The notion of infinitely generated tilting module arose as a formal generalization of tilting modules from representation theory of finite-dimensional algebras. The latter tilting modules were known to represent derived equivalences [Hap], while the latter were believed not to until [Baz]. In the talks, I will explain a recent joint work with L. Positselski [PS1], [PS2], where we develop a systematic framework for derived equivalences induced by infinitely generated tilting modules and their generalizations. A crucial insight comes from [Pos] and the comodule-contramodule correspondence described there.
[Baz] S. Bazzoni: Equivalences induced by infinitely generated tilting modules, Proc. Amer. Math. Soc. 138, 533-544, 2010.
[Hap] D. Happel: On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62, 339-389, 1987.
[Pos] L. Positselski: Homological algebra of semimodules and semicontramodules (semi-infinite homological algebra of associative algebraic structures); Appendix C in collaboration with D. Rumynin; Appendix D in collaboration with S. Arkhipov. Monografie Matematyczne vol. 70, Birkhauser/Springer Basel, 2010.
[PS1] L. Positselski, J. Stovicek: The tilting-cotilting correspondence, preprint, arXiv:1710.02230.
[PS2] L. Positselski, J. Stovicek: Infinity-tilting theory, preprint, arXiv:1711.06169.