  We start the general structure theory of not necessarily semisimple finite
tensor categories, generalizing the results in the semisimple case (i.e. for
fusion categories), obtained recently in our joint work with D.Nikshych. In
particular, we generalize to the categorical setting the Hopf and quasi-Hopf
algebra freeness theorems due to Nichols-Zoeller and Schauenburg, respectively.
We also give categorical versions of the theory of distinguished group-like
elements in a finite dimensional Hopf algebra, of Lorenz's result on degeneracy
of the Cartan matrix, and of the absence of primitive elements in a finite
dimensional Hopf algebra in zero characteristic. We also develop the theory of
module categories and dual categories for not necessarily semisimple finite
tensor categories; the crucial new notion here is that of an exact module
category. Finally, we classify indecomposable exact module categories over the
simplest finite tensor categories, such as representations of a finite group in
positive characteristic, representations of a finite supergroup, and
representations of the Taft Hopf algebra.
