|Location:||MSRI: Simons Auditorium|
The background motivation for this talk is the lack of understanding of stable module categories of finite group
algebras. Since finite group algebras are self-injective, their stable module categories are triangulated categories.
We do not know in general how to deduce even some of the most basic invariants of finite group algebras from the structure of their stable module categories, such as the numbers of nonprojective simple modules of the group algebra. Abelian subcategories of triangulated categories would at least yield a mechanism to attach some numerical invariants, such as their numbers of simple objects, for instance.
Stable module categories of self-injective algebras have in general no t-structures, and so in particular, their stability spaces would be empty in these cases. Stable module categories do have in many cases abelian subcategories which are not the heart of any t-structure, but which are compatible with the triangulated structure in a precise sense. While stability spaces are manifolds, there is evidence that the abelian subcategories of stable module categories can be organised as varieties.