  We introduce the class of partially invertible modules and show that it is an
inverse category which we call the Picard inverse category. We use this
category to generalize the classical construction of crossed products to, what
we call, generalized epsilon-crossed products and show that these coincide with
the class of epsilon-strongly groupoid graded rings. We then use generalized
epsilon-crossed groupoid products to obtain a generalization, from the group
graded situation to the groupoid graded case, of the bijection from a certain
second cohomology group, defined by the grading and the functor from the
groupoid in question to the Picard inverse category, to the collection of
equivalence classes of rings epsilon-strongly graded by the groupoid.
