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

Part of: Rings and algebras with additional structure Modules, bimodules and ideals Homological methods Cycles and subschemes

Published online by Cambridge University Press:  12 March 2019

PATRIK NYSTEDT ,
JOHAN ÖINERT  and
HÉCTOR PINEDO
PATRIK NYSTEDT
Affiliation:
Department of Engineering Science, University West, SE-46186 Trollhättan, Swedene-mail:patrik.nystedt@hv.se
JOHAN ÖINERT*
Affiliation:
Department of Mathematics and Natural Sciences, Blekinge Institute of Technology, SE-37179 Karlskrona, Swedene-mail:johan.oinert@bth.se
HÉCTOR PINEDO
Affiliation:
Escuela de Matemáticas, Universidad Industrial de Santander, Carrera 27 Calle 9, Edificio Camilo Torres Apartado de correos 678, Bucaramanga, Colombiae-mail:hpinedot@uis.edu.co
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Abstract

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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.

MSC classification

Primary: 16W50: Graded rings and modules
Secondary: 16E99: None of the above, but in this section 16D99: None of the above, but in this section 14C22: Picard groups
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 1 , January 2020 , pp. 233 - 259
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Glasgow Mathematical Journal Trust 2019

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