Auslander's formula and correspondence for exact categories

The Auslander correspondence is a fundamental result in Auslander-Reiten theory. In this paper we introduce the category mod_adm(E) of admissibly finitely presented functors and use it to give a version of Auslander correspondence for any exact category E. An important ingredient in the proof is the localization theory of exact categories. We also investigate how properties of E are reflected in mod_adm(E), for example being (weakly) idempotent complete or having enough projectives or injectives. Furthermore, we describe mod_adm(E) as a subcategory of mod(E) when E is a resolving subcategory of an abelian category. This includes the category of Gorenstein projective modules and the category of maximal Cohen-Macaulay modules as special cases. Finally, we use mod_adm(E) to give a bijection between exact structures on an idempotent complete additive category C and certain resolving subcategories of mod(C).