On Ext-indices of ring extensions

TL;DR

This paper investigates the finiteness properties of Ext-indices in ring extensions, introduces related conjectures, and proves specific cases including the finite Ext-index of certain Artinian local rings and the validity of the Auslander-Reiten conjecture.

Contribution

It introduces conjectures on Ext-indices of ring extensions and proves their validity in special cases, including trivial extensions of Artinian local rings.

Findings

Trivial extension of an Artinian local ring by its residue field has finite Ext-index.

The Auslander-Reiten conjecture holds for these trivial extension rings.

Confirmed the conjectures in specific cases.

Abstract

In this paper we are concerned with the finiteness property of Ext-indices of several ring extensions. In this direction, we introduce some conjectures and discuss the relationship of them. Also we give affirmative answers to these conjectures in some special cases. Furthermore, we prove that the trivial extension of an Artinian local ring by its residue class field is always of finite Ext-index and we show that the Auslander-Reiten conjecture is true for this type of rings.