论文标题
Serre的张量产品和$ n $ tor-rigities的状况
Serre's condition for tensor products and $n$-Tor-rigidity of modules
论文作者
论文摘要
在本文中，我们研究了Serre的状况$（S_N）$，用于张量的Noetherian本地环上的模块张量产品。该论文旨在显示以下内容。让$ m $和$ n $是有限生成的模块，这是noetherian本地环$ r $，其中任何一个是$（n+1）$ - tor-rigid。如果张量产品$ m \ otimes_r n $满足$（s_ {n+1}）$，则在某些假设下$ \ mathrm {tor} _ {i} _ {i}^r（m，n）= 0 $ for All $ i \ i \ ge ge 1 $。关键角色由$（n+1）$ - 模块的tor-rigity扮演。作为应用程序，我们将证明结果恢复了几个已知结果。
In this paper, we study Serre's condition $(S_n)$ for tensor products of modules over a commutative noetherian local ring. The paper aims to show the following. Let $M$ and $N$ be finitely generated module over a commutative noetherian local ring $R$, either of which is $(n+1)$-Tor-rigid. If the tensor product $M \otimes_R N$ satisfies $(S_{n+1})$, then under some assumptions $\mathrm{Tor}_{i}^R(M, N) = 0$ for all $i \ge 1$. The key role is played by $(n+1)$-Tor-rigidity of modules. As applications, we will show that the result recovers several known results.
