# 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}
}
• Published 2019
• Mathematics
• Glasgow Mathematical Journal
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
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#### References

SHOWING 1-10 OF 23 REFERENCES
The category of groupoid graded modules
We introduce the abelian category R-gr of groupoid graded modules and give an answer to the following general question: If U : R-gr→ R-mod denotes the functor which associates to any graded leftExpand
Epsilon-strongly graded rings, separability and semisimplicity
• Mathematics
• 2016
We introduce the class of epsilon-strongly graded rings and show that it properly contains both the class of strongly graded rings and the class of unital partial crossed products. We determineExpand
Circle actions on C*-algebras, partial automorphisms, and a generalized Pimsner-Voiculescu exact sequence
Abstract We introduce a method to study C *-algebras possessing an action of the circle group, from the point of view of their internal structure and their K -theory. Under relatively mild conditionsExpand
Partial category actions on sets and topological spaces
ABSTRACT We introduce (continuous) partial category actions on sets (topological spaces) and show that each such action admits a universal globalization. Thereby, we obtain a simultaneousExpand
Associativity of crossed products by partial actions, enveloping actions and partial representations
• Mathematics
• 2002
Given a partial action a of a group G on an associative algebra A, we consider the crossed product A × α G. Using the algebras of multipliers, we generalize a result of Exel (1997) on theExpand
PARTIAL ACTIONS OF ORDERED GROUPOIDS ON RINGS
• Mathematics
• 2010
In this paper, we introduce the notion of a partial action of an ordered groupoid on a ring and we construct the corresponding partial skew groupoid ring. We present sufficient conditions under whichExpand
Partial projective representations and partial actions II
• Mathematics
• 2010
Abstract This paper is a continuation of Dokuchaev and Novikov (2010) [8] . The interaction between partial projective representations and twisted partial actions of groups considered in DokuchaevExpand
Partial Groupoid Actions: Globalization, Morita Theory, and Galois Theory
• Mathematics
• 2012
In this article, we introduce the notion of a partial action of a groupoid on a ring as well as we give a criteria for the existence of a globalization of it. We construct a Morita context associatedExpand
Free path groupoid grading on Leavitt path algebras
• Computer Science, Mathematics
• Int. J. Algebra Comput.
• 2016
Using this grading, free path groupoid graded isomorphisms of Leavitt path algebras that preserves generators are characterized. Expand
Partial cohomology of groups
• Mathematics
• 2013
We develop a cohomology theory of groups based on partial actions and explore its relation with the partial Schur multiplier as well as with cohomology of inverse semigroups.