Villamayor–Zelinsky Sequence for Symmetric Finite Tensor Categories

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 à la Villamayor–Zelinsky for C. It consists of the corresponding cohomology groups evaluated at three types of coefficients which repeat periodically in the sequence. We compute some subgroups of the groups appearing in the sequence.