Title: Nilpotent operators, categories of modules, and Auslander-Reiten Theory

Description:

The theory of rank varieties, expounded in the early 1980s by Jon Carlson for modular representations of finite groups, was designed as a tool to understand cohomological support varieties. Subsequent work by Jantzen, Friedlander-Parshall and Suslin-Friedlander-Bendel showed that similar methods can be introduced for representations of restricted Lie algebras and infinitesimal group schemes. About five years ago, Friedlander and Pevtsova have unified and extended the afore-mentioned approaches by introducing representation-theoretic supports for finite group schemes. The elements of these topological spaces are equivalence classes of at algebra homomorphisms α: kZ/(p) → kG between the group algebras of the cyclic group with p-elements (p = char(k)) and a finite group scheme G. These maps give rise to p-nilpotent operators on G-modules that have been used to define new invariants on the one hand, while leading to new full subcategories of mod G on the other. Starting with a review of the historical origins, we first look at rank varieties in the abovementioned contexts, briey touching upon their cohomological aspects. The second lecture introduces the notion of a p-point and gives the connection between rank varieties and the space of equivalence classes of p-points. In preparation for our applications, we also discuss the extent to which G-modules are determined by the action of their p-nilpotent operators. The last two lectures are concerned with new invariants arising via p-points. Being finer than those associated with rank varieties, they actually provide new insight even within the classical" contexts of finite groups and restricted Lie algebras. Operators of p-points have been employed to identify three full subcategories of mod G, consisting of modules of constant Jordan type (Carlson-Friedlander-Pevstova, 2008), modules of constant rank (Friedlander-Pevtsova, 2010), and modules satisfying the equal images property (Carlson-Friedlander-Suslin, 2011). We explain how the structure and the relationship between these categories depends on the underlying group scheme.

The series ends with applications concerning Auslander-Reiten theory. It turns out that p-points give rise to new invariants for the connected components of the stable Auslander-Reiten quiver Γs(G) of mod G. These are exploited in the study of the interplay between s(G) and the abovementioned subcategories.