The Lattice of Subobject Closed Subcategories and Colocal Type

TL;DR

This paper studies the structure of subcategories in abelian length categories of colocal type, showing the lattice of subobject closed subcategories is distributive and providing a characterization and description of these categories.

Contribution

It establishes the distributivity of the lattice of subobject closed subcategories and characterizes abelian length categories of colocal type, especially over algebraically closed fields.

Findings

Lattice of subobject closed subcategories is distributive.

Characterization of abelian length categories of colocal type.

Explicit description of the lattice for algebras over algebraically closed fields.

Abstract

We consider abelian length categories, a generalization of module categories over Artin algebras. Let $\mathcal{A}$ be an abelian length category of colocal type. We show that the lattice $\mathsf{S}(\mathcal{A})$ of full additive subobject closed subcategories of $\mathcal{A}$ is distributive. Furthermore, we give a characterization of abelian length categories of colocal type. If $A$ is an algebra of colocal type over an algebraically closed field, then this characterization is especially simple and we can describe the lattice $\mathsf{S(mod} ~A)$ up to isomorphism.