Classification of exact structures using the Ziegler spectrum
Julia Sauter
TL;DR
The paper constructs a topological framework linking exact structures on a small idempotent complete additive category to closed subsets of a Ziegler-style spectrum, yielding an explicit anti-isomorphism between the lattice of exact structures and Ziegler-closed sets. It develops the Ind-completion machinery, defines fp-injective and pure-injective objects, and proves that the set of indecomposable injectives in the ind-completion captures all exact structures via U-closures. In representation-finite cases, Enomoto’s results provide a clean lattice-theoretic classification in terms of generators, with concrete global-dimension consequences for quivers. The Ziegler spectrum is then used to parametrize exact structures in broader settings (e.g., Kronecker quiver, DVRs, Dedekind domains), and the DRSS construction gives a versatile method to generate new exact structures from functors. Collectively, the work integrates model-theoretic, homological, and lattice-theoretic viewpoints to classify exact substructures and their homological dimensions across a range of categorical contexts.
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.