Tirdad Sharif, Siamak Yassemi
Let $(R,\fm)$ be commutative Noetherian local ring. It is shown that $R$ is Cohen--Macaulay ring if there exists a Cohen--Macaulay finite (i.e. finitely generated) $R$--module with finite upper Gorenstein dimension. In addition, we show that, in the Intersection Theorem, projective dimension can be replaced by quasi--projective dimension.
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For finitely generated module $M$ over a local ring $R$, the conventional notions of complete intersection dimension $\cid_R M$ and Cohen-Macaulay dimension $\cmdim_R M$ do not extend to cover the case of infinitely generated modules. In this paper we introduce similar invariants for not necessarily finitely generated modules, (namely, complete intersection flat and Cohen-Macaulay flat dimensions) which for finitely generated modules, coincide with the corresponding classical ones.
Regularity, complete intersection and Gorenstein properties of a local ring can be characterized by homological conditions on the canonical homomorphism into its residue field (Serre, Avramov, Auslander). It is also known that in positive characteristic, the Frobenius endomorphism can also be used for these characterizations (Kunz, ...), and more generally any contracting endomorphism. We introduce here a class of local homomorphisms, in some sense larger than all above, for which these characterizations still hold, providing an unified treatment for this class of homomorphisms.
Complete intersection and quasi-homological dimensions
本文证明交换诺特环上有限完全交维数模具有有限拟射影维数，改编Bergh技术并给出内射版本及局部化行为的回答。
We study homological properties of a locally complete intersection ring by importing facts from homological algebra over exterior algebras. One application is showing that the thick subcategories of the bounded derived category of a locally complete intersection ring are self-dual under Grothendieck duality. This was proved by Stevenson when the ring is a quotient of a regular ring modulo a regular sequence; we offer two independent proofs in the more general setting. Second, we use these techniques to supply new proofs that complete intersections possess symmetry of complexity.
In this article, applying the quasi-Gorenstein analogous of the Ulrich's deformation of certain Gorenstein rings we show that some homological conjectures, including the Monomial Conjecture, Big Cohen-Macaulay Algebra Conjecture as well as the Small Cohen-Macaulay Conjecture reduce to the excellent unique factorization domains. Some other reductions of the Monomial Conjecture are also proved. We, moreover, show that certain almost complete intersections adhere the Monomial Conjecture.
Homological properties of the relative Frobenius morphism
本文研究了正特征域上交换noetherian局部环间映射的相对弗罗贝尼乌斯同调性质，重点探讨完全交換和戈雷诺斯基性质。
An ideal I of a local Cohen-Macaulay ring R is called a cohomologically complete intersection if H^i_I(R) = 0 for all i \neq c = height(I). Here H^i_I(R), i \in Z denotes the local cohomology of R with respect to I. For instance, a set-theoretic complete intersection is a cohomologically complete intersection. Here we study cohomologically complete intersections from various homological points of view. As a main result it is shown that the vanishing H^iI_(M) = 0 for all i \neq c is completely encoded in homological properties of H^cI_(M). These results extend those of Hellus and Schenzel (see …
Finiteness of homological dimensions of RHom complexes
本文研究了RHom复形和Ext模块的有限完备交集维度，证明了相关稳定性结果和标准，改进了Ghosh和Samanta的结果，并证明了Auslander-Reiten猜想。
Let $R$ be a commutative Noetherian local ring with residue field $k$. Using the structure of Vogel cohomology, for any finitely generated module $M$, we introduce a new dimension, called $ζ$-dimension, denoted by $ζ-dim_R M$. This dimension is finer than Gorenstein dimension and has nice properties enjoyed by homological dimensions. In particular, it characterizes Gorenstein rings in the sense that: a ring $R$ is Gorenstein if and only if every finitely generated $R$-module has finite $ζ$-dimension. Our definition of $ζ$-dimension offer a new homological perspective on the projective dimensio…
We introduce the notion of residual intersections of modules and prove their existence. We show that projective dimension one modules have Cohen-Macaulay residual intersections, namely they satisfy the relevant Artin-Nagata property. We then establish a formula for the core of orientable modules satisfying certain homological conditions, extending previous results of Corso, Polini, and Ulrich on the core of projective one modules. Finally, we provide examples of classes of modules that satisfy our assumptions.
We prove the depth formula, for homologically bounded complexes $X, Y$ provided that the complete intersection flat dimension of $X$ is finite and $\sup(X\utp_RY)<\infty$. In particular, let $M$ and $N$ are two $R$-modules and the complete intersection flat dimension of $M$ is finite. Then $M$ and $N$ satisfies the depth formula, provided $\Tor^R_i(M,N)=0$ for all $i\ge 1$.
An ideal $I$ of a local Gorenstein ring $(R, \mathfrak m)$ is called cohomologically complete intersection whenever $H^i_I(R) = 0$ for all $i \not= \height I.$ Here $H^i_I(R), i \in \mathbb Z,$ denotes the local cohomology of $R$ with respect to $I.$ For instance, a set-theoretic complete intersection is a cohomologically complete intersection. Here we study cohomologically complete intersections from various homological points of view, in particular in terms of their Bass numbers of $H^c_I(R), c = \height I.$ As a main result it is shown that the vanishing $H^i_I(R) = 0$ for all $i \not= c$ i…
This paper main goal is to measure the defect of Cohen-Macaulayness, Gorensteiness, complete intersection and regularity for the tensor product of algebras over a ring. For this sake, we determine the homological invariants which are inherent to these notions, such as the Krull dimension, depth, injective dimension, type and embedding dimension of the tensor product constructions in terms of those of their components.
Relative homological rings and modules
本文针对交换 Noether 环及其真理想，发展相对同调环与模理论，引入 $\ rak a$-相对正则、完全交和 Gorenstein 环与模，并推广相关经典结论。
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 $H^c_{I}(R)$.
Let $R$ be commutative Noetherian ring and let $\fa$ be an ideal of $R$. For complexes $X$ and $Y$ of $R$--modules we investigate the invariant $\inf{\mathbf R}Γ_{\fa}({\mathbf R}\Hom_R(X,Y))$ in certain cases. It is shown that, for bounded complexes $X$ and $Y$ with finite homology, $\dim Y\le\dim{\mathbf R}\Hom_R(X,Y)\le\pd X+\dim(X\otimes^{\mathbf L}_RY)+\sup X$ which strengthen the Intersection Theorem. Here $\inf X$ and $\sup X$ denote the homological infimum, and supremum of the complex $X$, respectively.
Homological properties of the module of differentials
本文整理 Herzog 1980 年讲义，围绕 Vasconcelos 关于余正规模和微分模有限投射维数刻画完全交的猜想，总结相关结果并提出余切复形有界同调的猜想。
Homological dimensions, the Gorenstein property, and special cases of some conjectures
本文研究 Noetherian 环上有限模的同调维数有限性判据及其在局部 Cohen-Macaulay 环中的影响，并利用反典范模刻画 Gorenstein 性质，进而证实 Vasconcelos 正规模猜想等若干猜想的特殊情形。
Let (V,0) be a germ of a complete intersection variety in \CC^{n+k}, n>0, having an isolated singularity at 0 and X be the germ of a holomorphic vector field on \CC^{n+k} tangent to V and having on V an isolated zero at 0. We show that in this case the homological index and the GSV-index coincide. In the case when the zero of X is also isolated in the ambient space \CC^{n+k} we give a formula for the homological index in terms of local linear algebra.
We prove versions of results of Foxby and Holm about modules of finite (Gorenstein) injective dimension and finite (Gorenstein) projective dimension with respect to a semidualizing module. We also verify special cases of a question of Takahashi and White.
Let $Λ$ be a left and right noetherian ring. First, for $m,n\in\mathbb{N}\cup\{\infty\}$, we give equivalent conditions for a given $Λ$-module to be $n$-torsionfree and have $m$-torsionfree transpose. Using them, we investigate totally reflexive modules and reducing Gorenstein dimension. Next, we introduce homological invariants for $Λ$-modules which we call upper reducing projective and Gorenstein dimensions. We provide an inequality of upper reducing projective dimension and complexity when $Λ$ is commutative and local. Using it, we consider how upper reducing projective dimension relates to…
A theory of cohomological support for pairs of DG modules over a Koszul complex is investigated. These specialize to the support varieties of Avramov and Buchweitz defined over a complete intersection ring, as well as support varieties over an exterior algebra. The main objects of study are certain DG modules over a polynomial ring; these determine the aforementioned cohomological supports and are shown to encode (co)homological information about pairs of DG modules over a Koszul complex. The perspective in this article leads to new proofs of well-known results for pairs of complexes over a co…
In this paper we study homological dimensions of finitely generated modules over commutative Noetherian local rings, called reducing homological dimensions. We obtain new characterizations of Gorenstein and complete intersection local rings via reducing homological dimensions. For example, we extend a classical result of Auslander and Bridger, and prove that a local ring is Gorenstein if and only if each finitely generated module over it has finite reducing Gorenstein dimension. Along the way, we prove various connections between complexity and reducing projective dimension of modules.
Let R be a locally finitely generated algebra over a discrete valuation ring V of mixed characteristic. For any of the homological properties, the Direct Summand Theorem, the Monomial Theorem, the Improved New Intersection Theorem, the Vanishing of Maps of Tors and the Hochster-Roberts Theorem, we show that it holds for R and possibly some other data defined over R, provided the residual characteristic of V is sufficiently large in terms of the complexity of the data, where the complexity is primarily given in terms of the degrees of the polynomials over V that define the data, but possibly al…
It is proved that a module $M$ over a Noetherian local ring $R$ of prime characteristic and positive dimension has finite flat dimension if Tor$_i^R({}^e R, M)=0$ for dim $R$ consecutive positive values of $i$ and infinitely many $e$. Here ${}^e R$ denotes the ring $R$ viewed as an $R$-module via the $e$th iteration of the Frobenius endomorphism. In the case $R$ is Cohen-Macualay, it suffices that the Tor vanishing above holds for a single $e\geq \log_p e(R)$, where $e(R)$ is the multiplicity of the ring. This improves a result of D. Dailey, S. Iyengar, and the second author, as well as genera…
Quasi-projective dimension for complexes via filtrations
本文通过导出范畴中的有限滤过，将拟射影维数推广到同调有限复形，建立Auslander-Buchsbaum公式等结果，并证明完全交环上复形具有有限维数及Tor、Ext消失性质。
Homological properties of pinched Veronese rings
本文用半群方法研究捏合 Veronese 环的同调性质，重新证明并修正了 Greco-Martino 关于 Cohen-Macaulay 性的分类，并在素数特征下刻画了其 F-奇点及 Frobenius 测试指数上界。
Homological properties of rings defined by $n+1$ general quadrics in $n$ variables
研究由n+1个一般二次曲面定义的环R和其链接Gorenstein环A的同调性质，证明其具有扩展Koszul环的同调性质，并给出Poincaré级数的具体公式。
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。
Counterexamples to Peskine-Szpiro's conjecture on modules of finite projective dimension
该研究针对诺特局部环上有限投射维数模的Peskine-Szpiro同调猜想，给出反例并证明其中的维数不等式、强相交猜想和等级猜想在一般情况下不成立。
An application of the almost purity theorem to the homological conjectures
本文利用几乎纯度定理证明了在特定条件下混合特征中大Cohen-Macaulay代数的存在性。
Characteristic modules over a local ring
本文定义了有限生成模M的特征模T_M和余特征模E_M，利用它们给出了Cohen-Macaulay环和Gorenstein环的刻画，并证明了若几乎完全交环上余法模的内射维数有限，则该环是完全交环。
In this paper, various Homological Conjectures are studied for local rings which are locally finitely generated over a discrete valuation ring $V$ of mixed characteristic. Typically, we can only conclude that a particular Conjecture holds for such a ring provided the residual characteristic of $V$ is sufficiently large in terms of the complexity of the data, where the complexity is primarily given in terms of the degrees of the polynomials over $V$ that define the data, but possibly also by some additional invariants such as (homological) multiplicity. Thus asymptotic versions of the Improved …
Quasi-homological dimensions with respect to semidualizing modules
本文定义并研究了模关于半对偶化模的拟投射维数与拟内射维数，统一推广了经典同调维数理论，并推广了多个经典公式，还证明了Auslander-Reiten猜想的特殊情形。
The use of homological and homotopical devices, such as Tor and André-Quillen homology, have found substantial use in characterizing commutative algebras. The primary category setting has been differentially graded algebras and modules, but recently simplicial categories have also proved to be useful settings. In this paper, we take this point of view up a notch by extending some recent uses of homological algebra in characterizing Noetherian commutative algebras to characterizing simplicial commutative algebras having finite Noetherian homotopy through the use of simplicial homotopy theory. T…
Let $R$ be a local ring and $M$ a finitely generated $R$-module. The complete intersection dimension of $M$--defined by Avramov, Gasharov and Peeva, and denoted $\cidim_R(M)$--is a homological invariant whose finiteness implies that $M$ is similar to a module over a complete intersection. It is related to the classical projective dimension and to Auslander and Bridger's Gorenstein dimension by the inequalities $\gdim_R(N)\leq\cidim_R(N)\leq\pd_R(N)$. Using Blanco and Majadas' version of complete intersection dimension for local ring homomorphisms, we prove the following generalization of a the…
Lower bounds on Hilbert-Samuel multiplicity are known for several types of commutative noetherian local rings, and rings with multiplicities which achieve these lower bounds are said to have minimal multiplicity. The first part of this paper gives characterizations of rings of minimal multiplicity in terms of the Ext-algebra of the ring; in particular, we show that minimal multiplicity can be detected via an Ext-algebra which is Gorenstein or Koszul AS-regular. The second part of this paper characterizes rings of minimal multiplicity via a numerical homological invariant introduced by J. Herzo…
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.
A ring is called $n$-perfect ($n\geq 0$), if every flat module has projective dimension less or equal than $n$. In this paper, we show that the $n$-perfectness relate, via homological approach, some homological dimension of rings. We study $n$-perfectness in some known ring constructions. Finally, several examples of $n$-perfect rings satisfying special conditions are given.
Let $M,N$ be finitely generated modules over a local complete intersection $R$. Assume that for each $i>0$, $\mathrm{Tor}^R_i(M,N)=0$. We prove that the cohomological support of $M\otimes_R N$ (in the sense of Avramov-Buchweitz) is equal to the geometric join of the cohomological supports of $M,N$. Such result gives a new connection between two active areas or research, and immediately produces several surprising corollaries. Naturally, it also raises many intriguing new questions about the homological properties of modules over a complete intersection, some of those are investigated in the se…
New homotopy invariant finiteness conditions on modules over commutative rings are introduced, and their properties are studied systematically. A number of finiteness results for classical homological invariants like flat dimension, injective dimension, and Gorenstein dimension, are established. It is proved that these specialize to give results concerning modules over complete intersection local rings. A noteworthy feature is the use of techniques based on thick subcategories of derived categories.
The local homological structure of generalized splines
本文研究了广义样条的局部同调结构，通过两个精确序列揭示其与局部环同调机器的协调性，并证明了特定条件下广义样条模块的自由性。
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