When are KE-closed subcategories torsion-free classes?

TL;DR

This paper characterizes KE-closed subcategories of finitely generated modules over commutative noetherian rings, showing they are torsion-free classes, and classifies them for rings of dimension at most one and certain two-dimensional domains.

Contribution

It provides a characterization of KE-closed subcategories as torsion-free classes and classifies these subcategories for specific classes of rings, offering a complete answer to the posed question.

Findings

KE-closed subcategories are torsion-free classes in certain contexts.

Classification of KE-closed subcategories for rings with dimension ≤ 1.

Complete characterization for two-dimensional normal domains.

Abstract

Let $R$ be a commutative noetherian ring and denote by $\mathsf{mod} R$ the category of finitely generated $R$-modules. In this paper, we study KE-closed subcategories of $\mathsf{mod} R$, that is, additive subcategories closed under kernels and extensions. We first give a characterization of KE-closed subcategories: a KE-closed subcategory is a torsion-free class in a torsion-free class. As an immediate application of the dual statement, we give a conceptual proof of Stanley-Wang's result about narrow subcategories. Next, we classify the KE-closed subcategories of $\mathsf{mod} R$ when $\mathrm{dim} R \le 1$ and when $R$ is a two-dimensional normal domain. More precisely, in the former case, we prove that KE-closed subcategories coincide with torsion-free classes in $\mathsf{mod} R$. Moreover, this condition implies $\mathrm{dim} R \le 1$ when $R$ is a homomorphic image of a Cohen-Macaulay ring (e.g. a finitely generated algebra over a regular ring). Thus, we give a complete answer for the title.