Vanishing of Tor over fiber products

Abstract

Let $(S,\mathfrak{m},k)$ and $(T,\mathfrak{n},k)$ be local rings, and let $R$ denote their fiber product over their common residue field $k$. Inspired by work of Naseh and Sather-Wagstaff, we explore consequences of vanishing of ${\mathrm{Tor}}_m^R(M,N)$ for various values of $m$, where $M$ and $N$ are finitely generated $R$-modules.