Tilting Theory in exact categories

TL;DR

This paper generalizes tilting theory to arbitrary exact categories, establishing foundational results, characterizations, and conditions for derived equivalences, extending classical theorems to a broader categorical context.

Contribution

It unifies existing tilting subcategory definitions in exact categories and introduces the concept of ideq tilting, along with a generalized Miyashita's theorem.

Findings

Generalization of Miyashita's theorem

Characterization of exact categories with ideq tilting

Conditions for derived equivalences in exact categories

Abstract

We define tilting subcategories in arbitrary exact categories to archieve the following. Firstly: Unify existing definitions of tilting subcategories to arbitrary exact categories. Discuss standard results for tilting subcategories: Auslander correspondence, Bazzoni description of the perpendicular category. Secondly: We treat the question of induced derived equivalences separately - given a tilting subcategory T, we ask if a functor on the perpendicular category induces a derived equivalence to a (certain) functor category over T. If this is the case, we call the tilting subcategory ideq tilting. We prove a generalization of Miyashita's theorem (which is itself a generalization of a well-known theorem of Brenner-Butler) and characterize exact categories with enough projectives allowing ideq tilting subcategories. In particular, this is always fulfilled if the exact category is abelian with enough projectives.