For a compactly generated triangulated category there is a notion of stratification via the central action of an appropriate cohomology ring. I will explain this concept and indicate how it works for the stable module category of a finite group. Globally, compact and dualisable objects coincide; so, the stable module category is rigid. Locally, the story is more complicated. So, for each stratum I will describe its dualisable objects in cohomological terms (and the analogue for the derived category of a commutative ring will also be mentioned). All this is based on joint work with Benson, Iyengar, and Pevtsova.
