Classification of exact structures using the Ziegler spectrum

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.