Epsilon-strongly groupoid graded rings, the Picard inverse category and cohomology

TL;DR

This paper introduces the Picard inverse category and generalizes the concept of crossed products to epsilon-strongly groupoid graded rings, establishing a cohomological classification in this broader context.

Contribution

It defines the Picard inverse category using partially invertible modules and extends classical crossed product constructions to groupoid gradings, linking them to second cohomology groups.

Findings

Introduction of the Picard inverse category.

Generalization of epsilon-crossed products to groupoid gradings.

Establishment of a cohomological classification of epsilon-strongly groupoid graded rings.

Abstract

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.