Reduction of exact structures

TL;DR

This paper investigates the structure and properties of all exact structures on a fixed additive category, revealing that the poset of these structures forms a complete lattice and exploring their invariants.

Contribution

It introduces a detailed study of the poset of exact structures, showing it is a complete lattice and analyzing how invariants change with different structures.

Findings

The poset of all exact structures is a complete lattice.

Under certain conditions, the poset is boolean.

The paper generalizes notions like length and quiver to exact categories.

Abstract

Examples of exact categories in representation theory are given by the category of Delta-filtered modules over quasi-hereditary algebras, but also by various categories related to matrix problems, such as poset representations or representations of bocses. Motivated by the matrix problem background, we study in this article the reduction of exact structures, and consider the poset Ex(A) of all exact structures on a fixed additive category A. This poset turns out to be a complete lattice, and under suitable conditions results of Enomoto's imply that it is boolean. We initiate in this article a detailed study of exact structures E by generalizing notions from abelian categories such as the length of an object relative to E and the quiver of an exact category (A,E). We investigate the Gabriel-Roiter measure for (A,E), and further study how these notions change when the exact structure varies.