外幂与Tor-持续性
Exterior powers and Tor-persistence
AI总结：
本文利用模与复形的外幂性质，证明了满足$m^3=0$的局部环具有Tor-persistent性质，并为特征不为2的域上正分次环的Tachikawa猜想提供了新证明。
摘要：
一个交换Noetherian环$R$被称为Tor-persistent，如果对于任意有限生成$R$-模$M$，当$i\gg 0$时$\operatorname{Tor}_i^R(M,M)$的消失意味着$M$具有有限投射维数。Avramov等人提出了一个公开问题，询问是否所有此类环$R$都是Tor-persistent的。在本文中，我们利用模与复形的外幂性质，为该问题提供了几个部分解答；特别地，我们证明了每个满足$\mathfrak{m}^3=0$的局部环$(R,\mathfrak{m})$都是Tor-persistent的。作为我们方法的推论，我们为特征不为2的域上正分次环的Tachikawa猜想提供了一个新证明。
英文摘要：
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 Avramov, et. al. asks whether any such $R$ is Tor-persistent. In this work, we exploit properties of exterior powers of modules and complexes to provide several partial answers to this question; in particular, we show that every local ring $(R,\mathfrak{m})$ with $\mathfrak{m}^3=0$ is Tor-persistent. As a consequence of our methods, we provide a new proof of the Tachikawa Conjecture for positively graded rings over a field of characteristic different from 2.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
