Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T07:15:10.564Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

Bagio, D., Flôres, D. and Paques, A., Partial actions of ordered groupoids on rings. J. Algebra Appl. 9(3) (2010), 501–517.CrossRefGoogle Scholar
Bagio, D. and Paques, A., Partial groupoid actions: globalization, Morita theory and Galois theory, Comm. Algebra 40(10) (2012), 3658–3678.CrossRefGoogle Scholar
Bass, H., Algebraic K-theory (Benjamin, New York, 1968).Google Scholar
Dokuchaev, M. and Exel, R., Associativity of crossed products by partial actions, enveloping actions and partial representations, Trans. Amer. Math. Soc. 357(5) (2005), 1931–1952.CrossRefGoogle Scholar
Dokuchaev, M. and Khrypchenko, M., Partial cohomology of groups, J. Algebra 427 (2015), 142–182.CrossRefGoogle Scholar
Dokuchaev, M. and Novikov, B., Partial projective representations and partial actions, J. Pure Appl. Algebra 214(3) (2010), 251–268.CrossRefGoogle Scholar
Dokuchaev, M. and Novikov, B., Partial projective representations and partial actions II, J. Pure Appl. Algebra 216(2) (2012), 438–455.CrossRefGoogle Scholar
Dokuchaev, M., Paques, A. and Pinedo, H., Partial Galois cohomology, and related homomorphisms, To appear in Quart. J. Math. (2019).CrossRefGoogle Scholar
Exel, R., Circle actions on C*-algebras, partial automorphisms and generalized Pimsner–Voiculescu exact sequences, J. Funct. Anal. 122(3) (1994), 361–401.CrossRefGoogle Scholar
Gonçalves, D. and Yoneda, G., Free path groupoid grading on Leavitt path algebras, Int. J. Algebra Comput. 26(6) (2016), 1217–1235.CrossRefGoogle Scholar
Hazrat, R., The graded structure of Leavitt path algebras, Israel J. Math. 195(2) (2013), 833–895.CrossRefGoogle Scholar
Kanzaki, T., On generalized crossed product and Brauer group, Osaka J. Math. 5, 175–188 (1968).Google Scholar
Lundström, P., The category of groupoid graded modules, Colloq. Math. 100(4) (2004), 195–211.CrossRefGoogle Scholar
Lundström, P., Strongly groupoid graded rings and cohomology, Colloq. Math. 106(1)(2006), 1–13.CrossRefGoogle Scholar
Nǎstǎsescu, C. and Van Oystaeyen, F., Graded ring theory (North-Holland Publishing Co., Amsterdam-New York, 1982).Google Scholar
Nystedt, P., Partial category actions on sets and topological spaces, Comm. Algebra 46(2) (2018), 671–683.CrossRefGoogle Scholar
Nystedt, P., Öinert, J. and Pinedo, H., Epsilon-strongly graded rings, separability and semisimplicity, J. Algebra 514 (2018), 1–24.CrossRefGoogle Scholar
Pinedo, H., Partial projective representations and the partial Schur multiplier: a survey, Bol. Mat. 22(2) (2015), 167–175.Google Scholar
Renault, J., A groupoid approach to C*-algebras, Lecture Notes in Mathematics, vol. 793 (Springer, Berlin, 1980).CrossRefGoogle Scholar
Westman, J., Groupoid theory in algebra, topology and analysis, (University of California, Irvine, 1971).Google Scholar