When is there a nontrivial extension-closed subcategory?

TL;DR

This paper investigates conditions under which the category of finitely generated modules over a commutative Noetherian local ring contains nontrivial extension-closed subcategories, focusing on specific algebraic structures.

Contribution

It establishes new criteria for the existence of nontrivial extension-closed subcategories in mod R based on properties of the ring's generators and specific classes of Artinian Gorenstein rings.

Findings

Existence of nontrivial extension-closed subcategories when xy=0 for generators x,y.

Such subcategories exist in non-hypersurface stretched Artinian Gorenstein rings.

Provides algebraic conditions linking ring structure to subcategory properties.

Abstract

Let R be a commutative Noetherian local ring, and denote by mod R the category of finitely generated R-modules. In this paper, we consider when mod R has a nontrivial extension-closed subcategory. We prove that this is the case if there are part of a minimal system of generators x,y of the maximal ideal with xy=0, and that it holds if R is a stretched Artinian Gorenstein local ring which is not a hypersurface.