A Morita theory for permutation modules
It is known that the trivial source modules (or p-permutation modules) over a block of group algebra share a lot of similarities with the projective modules. For example, there are only finitely many of them, and working over a p-modular system, any trivial source module over the field of positive characteristic lifts uniquely to a trivial source module over the valuation ring. It has also been shown by Arnold that it is possible to do homological algebra with this family of modules.
The aim of the talk is to explain what happens when you replace projective modules by trivial source modules in the classical Morita theory between blocks of group algebras. This is a joint work with Markus Linckelmann.
This talk is based on the article On Morita and derived equivalences for cohomological Mackey algebras (ArXiv). It is also related to the work of Fang and Koenig on Gendo-Symmetric algebras (for more details).