An algorithm for computing the integral closure

一维相干整环，其整闭包非Prüfer

A one-dimensional coherent domain with non-Prüfer integral closure

本文构造了一个一维相干局部整环，其整闭包非Prüfer，否定了Vasconcelos在1978年提出的问题。

Absolute Integral Closure

This paper summarizes three talks given by the author during a PASI conference in Olinda, Brasil

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.

Specialization of Integral Closure of Ideals by General Elements

In this paper, we prove a result similar to results of Itoh and Hong-Ulrich, proving that integral closure of an ideal is compatible with specialization by a general element of that ideal for ideals of height at least two in a large class of rings. Moreover, we show integral closure of sufficiently large powers of the ideal is compatible with specialization by a general element of the original ideal. In a polynomial ring over an infinite field, we give a class of squarefree monomial ideals for which the integral closure is compatible with specialization by a general linear form.

On The Integral Closure of Radical Towers in Mixed Characteristic

We study the Cohen-Macaulay property of a particular class of radical extensions of an unramified regular local ring having mixed characteristic.

Prüfer 整环的完全整闭包是一个拓扑性质

The complete integral closure of a Prüfer domain is a topological property

该研究证明了Prüfer整环的完全整闭包的素谱完全由其自身素谱上的Zariski拓扑决定，揭示了整环代数性质与拓扑结构的关联。

Variation of the first Hilbert coefficients of parameters with a common integral closure

A problem posed by Wolmer V. Vasconcelos on the variation of the first Hilbert coefficients of parameter ideals with a common integral closure in a local ring is studied. Affirmative answers are given and counterexamples are explored as well.

Complete integral closure and strongly divisorial prime ideals

It is well known that a domain without proper strongly divisorial ideals is completely integrally closed. In this paper we show that a domain without {\em prime} strongly divisorial ideals is not necessarily completely integrally closed, although this property holds under some additional assumptions.

Separable integral extensions and plus closure

Let R be an excellent local domain of positive characteristic, and R^+ denote the integral closure of R in an algebraic closure of its fraction field. Hochster and Huneke proved that R^+ is a big Cohen-Macaulay algebra for R, and asked if there is a smaller R-algebra with the Cohen-Macaulay property. In this paper we establish the existence of a smaller big Cohen-Macaulay algebra which is, moreover, a separable extension.

Integral closure of ideals and annihilators of homology

This article outgrew from an effort to understand our basic question: Are the annihilators of the non-zero Koszul homology modules $H_i$ of an unmixed ideal $I$ contained in the integral closure $\bar{I}$ of $I$? We also obtain some variations on a result of Burch, which continue the theme of the paper in that they deal with annihilators of homology and integrally closed ideals.

Integral closure of ideals in excellent local rings

Let R be an excellent local ring, m its maximal ideal and I an ideal. Then there exists a positive integer c such that for all integers n, the integral closure of (I + m^n) is contained in m^(n/c) + the integral closure of I. In the proof, a version of the linear Artin approximation theorem is proved.

t-Reductions and t-integral closure of ideals

Let R be an integral domain and I a nonzero ideal of R. A sub-ideal J of I is a t-reduction of I if (JI^{n})_{t}=(I^{n+1})_{t} for some positive integer n. An element x in R is t-integral over I if there is an equation x^{n} + a_{1}x^{n-1} +...+ a_{n-1}x + a_{n} = 0 with a_{i} in (I^{i})_{t} for I = 1,...,n. The set of all elements that are t-integral over I is called the t-integral closure of I. This paper investigates the t-reductions and t-integral closure of ideals. Our objective is to establish satisfactory t-analogues of well-known results, in the literature, on the integral closure of i…

On the vanishing of Tor of the absolute integral closure

Let $R$ be an excellent local domain of positive characteristic with residue field $k$ and let $R^+$ be its absolute integral closure. If $\text{Tor}^R_1(R^+,k)$ vanishes, then $R$ is Cohen-Macaulay, normal, F-rational and F-pure. If $R$ has at most an isolated singularity or has dimension at most two, then $R$ is regular.

一个幂及其整闭包与符号幂的正则性差异

Differences in regularities of a power and its integral closure and symbolic power

本文证明无平方单项式理想的符号二次幂和二次幂整闭包的正则性可与普通二次幂相差任意大整数。

一维环的积分闭包

Integral closure of 1-dimensional rings

本文研究了一维局部积分域的模性质，探讨了导出理想阶数、长度为二的理想反射性以及DVR绝对积分闭包的自由性问题。

关于 $p$-adic 完成的绝对整闭性的完整性

On complete integral closedness of the $p$-adic completion of absolute integral closure

本文研究了 $p$-adic 完成的绝对整闭性，证明了在特定条件下其在某些扩张中是完全整闭的，但不总是如此。

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。

Associated primes and integral closure of Noetherian rings

Let A be a Noetherian ring and B be a finitely generated A-algebra. Denote by A' the integral closure of A in B. We give necessary and sufficient conditions for prime ideals to be in Ass_{A}(B/A') and Ass_{A'}(B/A') generalizing and strengthening classical results for rings of special type.

F-thresholds, integral closure and convexity

The purpose of this note is to revisit the results of arXiv:1407.4324 from a slightly different perspective, outlining how, if the integral closures of a finite set of prime ideals abide the expected convexity patterns, then the existence of a peculiar polynomial allows to compute the F-jumping numbers of all the ideals formed by taking sums of products of the original ones. The note concludes with the suggestion of a possible source of examples falling in such a framework.

An Algorithm for Computing the Ratliff-Rush Closure

Let I\subset K[x,y] be a <x,y>-primary monomial ideal where K is a field. This paper produces an algorithm for computing the Ratliff-Rush closure I for the ideal I=<m_0,...,m_{n}> whenever m_{i} is contained in the integral closure of the ideal <x^{a_{n}},y^{b_0}>. This generalizes of the work of Crispin \cite{Cri}. Also, it provides generalizations and answers for some questions given in \cite{HJLS}, and enables us to construct infinite families of Ratliff-Rush ideals.

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…

理想的新弗罗贝尼乌斯和积分闭包版本

New versions of Frobenius and integral closure of ideals

本文提出三种新的理想闭包操作，用于推广积分闭包和弗罗贝尼乌斯闭包，并应用于代数几何领域。

单项式理想及其整数闭包的正则性

The regularity of monomial ideals and their integral closures

本文研究了单项式理想及其整数闭包的正则性，证明了闭包的正则性不高于原理想，并给出了正则性等于度数的充要条件。

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…

On the integral closure of ideals

Among the several types of closures of an ideal $I$ that have been defined and studied in the past decades, the integral closure $\bar{I}$ has a central place being one of the earliest and most relevant. Despite this role, it is often a difficult challenge to describe it concretely once the generators of $I$ are known. Our aim in this note is to show that in a broad class of ideals their radicals play a fundamental role in testing for integral closedness, and in case $I\neq \bar{I}$, $\surd{I}$ is still helpful in finding some fresh new elements in $\bar{I}\setminus I$. Among the classes of id…

Homological properties of the perfect and absolute integral closures of Noetherian domains

For a Noetherian local domain $R$ let $R^+$ be the absolute integral closure of $R$ and let $R_{\infty}$ be the perfect closure of $R$, when $R$ has prime characteristic. In this paper we investigate the projective dimension of residue rings of certain ideals of $R^+$ and $R_{\infty}$. In particular, we show that any prime ideal of $R_{\infty}$ has a bounded free resolution of countably generated free $R_{\infty}$-modules. Also, we show that the analogue of this result is true for the maximal ideals of $R^+$, when $R$ has residue prime characteristic. We compute global dimensions of $R^+$ and …

Computing the closure of a support

When $E$ is an $R$-module over a commutative unital ring $R$, the Zariski closure of its support is of the form $\mathrm V(\mathcal O(E))$ where $\mathcal O(E)$ is a unique radical ideal. We give an explicit form of $\mathcal O(E)$ and study its behavior under various operations of algebra. Applications are given, in particular for ring extensions of commutative unital rings whose supports are closed. We provide some applications to crucial and critical ideals of ring extensions.

某些边加权图的边理想的整闭性与正规性

Integral closure and normality of edge ideals of some edge-weighted graphs

本文完整刻画了边加权简单图中边理想为整闭的条件，并证明对星图、路径和圈，整闭性蕴含正规性。

Integral closure of ideals

The Qth-power algorithm for computing structured global presentations of integral closures of affine domains over finite fields is modified to compute structured presentations of integral closures of ideals in affine domains over finite fields relative to a local monomial ordering. A non-homogeneous version of the standard (homogeneous) Rees algebra is introduced as well.

Coherence of absolute integral closures

We prove that the absolute integral closure $R^{+}$ of an equicharacteristic zero noetherian complete local domain $R$ is not coherent, provided $\dim(R)\geq 2$. As a corollary, we give an elementary proof of the mixed characteristic version of the result due to Asgharzadeh and extend it to dimension $3$. Furthermore, we apply the methods of Aberbach and Hochster used to prove the positive characteristic version of this result to study F-coherent rings and our work naturally suggests a mixed characteristic analogue of a result of Smith.

Integral closures, Primary Hyperideals and Hypervaluation Hyperideals of Kranser Hyperrings

In this paper, the notions of integral closure of hyperrings and hyperideals in a Krasner hyperring $(R, +, \cdot)$ are defined and some basics properties of them are studied. We define also the notion of hypervaluation hyperideals and then a relations between hypervaluations, integral closure of hyperideals and primary hyperideals are studied. In fact it is shown that the integral closure of a hyperideal is determined by the hypervaluation Krasner hyperrings.

Integral Closures of Cohen-Macaulay monomial ideals

The purpose of this paper is to present a family of Cohen-Macaulay monomial ideals such that their integral closures have embedded components and hence are not Cohen-Macaulay.

某些类单项式理想及其整闭的正则性比较

A comparison of the regularity of certain classes of monomial ideals and their integral closures

本文针对某些类的单项式理想，证明了 Küronya 和 Pintye 关于齐次理想整闭的 Castelnuovo-Mumford 正则性不超过原理想正则性的猜想。

Stability of Associated Primes and Depth of Integral Closures of Powers of Edge Ideals

In this paper, we study associated primes and depth of integral closures of powers of edge ideals. We provide sharp bounds on how big of powers for which the set of associated primes and the depth of integral closures of powers of edge ideals are stable.

Integral closures of monomial ideals and Fulkersonian hypergraphs

We prove that the integral closures of the powers of a squarefree monomial ideal I equal the symbolic powers if and only if I is the edge ideal of a Fulkersonian hypergraph.

关于Küronya和Pintye关于正则性和积分闭包猜想的反例

Regularity under Integral Closure: A General Criterion and a Sharp Counterexample

研究提出一个等生成的单项式理想，其积分闭包的正则性大于原理想，挑战了Küronya-Pintye猜想的多项式环形式。

Absolute integral closures of commutative rings

Absolute integral closures of general commutative unital rings are explored. All rings admit absolute integral closures, but in general they are not unique. Among the reduced rings with finitely many minimal prime ideals, finite products of domains are the only rings for which they are unique. Arguments using model theory suggest that the same holds for all infinite rings that are finite products of connected rings. Universal absolute integral closures, which contain every aic of a given ring, are shown to exist for certain subrings of products of domains.

Stability of Depth and Cohen-Macaulayness of Integral Closures of Powers of Monomial Ideals

Let $I$ be a monomial ideal $I$ in a polynomial ring $R = k[x_1,...,x_r]$. In this paper we give an upper bound on $\overline{\dstab} (I)$ in terms of $r$ and the maximal generating degree $d(I)$ of $I$ such that $\depth R/\overline{I^n}$ is constant for all $n\geqslant \overline{\dstab}(I)$. As an application, we classify the class of monomial ideals $I$ such that $\overline{I^n}$ is Cohen-Macaulay for some integer $n\gg 0$.

F-rational rings and the integral closures of ideals

We show in this paper that the Briancon-Skoda theorem holds for all ideals in F-rational rings of positive prime characteristic, and also in rings with rational singularities which are of finite type over a field of characteristic 0. Moreover, in Gorenstein F-rational rings of characteristic p we show that in many cases the bound given in the Briancon-Skoda theorem may be improved by at least one.

Regularity comparison of symbolic powers, integral closure of powers and powers of edge ideals

We study the regularity of small symbolic powers and integral closure of small powers of edge ideals. We also prove that the regularity of integral closure of powers of edge ideals of graphs with at most two odd cycles is the same as the regularity of their powers.

极值理想与积分闭包 via 极端理想

Symbolic powers and integral closures via extremal ideals

本文通过极值理想计算平方自由单项式理想幂和符号幂的积分闭包，并提供精确的不变量界。

单项式理想的整闭包滤过的$\operatorname{v}$-数

$\operatorname{v}$-numbers of integral closure filtrations of monomial ideals

本文研究了多项式环中单项式理想幂及其整闭包的$\operatorname{v}$-数，分析了不可约和完全交单项式理想整闭包滤过的$\operatorname{v}$-数与Castelnuovo-Mumford正则性的关系，并给出了相关公式与上界。

整闭包滤过的Itoh猜想的一个解

A solution to Itoh's conjecture for integral closure filtration

针对解析无分歧Cohen-Macaulay局部环上整闭包滤过的相伴分次环的Itoh猜想，研究者证明了在更一般的解析无分歧Cohen-Macaulay局部环条件下，当第三个希尔伯特系数为0时该分次环是Cohen-Macaulay的，解决了Itoh猜想。

（加法幂等）半环的整闭包

Integral closure for (additively idempotent) semirings

研究幂等半环整性的不同定义及关系，通过证明加法幂等半环上的凯莱 - 哈密顿定理作为工具，计算坐标半环及其子半环的整闭包，为热带簇正规化等研究提供途径。

An algorithm for computing the Arf closure of an algebroid curve with more than one branch

In this paper, we give a fast algorithm for the computation of the Arf closure of an algebroid curve with more than one branch, generalizing an algorithm presented by Arslan and Sahin for the algebroid branch case.

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。

F-thresholds, tight closure, integral closure, and multiplicity bounds

The F-threshold $c^J(\a)$ of an ideal $\a$ with respect to the ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\a$ with the Frobenius powers of $J$. We show that under mild assumptions, we can detect the containment in the integral closure or the tight closure of a parameter ideal using F-thresholds. We formulate a conjecture bounding $c^J(\a)$ in terms of the multiplicities $e(\a)$ and $e(J)$, when $\a$ and $J$ are zero-dimensional ideals, and $J$ is generated by a system of parameters. We prove the conjecture when $J$ is a monomial ideal in a polynomial …

Cohen-Macaulayness of absolute integral closures

We prove that, modulo any power of a prime $p$, the absolute integral closure of an excellent noetherian domain is Cohen-Macaulay. A graded analog is also established, yielding variants of Kodaira vanishing "up to finite covers" in mixed characteristic. Our main tools are (log) prismatic cohomology (which yields a Frobenius action in mixed characteristic) and the $p$-adic Riemann-Hilbert functor for constructible étale $\mathbf{F}_p$-sheaves on varieties over a $p$-adic field (which almost controls perfectified prismatic cohomology).

Results on the Ratliff-Rush Closure and the Integral Closedness of Powers of Certain Monomial Curves

Starting from \cite{Ayy2} we compute the Groebner basis for the defining ideal, P, of the monomial curves that correspond to arithmetic sequences, and then give an elegant description of the generators of powers of the initial ideal of P, inP. The first result of this paper introduces a procedure for generating infinite families of Ratliff-Rush ideals, in polynomial rings with multivariables, from a Ratliff-Rush ideal in polynomial rings with two variables. The second result is to prove that all powers of inP are Ratliff-Rush. The proof is through applying the first result of this paper combin…

On the computation of Ratliff-Rush closure

Let $R$ be a Cohen-Macaulay local ring with maximal ideal $\max$. In this paper we present a procedure for computing the Ratllif-Rush closure of a $\max-$primary ideal $I \subset R$.

Integral closure and Hilbert series of a special monomial ideal

Let $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$ and let $M_{n,t}=(x^{e_1},\ldots, x^{e_n})$ be a monomial ideal of $R$, where $x^{e_i}=x_1^t\ldots x_{i-1}^tx_{i+1}^t\ldots x_n^t$. We study the unmixedness of its integral closure. Furthermore, we compute the Hilbert series of this ideal and we show that this ideal is Freiman.

整闭包与局部上同调

Integral closure and local cohomology

本文研究理想乘法滤过的局部上同调消失性质与其各项包含于相应幂理想整闭包之间的等价关系。

Computing the tight closure in dimension two

We study computational aspects of the tight closure of a homogeneous primary ideal in a two-dimensional normal standard-graded domain. We show how to use slope criteria for the sheaf of syzygies for generators of the ideal to compute the tight closure. Our method gives in particular an algorithm to compute the tight closure of three elements under the condition that we are able to compute the Harder-Narasimhan filtration. We apply this to the computation of (x^a,y^a,z^a)^* in K[x,y,z]/(F), where F is a homogeneous polynomial.

Nilpotents, Integral Closure and Equisingularity conditions

In earlier work, the author described various stratification conditions for a complex analytic set X in terms of the theory of integral closure of modules. However, even if an analytic set has a reduced structure, often geometric operations like intersection give a non-reduced structure on the resulting analytic space. In this note we show that it is not necessary for X to have a reduced structure to apply the earlier integral closure results. As an application we give a simple proof of a necessary and sufficient condition for the Whitney conditions to pass to the intersection of X with a hype…