David A. Jorgensen, Liana M. Sega
We construct a class of Gorenstein local rings $R$ which admit minimal complete $R$-free resolutions $\bd C$ such that the sequence $\{\rank_R C_i\}$ is constant for $i< 0$, and grows exponentially for all $i>0$. Over these rings we show that there exist finitely generated $R$-modules $M$ and $N$ such that $\Ext^i_R(M,N)=0$ for all $i> 0$, but $\Ext^i_R(N,M)\ne 0$ for all $i>0$.
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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 $i\in\mathbb{Z}$.
We prove that if M, N are finite modules over a Gorenstein local ring R of codimension at most 4, then the vanishing of Ext^n_R(M,N) for n\gg 0 is equivalent to the vanishing of Ext^n_R(N,M) for n\gg 0. Furthermore, if the completion of $R$ has no embedded deformation, then such vanishing occurs if and only if M or N has finite projective dimension.
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 $i >> 0$.
Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring which contains a regular sequence $ \underline{x} = x_1,\ldots,x_d \in \mathfrak{m} \smallsetminus \mathfrak{m}^2 $ such that $ \mathfrak{m}^3 \subseteq (\underline{x}) $. Let $ M $ be a finite $ R $-module with maximal complexity or curvature, e.g., $ M $ can be a nonzero direct summand of some syzygy module of the residue field $ R/\mathfrak{m} $. It is shown that the following are equivalent: (1) $R$ is Gorenstein, (2) $\mathrm{Ext}_R^{\gg 0}(M,R)=0$, and (3) $\mathrm{Tor}_{\gg 0}^R(M,ω) = 0$, where $ω$ denotes a canonical module…
Auslander's depth formula for pairs of Tor-independent modules over a regular local ring, depth(M \otimes N) = depth(M) + depth(N) - depth(R), has been generalized in several directions over a span of four decades. In this paper we establish a depth formula that holds for every pair of Tate Tor-independent modules over a Gorenstein local ring. It subsumes previous eneralizations of Auslander's formula and yields exact bounds for vanishing of cohomology over certain Gorenstein rings.
We prove several characterizations of Gorenstein rings in terms of vanishings of derived functors of certain modules or complexes whose scalars are restricted via contracting endomorphisms. These results can be viewed as analogues of results of Kunz (in the case of the Frobenius) and Avramov-Hochster-Iyengar-Yao (in the case of general contracting endomorphisms).
Gorenstein rings via homological dimensions, and symmetry in vanishing of Ext and Tate cohomology
本文通过张量-同构伴随谱序列研究交换Noetherian局部环的Gorenstein性，给出用有限Gorenstein维数CM模、内射维数有限性的刻画，还证明CM环上Ext与Tor消失的对称性及Gorenstein环上Tate上同调的类似对称结果。
Let $ R $ be a Cohen-Macaulay local ring. We prove that the $ n $th syzygy module of a maximal Cohen-Macaulay $ R $-module cannot have a semidualizing direct summand for every $ n \ge 1 $. In particular, it follows that $ R $ is Gorenstein if and only if some syzygy of a canonical module of $ R $ has a non-zero free direct summand. We also give a number of necessary and sufficient conditions for a Cohen-Macaulay local ring of minimal multiplicity to be regular or Gorenstein. These criteria are based on vanishing of certain Exts or Tors involving syzygy modules of the residue field.
Generalized symmetry in the vanishing of Ext
研究具有对偶复形的诺特环上同调有限复形 Ext 消失的广义对称性，通过额外同调条件扩展并统一局部戈伦斯坦环上相关对称结果，揭示对偶复形在其中的作用。
It is proved that if one of the finite modules M and N, over a local ring R, has reducible complexity and has finite Gorenstein dimension then the depth formula holds, provided TorR_i(M,N) = 0 for i>>0. We also study the vanishing of cohomology of a module of finite complete intersection dimension.
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.
Indicator functions, v-numbers and Gorenstein rings in the theory of projective Reed-Muller-type codes
本文研究射影Reed-Muller型码的对偶性，利用v-数、Hilbert函数和指示函数给出全局对偶判据，推广了Gorenstein消失理想上的对偶定理，并分类了自对偶码及计算广义Hamming权函数的正则度指标。
Auslander conjectured that every Artin algebra satisfies a certain condition on vanishing of cohomology of finitely generated modules. The failure of this conjecture - by a 2003 counterexample due to Jorgensen and Sega - motivates the consideration of the class of rings that do satisfy Auslander's condition. We call them AC rings and show that an AC Artin algebra that is left-Gorenstein is also right-Gorenstein. Furthermore, the Auslander-Reiten Conjecture is proved for AC rings, and Auslander's G-dimension is shown to be functorial for AC rings that are commutative or have a dualizing complex…
Let $\fa$ denote an ideal of a $d$-dimensional Gorenstein local ring $R$ and $M$ and $N$ two finitely generated $R$-modules with $\pd M< \infty$. It is shown that $H^d_{\fa}(M,N)=0$ if and only if $\dim \hat{R}\big/ \fa\hat{R}+\fp>0$ for all $\fp\in\Ass_{\hat{R}}\hat{M}\cap\Supp_{\hat{R}}\hat{N}$.
The Auslander-Reiten conjecture is a notorious open problem about the vanishing of Ext modules. In a Cohen-Macaulay complete local ring $R$ with a parameter ideal $Q$, the Auslander-Reiten conjecture holds for $R$ if and only if it holds for the residue ring $R/Q$. In the former part of this paper, we study the Auslander-Reiten conjecture for the ring $R/Q^\ell$ in connection with that for $R$, and prove the equivalence of them for the case where $R$ is Gorenstein and $\ell\le \dim R$. In the latter part, we generalize the result of the minimal multiplicity by J. Sally. Due to these two of our…
Let $I$ be ideal of an $n$-dimensional local Gorenstein ring $R$. In this paper we will describe several necessary and sufficient conditions such that the ideal $I$ becomes cohomologically complete intersections. In fact, as a technical tool, it will be shown that the vanishing $H^i_{I}(R)= 0$ for all $i\neq c= \grade (I)$ is equivalent to the vanishing of the Betti numbers of $H^c_{I}(R)$. This gives a new characterization to check the cohomologically complete intersections property with the homological properties of the vanishing of Tor modules of $H^c_{I}(R)$.
Motivated by a recent result of Yoshino, and the work of Bergh on reducible complexity, we introduce reducing versions of invariants of finitely generated modules over commutative Noetherian local rings. Our main result considers modules which have finite reducing Gorenstein dimension, and determines a criterion for such modules to be totally reflexive in terms of the vanishing of Ext. Along the way we give examples and applications, and in particular, prove that a Cohen-Macaulay local ring with canonical module is Gorenstein if and only if the canonical module has finite reducing Gorenstein d…
The depth of tensor product of modules over a Gorenstein local ring is studied. For finitely generated modules M and N over a Gorenstein local ring R, under some assumptions on the vanishing of finite number of Tate and relative homology modules, the depth($M\otimes N$) is determined in terms of the depth(M) and depth(N).
The celebrated Auslander-Reiten Conjecture, on the vanishing of self extensions of a module, is one of the long-standing conjectures in ring theory. Although it is still open, there are several results in the literature that establish the conjecture over Gorenstein rings under certain conditions. The purpose of this article is to obtain extensions of such results over Cohen-Macaulay local rings that admit canonical modules. In particular, our main result recovers theorems of Araya, and Ono and Yoshino simultaneously.
In this paper, we aim to obtain some results under the condition that the dual of a module over a commutative Noetherian ring has finite Gorenstein dimension. In this direction, we derive results involving vanishing of Ext as well as the freeness or totally reflexivity of modules. For instance, we provide a generalization of a celebrated theorem by Auslander and Bridger, obtain criteria for the totally reflexivity of modules over Cohen-Macaulay rings as well as of locally totally reflexive modules on the punctured spectrum, and recover a result by Araya. Moreover, we prove that the Auslander-R…
Let $(R,\fm)$ be a local ring and $(-)^{\vee}$ denote the Matlis duality functor. We investigate the relationship between Foxby equivalence and local duality through generalized local cohomology modules. Assume that $R$ possesses a normalized dualizing complex $D$ and $X$ and $Y$ are two homologically bounded complexes of $R$-modules with finitely generated homology modules. We present several duality results for $\fm$-section complex ${\bf R}Γ_{\fm}({\bf R}\Hom_R(X,Y))$. In particular, if G-dimension of $X$ and injective dimension of $Y$ are finite, then we show that $${\bf R}Γ_{\fm}({\bf R}\…
We investigate stable homology of modules over a commutative noetherian ring $R$ with respect to a semidualzing module $C$, and give some vanishing results that improve/extend the known results. As a consequence, we show that the balance of the theory forces $C$ to be trivial and $R$ to be Gorenstein.
On the vanishing of Ext modules over a local unique factorization domain with an isolated singularity
本文从诺特完备等特征局部环构造具有孤立奇点的局部 UFD，并据此构造反例说明 Ext 或 Tor 的消失性不能推出对称消失性、深度公式或模的全自反性。
We analyze stable homology over associative rings and obtain results over Artin algebras and commutative noetherian rings. Our study develops similarly for these classes; for simplicity we only discuss the latter here. Stable homology is a broad generalization of Tate homology. Vanishing of stable homology detects classes of rings---among them Gorenstein rings, the original domain of Tate homology. Closely related to gorensteinness of rings is Auslander's G-dimension for modules. We show that vanishing of stable homology detects modules of finite G-dimension. This is the first characterization…
Let \fa be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let \cd_{\fa}(M,N) denote the supremum of the i's such that H^i_{\fa}(M,N)\neq 0. First, by using the theory of Gorenstein homological dimensions, we obtain several upper bounds for \cd_{\fa}(M,N). Next, over a Cohen-Macaulay local ring (R,\fm), we show that \cd_{\fm}(M,N)=\dim R-\grade(\Ann_RN,M), provided that either projective dimension of M or injective dimension of N is finite. Finally, over such rings, we establish an analogue of the Hartshorne-Lichtenbaum Vanishing Theorem in the context…
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.
In this paper, we compare the Gorenstein homological dimension of a ring $R$ and of its trivial ring extension by an module $E$.
Quasi-Gorenstein extended Rees algebras associated with filtrations
本文研究诺特局部环上Hilbert滤链对应的扩张Rees代数的拟Gorenstein性质，利用局部上同调模的Matlis对偶的Cohen-Macaulay性给出变形条件，并借助局部上同调长度给出拟Gorenstein性质的刻画。
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.
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.
We establish new results on (co)homology vanishing and Ext-Tor dualities, and derive a number of freeness criteria for finite modules over Cohen-Macaulay local rings. In the main application, we settle the long-standing Auslander-Reiten conjecture for the class of Cohen-Macaulay Burch rings, among other results toward this and related problems, e.g., the Tachikawa and Huneke-Wiegand conjectures. We also derive results on further topics of interest such as Cohen-Macaulayness of tensor products and Tor-independence, and inspired by a paper of Huneke and Leuschke we obtain characterizations of wh…
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 …
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…
A theorem of Gordan and Noether via Gorenstein rings
本文以标准Artinian Gorenstein代数与Lefschetz性质为研究视角，给出Gordan-Noether超曲面锥面判定定理的新证法，并证明二次型正则序列生成理想的四维Gorenstein代数具强Lefschetz性质，可应用于光滑三次三维簇雅可比环。
In this note we characterize the (resp., weak) Gorenstein global dimension for an arbitrary ring. Also, we extend the well-known Hilbert's syzygy Theorem to the weak Gorenstein global dimension and we study the weak Gorenstein homological dimensions of direct product of rings, which gives examples of non-coherent rings of finite Gorenstein dimensions $>0$ and infinite classical weak dimension.
On Tor-vanishing of local rings
本文研究了局部环上 Tor-消没性质的变体，证明了广义 Golod 环及相关环满足该性质，从而满足 Auslander-Reiten 猜想，并提供了统一的方法。
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.
Huneke and Wiegand conjectured that, if $M$ is a finitely generated, non-free, torsion-free module with rank over a one-dimensional Cohen-Macaulay local ring $R$, then the tensor product of $M$ with its algebraic dual has torsion. This conjecture, if $R$ is Gorenstein, is a special case of a celebrated conjecture of Auslander and Reiten on the vanishing of self extensions that stems from the representation theory of finite-dimensional algebras. If $R$ is a one-dimensional Cohen-Macaulay ring such that $R=S/(f)$ for some local ring $(S, \mathfrak{n})$, and a non zero-divisor $f \in \mathfrak{n}…
We investigate the existence of ideals $I$ in a one-dimensional Gorenstein local ring $R$ satisfying $\mathrm{Ext}^{1}_{R}(I,I)=0$.
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。
Nearly Gorenstein blow-up algebras over two-dimensional regular local rings
研究二维正则局部环上与理想相关的爆破代数（里斯代数、扩展里斯代数和相伴分次环）的近戈伦斯坦性质，通过扩展框架建立刻画及关系，得到整闭理想的完全分类。
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…
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.
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。
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。
Over an arbitrary commutative ring, correspondences among three sets, the set of trace ideals, the set of stable ideals, and the set of birational extensions of the base ring, are studied. The correspondences are well-behaved, if the base ring is a Gorenstein ring of dimension one. It is shown that with one extremal exception, the surjectivity of one of the correspondences characterizes the Gorenstein property of the base ring, provided it is a Cohen-Macaulay local ring of dimension one. Over a commutative Noetherian ring, a characterization of modules in which every submodule is a trace modul…
The so-called 'change-of-ring' results are well-known expressions which present several connections between projective, injective and flat dimensions over the various base rings. In this note we extend these results to the Gorenstein dimensions over Cohen-Macaulay local rings.
Let $ R $ be a $ d $-dimensional Cohen-Macaulay (CM) local ring of minimal multiplicity. Set $ S := R/({\bf f}) $, where $ {\bf f} := f_1,\ldots,f_c $ is an $ R $-regular sequence. Suppose $ M $ and $ N $ are maximal CM $ S $-modules. It is shown that if $ \mathrm{Ext}_S^i(M,N) = 0 $ for some $ (d+c+1) $ consecutive values of $ i \geqslant 2 $, then $ \mathrm{Ext}_S^i(M,N) = 0 $ for all $ i \geqslant 1 $. Moreover, if this holds true, then either $ \mathrm{projdim}_R(M) $ or $ \mathrm{injdim}_R(N) $ is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, …
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.
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…
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.
We give counterexamples to the following conjecture of Auslander: given a finitely generated module $M$ over an Artin algebra $Λ$, there exists a positive integer $n_M$ such that for all finitely generated $Λ$-modules $N$, if $\Ext_Λ^i(M,N)=0$ for all $i\gg 0$, then $\Ext_Λ^i(M,N)=0$ for all $i\geq n_M$. Some of our examples moreover yield homologically defined classes of commutative local rings strictly between the class of local complete intersections and the class of local Gorenstein rings.
An integral domain R is said to be a splinter if it is a direct summand, as an R-module, of every module-finite extension ring. Hochster's direct summand conjecture is precisely the conjecture that every regular local ring is a splinter. An integral domain containing the rational numbers is a splinter if and only if it is a normal ring, but the notion is more subtle for rings of positive characteristic: F-regular rings are splinters, and Hochster and Huneke proved that the converse is true for Gorenstein rings. We extend their result by showing that Q-Gorenstein splinters of positive character…
Almost Gorenstein simplicial semigroup rings
本文给出单纯半群环具有几乎Gorenstein性质的判别准则，将Nari定理推广到高秩单纯半群，并据此分类了具有Ulrich元素的二维正规半群环。
In this paper, we study the pair $(\GP(R),\GP(R)^{\perp})$ where $\GP(R)$ is the class of all Gorenstein projective modules. We prove that it is complete hereditary cotorsion theory provided $l.\Ggldim(R)<\infty$. We discuss also, when every Gorenstein projective module is Gorenstein flat.
Given a semidualizing module $C$ over a commutative noetherian ring, Holm and Jørgensen investigate some connections between $C$-Gorenstein dimensions of an $R$-complex and Gorenstein dimensions of the same complex viewed as a complex over the "trivial extension" $R \ltimes C$. We generalize some of their results to a certain type of retract diagram. We also investigate some examples of such retract diagrams, namely D'Anna and Fontana's amalgamated duplication and Enescu's pseudocanonical cover.
Bounds on Gorenstein Dimensions and Exceptional Complete Intersection Maps
本文将局部同态下模的Gorenstein平坦维数界与例外完全交映射的结论相关联，并推广了已有结果，同时证明了涉及Gorenstein内射维数的类似结论。
A Fitting criterion for quasi-Gorenstein local rings
该研究针对具有典范模的诺特局部环，给出拟戈伦斯坦环的一个等价刻画条件，证实了相关学者的预期并推广了其已有结果。
Suppose $R$ is a $\mathbb{Q}$-Gorenstein $F$-finite and $F$-pure ring of prime characteristic $p>0$. We show that if $I\subseteq R$ is a compatible ideal (with all $p^{-e}$-linear maps) then there exists a module finite extension $R\to S$ such that the ideal $I$ is the sum of images of all $R$-linear maps $S\to R$.
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