Shan Chang
Let $A$ be a commutative augmented ring and $I$ be its augmentation ideal. This paper shows that the sequence $\{I^n/I^{n+1}\}$ becomes stationary up to isomorphism. The result yields stability in the associated graded ring of $A$ along $I$.
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Simple graded commutative algebras
研究了任意阿贝尔群$Γ$的$Γ$ -分次交换代数的概念。主要的例子是克利福德代数已经处理的阿尔布开克和马吉德。证明了Clifford代数是$\mathbb{C}$或$\mathbb{R}$上唯一的有限维结合分次交换代数.我们的方法也导致非结合分次交换代数扩展的Clifford代数。
Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring, $M$ a graded $R$-module and $A\subseteq h(R)$ a multiplicatively closed subset of $R$. In this paper, we introduce the concept of graded $A$-2-absorbing submodules of $M$ as a generalization of graded 2-absorbing submodules and graded $A$-prime submodules of $M.$ We investigate some properties of this class of graded submodules.
Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper, we introduce the concept of graded classical and graded strongly classical 2-absorbing second submodules of graded modules over a graded commutative rings. A number of results concerning these classes of graded submodules and their homogeneous components are given.
Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper, we introduce the concept of graded primary-like submodules as a new generalization of graded primary ideals and give some basic results about graded primary-like submodules of graded modules. Special attention has been paid, when graded submodules satisfies the gr-primeful property, to find extra properties of these graded submodules.
Differential graded (DG) algebras are powerful tools from rational homotopy theory. We survey some recent applications of these in the realm of homological commutative algebra.
Comparing commutative and associative unbounded differential graded algebras over Q from homotopical point of view
本文在有理数域$\mathbb{Q}$上建立了增广微分分次交换代数CDGA的模型范畴的一个子范畴与增广微分分次代数代数DGA的模型范畴的一个子范畴在同伦意义下的忠实性结果.
This paper deals with the graded commutative rings in which every homogeneous prime ideal is contained in a unique homogeneous maximal ideal called Gelfand graded ring. The purpose is to establish some topological and algebraic characterizations of these rings, one of which is the algebraic analogue of the Urysohn's lemma. Finally we look at a special class of those graded rings called pm$^+$ graded rings which can be viewed as graded ring with a Gelfand strong property.
A somewhat gentle introduction to differential graded commutative algebra
微分分次交换代数为证明交换环上模的定理提供了有力的工具。这些笔记是对这些技术的一个有点口语化的介绍。为了提供一些动力交换代数谁想知道的好处，学习和使用这些技术，我们提出了他们的背景下，最近的结果Nasseh和萨瑟-瓦格斯塔夫。这些笔记用于课程“微分分级交换代数“，这是在里贾纳大学举行的代数和几何之间的连接研讨会的一部分，2012年5月29日至6月1日。
Let $R$ be a $G$-graded ring and M be a $G$-graded $R$-module. We define the graded primary spectrum of $M$, denoted by $\mathcal{PS}_G(M)$, to be the set of all graded primary submodules $Q$ of M such that $(Gr_M(Q):_R M)=Gr((Q:_R M))$. In this paper, we define a topology on $\mathcal{PS}_G(M)$ having the Zariski topology on the graded prime spectrum $Spec_G(M)$ as a subspace topology, and investigate several topological properties of this topological space.
Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring with identity and $M$ a graded $R$-module. In this paper, we introduce the concept of graded $I_{e}$-prime submodule as a generalization of a graded prime submodule for $I=\oplus_{g\in G}I_{g}$ a fixed graded ideal of $R$. We give a number of results concerning of these classes of graded submodules and their homogeneous components. A proper graded submodule $N$ of $M$ is said to be a graded $I_{e}$-prime submodule of $M$ if whenever $% r_{g}\in h(R)$ and $m_{h}\in h(M)$ with $r_{g}m_{h}\in N-I_{e}N,$ then either $r_…
Computing Small 1-Homological Models for Commutative Differential Graded Algebras
我们使用同调扰动机械具体的代数范畴[P。同调扰动理论与结合论。Homology，Homotopy and Applications第2卷，n. 5（2000）51-88]给出了计算交换微分分次代数（简称CDGA）的小1-同调模型的微分结构的算法。的过程的复杂性进行了研究，并在Mathematica的计算机软件包中描述用于确定这样的模型。
In this work, we prove that if a graded, commutative algebra $R$ over a field $k$ is not Koszul then, denoting by $\mathfrak{m}$ the maximal homogeneous ideal of $R$ and by $M$ a finitely generated graded $R$-module, the nonzero modules of the form $\mathfrak{m} M$ have infinite Castelnuovo-Mumford regularity. We also prove that over complete intersections which are not Koszul, a nonzero direct summand of a syzygy of $k$ has infinite regularity. Finally, we relate the vanishing of the graded deviations of $R$ to having a nonzero direct summand of a syzygy of $k$ of finite regularity.
Let $\Bbb Z_2:=\Bbb Z/2\Bbb Z$ be the additive group with two elements. In this article, we focus only on $\Bbb Z_2$-graded commutative ring i.e commutative ring $R$ such that $R=R_0\oplus R_1$ as Abelian group and $R_iR_j\subseteq R_{i+j}$ for all $i,j\in \Bbb Z_2$. Our main goals is to establish a strong relation between $\Bbb Z_2$-graded prime ( maximal ) ideals of $R$ and prime ( maximal) ideals of $R_0$, for instance, it is showed that, the $\Bbb Z_2$-graded spectrum of $R$ is homeomorphic to the spectrum of $R_0$ with respect to the Zariski topologies.
This paper is a review of concepts from graded commutative algebra with specific attention given to length and multiplicity. The author's motivation for this paper comes from the study of equivariant cohomology in algebraic topology where the graded commutative algebra of the module is intimately connected to topological properties of the space (as shown by Quillen). Results (and their proofs) which are often left as exercises, or considered 'folklore' in the commutative algebra community are included in this paper, as are references to relevant applications in topology. As such, this paper is…
Even more spectra: tensor triangular comparison maps via graded commutative 2-rings
我们提出了分次交换2-环的理论，它是分次交换环的一个分类。我们的目标是提供一个系统的推广保罗巴尔默的比较地图之间的频谱张量三角范畴和Zebraki频谱的中心环。通过应用我们的构造，我们计算了任何分次交换环上完全复形的导出范畴的谱，并且我们将每一个具有丰富的线丛族的方案嵌入到相应的分次交换2-环的谱中.
Regular $\mathbb{Z}$-graded local rings and Graded Isolated Singularities
本文研究正则Z-分次局部环的刻画，并用非交换射影概形的整体维数刻画分次孤立奇点，得到其与光滑性及凝聚层范畴有限整体维数的等价条件。
The Quasi-Zariski topology on the graded quasi-primary spectrum of a graded module over a graded commutative ring
本文定义分次交换环上分次模的分次准素谱，并在其上引入拟-Zariski拓扑，研究其拓扑性质及成为Noether谱空间的条件。
In this paper we undertake a basic study on connective commutative differential graded algebras (CDGA), more precisely, piecewise Noetherian CDGA, which is a DG-counter part of commutative Noetherian algebra. We establish basic results for example, Auslaner-Buchsbaum formula and Bass formula without any unnecessary assumptions. The key notion is the sup-projective (sppj) and inf-injective (ifij) resolutions introduced by the author, which are DG-versions of the projective and injective resolution for ordinary modules. These are different from DG-projective and DG-injective resolutions which is…
Graded commutative algebras: examples, classification, open problems
考虑$\G$ -分次交换代数，其中$\G$是交换群.从一个显着的例子，古典代数的四元数，更一般地说，一个任意的Clifford代数，我们开发一个一般的观点对这个问题。然后，我们给出了最近的分类结果，并制定了一个开放的问题。
On the derived category of a graded commutative noetherian ring
对于任何分次交换诺特环，其中分次群是交换的，并且交换性在相当一般的意义上是允许的，我们建立了一方面，导出范畴的扭闭局部化子范畴，另一方面，环的素理想的齐次谱的子集之间的包含保持双射。我们提供了一个应用加权射影概形。
Lying-Over Theorem on Left Commutative Rngs
引入分次整元的概念，证明了交换代数上的lying-over定理在左交换环中的对应定理，并利用Hu-Liu积选取了一类非交换环.
We show that the infinite symmetric product of a connected graded-commutative algebra over the rationals is naturally isomorphic to the free graded-commutative algebra on the positive degree subspace of the original algebra. In particular, the infinite symmetric product of a connected commutative (in the usual sense) graded algebra over the rationals is a polynomial algebra. Applied to topology, we obtain a quick proof of the Dold-Thom theorem in rational homotopy theory for connected spaces of finite type. We also show that finite symmetric products of certain simple free graded commutative a…
The André-Quillen cohomology of commutative monoids
本文指出交换幺半群的 Beck 模块等价于相关分次交换环上的模，证明其 Quillen 同调是 André-Quillen 同调的特例，并发展分次形式化理论验证了 Grillet 部分上链复形与 Barr 修正的 Harrison 上链复形的联系。
Differential Calculus over Graded Commutative Algebras and Vector Bundles with Inner Structures
本文借助 Vinogradov 微分学函子并适配分次交换代数，统一解释多种几何结构，证明 triole 代数适合刻画带向量值纤维度量的向量丛微分学。
In this article, we introduce a generalization of the concept of graded $r$-ideals in graded commutative rings with nonzero unity. Let $G$ be a group, $R$ be a $G$-graded commutative ring with nonzero unity and $GI(R)$ be the set of all graded ideals of $R$. Suppose that $ϕ: GI(R)\rightarrow GI(R)\bigcup\{\emptyset\}$ is a function. A proper graded ideal $P$ of $R$ is called a graded $ϕ$-$r$-ideal of $R$ if whenever $x, y$ are homogeneous elements of $R$ such that $xy\in P-ϕ(P)$ and $Ann(x) =\{0\}$, then $y\in P$. Several properties of graded $ϕ$-$r$-ideals have been examined.
Homology over trivial extensions of commutative DG algebras
当$M$和$N$是无限投射维数的单生$R$ -模时，Noether局部环$R$的Koszul复形上的条件保证了对于无限多个$i$，$\mathrm{Tor}^{R}_{i}(M,N)$是非零的.这些条件是由交换微分分次代数的某些平凡扩张上的微分分次模的Tor的结果得到的。
Let $R$ be a commutative $G$-graded ring with a nonzero unity. In this article, we introduce the concept of graded radically principal ideals. A graded ideal $I$ of $R$ is said to be graded radically principal if $Grad(I)=Grad(\langle c\rangle)$ for some homogeneous $c\in R$, where $Grad(I)$ is the graded radical of $I$. The graded ring $R$ is said to be graded radically principal if every graded ideal of $R$ is graded radically principal. We study graded radically principal rings. We prove an analogue of the Cohen theorem, in the graded case, precisely, a graded ring is graded radically princ…
The main goal of this article is to introduce the concept of $EM-G-$graded rings. This concept is an extension of the notion of $EM-$rings. Let $G$ be a group and $R$ be a $G-$graded commutative ring. The $G-$gradation of $R$ can be extended to $R[x]$ by taking the components $(R[x])_σ=R_σ[x]$. We define $R$ to be $EM-G-$graded ring if every homogeneous zero divisor polynomial has an annihilating content. We provide examples of $EM-G-$graded rings that are not $EM-$rings and we prove some interesting results regarding these rings.
The volume intrinsic to a commutative graded algebra
本文研究交换分次代数中典范模在 Artinian 约化模空间形变下的行为，通过构造有效几何视角解释无环交点数，并探讨规范化性质。
Let $G$ be a group with identity $e$ and $R$ a commutative $G$-graded ring with a nonzero unity $1$. In this article, we introduce the concepts of graded $r$-submodules and graded special $r$-submodules, which are generalizations for the notion of graded r-ideals. For a nonzero $G$-graded $R$-module $M$, a proper graded $R$-submodule $K$ of $M$ is said to be graded $r$-submodule (resp., graded special $r$-submodule) if whenever $a\in h(R)$ and $x\in h(M)$ such that $ax\in K$ with $Ann_{M}(a)=\{0\}$ (resp., $Ann_{R}(x)=\{0\}$), then $x\in K$ (resp., $a\in (K:_{R}M)$). We study various propertie…
In this note, we consider a situation that is generally used as an intermediate technical step in proving the Artin-Rees lemma but otherwise is not much discussed in introductory accounts of commutative algebra. I hope to show in this note that such technical step deserves more recognition and emphasis in any introduction to commutative algebra because it can be used to prove some significant results in a straight-forward manner, including a generalization of a theorem by Pierre Cartier and John Tate.
Stable cohomology is a generalization of Tate cohomology to associative rings, first defined by Pierre Vogel. For a commutative local ring $R$ with residue field $k$, stable cohomology modules $\widehat{\mathrm{Ext}}{\vphantom E}^{n}_R\;(k,k)$, defined for $n\in\mathbb{Z}$, have been studied by Avramov and Veliche. Stable cohomology carries a structure of $\mathbb{Z}$-graded $k$-algebra. One of the main goals of this paper is to prove that, for a class of Gorenstein rings, this algebra is a trivial extension of absolute cohomology $\mathrm{Ext}_R(k,k)$ and a shift of $\mathrm{Hom}_k(\mathrm{Ex…
Let $G$ be an abelian group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper we will obtain some results concerning the graded generalized 2-absorbing submodules and their homogeneous components. Special attention has been paid, when graded rings are graded gr-Noetherian, to find extra properties of these graded submodules.
On the $h$-polynomials of cyclotomic standard graded commutative algebras
本文研究分圆标准分次交换代数的 h-多项式何时为 CI 型，证明了当 h_R(1) 为 1、4、6 或素数时必为 CI 型，并构造了 h_R(1) 为不小于 8 的非素数时的非 CI 型反例。
Homogeneous Buchberger algorithms and Sullivant's computational commutative algebra challenge
我们给出了正分次格理想的齐次Buchberger算法的一个变种。使用该算法，我们解决了Sullivant计算交换代数的挑战。
GBNL: Graded Betti Number Learning of Complex Biological Data
GBNL首次将交换代数分次贝蒂数用于蛋白质-核酸结合预测，结合k-mer集合与Transformer嵌入，有效检测突变，建立比较序列分析新范式。
Let $R$ be a commutative Noetherian ring graded by a torsionfree abelian group $G$. We introduce the notion of $G$-graded irreducibility and prove that $G$-graded irreducibility is equivalent to irreducibility in the usual sense. This is a generalization of Chen and Kim's result in the $\mathbb{Z}$-graded case. We also discuss the concept of the index of reducibility and give an inequality for the indices of reducibility between any radical non-graded ideal and its largest graded subideal.
Homological Associativity of Differential Graded Algebras and Gröbner Bases
本文从结合子子复形的同调检测和Gröbner基计算两个视角，研究交换noetherian环上链复形的乘法结合性。
Matijevic-Roberts type theorems, Rees rings and associated graded rings
本文研究交换代数中提升问题、沿相伴分次环的上升/下降问题与Matijevic-Roberts型问题这三个显著问题之间的相互关联结构。
Let $R$ be a $G$ graded commutative ring and $M$ be a $G$-graded $R$-module. The set of all graded second submodules of $M$ is denoted by $Spec_G^s(M)$ and it is called the graded second spectrum of $M$. In this paper, we discuss graded rings with Noetherian graded prime spectrum and obtain some conclusions. In addition, we introduce the notion of the graded Zariski socle of graded submodules and explore their properties. Using these conclusions and properties, we also investigate $Spec_G^s(M)$ with the Zariski topology from the viewpoint of being a Noetherian space and give some related outco…
Cohomology of finite graded group varieties
证明了：如果$A$是正分次、分次交换的局部有限霍普夫代数，则它的上同调是非生成的，从而统一了Wilkerson和Hopkins-Smith以及Friedlander-Suslin的经典结果.我们通过证明余正规初等代数的存在性来做到这一点。
Invertible modules for commutative $\mathbb{S}$-algebras with residue fields
本文通过取同伦群，研究交换S-代数上Elmendorf，Kriz，Mandell和May意义下的可逆模在什么条件下产生系数环上可逆分次模的代数Picard群中的元素.若连通交换S-代数R对R * 中的每个极大理想m都有凝聚局部化（R *）m，则对每个可逆R-模U，U * 是可逆分次R *-模.在某些非连通的情形下，如果R_* 的整体维数很小，或者R是2-周期的，其下有Noether完备局部正则环R_0，则在交换S-代数对R_* 中的所有极大理想m都有“剩余域”的条件下，我们可以推广这个结果.
It is a fundamental result in commutative algebra and invariant theory that a finitely generated graded module over a commutative finitely generated graded algebra has rational Hilbert series, and consequently the Hilbert series of the algebra of polynomial invariants of a group of linear transformations is rational, whenever this algebra is finitely generated. This basic principle is applied here to prove rationality of Hilbert series of algebras of invariants that are neither commutative nor finitely generated. Our main focus is on linear groups acting on certain factor algebras of the tenso…
On graded Going down domains
本文针对无挠交换可消幺半群上的分次整环，引入分次下降整环概念，给出分次Prüfer整环的等价刻画，借助分次可除整环刻画该类整环，并证明其下降性质在商整环下稳定。
Hasse--Schmidt derivations, divided powers and differential smoothness
设$k$是交换环，$A$是交换$k$ -代数，$D$是$A$的$k$ -线性微分算子的滤环.我们证明：（1）分次环$\gr D$允许$θ$在$k$上的$A$的微分模$Ω_{A/k}$的对称代数的分次对偶中有一个标准嵌入，它具有标准的可分幂结构. (2)从$k$的模的除幂代数-$A$的线性Hasse-Schmidt可积导子到$\gr D$存在一个典范态射$\vartheta$. (3)态射$θ$和$\vartheta$适合于规范的交换图表。
Let $G$ be a group with identity $e$ and $R$ be a $G$-graded commutative ring with nonzero unity $1$. In this article, we introduce the concept of graded strongly $1$-absorbing primary ideals. A proper graded ideal $P$ of $R$ is said to be a graded strongly $1$-absorbing primary ideal of $R$ if whenever nonunit homogeneous elements $x, y, z\in R$ such that $xyz\in P$, then either $xy\in P$ or $z\in Grad(\{0\})$ (the graded radical of $\{0\}$). Several properties of graded strongly $1$-absorbing primary ideals are investigated. Many results are given to disclose the relations between this new c…
We show that the mixed volumes of arbitrary convex bodies are equal to mixed multiplicities of graded families of monomial ideals, and to normalized limits of mixed multiplicities of monomial ideals. This result evinces the close relation between the theories of mixed volumes from convex geometry and mixed multiplicities from commutative algebra.
Let $G$ be a group, $R$ be a $G$-graded commutative ring with nonzero unity and $GI(R)$ be the set of all graded ideals of $R$. Suppose that $ϕ:GI(R)\rightarrow GI(R)\cup\{\emptyset\}$ is a function. In this article, we introduce and study the concept of graded $ϕ$-$1$-absorbing prime ideals. A proper graded ideal $I$ of $R$ is called a graded $ϕ$% -$1$-absorbing prime ideal of $R$ if whenever $a,b,c$ are homogeneous nonunit elements of $R$ such that $abc\in I-ϕ(I)$, then $ab\in I$ or $c\in I$. Several properties of graded $ϕ$-$1$-absorbing prime ideals have been examined.
On commutative algebra associated to $t$-labeled subforests of a graph
对于给定的图$G$，我们构造了一个相伴的交换代数，其维数等于$G$的$t$ -标号森林的个数。证明了该代数的$k$次分次分支的维数也具有组合意义，并且其Hilbert多项式可以用$G$的Tutte多项式表示.
Class and rank of differential modules
微分模是一个具有平方零自同态的模。这种结构支持环上模的复形，以及分次环上的微分分次模。我们建立下界的类-一个替代的长度的一个自由复杂的-和排名的微分模块的不变量的同源性。这些结果专门用于交换代数和代数拓扑学中的基本定理。一个例子是关于诺特交换环上的复形的Hochster、Peskine、P. Roberts和Szpiro的新交定理的等特征情形和G.关于分次多项式环上的微分分次模。
Supporting degrees of multi-graded local cohomolgoy modules
对于正分次交换诺特环$R$上的分次生成分次模$M$，第二作者在1999年建立了一些限制，这些限制可以用$M$的Castelnuovo正则性或$M$的所谓$a^*$ -不变量来表示，它们是关于分次不可分解分次内射直和项的支撑度的，对于$M$的最小分次内射分解中的任一项，其相相伴素理想包含$R$的无关理想。早在1995年，T. Marley建立了$M$的分次局部上同调模和$M$在$\Proj(R)$上的局部行为之间的联系。本文的目的是提出一些多级类似的上述工作。
On the asymptotic linearity of Castelnuovo-Mumford regularity
设R是单位交换诺特环上的标准分次环，I是R的分次理想.设M是M生成的分次R-模。证明了存在整数e和ρ_M（I）使得对所有大的n，reg（I^nM）= ρ_M（I）n+e.
A Fitting Lemma for Z/2-graded modules
研究了特征为0的Z/2-分次斜交换代数上自由Z/2-分次模映射的上核的零化子，并定义了它的Fitting理想的类似.我们表明，在“一般”的情况下，零化子是由拟合理想，并解释拟合理想和零化子之间的关系，一般情况下。我们的结果推广了经典的Fitting引理，推广了Green [1999]的关键结果.它们依赖于一般线性李超代数的表示的Behren-Regev理论。
On the Graded Annihilators of Right Modules Over The Frobenius Skew Polynomial Ring
设R是素特征的交换诺特环，M是x-可分右R[x，f]-模，且R是诺特模.在R是半局部的情况下，我们肯定地回答了Sharp和Yoshino的问题，并证明了M的R[x，f]-同态像的分次零化子集是有限的.在一般情况下，我们也给出了一个反例。
Free resolutions over commutative Koszul algebras
对于R=Q/J，Q是域上的交换分次代数，J非零，我们将R在Q上的最小分解的斜率与k=R/ R_{+}在R上的最小分解的斜率联系起来.当Q和R是Koszul且J_1=0时，我们证明了Tor^Q_i（R，k）_j=0，对于j> 2 i，对于每个非负整数i，以及对于j= 2 i，当i>dim Q-dim R且pd_QR是有限的。
Let $R$ be a positively graded algebra over a field. We say that $R$ is Hilbert-cyclotomic if the numerator of its reduced Hilbert series has all of its roots on the unit circle. Such rings arise naturally in commutative algebra, numerical semigroup theory and Ehrhart theory. If $R$ is standard graded, we prove that, under the additional hypothesis that $R$ is Koszul or has an irreducible $h$-polynomial, Hilbert-cyclotomic algebras coincide with complete intersections. In the Koszul case, this is a consequence of some classical results about the vanishing of deviations of a graded algebra.
Triangulations of projective modules
我们证明了分次交换环上的投射模范畴允许关于模悬置的三角剖分当且仅当该环是分次域上的一个生成元上的分次域和外代数的有限积（具有适当次数的单位）.我们还对投射模范畴允许关于单位悬置的三角剖分的未分次交换环进行了分类。给出了代数拓扑中生成假设的两个类似物的应用，并将我们的结果转化为对称环谱或S-代数上模的设置。
Let $R$ be a graded commutative ring with non-zero unity $1$ and $M$ be a graded unitary $R$-module. Let $GS(M)$ be the set of all graded $R$-submodules of $M$ and $ϕ: GS(M)\rightarrow GS(M)\bigcup\{\emptyset\}$ be a function. A proper graded $R$-submodule $K$ of $M$ is said to be a graded $ϕ-$prime $R$-submodule of $M$ if whenever $r$ is a homogeneous element of $R$ and $m$ is a homogeneous element of $M$ such that $rm\in K-ϕ(K)$, then either $m\in K$ or $r\in (K:_{R}M)$. If $ϕ(K)=\emptyset$ for all $K\in GS(M)$, then a graded $ϕ-$prime submodule is exactly a graded prime submodule. If $ϕ(K)=…
Koszul Graded Möbius Algebras and Strongly Chordal Graphs
本文研究了拟阵分次 Möbius 代数何时为 Koszul 代数，基于图的性质刻画了图圈拟阵的 Koszul 分次 Möbius 代数，并通过边排序给出了强弦图的新刻画。
On Simplicial Commutative Rings with Vanishing André-Quillen Homology
我们提出了D. Quillen，关于André-Quillen同调的消失，到单纯交换环。该猜想刻画了局部完全交的概念，并在局部特征的适当假设下，推广到单纯集。进一步，在有限型同调的条件下，我们证明了在非零特征域上增广的单纯交换代数的情况下的猜想。作为结果，我们得到了一个证明奎伦猜想的诺特交换代数-再次扩大了一个领域的非零特征。
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