Vanishing of Ext and Tor over fiber products

TL;DR

This paper investigates the vanishing patterns of Ext and Tor over fiber product rings, revealing conditions under which modules have projective dimension at most one, and confirming the Auslander-Reiten Conjecture for such rings.

Contribution

It establishes new vanishing criteria for Tor modules over fiber products, leading to implications for module projective dimensions and conjectures.

Findings

Vanishing of two consecutive Tor modules implies low projective dimension

Fiber product rings satisfy the Auslander-Reiten Conjecture

Conditions for modules to have projective dimension at most one

Abstract

Consider a non-trivial fiber product $R=S\times_kT$ of local rings $S$, $T$ with common residue field $k$. Given two finitely generate $R$-modules $M$ and $N$, we show that if $\operatorname{Tor}^R_i(M,N)=0=\operatorname{Tor}^R_{i+1}(M,N)$ for some $i\geq 5$, then $\operatorname{pd}_R(M)\leq 1$ or $\operatorname{pd}_R(N)\leq 1$. From this, we deduce several consequence, for instance, that $R$ satisfies the Auslander-Reiten Conjecture.