Finite generation of cohomology and self-Ext in finite tensor categories (Etingof–Ostrik conjecture)

Establish that for every finite tensor category C over an algebraically closed field k, the cohomology ring H^*(C) = Ext_C^*(1,1) is finitely generated as a k-algebra, and that for every object X in C, Ext_C^*(X,X) is finitely generated as a module over H^*(C).

Background

The paper develops representation type results for finite tensor categories and their exact module categories under a finiteness condition (Fg) on cohomology. This condition assumes finite generation of the cohomology ring H^*(C) and of self-Ext modules Ext_C^*(X,X) over H^*(C).

The authors explicitly note that Etingof and Ostrik conjectured these finiteness properties for all finite tensor categories. Because this conjecture remains open, the authors adopt Fg as a hypothesis to obtain support variety theory and the main wildness results presented in the paper.