Villamayor-Zelinsky sequence for symmetric finite tensor categories

TL;DR

This paper establishes a Villamayor-Zelinsky sequence for symmetric finite tensor categories, showing their bimodule categories form an abelian group and introducing a related cohomology theory.

Contribution

It proves the symmetry of bimodule categories in symmetric finite tensor categories and constructs a Villamayor-Zelinsky type exact sequence for their cohomology.

Findings

The monoidal category of one-sided bimodule categories is symmetric.

The Picard group of a symmetric finite tensor category is abelian.

An infinite exact sequence analogous to Villamayor-Zelinsky is constructed.

Abstract

We prove that if a finite tensor category $\C$ is symmetric, then the monoidal category of one-sided $\C$-bimodule categories is symmetric. Consequently, the Picard group of $\C$ (the subgroup of the Brauer-Picard group introduced by Etingov-Nikshych-Gelaki) is abelian in this case. We then introduce a cohomology over such $\C$. An important piece of tool for this construction is the computation of dual objects for bimodule categories and the fact that for invertible one-sided $\C$-bimodule categories the evaluation functor involved is an equivalence, being the coevaluation functor its quasi-inverse, as we show. Finally, we construct an infinite exact sequence a la Villamayor-Zelinsky for $\C$. It consists of the corresponding cohomology groups evaluated at three types of coefficients which repeat periodically in the sequence.