Cohomology of finite tensor categories: duality and Drinfeld centers

TL;DR

This paper investigates the cohomological finiteness properties of finite tensor categories, demonstrating their preservation under duality and Drinfeld center operations, and providing new examples including quantum groups and infinitesimal group schemes.

Contribution

It establishes the preservation of cohomological finiteness under duality and Drinfeld center, and introduces new classes of finite tensor categories with finitely generated cohomology.

Findings

Cohomological finiteness is preserved under duality and Drinfeld center operations.

Finite tensor categories of odd Frobenius-Perron dimension have finitely generated cohomology.

Dynamical quantum groups at roots of unity have finitely generated cohomology in characteristic 0.

Abstract

We consider the finite generation property for cohomology of a finite tensor category C, which requires that the self-extension algebra of the unit Ext*_C(1,1) is a finitely generated algebra and that, for each object V in C, the graded extension group Ext*_C(1,V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on C. For example, the stated result holds when C is a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0, we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes.