  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.
