When are KE-closed subcategories torsion-free classes?

Abstract

Let R be a commutative noetherian ring and denote by \({{\,\mathrm{\textsf{mod}}\,}}R\) the category of finitely generated R-modules. In this paper, we study KE-closed subcategories of \({{\,\mathrm{\textsf{mod}}\,}}R\), 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 \({{\,\mathrm{\textsf{mod}}\,}}R\) when \(\dim R \le 1\) 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 \({{\,\mathrm{\textsf{mod}}\,}}R\). Moreover, this condition implies \(\dim R \le 1\) 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.