The vanishing of Tor_1^R(R^+,k) implies that R is regular
Ian M. Aberbach
设（R，m，k）是具有正素特征的优良局部环.证明了若Tor_1^R（R^+，k）= 0，则R是正则的.这改进了Schoutens的一个结果，在Schoutens的证明中需要附加一个假设，即R是一个孤立奇点。
Let (R,m,k) be an excellent local ring of positive prime characteristic. We show that if Tor_1^R(R^+,k) = 0 then R is regular. This improves a result of Schoutens, in which the additional hypothesis that R was an isolated singularity was required for the proof.
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We consider vanishing of Ext and Tor, especially over Artinian rings. In particular, we prove the Auslander-Reiten conjecture for all commutative local rings in which the cube of the maximal ideal is zero.
In this paper we exploit properties of Dao's eta-pairing as well as techniques of Huneke, Jorgensen, and Wiegand to study the vanishing of Tor_i(M,N) of finitely generated modules M, N over complete intersections. We prove vanishing of Tor_i(M, N) for all positive integers i under depth conditions on M, N, and their tensor product. Our arguments improve a result of Dao and establish a new connection between the vanishing of Tor and the depth of tensor products.
Vanishing of Tor Over Complete Intersections
本文研究了完备交环上$\textnormal{Tor}$的消失问题。基于C. Huneke，D. Jorgensen和R. Wiegand和最近的H.道，我们得到了新的结果表明，$R$ -模块$M$，$N$和$M\otimes_RN$上的良好的深度性质的力量消失的$\textnormal{Tor}^{R}_{i}(M,N)$所有$i\geq 1$。
Freeness criteria via vanishing of Tor
本文研究满足Tor消没可推出模自由这一性质的模L，并给出相关应用。
Let $(S,\mathfrak{m},k)$ and $(T,\mathfrak{n},k)$ be local rings, and let $R$ denote their fiber product over their common residue field $k$. Inspired by work of Naseh and Sather-Wagstaff, we explore consequences of vanishing of ${\rm Tor}^R_m(M,N)$ for various values of $m$, where $M$ and $N$ are finitely generated $R$-modules.
Vanishing of Tor, and why we care about it
给定局部环$R$上有限生成模$M$和$N$，张量积$M\otimes_RN$通常具有非零挠率。实际上，张量积无挠的假设影响了所有$i\geq 1$的模$\Tor^R_i(M,N)$的结构和消失。进而，$\Tor^R_i(M,N)$的消失对模块$M$和$N$的深度属性施加限制。这些联系首次出现在Auslander的1961年论文“Modules over unramified正则局部环“中。我们将调查这些主题的文献，重点是在过去二十年的进展。
The long-standing Auslander and Reiten Conjecture states that a finitely generated module over a finite-dimensional algebra is projective if certain Ext-groups vanish. Several authors, including Avramov, Buchweitz, Iyengar, Jorgensen, Nasseh, Sather-Wagstaff, and Şega, have studied a possible counterpart of the conjecture, or question, for commutative rings in terms of vanishing of Tor. This has led to the notion of Tor-persistent rings. Our main result shows that the class of Tor-persistent local rings is closed under a number of standard procedures in ring theory.
On the vanishing of Tor of the absolute integral closure
设$R$是一个具有正特征的优良局部区域，其剩余域为$k$，$R^+$是它的绝对整闭包.如果$\text{Tor}^R_1(R^+,k)$为零，则$R$是Cohen-Macaulay、正规的、F-有理数和F-纯的。如果$R$至多有一个孤立奇点或维数至多为2，则$R$是正则的。
Consider a non-trivial fiber product $R=S\times_kT$ of local rings $S$, $T$ with common residue field $k$. Given two finitely generate $R$-modules $M$ and $N$, we show that if $\operatorname{Tor}^R_i(M,N)=0=\operatorname{Tor}^R_{i+1}(M,N)$ for some $i\geq 5$, then $\operatorname{pd}_R(M)\leq 1$ or $\operatorname{pd}_R(N)\leq 1$. From this, we deduce several consequence, for instance, that $R$ satisfies the Auslander-Reiten Conjecture.
Test properties of some Cohen-Macaulay modules and criteria for local rings via finite vanishing of Ext or Tor
本文通过Ext或Tor的有限消失性质，测试Cohen-Macaulay模的特性，并用于刻画局部环，验证了Auslander-Reiten猜想的扩展结果。
For finitely generated modules $M$ and $N$ over a Gorenstein local ring $R$, one has $depth M + depth N= depth(M\otimes N) +depth R$, i.e., the depth formula holds, if $M$ and $N$ are Tor-independent and Tate homology $\hat{Tor}_{i}^{R}(M,N)$ vanishes for all $i\in\mathbb{Z}$. We establish the same conclusion under weaker hypotheses: if $M$ and $N$ are $\mathcal{G}$-relative Tor-independent, then the vanishing of $\hat{Tor}_{i}^{R}(M,N)$ for all $i\le 0$ is enough for the depth formula to hold. We also analyze the depth of tensor products of modules and obtain a necessary condition for the dep…
We say an excellent local domain $(S,n)$ satisfies the vanishing conditions for maps of Tor, if for every $A\to R\to S$ with $A$ regular and $A\to R$ module-finite torsion-free extension, and every $A$-module $M$, the map $Tor^A_i(M, R)\to Tor_i^A(M, S)$ vanishes for every $i\geq 1$. Hochster-Huneke's conjecture (theorem in equal characteristic) thus states that regular rings satisfy such vanishing conditions. The main theorem of this paper shows that, in equal characteristic, rings that satisfy the vanishing conditions for maps of Tor are exactly derived splinters in the sense of Bhatt. In pa…
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…
We prove that each positive power of the maximal ideal of a commutative Noetherian local ring is Tor-rigid, and strongly-rigid. This gives new characterizations of regularity and, in particular, shows that such ideals satisfy the torsion condition of a long-standing conjecture of Huneke and Wiegand.
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。
Local vanishing for toric varieties
本文研究了环面簇的局部消没问题，建立了特定层的消没与非消没结果，推广了已有定理，并据此提出了环面簇具有k-有理奇点的判据。
The Degree and regularity of vanishing ideals of algebraic toric sets over finite fields
设X* 是有限域K上的仿射空间A^s的子集，K由杂波的边缘参数化。设X和Y分别是X* 在映射x -> [x]和x -> [（x，1）]下的像，其中[x]和[（x，1）]分别是射影空间P^{s-1}和P^s中的点.对于某些杂波和连通图，我们能够将消失理想I（X）和I（Y）的代数不变式和性质联系起来。在一些有趣的情况下，我们计算它的度和正则性。对于Hamilton二部图，证明了Eisenbud-Goto正则性猜想.当图是二部图时，我们给出了正则性的最优界。证明了X* 是仿射环面当且仅当I（Y）是完全交。我们提出了一些应用程序的编码理论，并显示了一些界限的最小距离的参数化线性码的连接二部图。
We prove a version of weakly functorial big Cohen-Macaulay algebras that suffices to establish Hochster-Huneke's vanishing conjecture for maps of Tor in mixed characteristic. As a corollary, we prove an analog of Boutot's theorem that direct summands of regular rings are pseudo-rational in mixed characteristic. Our proof uses perfectoid spaces and is inspired by the recent breakthroughs on the direct summand conjecture by André and Bhatt.
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 show that a ring $R$ is regular if $Tor_{i}^{R}(R^{+},k) = 0$ for some $i\geq 1$ assuming further that $R$ is a $\mathbb{N}$-graded ring of dimension $2$ finitely generated over an equi-characteristic zero field $k$. This answers a question of Bhatt, Iyengar, and Ma. We use almost mathematics over $R^{+}$ to deduce properties of the noetherian ring $R$ and rational surface singularities. Moreover we show that $R^{+}$ in equi-characteristic zero is $m$-adically ideal(wise) separated, a condition which appears in the proof of local criterion for flatness. In dimension $2$ it is Ohm-Rush and i…
Vanishing of Tate homology and depth formulas over local rings
Auslander关于正则局部环上的Tor-无关模对的深度公式，深度（M \otimes N）=深度（M）+深度（N）-深度（R），在过去的四十年中得到了几个方向的推广.本文建立了Gorenstein局部环上Tate Tor-独立模对的深度公式。它包含以前的推广Auslander的公式，并产生确切的界限消失的上同调在某些Gorenstein环。
Decent intersection and Tor-rigidity for modules over local hypersurfaces
本文研究了局部超曲面$R$上模的两个性质：体面性和刚性。我们表明，消失的Hochster的功能$θ^R(M,N)$，已知意味着体面的交叉，也意味着刚性。研究了超曲面上$θ^R(M,N)$的消失问题，得到了关于超曲面上的体面和刚性的新结果.我们采用了混合技术从交换代数和相交理论的代数循环。
For finitely generated modules M and N over a complete intersection R, the vanishing of Tor_i^R(M,N) for all i> 0 gives a tight relationship among depth properties of M, N and their tensor product. Here we concentrate on the converse and show, under mild conditions, that the tensor product of M and N being torsion-free (or satisfying higher Serre conditions) forces vanishing of Tor. Special attention is paid to the case of tensor powers of a single module.
On the vanishing ideal of an algebraic toric set and its parameterized linear codes
设K是一个有限域，X是域K上的一个射影空间的子集，它由杂波边缘产生的单项式参数化。我们给出了X的消失理想I（X）的度复杂度的一些估计，关于Revlex阶。如果杂波是均匀的，我们用线性代数对I（X）的完全相交性进行分类。我们显示了一定的参数化线性码的最小距离的上界，随着一定的估计的代数不变式的I（X）。
A note on the depth formula and vanishing of cohomology
证明了：若局部环R上的有限模M和N之一具有可约复杂性和有限Gorenstein维数，则当i>>0时，若TorRi（M，N）= 0，则深度公式成立.我们还研究了有限完全交维数模的上同调消失问题。
Tor and Ext vanishing results for commutative Artinian rings
该研究针对交换阿廷环领域的三个猜想，通过构造特定的代数与模，对Avramov等人、Huneke等人的猜想给出反例，回答了Dao的问题，推进了Tachikawa猜想的相关研究。
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…
Cohomologically complete intersections with vanishing of Betti numbers
设$I$是$n$维局部Gorenstein环$R$的理想.本文给出了理想$I$成为上同调完全交的几个充要条件。事实上，作为一种技术工具，它将表明，所有$i\neq c= \grade (I)$的$H^i_{I}(R)= 0$消失等价于$H^c_{I}(R)$的贝蒂数消失。这给出了一个新的刻画，用$H^c_{I}(R)$的Tor模的消失的同调性质来检验其上同调完全交性质.
Bounds on depth of tensor products of modules
设$R$是局部完备交环，$M$和$N$是非零的可生成$R$ -模.我们采用Auslander的转置的消失的Tor的研究，并获得有用的界限的深度的张量积$M\otimes_{R}N$。我们的主要论点的应用表明，如果$M$是局部自由的$R$的打孔频谱上，那么要么$\depth(M\otimes_{R}N)\geq \depth(M)+\depth(N)-\depth(R)$，或$\depth(M\otimes_{R}N)\leq \cod(R)$。同时推广了D. A. Jorgensen，并确定确保所有更高$\Tor_i^R(M,N)$消失所需的$\Tor_i^R(M,N)$连续消失的数量。
Asymptotic vanishing of cohomology in triangulated categories
本文研究了在三角范畴中中心作用的分层交换环下，上同调的渐近消失性质及其应用。
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