论文标题
外能力
Exterior powers and Tor-persistence
论文作者
论文摘要
如果任何有限生成的$ r $ -Module $ m $，$ \ operatorAtorname {tor} _i _i^r（m，m）$ for $ i \ gg 0 $ m $ mive $ m $ mim m $ M $具有有限的投射增长。 Avramov等的一个公开问题。 al。询问是否有任何这样的$ r $是torpersistent。在这项工作中，我们利用模块和复合物的外部力量的属性为这个问题提供了一些部分答案。特别是，我们表明每个本地戒指$（r，\ mathfrak {m}）$带有$ \ mathfrak {m}^3 = 0 $ is tor-persistent。由于我们的方法，我们提供了tachikawa猜想的新证明，用于在不同的特征区域上与2个特征的领域进行积极分级的环。
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.
