Epsilon-Strongly Groupoid-Graded Rings, the Picard Inverse Category and Cohomology

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.

Ring theoretic properties of partial skew groupoid rings with applications to Leavitt path algebras

“…Also, if A g is a unital ring for each g ∈ G, then B is an associative ring. Partial skew groupoid rings appear naturally in the partial Galois theory of groupoids [7] and in the context of groupoid graded rings [23]. Also, it was proved in [16] that every Leavitt path algebra is a partial skew groupoid ring.…”

Isomorphism Theorems for Groupoids and Some Applications

“…Recently, some applications of groupoids to the study of partial actions are presented in different branches, for instance, in [6] the author constructs a Birget-Rhodes expansion G BR associated with an ordered groupoid G and shows that it classifies partial actions of G on sets, in the topological context in [7] is treated the globalization problem, connections between partial actions of groups and groupoids are given in [8,9]. Also, ring theoretic and cohomological results of global and partial actions of groupoids on algebras are obtained in [10][11][12][13][14][15][16]. Galois theoretic results for groupoid actions are obtained in [12,[17][18][19].…”

Object-unital groupoid graded rings, crossed products and separability

“…There are relevant classes of rings that can be graded by groupoids, such as matrix rings, crossed product algebras defined by separable extensions and partial skew groupoid rings that are not, in a natural way, graded by groups (see e.g. [12,14,23]).…”