Algebra seminar: Tilting theory in exact categories

Zoomlink (meeting ID: 645 5572 6999, for passcode: please contact the organizer).

Abstract: 

Part 1: We introduce exact categories (in the sense of Quillen) as a generalization of abelian categories as the natural enviroment for homological algebra since many concepts for abelian categories have straight-forward generalization to exact categories.

We can define extension groups, derived categories, Hall algebras and many more. But the bounded derived category of an exact category is still very little understood in general (e.g. what is the analogue of "t-structures" which gives the exact category?).

Secondly, I will give a quick intro to tilting theory. Starting with artin algebras and how easy it gives us derived equivalences. I will discuss some other situations where we get these derived equivalences (e.g. tilting sheaves and relative tilting modules).

Part 2: We explore tilting theory for arbitrary exact categories.

Here we have to pass from tilting objects to tilting subcategories (because if an exact category has enough projectives we want that the projectives are an example of a tilting subcategory). The main open question is when do they induce triangle equivalences on bounded derived categories (and of course on which derived categories)?

I choose a specific functor to a functor category over the tilting subcategory (with enough projectives) because it is a generalization of our well-known tilting situations (cp part 1). My main result is for exact categories with enough projectives: They induce derived equivalences if and only if they are finitely resolving subcategories in an arbitrary functor category with enough projectives - all instances of part 1 are special situations of this. At the end I talk about examples.

This is a seminar in our seminar series about algebra.