Tight closure and linkage classes in Gorenstein rings

Some characterizations of special rings by delta-invariant

This paper is devoted to present some characterizations for a local ring to be generically Gorenstein and Gorenstein by means of $δ$-invariant and linkage theory.

Invariants of Linkage of modules

Let $(A,\mathfrak{m})$ be a Gorenstein local ring and let $M, N$ be two Cohen-Macaulay \ $A$-modules with $M$ linked to $N$ via a Gorenstein ideal $\mathfrak{q}$. Let $L$ be another finitely generated $A$-module. We show that $Ext^i_A(L,M) = 0 $ for all $i \gg 0$ if and only if $Tor^A_i(L,N) = 0$ for all $i \gg 0$. If $D$ is Cohen-Macaulay then we show that $Ext^i_A(M, D) = 0 $ for all $i \gg 0$ if and only if $Ext^i_A(D^\dagger, N) = 0$ for all $i \gg 0$, where $D^\dagger = Ext^r_A(D,A)$ and $r = codim \ D$. As a consequence we get that $Ext^i_A(M, M) = 0 $ for all $i \gg 0$ if and only if $E…

Linkage Extensions

Given two equidimensional Cohen-Macaulay local rings of the same dimension, one shows that a simultaneous extension of each of them by a dualizing module of the other is Gorenstein. This generalizes a theorem of Fossum. The geometrical analogue is also considered. The pairs of double lines in the projective space which are algebraically linked are classified.

完美闭包检测内射维数

Perfect closure detects injective dimension

本文证明局部环R的完美闭包R^∞可作为单个测试模检测有限生成R-模N的有限内射维数，改进了Herzog定理需要无穷多个Frobenius扭曲模的结果。

关于近 Gorenstein 单纯半群代数

On Nearly Gorenstein Simplicial Semigroup Algebras

本文研究数值半群的近 Gorenstein 射影闭包及单纯仿射半群相伴分次环的相应性质，并在粘合情形下回答 Herzog-Hibi-Stamate 的问题。

模与理想联系下的复杂性、曲率与同调维数

Complexity, curvature and homological dimension of modules under linkage

本文研究在Gorenstein局部环上，通过Gorenstein完美理想联系的Cohen-Macaulay模的投射与内射复杂性、曲率以及完全交维数之间的关系，并推广了Peskine和Szpiro的经典结果。

Higher structure maps for free resolutions of length 3 and linkage

Let $I$ be a perfect ideal of height 3 in a Gorenstein local ring $R$. Let $\mathbb{F}$ be the minimal free resolution of $I$. A sequence of linear maps, which generalize the multiplicative structure of $\mathbb{F}$, can be defined using the generic ring associated to the format of $\mathbb{F}$. Let $J$ be an ideal linked to $I$. We provide formulas to compute some of these maps for the free resolution of $J$ in terms of those of the free resolution of $I$. We apply our results to describe classes of licci ideals, showing that a perfect ideal with Betti numbers $(1,5,6,2)$ is licci if and only…

Subcanonical coordinate rings are Gorenstein

Using a classical result of Avramov-Golod we strengthen a recent result of Gorodentsev, Khoroshkin and Rudakov on syzygies of highest weight orbit closure.

Twenty Points in P^3

Using the possibility of computationally determining points on a finite cover of a unirational variety over a finite field, we determine all possibilities for direct Gorenstein linkages between general sets of points in P^3 over an algebraically closed field of characteristic 0. As a consequence we show that a general set of d points is glicci (that is, in the Gorenstein linkage class of a complete intersection) if d <= 33 or d=37,38. Computer algebra plays an essential role in the proof. The case of 20 points had been an outstanding problem in the area for a dozen years.

Computing Gorenstein Colength

Given an Artinian local ring $R$, we define its Gorenstein colength $g(R)$ to measure how closely we can approximate $R$ by a Gorenstein Artin local ring. In this paper, we show that $R = T/I$ satisfies the inequality $g(R) \leq λ(R/\soc(R))$ in the following two cases: (a) $T$ is a power series ring over a field of characteristic zero and $I$ an ideal that is the power of a system of parameters or (b) $T$ is a 2-dimensional regular local ring with infinite residue field and $I$ is primary to the maximal ideal of $T$. In the first case, we compute $g(R)$ by constructing a Gorenstein Artin loca…

Frobenius maps on injective hulls and their applications to tight closure

This paper studies Frobenius maps on injective hulls of residue fields of complete local rings with a view toward providing constructive descriptions of objects originating from the theory of tight closure. Specifically, the paper describes algorithms for computing parameter test ideals, and tight closure of certain submodules of the injective hull of residue fields of a class of well-behaved rings which includes all quasi-Gorenstein complete local rings.

A constructive approach to one-dimensional Gorenstein $k$-algebras

Let $R$ be the power series ring or the polynomial ring over a field $k$ and let $I $ be an ideal of $R.$ Macaulay proved that the Artinian Gorenstein $k$-algebras $R/I$ are in one-to-one correspondence with the cyclic $R$-submodules of the divided power series ring $Γ. $ The result is effective in the sense that any polynomial of degree $s$ produces an Artinian Gorenstein $k$-algebra of socle degree $s.$ In a recent paper, the authors extended Macaulay's correspondence characterizing the $R$-submodules of $Γ$ in one-to-one correspondence with Gorenstein d-dimensional $k$-algebras. However, th…

Absolute integral closure in positive characteristic

Let R be a local Noetherian domain of positive characteristic. A theorem of Hochster and Huneke (1992) states that if R is excellent, then the absolute integral closure of R is a big Cohen-Macaulay algebra. We prove that if R is the homomorphic image of a Gorenstein local ring, then all the local cohomology (below the dimension) of such a ring maps to zero in a finite extension of the ring. There results an extension of the original result of Hochster and Huneke to the case in which R is a homomorphic image of a Gorenstein local ring, and a considerably simpler proof of this result in the case…

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.

On reduced G-perfection and horizontal linkage relative to a semidualizing module

In their investigation of horizontal linkage of modules of finite Gorenstein dimension over a commutative, Noetherian, semiperfect (e.g., local) ring, Dibaei and Sadeghi introduced the class of reduced G-perfect modules, making use of Bass' concept of reduced grade. A few years later, the same authors extended this class by considering the relative property of reduced G$_C$-perfection, where $C$ is a semidualizing module, and studied linkage even further. In the present paper, we contribute to their theory and also generalize results of Auslander and Bridger as well as of Martsinkovsky and Str…

长度为3的自由分解的映射 II——模格式

Mapping free resolutions of length three II -- Module formats

本文研究投射维数为3的完美模的极小自由分解格式，借助泛型环定义高阶结构映射，并提出基于 Buchsbaum–Rim 复形的模链接推广，给出特定格式下的链接类判据。

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…

Cyclic covers of rings with rational singularities

We examine some recent work of Phillip Griffith on étale covers and fibered products from the point of view of tight closure theory. While it is known that cyclic covers of Gorenstein rings with rational singularities are Cohen-Macaulay, we show this is not true in general in the absence of the Gorenstein hypothesis. Specifically, we show that the canonical cover of a $\mathbb Q$-Gorenstein ring with rational singularities need not be Cohen-Macaulay.

关于半群环的特定粘合与半群环的不可或缺的分解

On Certain Gluing of semigroup rings and indispensable resolution of semigroup rings

本文通过粘合半群技术构造了无限多个具Cohen-Macaulay性质的射影闭包族和一维Gorenstein局部环族以支持Rossi问题，并研究仿射半群连接操作下的强不可或缺极小自由分解以支持Charalambous和Thoma问题。

作为联络不变量的F-纯度与F-纯阈值

$F$-purity and the $F$-pure threshold as invariants of linkage

研究无混合根式理想一般联络下相关性质，通过识别无混合理想沿一般联络传播的性质，证明初始理想无平方性、F-纯度和F-纯阈值在特定情形下被保留，还建立了F-正则性并研究了完全交理想一般剩余交的F-纯阈值。

符号幂的通用连通性最大余子式

Symbolic powers of the generic linkage of maximal minors

本文研究了最大余子式生成的理想J的符号幂与普通幂相等性，通过格罗布纳退化描述生成元的首项，并证明关联格雷德环的格雷索斯性质及正特征下的Rees代数F-理性性。

Nakayama closures, interior operations, and core-hull duality

Exploiting the interior-closure duality developed by Epstein and R.G., we show that for the class of Matlis dualizable modules $\mathcal{M}$ over a Noetherian local ring, when cl is a Nakayama closure and i its dual interior, there is a duality between cl-reductions and i-expansions that leads to a duality between the cl-core of modules in $\mathcal{M}$ and the i-hull of modules in $\mathcal{M}^\vee$. We further show that many algebra and module closures and interiors are Nakayama and describe a method to compute the interior of ideals using closures and colons. We use our methods to give a un…

Big tight closure test elements for some non-reduced excellent rings

This paper is concerned with existence of big tight closure test elements for a commutative Noetherian ring $R$ of prime characteristic $p$. Let $R^{\circ}$ denote the complement in $R$ of the union of the minimal prime ideals of $R$. A big test element for $R$ is an element of $R^{\circ}$ which can be used in every tight closure membership test for every $R$-module, and not just the finitely generated ones. The main results of the paper are that, if $R$ is excellent and satisfies condition $(R_0)$, and $c \in R^{\circ}$ is such that $R_c$ is Gorenstein and weakly $F$-regular, then some power …

关于偏序集、单项式理想、Gorenstein 理想及其组合学

On posets, monomial ideals, Gorenstein ideals and their combinatorics

本文从组合角度比较单项式理想的 socle 元素与 Macaulay 逆系统，建立相关偏序集的闭包性质，推导 Γ-分次环的代数命题及其组合推论，并对零维 Gorenstein 理想进行直接计算，揭示多项式环与局部环之间的关键差异。

Resolutions of defining ideals of orbit closures for quivers of type $A_3$

We investigate the properties of coordinate rings of orbit closures for quivers of type $A_3$ by considering the desingularization given by Reineke. We construct explicit minimal free resolutions of the defining ideals of the orbit closures thus giving us a minimal set of generators for the defining ideal. The resolution allows us to read off some geometric properties of the orbit closure. In addition, we give a characterization for the orbit closure to be Gorenstein.

An interpretation of multiplier ideals via tight closure

Hara and Smith independently proved that in a normal $\mQ$-Gorenstein ring of characteristic $p \gg 0$, the test ideal coincides with the multiplier ideal associated to the trivial divisor. We extend this result for a pair $(R, Δ)$ of a normal ring $R$ and an effective $\mQ$-Weil divisor $Δ$ on $\Spec R$. As a corollary, we obtain the equivalence of strongly F-regular pairs and klt pairs.

Links of prime ideals and their Rees algebras

In a previous paper we exhibited the somewhat surprising property that most direct links of prime ideals in Gorenstein rings are equimultiple ideals with reduction number $1$. This led to the construction of large families of Cohen--Macaulay Rees algebras. The first goal of this paper is to extend this result to arbitrary Cohen--Macaulay rings. The means of the proof are changed since one cannot depend so heavily on linkage theory. We then study the structure of the Rees algebra of these links, more specifically we describe their canonical module in sufficient detail to be able to characterize…

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。

Homogeneous Liaison and the Sequentially Bounded Licci Property

In CI-Liaison, significant effort has been made to study ideals that are in the linkage class of a complete intersection, which are called licci ideals. In a polynomial ring, recently E. Chong defined a "sequentially bounded" condition on the degrees of the forms generating the regular sequences, and used this condition to find a large class of licci ideals satisfying the Eisenbud-Green-Harris Conjecture (among them, grade $3$ homogeneous Gorenstein ideals). He raised the question of whether all homogeneous licci ideals are sequentially bounded licci. In this paper we construct a class of exam…

关于Gorenstein内射覆盖的存在性

On the existence of Gorenstein injective covers

本文刻画了Gorenstein内射覆盖性质，构造反例否定Iacob的问题，证明Noetherian概型的覆盖性质是Zariski局部的，还给出了茎局部不成立的反例。

A generalization of tight closure and multiplier ideals

We introduce a new variant of tight closure associated to any fixed ideal $\a$, which we call $\a$-tight closure, and study various properties thereof. In our theory, the annihilator ideal $τ(\a)$ of all $\a$-tight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal $τ(\a)$ and the multiplier ideal associated to $\a$ (or, the adjoint of $\a$ in Lipman's sense) in normal $\Q$-Gorenstein rings reduced from characteristic zero to characteristic $p \gg 0$. Also, in fixed prim…

极性与戈林斯坦联络

Polarization and Gorenstein liaison

研究通过极化技术探讨G-联络构造，证明极化稳定Cohen-Macaulay单项式理想的Stanley-Reisner复形可顶点分解，并建立极化与G-联络的关系。

用平衡的大Cohen-Macaulay模进行余倾斜

Cotilting with balanced big Cohen-Macaulay modules

本文在带典范模的$d$维Cohen-Macaulay环上分类了含平衡大Cohen-Macaulay模的$d$-余倾斜类，确定其最小、最大类，并研究Gorenstein局部环上相关结构的相互作用。

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…

Test ideals in diagonal hypersurface rings II

Let $R=k[x_1, ..., x_n]/(x_1^d + ... + x_n^d)$, where $k$ is a field of characteristic $p$, $p$ does not divide $d$ and $n \geq 3$. We describe a method for computing the test ideal for these diagonal hypersurface rings. This method involves using a characterization of test ideals in Gorenstein rings as well as developing a way to compute tight closures of certain ideals despite the lack of a general algorithm. In addition, we compute examples of test ideals in diagonal hypersurface rings of small characteristic (relative to $d$) including several that are not integrally closed. These examples…

Ext的 annihilator

Annihilator of $Ext^g_R(R/I,R)$

本文研究了Ext的 annihilator 的结构和性质，探讨了其在不同理想类中的包含关系，并给出了相关应用。

二部图边环的 $h$-多项式的回文性准则

A palindromicity criterion for the $h$-polynomials of bipartite edge rings

研究二部图边环的 $h$-多项式的对称性，证明在伪Gorenstein且$h_1=h_{s-1}$条件下边环是Gorenstein，即$h$-多项式是回文的。

乘法理想通过超积的刻画

A characterization of multiplier ideals via ultraproducts

本文通过超积方法引入超紧闭包，证明了在正规Q-Gorenstein局部环上超测试理想与乘法理想一致，并应用于纯环扩张下乘法理想的性质研究。

Excellence, F-singularities, and solidity

An $R$-algebra $S$ is $R$-solid if there exists a nonzero $R$-linear map $S \rightarrow R$. In characteristic $p$, the study of $F$-singularities such as Frobenius splittings implicitly rely on the $R$-solidity of $R^{1/p}$. Following recent results of the first two authors on the Frobenius non-splitting of certain excellent $F$-pure rings, in this paper we use the notion of solidity to systematically study the notion of excellence, with an emphasis on $F$-singularities. We show that for rings $R$ essentially of finite type over complete local rings of characteristic $p$, reducedness implies t…

余模模的内射维数

On injective dimension of the conormal module

本文研究 Gorenstein 局部环中余模模有限内射维数迫使理想由正则序列生成的条件，覆盖链结类、正则环数值条件、正规域商及分次情形，并给出 Kunz 定理的逆命题。

One-dimensional stable rings

A commutative ring $R$ is stable provided every ideal of $R$ containing a nonzerodivisor is projective as a module over its ring of endomorphisms. The class of stable rings includes the one-dimensional local Cohen-Macaulay rings of multiplicity at most $2$, as well as certain rings of higher multiplicity, necessarily analytically ramified. The former are important in the study of modules over Gorenstein rings, while the latter arise in a natural way from generic formal fibers and derivations. We characterize one-dimensional stable local rings in several ways. The characterizations involve the …

The resolution of the bracket powers of the maximal ideal in a diagonal hypersurface ring

Let $k$ be a field. For each pair of positive integers $(n,N)$, we resolve $Q=R/(x^N,y^N,z^N)$ as a module over the ring $R=k[x,y,z]/(x^n+y^n+z^n)$. Write $N$ in the form $N=a n+r$ for integers $a$ and $r$, with $r$ between $0$ and $n-1$. If $n$ does not divide $N$ and the characteristic of $k$ is fixed, then the value of $a$ determines whether $Q$ has finite or infinite projective dimension. If $Q$ has infinite projective dimension, then value of $r$, together with the parity of $a$, determines the periodic part of the infinite resolution. When $Q$ has infinite projective dimension we give an…

准希尔伯特环与拉蒂夫-拉什滤尘

Quasi-Hilbert rings and Ratliff-Rush filtrations

本文研究了准希尔伯特环的性质，引入了准希尔伯特环的概念，并探讨了其在理想闭包和环完成中的应用。