SERRE’S CONDITION FOR TENSOR PRODUCTS AND N-TOR-RIGIDITY OF MODULES

Abstract

We study Serre’s condition $(S_n)$ for tensor products of modules over a commutative noetherian local ring $R$. Specifically, we consider the following question. For finitely generated $R$-modules $M$ and $N$, either of which is $(n+1)$-Tor-rigid, if the tensor product $M{\otimes }_{R}N$ satisfies $(S_{n+1})$, then does ${\text{Tor}}_i^R(M,N)=0$ hold for all $i\ge 1$? The aim of this paper is to give an affirmative answer to this question if we assume local freeness and Serre’s condition on modules. As applications, we recover several known results.