$ and $\overline{I(G)^k}/\overline{I(G)^{k+1}}$ satisfy Stanley's inequality for every integer $k\gg 0$. If $G$ is a non-bipartite graph, we show that the ideals $\overline{I(G)^k}$ satisfy Stanley's inequality for all $k\gg 0$. For every connected bipartite graph $G$ (with at least one edge), we prove that ${\rm sdepth}(I(G)^k)\geq 2$, for any positive integer $k\leq {\rm girth}…

Stanley depth of the integral closure of monomial ideals

Let $I$ be a monomial ideal in the polynomial ring $S=\mathbb{K}[x_1,...,x_n]$. We study the Stanley depth of the integral closure $\bar{I}$ of $I$. We prove that for every integer $k\geq 1$, the inequalities ${\rm sdepth} (S/\bar{I^k}) \leq {\rm sdepth} (S/\bar{I})$ and ${\rm sdepth} (\bar{I^k}) \leq {\rm sdepth} (\bar{I})$ hold. We also prove that for every monomial ideal $I\subset S$ there exist integers $k_1,k_2\geq 1$, such that for every $s\geq 1$, the inequalities ${\rm sdepth} (S/I^{sk_1}) \leq {\rm sdepth} (S/\bar{I})$ and ${\rm sdepth} (I^{sk_2}) \leq {\rm sdepth} (\bar{I})$ hold. In…

Itoh's conjecture for normal ideals

Let $(A,\mathfrak{m})$ be an analytically unramified Cohen-Macaulay local ring and let $\mathfrak{a}$ be an $\mathfrak{m}$-primary ideal in $A$. If $I$ is an ideal in $A$ then let $I^*$ be the integral closure of $I$ in $A$. Let $G_{\mathfrak{a}}(A)^* = \bigoplus_{n\geq 0 }(\mathfrak{a}^n)^*/(\mathfrak{a}^{n+1})^*$ be the associated graded ring of the integral closure filtration of $\mathfrak{a}$. Itoh conjectured that if $e_3^{\mathfrak{a}^*}(A) = 0$ and $A$ is Gorenstein then $G_{\mathfrak{a}}(A)^* $ is Cohen-Macaulay. In this paper we prove an important case of Itoh's conjecture: we show th…

Integral closure of rings of integer-valued polynomials on algebras

Let $D$ be an integrally closed domain with quotient field $K$. Let $A$ be a torsion-free $D$-algebra that is finitely generated as a $D$-module. For every $a$ in $A$ we consider its minimal polynomial $μ_a(X)\in D[X]$, i.e. the monic polynomial of least degree such that $μ_a(a)=0$. The ring ${\rm Int}_K(A)$ consists of polynomials in $K[X]$ that send elements of $A$ back to $A$ under evaluation. If $D$ has finite residue rings, we show that the integral closure of ${\rm Int}_K(A)$ is the ring of polynomials in $K[X]$ which map the roots in an algebraic closure of $K$ of all the $μ_a(X)$, $a\i…