Serre's condition for tensor products and $n$-Tor-rigidity of modules

Abstract

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.

9 pages