Afsaneh Esmaeelnezhad, Parviz Sahandi
There are nice relations between graded homological dimensions and ordinary homological dimensions. We study the Gorenstein injective dimension of a complex of graded modules denoted by $^*\Gid$, and derive its properties. In particular we prove the Chouinard's like formula for $^*\Gid$, and compare it with the usual Gorenstein injective dimension.
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The Chouinard's formula for injective dimension is extended to the Gorenstein injective dimension.
Let R be a commutative Noetherian ring and A an Artinian R-module. We prove that if A has finite Gorenstein injective dimension, then A possesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields that over a Gorenstein ring any Artinian module possesses a Gorenstein injective envelope which is special and Artinian.
In this paper, we study a particular case of Gorenstein projective, injective, and flat modules, which we call, respectively, strongly Gorenstein projective, injective, and flat modules. These last three modules give us a new characterization of the first modules, and confirm that there is an analogy between the notion of Gorenstein projective, injective, and flat modules and the notion of the usual projective, injective, and flat modules.
We give a sufficient condition for the class of Gorenstein injective modules be precovering: if $R$ is right noetherian and if the class of Gorenstein injective modules, $\mathcal{GI}$, is closed under filtrations, then $\mathcal{GI}$ is precovering in $R-Mod$. The converse is also true when we assume that $\mathcal{GI}$ is covering. We extend our results to the category of complexes. We prove that if the class of Gorenstein injective modules is closed under filtrations then the class of Gorenstein injective complexes is precovering in $Ch(R)$. We also give a sufficient condition for the exist…
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).
This paper generalize the idea of the authors in \cite{Bennis and Mahdou1}. Namely, we define and study a particular case of modules with Gorenstein projective, injective, and flat dimension less or equal than $n\geq 0$, which we call, respectively, strongly n-Gorenstein projective, injective and flat modules. These three classes of modules give us a new characterization of the first modules, and they are a generalization of the notions of strongly Gorenstein projective, injective, and flat modules respectively.
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。
In this note, we study commutative Noetherian local rings having finitely generated modules of finite Gorenstein injective dimension. In particular, we consider whether such rings are Cohen-Macaulay.
Direct limits of Gorenstein injective modules
本文研究Gorenstein内射模类对任意正向极限封闭的条件，证明其与左noetherian性、特征模Gorenstein平坦性、覆盖性及纯子模或纯商封闭性等价。
Let $(R,\frak m)$ be a commutative Noetherian local ring and let $M$ and $N$ be finitely generated $R$-modules of finite injective dimension and finite Gorenstein injective dimension, respectively. In this paper we prove a generalization of Ischebeck Formula, that is $\depth_RM+\sup\{i| {0.1cm}\Ext_R^i(M,N)\neq 0\}=\depth R.$
Complexes of finite Gorenstein flat and injective dimensions
本文证明若交换Noetherian局部环上存在深度、Gorenstein平坦维数和Gorenstein内射维数均有限的复形，则该环为Gorenstein环，从而肯定回答了Christensen、Foxby和Holm的原始问题并将其推广到复形。
Let $R \to S$ be a local ring homomorphism and $N$ a finitely generated $S$-module. We prove that if the Gorenstein injective dimension of $N$ over $R$ is finite, then it equals the depth of $R$.
Let R be a ring, X a class of R-modules and n>1 an integer. In this paper, via special finitely presented modules, we introduce the concepts of Gorenstein n-X-injective and n-X-flat modules. And aside, we obtain some equivalent properties of these modules on n-X-coherent rings. Then, we investigate the relations among Gorenstein n-X-injective, n-X-flat, injective and flat modules on X-FC-rings (i.e., self n-X-injective and n-X-coherent rings). Several known results are generalized to this new context
The class of Gorenstein injective modules is covering if and only if it is closed under direct limits
该研究证明Gorenstein内射模类的特殊预覆盖性、覆盖性与直极限封闭性三者等价，为Enochs猜想提供新例证，并刻画了使该模类为覆盖类的环的等价条件。
Let R be a ring. In this paper, Gorenstein (n,d)-flat and (n,d)-Gorenstein (n,d)-injective modules and some of their basic properties are studied. Moreover, some characterizations of rings over Gorenstein (n,d)-flat and (n,d)-Gorenstein (n,d)-injective modules are given. Also, some known results can be obtained as corollaries
Over a noetherian ring, it is a classic result of Matlis that injective modules admit direct sum decompositions into injective hulls of quotients by prime ideals. We show that over a Cohen-Macaulay ring admitting a dualizing module, Gorenstein injective modules admit similar filtrations. We also investigate Tor-modules of Gorenstein injective modules over such rings. This extends work of Enochs and Huang over Gorenstein rings. Furthermore, we give examples showing the following: (1) the class of Gorenstein injective $R$-modules need not be closed under tensor products, even when $R$ is local a…
We prove that the class of Gorenstein injective modules is both enveloping and covering over a two sided noetherian ring such that the character modules of Gorenstein injective modules are Gorenstein flat. In the second part of the paper we consider the connection between the Gorenstein injective modules and the strongly cotorsion modules. We prove that when the ring R is commutative noetherian of finite Krull dimension, the class of Gorenstein injective modules coincides with that of strongly cotorsion modules if and only if the ring R is in fact Gorenstein.
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。
The principle "Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra" is given in [3]. There is a remarkable body of evidence supporting this claim (cf. [2] and [3]). Perhaps one of the most glaring exceptions is provided by the fact that tensor products of Gorenstein projective modules need not be Gorenstein projective, even over Gorenstein rings. So perhaps it is surprising that tensor products of Gorenstein injective modules over Gorenstein rings of finite Krull dimension are Gorenstein injective. Our main result is in support of the princ…
In this paper we show a partial answer the a question of C. Huneke and G. Leuschke (2003): Let R be a standard graded Cohen-Macaulay ring of graded countable Cohen-Macaulay representation type, and assume that R has an isolated singularity. Is R then necessarily of graded finite Cohen-Macaulay representation type? In particular, this question has an affirmative answer for standard graded non-Gorenstein rings as well as for standard graded Gorenstein rings of minimal multiplicity. Along the way, we obtain a partial classification of graded Cohen-Macaulay rings of graded countable Cohen-Macaulay…
A central problem in the theory of Gorenstein dimensions over commutative noetherian rings is to find resolution-free characterizations of the modules for which these invariants are finite. Over local rings, this problem was recently solved for the Gorenstein flat and the Gorenstein projective dimensions; here we give a solution for the Gorenstein injective dimension. Moreover, we establish two formulas for the Gorenstein injective dimension of modules in terms of the depth invariant; they extend formulas for the injective dimension due to Bass and Chouinard.
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…
Let $(R,\fm)$ be a commutative Noetherian local ring and let $M$ be an $R$-module which is a relative Cohen-Macaulay with respect to a proper ideal $\fa$ of $R$ and set $n:=\h_{M}\fa$. We prove that $\ind M<\infty$ if and only if $\ind\H^{n}_\fa(M)<\infty$ and that $\ind\H^{n}_\fa(M)=\ind M-n$. We also prove that if $R$ has a dualizing complex and $\Gid_{R} M<\infty$, then $\Gid_{R}\H^{n}_\fa(M)<\infty$ and $\Gid_{R}\H^{n}_\fa(M)=\Gid_{R} M-n$. Moreover if $R$ and $M$ are Cohen-Macaulay, then it is proved that $\Gid_{R} M<\infty$ whenever $\Gid_{R}\H^{n}_\fa(M)<\infty$. Next, for a finitely ge…
Let $R$ be a local ring of positive characteristic and $X$ a complex with nonzero finitely generated homology and finite injective dimension. We prove that if derived base change of $X$ via the Frobenius (or more generally, via a contracting) endomorphism has finite injective dimension then $R$ is Gorenstein.
In this paper we study the Weak Lefschetz property of two classes of standard graded Artinian Gorenstein algebras associated in a natural way to the Apéry set of numerical semigroups. To this aim we also prove a general result about the transfer of Weak Lefschetz property from an Artinian Gorenstein algebra to its quotients modulo a colon ideal.
Betti numbers for connected sums of graded Gorenstein artinian algebras
本文确定了分次阿廷Gorenstein代数连通和的分次Betti数，并证明加倍代数的连通和等价于纤维积环的加倍，从而将连通和构造与加倍构造联系起来。
Weakly Ding injective complexes
在左凝聚环上定义弱Ding内射复形类，研究其生成的余挠对，并展示为阿贝尔模型结构的一部分；在Ding-Chen环上证明稳定链复形范畴紧生成且三角等价于四个Frobenius范畴的稳定范畴。
Let $R$ be a commutative Noetherian ring. In this paper, we study those finitely generated $R$-modules whose Cousin complexes provide Gorenstein injective resolutions. We call such a module a G-Gorenstein module. Characterizations of G-Gorenstein modules are given and a class of such modules is determined. It is shown that the class of G-Gorenstein modules strictly contains the class of Gorenstein modules. Also, we provide a Gorenstein injective resolution for a balanced big Cohen-Macaulay $R$-module. Finally, using the notion of a G-Gorenstein module, we obtain characterizations of Gorenstein…
Parseval-Rayleigh identities for graded Artinian Gorenstein algebras
本文在正特征域上推导并证明了分层Artinian Gorenstein代数的Parseval-Rayleigh恒等式，并通过特例证明了Stanley-Reisner环的定向简形球的通用Artinian约简的Parseval-Rayleigh恒等式
We give a result of Auslander-Ringel-Tachikawa type for Gorenstein-projective modules over a complete Gorenstein order. In particular, we prove that a complete Gorenstein order is of finite Cohen-Macaulay representation type if and only if every indecomposable pure-injective object in the stable category of Gorenstein-projective modules is compact.
Let $(R,\fm)$ be a local ring and let $C$ be a semidualizing $R$--module. In this paper, we are concerned in $C$--injective and $G_{C}$--injective dimensions of certain local cohomology modules of $R$. Firstly, the injective dimension of $C$ and the above quantities of dimensions is compared. Then, as an application of the above comparisons, a characterization of a dualizing module of $R$ is given. Finally, it is shown that if $R$ is Cohen-Macaulay of dimension $d$ such that $\H_{\fm}^{d}(C)$ is $C$--injective, then $R$ is Gorernstein. This is an answer to the question which was recently prese…
A generalization of Grothendieck's non-vanishing theorem is proved for a module which is finite over a local homomorphism. It is also proved that the Gorenstein injective dimension of such a module, if finite, is bounded below by its Krull dimension and is equal to the supremum of the depths of the localizations of the ring over primes in the support of the module.
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。
For a given class of modules $\A$, we denote by $\widetilde{\A}$ the class of exact complexes $X$ having all cycles in $\A$, and by $dw(\A)$ the class of complexes $Y$ with all components $Y_j$ in $\A$. We consider a two sided noetherian ring $R$ and we use the notations $\mathcal{GI}$ $(\mathcal{GF}, \mathcal{GP})$ for the class of Gorenstein injective (flat, projective respectively) $R$-modules. We prove (Theorem 1) that the following are equivalent: 1. Every exact complex of injective modules is totally acyclic. 2. Every exact complex of Gorenstein injective modules is in $\widetilde{\mathc…
The weak Lefschetz property for Standard graded, Artinian Gorenstein algebras of embedding dimension four and socle degree three
该研究针对嵌入维数4、基座次数3的标准分次Artinian Gorenstein代数的弱Lefschetz性质展开，证明除特征2下的唯一特例外均满足该性质，明确了例外的特征条件。
In this paper, some new characterizations on Gorenstein projective, injective and flat modules over commutative noetherian local ring are given.
Quasi-injective dimension
本文引入拟内射维数以推广内射维数，并证明其在局部环上关于深度、Cohen-Macaulay、Gorenstein 性质及与拟投射维数等价性的多个经典结论。
We introduce the cohomological blow up of a graded Artinian Gorenstein (AG) algebra along a surjective map, which we term BUG (Blow Up Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blow up of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are complete intersecti…
Gorenstein flat preenvelopes and weakly Ding injective covers
本文在凝聚环框架下研究Gorenstein平坦模的预包络存在性，引入弱Ding内射模类，证明三个等价命题并给出GF为预包络类的充分条件，拓展了Ding-Chen环上的相关结果。
Let $R$ be a Gorenstein local ring, $\frak{a}$ an ideal in $R$, and $M$ an $R$-module. The local cohomology of $M$ supported at $\frak{a}$ can be computed by applying the $\frak{a}$-torsion functor to an injective resolution of $M$. Since $R$ is Gorenstein, $M$ has a complete injective resolution, so it is natural to ask what one gets by applying the $\frak{a}$-torsion functor to it. Following this lead, we define stable local cohomology for modules with complete injective resolutions. This gives a functor to the stable category of Gorenstein injective modules. We show that in many ways this b…
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。
On modules with finite reducing Gorenstein dimension
本文针对交换Noether局部环上的非零有限生成模，探究兼具有限内射维数与有限约化Gorenstein维数时，是否仍能推出模的有限投射维数及环的Gorenstein性，其中约化Gorenstein维数是比经典Gorenstein维数更精细的不变量。
We prove that, for any $n \geq 2$, the classes of $\rm{FP}_{n}$-injective modules and of $\rm{FP}_n$-flat modules are both covering and preenveloping over any ring $R$. This includes the case of $\rm{FP}_{\infty}$-injective and $\rm{FP}_{\infty}$-flat modules (i.e. absolutely clean and, respectively, level modules). Then we consider a generalization of the class of (strongly) Gorenstein flat modules - the (strongly) Gorenstein AC-flat modules (cycles of exact complexes of flat modules that remain exact when tensored with any absolutely clean module). We prove that some of the properties of Gor…
Let $R$ be a general ring. Duality pairs of $R$-modules were introduced by Holm-Jorgensen. Most examples satisfy further properties making them what we call semi-complete duality pairs in this paper. We attach a relative theory of Gorenstein homological algebra to any given semi-complete duality pair $\mathfrak{D} = (\mathcal{L},\mathcal{A})$. This generalizes the homological theory of the AC-Gorenstein modules defined by Bravo-Gillespie-Hovey, and we apply this to other semi-complete duality pairs. The main application is that the Ding injective modules are the right side of a complete (perfe…
Let $R$ be a Noetherian local ring and $I$ an $R$-ideal. It is well-known that if the associated graded ring $\gr_I(R)$ is Cohen-Macaulay (Gorenstein), then so is $R$, but the converse is not true in general. In this paper we investigate the Cohen-Macaulayness and Gorensteinness of the associated graded ring $\gr_I(R)$ under the hypothesis of the extended Rees algebra $R[It,t^{-1}]$ is quasi-Gorenstein or the associated graded ring $\gr_I(R)$ is a domain.
For a commutative ring R and a faithfully flat R-algebra S we prove, under mild extra assumptions, that an R-module M is Gorenstein flat if and only if the left S-module S\otimes M is Gorenstein flat, and that an R-module N is Gorenstein injective if and only if it is cotorsion and the left S-module Hom(S,N) is Gorenstein injective. We apply these results to the study of Gorenstein homological dimensions of unbounded complexes. In particular, we prove two theorems on stability of these dimensions under faithfully flat (co-)base change.
Strongly $FP$-injective dimensions and Gorenstein projective precovers
本文研究了格伦斯坦投射预覆盖的存在性及完备性问题，通过开发的工具验证了格伦斯坦对称猜想，并探讨了 $\mathrm{silp}(R)$ 与 $\mathrm{spli}(R)$ 的有限性关系。
Let $R$ be a polynomial ring over a field and $M= \bigoplus_n M_n$ a finitely generated graded $R$-module, minimally generated by homogeneous elements of degree zero with a graded $R$-minimal free resolution $\mathbf{F}$. A Cohen-Macaulay module $M$ is Gorenstein when the graded resolution is symmetric. We give an upper bound for the first Hilbert coefficient, $e_1$ in terms of the shifts in the graded resolution of $M$. When $M = R/I$, a Gorenstein algebra, this bound agrees with the bound obtained in \cite{ES} in Gorenstein algebras with quasi-pure resolution. We conjecture a similar bound f…
A result of Foxby states that if there exists a complex with finite depth, finite flat dimension, and finite injective dimension over a local ring $R$, then $R$ is Gorenstein. In this paper we investigate some homological dimensions involving a semidualizing complex and improve on Foxby's result by answering a question of Takahashi and White. In particular, we prove for a semidualizing complex $C$, if there exists a complex with finite depth, finite $\mathcal{F}_C$-projective dimension, and finite $\mathcal{I}_C$-injective dimension over a local ring $R$, then $R$ is Gorenstein.
A linear generalization of the nearly Gorenstein property, with applications to Veronese subalgebras
本文提出了一种几乎Gorenstein性质的线性推广，应用于Veronese子代数，证明了其在特定条件下的性质。
In this paper, we prove that the global Gorenstein projective dimension of a ring $R$ is equal to the global Gorenstein injective dimension of $R$, and that the global Gorenstein flat dimension of $R$ is smaller than the common value of the terms of this equality.
In 2012, Ananthnarayan, Avramov and Moore gave a new construction of Gorenstein rings from two Gorenstein local rings, called their connected sum. In this article, we investigate conditions on the associated graded ring of a Gorenstein Artin local ring Q, which force it to be a connected sum over its residue field. In particular, we recover some results regarding short, and stretched, Gorenstein Artin rings. Finally, using these decompositions, we obtain results about the rationality of the Poincare series of Q.
Let $A$ be a Cohen-Macaulay local ring with $\operatorname{dim} A = d\ge 3$, possessing the canonical module ${\mathrm K}_A$. Let $a_1, a_2, \ldots, a_r$ $(3 \le r \le d)$ be a subsystem of parameters of $A$ and set $Q= (a_1, a_2, \ldots, a_r)$. It is shown that if the Rees algebra ${\mathcal R}(Q)$ of $Q$ is an almost Gorenstein graded ring, then $A$ is a regular local ring and $a_1, a_2, \ldots, a_r$ is a part of a regular system of parameters of $A$.
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。
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.
The question of when the Rees algebra ${\mathcal R} (I)= \bigoplus_{n \ge 0}I^n$ of $I$ is an almost Gorenstein graded ring is explored, where $R$ is a two-dimensional regular local ring and $I$ a contracted ideal of $R$. It is known that ${\mathcal R} (I)$ is an almost Gorenstein graded ring for every integrally closed ideal $I$ of $R$. The main results of the present paper show that if $I$ is a contracted ideal with $\mathrm{o}(I) \le 2$, then ${\mathcal R} (I)$ is an almost Gorenstein graded ring, while if $\mathrm{o}(I) \ge 3$, then ${\mathcal R} (I)$ is not necessarily an almost Gorenstei…
The Multiplicity Conjecture is a deep problem relating the multiplicity (or degree) of a Cohen-Macaulay standard graded algebra with certain extremal graded Betti numbers in its minimal free resolution. In the case of level algebras of codimension three, Zanello has proposed a stronger conjecture. We prove this conjecture for the case of codimension three graded Gorenstein algebras.
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。
Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical counterparts, these dimensions do not immediately come with practical and robust criteria for finiteness, not even over commutative noetherian local rings. In this paper we enlarge the class of rings known to admit good criteria for finiteness of Gorenstein dimensions: It now includes, for instance, the rings encountered in commutative algebraic geometry and, in the…
We introduce new homological dimensions, namely the Cohen-Macaulay projective, injective and flat dimensions for homologically bounded complexes. Among other things we show that (a) these invariants characterize the Cohen-Macaulay property for local rings, (b) Cohen-Macaulay flat dimension fits between the Gorenstein flat dimension and the large restricted flat dimension, and (c) Cohen-Macaulay injective dimension fits between the Gorenstein injective dimension and the Chouinard invariant.
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