# EPSILON-STRONGLY GROUPOID-GRADED RINGS, THE PICARD INVERSE CATEGORY AND COHOMOLOGY

@article{Nystedt2019EPSILONSTRONGLYGR, title={EPSILON-STRONGLY GROUPOID-GRADED RINGS, THE PICARD INVERSE CATEGORY AND COHOMOLOGY}, author={Patrik Nystedt and Johan {\"O}inert and Hector Pinedo}, journal={Glasgow Mathematical Journal}, year={2019}, volume={62}, pages={233 - 259} }

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… Expand

#### 9 Citations

Graded modules over object-unital groupoid graded rings

- Communications in Algebra
- 2021

In a previous article (see [6]), we introduced and analyzed ring-theoretic properties of object unital G-graded rings R, where G is a groupoid. In the present article, we analyze the category G-R-mod… Expand

On direct product, semidirect product of groupoids and partial actions

- Mathematics
- 2020

We present some constructions of groupoids as: direct product, semidirect product, and we give necessary and sufficient conditions for a groupoid to be embedded into a direct product of groupoids.… Expand

Isomorphism Theorems for Groupoids and Some Applications

- Computer Science, Mathematics
- Int. J. Math. Math. Sci.
- 2020

Inspired by the Ehresmann-Schein-Nambooripad theorem, a result of R. Exel concerning a one-to-one correspondence between partial actions of groups and actions of inverse semigroups is improved. Expand

Object-unital groupoid graded modules.

- Mathematics
- 2019

Given a groupoid $\mathcal{G}$ and an associative but not necessarily unital ring $R$, we introduce the notion of object unital graded ring and construct the category of object unital graded modules… Expand

Homology and cohomology via the partial group algebra

- Mathematics
- 2020

We study partial homology and cohomology from ring theoretic point of view via the partial group algebra $\mathbb{K}_{par}G$. In particular, we link the partial homology and cohomology of a group $G$… Expand

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

- Mathematics
- 2021

Let α = (Ag, αg)g∈G be a group-type partial action of a connected groupoid G on a ring A = ⊕ z∈G0 Az and B := A ⋆α G the corresponding partial skew groupoid ring. In the first part of this paper we… Expand

Object-unital groupoid graded rings, crossed products and separability

- Mathematics
- 2020

Abstract We extend the classical construction by Noether of crossed product algebras, defined by finite Galois field extensions, to cover the case of separable (but not necessarily finite or normal)… Expand

On the structure of nearly epsilon and epsilon-strongly graded rings

- Mathematics
- 2020

In this work we study the classes of epsilon and nearly epsilon-strongly graded rings by a group $G$. In particular, we extend Dade's theorem to the realm of nearly epsilon-strongly graded rings.… Expand

Groupoids: Substructures and homomorphisms

- Mathematics
- 2019

Using an algebraic point of view we present an introduction to the groupoid theory, that is, we give fundamental properties of groupoids as, uniqueness of inverses and properties of the identities,… Expand

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