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

2020/02/27 by Matsui, Hiroki
#13C12 #13D05 #13D07 #13H10 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2002.11951

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 ⊗_R N satisfies (S_n+1), then under some assumptions Tor_i^R(M, N) = 0 for all i ≥ 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.

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