Modules of G-dimension zero over local rings with the cube of maximal ideal being zero

TL;DR

This paper investigates modules of G-dimension zero over local rings with nilpotent maximal ideals, providing conditions for their existence, constructing families of such modules, and analyzing their categorical properties.

Contribution

It establishes a criterion for the existence of non-free G-dimension zero modules and constructs a family of non-isomorphic indecomposable modules over rings with 3=0.

Findings

Existence condition for non-free G-dimension zero modules

Construction of a parameterized family of indecomposable modules

The subcategory of G-dimension zero modules is not necessarily contravariantly finite

Abstract

Let $(R, \m)$ be a commutative Noetherian local ring with $\m^3 =(0)$. We give a condition for $R$ to have a non-free module of G-dimension zero. We shall also construct a family of non-isomorphic indecomposable modules of G-dimension zero with parameters in an open subset of projective space. We shall finally show that the subcategory consisting of modules of G-dimension zero over $R$ is not necessarily a contravariantly finite subcategory in the category of finitely generated $R$-modules.