Spectra of small abelian categories

Here we take the view that small abelian categories are certain categorical versions of rings and the collection of Serre subcategories of a small abelian category is then analogous to the set of ideals of a commutative ring. We consider two topologies on the collection of Serre subcategories; one is a very direct lift of the Zariski topology, the other, dual, topology appeared in the context of commutative rings in the work of Thomason [24] but, more generally, had already arisen, see [4] for the connection, in the work of Ziegler [26] on the model theory of modules. We investigate the primes of these topologies/locales and the corresponding notion of “local” abelian category. We compare with the usual Zariski spectrum on a commutative noetherian ring (which is essentially a special case) and also briefly describe and compare a number of topologies which have been put on the set of isomorphism types of indecomposable injective modules over a commutative ring.