Classification of exact structures using the Ziegler spectrum

TL;DR

This paper establishes a topological space framework to classify exact structures in additive categories, linking lattice structures to closed subsets and exploring module categories with known Ziegler spectra.

Contribution

It explicitly constructs a topological space for classifying exact substructures and relates it to the Ziegler spectrum in categories with weak cokernels.

Findings

Lattice of exact substructures is anti-isomorphic to closed subsets of the constructed space.

In categories with weak cokernels, the space is an open subset of the Ziegler spectrum.

Calculated global dimensions of exact substructures in certain module categories.

Abstract

Given an idempotent complete additive category, we show the there is an explicitly constructed topological space such that the lattice of exact substructures is anti-isomorphic to the lattice of closed subsets. In the special case that the additive category has weak cokernels, this topological space is an open subset of the Ziegler spectrum and this is a result of Kevin Schlegel. We also look at some module categories of rings where the Ziegler spectrum is known and calculate the global dimensions of the corresponding exact substructures. Second version contains minor changes to first version.