# Local rings with quasi-decomposable maximal ideal

## TL;DR

This paper investigates local rings with decomposable or quasi-decomposable maximal ideals, showing how such structures influence the properties of modules and leading to classifications of subcategories in the singularity category.

## Contribution

It establishes that for rings with decomposable maximal ideals, the maximal ideal appears as a direct summand in syzygies of modules with infinite projective dimension, and classifies subcategories in the singularity category.

## Key findings

- Maximal ideal is a direct summand in syzygies of modules with infinite projective dimension.
- Provides classifications of subcategories, including the thick subcategories of the singularity category.
- Extends results to rings with quasi-decomposable maximal ideals.

## Abstract

Let $(R,\frak m)$ be a commutative noetherian local ring. In this paper, we prove that if $\frak m$ is decomposable, then for any finitely generated $R$-module $M$ of infinite projective dimension $\frak m$ is a direct summand of (a direct sum of) syzygies of $M$. Applying this result to the case where $\frak m$ is quasi-decomposable, we obtain several classfications of subcategories, including a complete classification of the thick subcategories of the singularity category of $R$.