Vanishing of (co)homology of Burch and related submodules
Vanishing of (co)homology of Burch and related submodules
复制标题
DOI：
10.1215/00192082-10429128
复制
发表时间：
2022-01
影响因子：
0.6
通讯作者：
Souvik Dey;Toshinori Kobayashi
中科院分区：
文献类型：
--
作者：
Souvik Dey;Toshinori Kobayashi
引入局部环上模的Burch子模和弱$\mathfrak m $-满子模的概念，并研究它们的性质.我们的一个主要结果表明Burch子模满足2-Tor刚性和测试性质。我们还表明，在一个本地环$（R，\mathfrak m）$一个由R $生成的$R $-模$X $的子模$M $，使得$M =\mathfrak m X $或$M（\subseteq\mathfrak m X $）在$X $中弱$\mathfrak m $-满，是1-Tor刚性的，并且是测试模，条件是$X $是忠实的（并且当$M $是弱$\mathfrak m $-满的时，$X/M $具有有限长度）。作为应用，我们给出了一类新的环，使得Huneke和Wiegand的一个猜想在其上是肯定的.
We introduce the notion of Burch submodules and weakly $\mathfrak m$-full submodules of modules over local rings and study their properties. One of our main results shows that Burch submodules satisfy 2-Tor rigid and test property. We also show that over a local ring $(R,\mathfrak m)$ a submodule $M$ of a finitely generated $R$-module $X$, such that either $M=\mathfrak m X$ or $M(\subseteq \mathfrak m X$) is weakly $\mathfrak m$-full in $X$, is 1-Tor rigid and a test module provided that $X$ is faithful (and $X/M$ has finite length when $M$ is weakly $\mathfrak m$-full). As an application, we give a new class of rings such that a conjecture of Huneke and Wiegand is affirmative over them.