Craig Huneke, Adela Vraciu
We introduce classes of rings which are close to being Gorenstein. These rings arise naturally as specializations of rings of countable CM type. We study these rings in detail, and along the way generalize an old result of Teter which characterized Artinian rings which are Gorenstein rings modulo their socle.
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In this paper, we introduce and study the rings of Gorenstein homological dimensions small or equal than 1, which we call Gorenstein (semi)hereditary rings, specially particular cases of these rings, which we call strongly Gorenstein (semi)hereditary rings.
In this paper, we extend the well-known Hilbert's syzygy theorem to the Gorenstein homological dimensions of rings. Also, we study the Gorenstein homological dimensions of direct product of rings, which gives examples of non-Noetherian rings of finite Gorenstein dimensions and infinite classical weak dimension.
In this paper, we extend the well-known Hilbert's syzygy theorem to the Gorenstein homological dimensions of rings. Also, we study the Gorenstein homological dimensions of direct products of rings. Our results generate examples of non-Noetherian rings of finite Gorenstein dimensions and infinite classical weak dimension.
The notion of $2$-almost Gorenstein ring is a generalization of the notion of almost Gorenstein ring in terms of Sally modules of canonical ideals. In this paper, we deal with two different topics related to $2$-almost Gorenstein rings. The purposes are to determine all the Ulrich ideals in $2$-almost Gorenstein rings and to clarify the structure of minimal free resolutions of $2$-almost Gorenstein rings.
In this paper, we study the rings with zero Gorenstein weak dimensions, which we call them Gorenstein Von Neumann regular rings.
In this paper, we state criteria of a Hibi ring to be level, non-Gorenstein and almost Gorenstein and to be non-level and almost Gorenstein in terms of the structure of the partially ordered set defining the Hibi ring. We also state a criterion of a ladder determinantal ring defined by 2-minors to be non-Gorenstein and almost Gorenstein in terms of the shape of the ladder.
When do pseudo-Gorenstein rings become Gorenstein?
本文研究伪Gorenstein环成为Gorenstein环的条件，证明在温和假设下伪Gorenstein的近乎Gorenstein分次整环必为Gorenstein，并阐明相关概念关系及给出构造拟Gorenstein环的方法。
The notion of generalized Gorenstein local ring (GGL ring for short) is one of the generalizations of Gorenstein rings. In this article, there is given a characterization of GGL rings in terms of their canonical ideals and related invariants.
We prove that the class of Gorenstein projective modules is special precovering over any left GF-closed ring such that every Gorenstein projective module is Gorenstein flat and every Gorenstein flat module has finite Gorenstein projective dimension. This class of rings includes (strictly) Gorenstein rings, commutative noetherian rings of finite Krull dimension, as well as right coherent and left n-perfect rings. In section 4 we give examples of left GF-closed rings that have the desired properties (every Gorenstein projective module is Gorenstein flat and every Gorenstein flat has finite Goren…
Let $C$ be a semidualizing module. We first investigate the properties of finitely generated $G_C$-projective modules. Then, relative to $C$, we introduce and study the rings of Gorenstein (weak) global dimensions at most 1, which we call $C$-Gorenstein (semi)hereditary rings, and prove that every $C$-Gorenstein hereditary ring is both coherent and $C$-Gorenstein semihereditary.
Starting with a commutative ring $R$ and an ideal $I$, it is possible to define a family of rings $R(I)_{a,b}$, with $a,b \in R$, as quotients of the Rees algebra $\oplus_{n \geq 0} I^nt^n$; among the rings appearing in this family we find Nagata's idealization and amalgamated duplication. Many properties of these rings depend only on $R$ and $I$ and not on $a,b$; in this paper we show that the Gorenstein and the almost Gorenstein properties are independent of $a,b$. More precisely, we characterize when the rings in the family are Gorenstein, complete intersection, or almost Gorenstein and we …
The aim of this paper is the study of Gorenstein global and weak dimensions of semi-primary rings.
The far-flung Gorenstein numerical semigroup rings of type 4
本文研究了类型4的远距离Gorenstein数值半群环的分类问题。
One of the main results of this paper is the characterization of the rings over which all modules are strongly Gorenstein projective. We show that these kinds of rings are very particular cases of the well-known quasi-Frobenius rings. We give examples of rings over which all modules are Gorenstein projective but not necessarily strongly Gorenstein projective.
We consider the following question: Is Gorenstein homology a X-pure homology, in the sense defined by Warfield, for a class X of modules? Let GP denote the class of Gorenstein projective modules. We prove that over a commutative Noetherian ring R of finite Krull dimension, Gorenstein homology is a GP-pure homology if and only if R is virtually Gorenstein.
In this paper, we compare the Gorenstein homological dimension of a ring $R$ and of its trivial ring extension by an module $E$.
In this paper we investigate the question of when the determinantal ring $R$ over a field $k$ is an almost Gorenstein local/graded ring in the sense of Goto, Takahashi, and the author. As a consequence of the main result, we see that if $R$ is a non-Gorenstein almost Gorenstein local/graded ring, then the ring $R$ has a minimal multiplicity.
In this paper, we show that the Gorenstein global dimension of trivial ring extensions is often infinite. Also we study the transfer of Gorenstein properties between a ring and its trivial ring extensions. We conclude with an example showing that, in general, the transfer of the notion of Gorenstein projective module does not carry up to pullback constructions.
We characterize Cohen-Macaulay and Gorenstein rings obtained from certain types of convex body semigroups. Algorithmic methods to check if a polygonal or circle semigroup is Cohen-Macaulay/Gorenstein are given. We also provide some families of Cohen-Macaulay and Gorenstein rings.
This paper introduces and studies a particular subclasses of the class of commutative rings with finite Gorenstein global (resp., weak) dimensions.
This paper introduces and studies a particular subclass of the class of commutative rings with finite Gorenstein global dimension.
In this paper several quasi-Gorenstein counterparts to some known properties of Gorenstein rings are given. We, furthermore, give an explicit description of the attach prime ideals of certain local cohomology modules.
In this paper, we establish, as a generalization of a result on the classical homological dimensions of commutative rings, an upper bound on the Gorenstein global dimension of commutative rings using the global cotorsion dimension of rings. We use this result to compute the Gorenstein global dimension of some particular cases of trivial extensions of rings and of group rings.
How many ideals whose quotient rings are Gorenstein exist?
本文研究Gorenstein环中商环为Gorenstein的非主理想数量问题，证明对称数值半群环中此类分次理想个数等于半群导子，并给出完整列表。
A ring $R$ is called left GF-closed, if the class of all Gorenstein flat left $R$-modules is closed under extensions. The class of left GF-closed rings includes strictly the one of right coherent rings and the one of rings of finite weak dimension. In this paper, we investigate the Gorenstein flat dimension over left GF-closed rings. Namely, we generalize the fact that the class of all Gorenstein flat left modules is projectively resolving over right coherent rings to left GF-closed rings. Also, we generalize the characterization of Gorenstein flat left modules (then of Gorenstein flat dimensi…
In this paper, after giving a criterion for a Noetherian local ring to be quasi-Gorenstein, we obtain some sufficient conditions for a quasi- Gorenstein ring to be Gorenstein. In the course, we provide a slight generalization of a theorem of Evans and Griffith.
Motivated by their impact on homological algebra, the change of rings results have been the subject of several interesting works in Gorenstein homological algebra over Noetherian rings. In this paper, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results established, in this…
We compute the Gorenstein weak dimension of a coherent power series rings over a commutative rings and we show that, in general, $\gwd(R) \leq 1$ does not imply that $R$ is an arithmetical ring.
This paper applies G. Lyubeznik's notion of $F$-finite modules to describe in a very down-to-earth manner certain annihilator submodules of some top local cohomology modules over Gorenstein rings. As a consequence we obtain an explicit description of the test ideal of Gorenstein rings in terms of ideals in a regular ring.
Gorenstein injective filtrations over rings with dualizing complexes
本文将Gorenstein环上Gorenstein内射模的有限Gorenstein内射子模滤过结论，推广至带对偶复形的交换诺特环，并用Auslander范畴和截面函子实现了这类滤过。
We investigate the existence of ideals $I$ in a one-dimensional Gorenstein local ring $R$ satisfying $\mathrm{Ext}^{1}_{R}(I,I)=0$.
In this paper, we make the notion of approximating an Artinian local ring by a Gorenstein Artin local ring precise using the concept of Gorenstein colength. We also answer the question as to when the Gorenstein colength is at most two.
In this paper, we study the Gorenstein global dimension of an \emph{amalgamated duplication} of a coherent ring along a regular principal ideal.
Koszulness of Gorenstein quadratic algebras of small socle degree is studied. In this note, we construct non-Koszul Gorenstein quadratic toric ring such that its socle degree is more than 3 by using stable set polytopes.
Almost Gorenstein standard graded rings and their localizations
本文证明了域上Cohen-Macaulay标准分次整环为几乎Gorenstein分次环，等价于其在分次极大理想处局部化的完备化为几乎Gorenstein局部环。
We show that $F$-purity deforms in local $\mathbb{Q}$-Gorenstein rings of prime characteristic $p>0$. Furthermore, we show that $F$-purity is $\mathfrak{m}$-adically stable in local Cohen-Macaulay $\mathbb{Q}$-Gorenstein rings.
In this paper, for the development of the study of almost Gorenstein graded rings, we discuss some relations between almost Gorensteinness of Cohen--Macaulay homogeneous rings and their $h$-vectors. Concretely, for a Cohen--Macaulay homogeneous ring $R$, we give a sufficient condition for $R$ to be almost Gorenstein in terms of the $h$-vector of $R$ and we also characterize almost Gorenstein homogeneous domains with small socle degrees in terms of the $h$-vector of $R$. Moreover, we also provide the examples of almost Gorenstein homogeneous domains arising from lattice polytopes.
The main aim of this paper is to investigate rings over which all (finitely generated strongly) Gorenstein projective modules are projective. We consider this propriety under change of rings, and give various examples of rings with and without this propriety.
Gorenstein Endomorphism Rings on Curve Singularities
本文研究曲线奇点上分式理想自同态环的 Gorenstein 性，利用理想稳定性与值半群理想条件给出刻画，并分类相关 Gorenstein 代数型曲线。
The concept of Gorenstein dimension, defined by Auslander and Bridger for finitely generated modules over a Noetherian ring, is studied in the context of finitely presented modules over a coherent ring. A generalization of the Auslander-Bridger formula is established and is used as a cornerstone in the development of a theory of coherent Gorenstein rings.
The notion of almost Gorenstein ring given by Barucci and Fr{ö}berg \cite{BF} in the case where the local rings are analytically unramified is generalized, so that it works well also in the case where the rings are analytically ramified. As a sequel, the problem of when the endomorphism algebra $\m : \m$ of $\m$ is a Gorenstein ring is solved in full generality, where $\m$ denotes the maximal ideal in a given Cohen-Macaulay local ring of dimension one. Characterizations of almost Gorenstein rings are given in connection with the principle of idealization. Examples are explored.
Characterization of almost Gorenstein rings in terms of the trace ideal
本文用迹理想刻画一维几乎Gorenstein环，并应用于研究$\mathbb{Z}_2$-分次环的几乎Gorenstein性质。
In this paper we study generalized Gorenstein Arf rings; a class of one-dimensional Cohen-Macaulay local Arf rings that is strictly contained in the class of Gorenstein rings. We obtain new characterizations and examples of Arf rings, and give applications of our argument to numerical semigroup rings and certain idealizations. In particular, we generalize a beautiful result of Barucci and Fröberg concerning Arf numerical semigroup rings.
A commutative noetherian ring with a dualizing complex is Gorenstein if and only if every acyclic complex of injective modules is totally acyclic. We extend this characterization, which is due to Iyengar and Krause, to arbitrary commutative noetherian rings, i.e. we remove the assumption about a dualizing complex. In this context Gorenstein, of course, means locally Gorenstein at every prime.
The purpose of this paper is, as part of the stratification of Cohen-Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product $R \times_T S$ of Cohen-Macaulay local rings $R$, $S$ of the same dimension $d>0$ over a regular local ring $T$ with $\dim T=d-1$ is an almost Gorenstein ring if and only if so are $R$ and $S$. Besides, the other generalizations of Gorenstein properties are also explored.
Let $(R,\frak m, k)$ be a noetherian local ring. It is well-known that $R$ is regular if and only if the injective dimension of $k$ is finite. In this paper it is shown that $R$ is Gorenstein if and only if the Gorenstein injective dimension of $k$ is finite. On the other hand a generalized version of the so-called Bass formula is proved for finitely generated modules of finite Gorenstein injective dimension. It also improves Christensen's generalized Bass formula (cf. "Gorenstein dimensions", volume 1747 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000).
Remarks on almost Gorenstein rings
本文考察分次环与局部环几乎 Gorenstein 性质的传递关系，证明分次情形可推出极大理想局部化后的局部情形，并在一维分次整环的温和条件下给出逆命题，同时研究相关数值半群环的定义理想。
Let $(R, \mathfrak{m}) $ be a Gorenstein local ring of dimension $d > 0$ and let $I$ be an ideal of $R$ such that $(0) \ne I \subsetneq R$ and $R/I$ is a Cohen-Macaulay ring of dimension $d$. There is given a complete answer to the question of when the idealization $A = R \ltimes I$ of $I$ over $R$ is an almost Gorenstein local ring.
Let $R$ be a Noetherian local ring. We prove that $R$ is regular of dimension at most four if, and only if, every prime ideal, defining a Gorenstein quotient ring, is syzygetic. We deduce a characterization of these rings in terms of the André-Quillen homology.
In this note we study trace ideals of canonical modules. Characterizations of the trace ideals in terms of annihilators of certain Ext modules are given. We apply our results to study many classes of rings close to being Gorenstein that appear in recent literature. We discuss the behavior of nearly Gorensteinness, introduced by Herzog-Hibi-Stamate, under reductions by regular sequences. A nearly Gorenstein version of the Tachikawa conjecture is posed, and an affirmative answer given for type 2 rings. We also introduce the class of weakly almost Gorenstein rings using the canonical module, and …
We provide a characterization of the almost Gorenstein property of determinantal rings of a symmetric matrix of indeterminates over an infinite field. We give an explicit formula for ranks of the last two modules in the resolution of determinantal rings using Schur functors.
Given positive integers e and s we consider Gorenstein Artinian local rings R of embedding dimension e whose maximal ideal $\mathfrak{m}$ satisfies $\mathfrak{m}^s\ne 0=\mathfrak{m}^{s+1}$. We say that R is a compressed Gorenstein local ring when it has maximal length among such rings. It is known that generic Gorenstein Artinian algebras are compressed. If $s\ne 3$, we prove that the Poincare series of all finitely generated modules over a compressed Gorenstein local ring are rational, sharing a common denominator. A formula for the denominator is given. When s is even this formula depends on…
In this paper, the principal tool to describe transversal polymatroids with Gorenstein base ring is polyhedral geometry, especially the $Danilov-Stanley$ theorem for the characterization of canonical module. Also, we compute the $a-invariant$ and the Hilbert series of base ring associated to this class of transversal polymatroids.
We investigate symmetry in the vanishing of Ext for finitely generated modules over local Gorenstein rings. In particular, we define a class of local Gorenstein rings, which we call AB rings, and show that for finitely generated modules $M$ and $N$ over an AB ring $R$, $Ext^i_R(M,N)=0$ for all $i >> 0$ if and only if $Ext^i_R(N,M)=0$ for all $i >> 0$.
A question of Conca, Rossi, and Valla asks whether every quadratic Gorenstein ring $R$ of regularity three is Koszul. In a previous paper, we use idealization to answer their question, proving that in nine or more variables there exist quadratic Gorenstein rings of regularity three which are not Koszul. In this paper, we study the analog of the Conca-Rossi-Valla question when the regularity of $R$ is four or more. Let $R$ be a quadratic Gorenstein ring having $\mathrm{codim}\, R = c$ and $\mathrm{reg}\, R = r \ge 4$. We prove that if $c = r+1$ then $R$ is always Koszul, and for every $c \geq r…
In \cite{Ouarghi}, the authors discuss the rings over which all modules are strongly Gorenstein projective. In this paper, we are interesting to an extension of this idea. Thus, we discuss the rings over which every Gorenstein projective (resp. flat) module is strongly Gorenstein projective (resp, flat). Our aim is to give examples of rings with different Gorenstein global dimension satisfied this condition.
This note proves that if S is an unramified regular local ring and I and J are proper ideals of height at least two, then S/IJ is never Gorenstein.
The structure of the complex $\operatorname{\mathbf{R}Hom}_R(R/I,R)$ is explored for an Ulrich ideal $I$ in a Cohen-Macaulay local ring $R$. As a consequence, it is proved that in a one-dimensional almost Gorenstein but non-Gorenstein local ring, the only possible Ulrich ideal is the maximal ideal. It is also studied when Ulrich ideals have the same minimal number of generators.
We provide algorithmic methods to check the Cohen--Macaulayness, Buchsbaumness and/or Gorensteiness of some families of semigroup rings that are constructed from the dilation of bounded convex polyhedrons of $\R^3_{\geq}$. Some families of semigroup rings are given satifying these properties.
We investigate symmetry in the vanishing of Tate cohomology for finitely generated modules over local Gorenstein rings. For finitely generated R-modules M and N over Gorenstein local ring R, it is shown that $\widehat{Ext}^i_R(M,N)=0$ for all $i\in\mathbb{Z}$ if and only if $\widehat{Ext}^i_R(N,M)=0$ for all $i\in\mathbb{Z}$.
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