This is a report on recent and ongoing joint work with Julia Plavnik and Sarah Witherspoon, where we have developed a theory of cohomological support varieties for finite tensor categories. Under suitable finite generation conditions – conjectured to hold for all finite tensor categories – the varieties encode homological properties of the objects, as in the classical case for group algebras. For example, the dimension of a variety equals the complexity of the corresponding object, so that the objects having trivial support varieties are precisely the projective ones. Moreover, every potential variety is actually the support variety of some object, and the support variety of an indecomposable object is connected. I will also discuss the so-called tensor product property for varieties.
This video is part of the New Directions in Group Theory and Triangulated Categories seminar series.
