Villamayor-Zelinsky Sequence for Symmetric Finite Tensor Categories.

We prove that if a finite tensor category $${\mathcal C}$$ is symmetric, then the monoidal category of one-sided $${\mathcal C}$$ -bimodule categories is symmetric. Consequently, the Picard group of $${\mathcal C}$$ (the subgroup of the Brauer-Picard group introduced by Etingov-Nikshych-Gelaki) is abelian in this case. We then introduce a cohomology over such $${\mathcal C}$$ . 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 $${\mathcal C}$$ -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 $${\mathcal C}$$ . 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. [ABSTRACT FROM AUTHOR]