Auslander-Reiten and Huneke-Wiegand conjectures over quasi-fiber product rings

TL;DR

This paper investigates the properties of quasi-fiber product rings, demonstrating they satisfy a strengthened form of the Auslander-Reiten Conjecture and exploring implications for the Huneke-Wiegand conjecture.

Contribution

It establishes that quasi-fiber product rings fulfill a sharpened version of the Auslander-Reiten Conjecture and provides new insights into the Huneke-Wiegand conjecture for these rings.

Findings

Quasi-fiber product rings satisfy a sharpened Auslander-Reiten Conjecture.

Gorenstein quasi-fiber product rings are AB-rings and Ext-bounded.

Insights into the Huneke-Wiegand conjecture for quasi-fiber product rings.

Abstract

In this paper we explore consequences of the vanishing of ${\rm Ext}$ for finitely generated modules over a quasi-fiber product ring $R$; that is, $R$ is a local ring such that $R/(\underline x)$ is a non-trivial fiber product ring, for some regular sequence $\underline x$ of $R$. Equivalently, the maximal ideal of $R/(\underline x)$ decomposes as a direct sum of two nonzero ideals. Gorenstein quasi-fiber product rings are AB-rings and are Ext-bounded. We show in Theorem 3.31 that quasi-fiber product rings satisfy a sharpened form of the Auslander-Reiten Conjecture. We also make some observations related to the Huneke-Wiegand conjecture for quasi-fiber product rings.