Jan Stovicek (Prague), A tilting-cotilting correspondence and tilting in infinite projective dimension

Abstract:

This is an account on joint work with Leonid Positselski. We show that there is a very general version of Rickard's tilting theorem which involves cotilting objects in Grothendieck-like abelian categories on one hand and tilting objects in categories of contramodules on the other hand. Contramodules are roughly speaking modules over rings with infinitary addition - complete topological modules over complete topological rings are contramodules, but the notion is broader than that. If the tilting and cotilting modules have finite homological dimensions, one obtains usual derived equivalences. In case of infinite homological dimensions, these concepts generalize Wakamatsu tilting modules and there is still a triangulated equivalence. This equivalence, however, is not necessarily between ordinary derived categories, but between Positselski's "exotic" derived categories.