Ring homomorphisms and local rings with quasi-decomposable maximal ideal

TL;DR

This paper explores the homological properties of local rings with quasi-decomposable maximal ideals, introduces new classes from a combinatorial perspective, and examines the behavior of these properties under local ring homomorphisms.

Contribution

It advances understanding of homological characteristics of these rings and introduces new classes from a combinatorial viewpoint, extending previous work.

Findings

Rings with quasi-decomposable maximal ideal have rigid homological properties.

New classes of such rings are introduced from a combinatorial perspective.

Homological properties are analyzed under diagrams of local ring homomorphisms.

Abstract

The notion of local rings with quasi-decomposable maximal ideal was formally introduced by Nasseh and Takahashi. In separate works, the authors of the present paper show that such rings have rigid homological properties; for instance, they are both Ext- and Tor-friendly. One point of this paper is to further explore the homological properties of these rings and also introduce new classes of such rings from a combinatorial point of view. Another point is to investigate how far some of these homological properties can be pushed along certain diagrams of local ring homomorphisms.