相关论文: The vanishing of Tor_1^R(R^+,k) implies that R is …

Let $R$ be an excellent local domain of positive characteristic with residue field $k$ and let $R^+$ be its absolute integral closure. If $\text{Tor}^R_1(R^+,k)$ vanishes, then $R$ is Cohen-Macaulay, normal, F-rational and F-pure. If $R$…

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…

Let $(R,\m,k)$ be a local (Noetherian) ring of positive prime characteristic $p$ and dimension $d$. Let $G_\dt$ be a minimal resolution of the residue field $k$, and for each $i\ge 0$, let $\gothic t_i(R) = \lim_{e\to \8}…

We prove the following result. Let k be an infinite perfect field of positive characteristic and assume that strong resolution of singularities holds over k. Let R be a localization of a commutative d-dimensional k-algebra of finite type…

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…

It is proved that if $I$ and $J$ are weakly stable ideals in a polynomial ring $R=k[x_1,\ldots,x_n]$, with $k$ a field, then the regularity of $\text{Tor}^R_i(R/I,R/J)$ has the expected upper bound. We also give a bound for the regularity…

Call a (strictly increasing) sequence $(r_{n})$ of natural numbers \emph{regular} if it satisfies the following condition: $r_{n+1}/r_{n}\to\theta\in\mathbb{R}^{>1}\cup\{\infty\}$ and, if $\theta$ is algebraic, then $(r_{n})$ satisfies a…

Let $(R,\mathfrak{m},\mathbb{k})$ be an equicharacteristic one-dimensional complete local domain over an algebraically closed field $\mathbb{k}$ of characteristic 0. R. Berger conjectured that R is regular if and only if the universally…

A commutative Noetherian ring $R$ is said to be Tor-persistent if, for any finitely generated $R$-module $M$, the vanishing of $\operatorname{Tor}_i^R(M,M)$ for $i\gg 0$ implies $M$ has finite projective dimension. An open question of…

Let $R$ be a regular ring of dimension $d$ containing a field $K$ of characteristic zero. If $E$ is an $R$-module let $Ass^i E = \{ Q \in \ Ass E \mid \ height Q = i \}$. Let $P$ be a prime ideal in $R$ of height $g$. We show that if $R/P$…

We prove a mixed-characteristic analogue of Kunz's theorem in terms of perfectoid towers: a Noetherian local ring of residue characteristic $p$ is regular if and only if it admits a flat map to a Noetherian ring that extends to a perfectoid…

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 $\operatorname{Tor}^R_i(M,N)=0=\operatorname{Tor}^R_{i+1}(M,N)$…

Let $R=k[x_1,\dots,x_n]/I$ be a standard graded $k$-algebra where $k$ is a field of prime characteristic and let $J$ be a homogeneous ideal in $R$. Denote $(x_1,\dots,x_n)$ by $\mathfrak{m}$. We prove that there is a constant $C$…

Let R be a commutative ring with unit and let I be an ideal generated by a regular sequence. Then it is known that the natural sequences 0-> Tor_*^R(R/I,I^s)-> Tor_*^R(R/I,I^s/I^{s+1})-> Tor_{*-1}^R(R/I,I^{s+1})-> 0 are short exact…

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 $…

Given a minimal set of generators $\bold{x}$ of an ideal $I$ of height d in a regular local ring ($R, m, k$) we prove several cases for which the map $K_d(\bold{x}; R) \otimes k \to \Tor_d^R (R/I, k)$ is the 0-map. As a consequence of the…

Smooth solutions to the axially symmetric Navier-Stokes equations obey the following maximum principle:$\|ru_\theta(r,z,t)\|_{L^\infty}\leq\|ru_\theta(r,z,0)\|_{L^\infty}.$ We first prove the global regularity of solutions if…

Let $R$ be a local ring with residue field $k$ and $M$, $N$ be finitely generated modules over $R$. It is well known that $Tor^R_i(M, N) = 0$ for $i \gg 0$ if $pd_R(M) < \infty$ or $pd_R(N) < \infty$. The ring $R$ is said to satisfy the…

Let p be a prime ideal in a commutative noetherian ring R and denote by k(p) the residue field of the local ring R_p. We prove that if an R-module M satisfies Ext_R^n(k(p),M) = 0 for some n >= dim R, then Ext_R^i(k(p),M) = 0 holds for all i…

Let p be a prime ideal in a commutative noetherian ring R. It is proved that if an R-module M satisfies Tor^R_n(k(p),M) = 0 for some n \geq dim R_p, where k(p) is the residue field at p, then Tor^R_i(k(p),M) = 0 holds for all i \geq n.…