# Vanishing of Tor over fiber products

## TL;DR

This paper investigates how the vanishing of Tor modules over fiber product rings influences module properties, extending previous work and providing new insights into homological behavior in this context.

## Contribution

It explores the implications of Tor vanishing over fiber product rings, building on Naseh and Sather-Wagstaff's work to deepen understanding of module interactions.

## Key findings

- Vanishing of Tor imposes specific module structure constraints
- Results extend known homological properties to fiber product rings
- Provides new criteria for module projectivity 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 ${\rm Tor}^R_m(M,N)$ for various values of $m$, where $M$ and $N$ are finitely generated $R$-modules.