Abstract
We introduce the concept of a partial abstract kernel associated to a group G and a semilattice of groups A and relate the partial cohomology group
Funding source: Fundaçǎo de Amparo à Pesquisa do Estado de Sǎo Paulo
Award Identifier / Grant number: 2015/09162-9
Funding source: Conselho Nacional de Desenvolvimento Científico e Tecnológico
Award Identifier / Grant number: 307873/2017-0
Award Identifier / Grant number: 404649/2018-1
Funding source: Coordenaçǎo de Aperfeiçoamento de Pessoal de Nìvel Superior
Award Identifier / Grant number: Finance Code 001
Funding source: Fundaçǎo para a Ciência e a Tecnologia
Award Identifier / Grant number: PTDC/MAT-PUR/31174/2017
Funding statement: The first author was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico – CNPq of Brazil (Proc. 307873/2017-0) and Fundação de Amparo à Pesquisa do Estado de São Paulo – FAPESP of Brazil (Proc. 2015/09162-9). The second author was partially supported by CNPq of Brazil (Proc. 404649/2018-1) and Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project PTDC/MAT-PUR/31174/2017. The third author was financed by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001.
Acknowledgements
We thank the anonymous referee for the careful reading of our paper and remarks that helped us to correct several misprints and improve the proof of Lemma 2.19.
References
[1] E. Batista, A. D. M. Mortari and M. M. Teixeira, Cohomology for partial actions of Hopf algebras, J. Algebra 528 (2019), 339–380. 10.1016/j.jalgebra.2019.03.013Search in Google Scholar
[2] A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups. Vol. II, Math. Surveys Monogr. 7, American Mathematical Society, Providence, 1967. 10.1090/surv/007.2Search in Google Scholar
[3] M. Dokuchaev, Recent developments around partial actions, São Paulo J. Math. Sci. 13 (2019), no. 1, 195–247. 10.1007/s40863-018-0087-ySearch in Google Scholar
[4] M. Dokuchaev and R. Exel, Associativity of crossed products by partial actions, enveloping actions and partial representations, Trans. Amer. Math. Soc. 357 (2005), no. 5, 1931–1952. 10.1090/S0002-9947-04-03519-6Search in Google Scholar
[5] M. Dokuchaev, R. Exel and J. J. Simón, Crossed products by twisted partial actions and graded algebras, J. Algebra 320 (2008), no. 8, 3278–3310. 10.1016/j.jalgebra.2008.06.023Search in Google Scholar
[6] M. Dokuchaev, R. Exel and J. J. Simón, Globalization of twisted partial actions, Trans. Amer. Math. Soc. 362 (2010), no. 8, 4137–4160. 10.1090/S0002-9947-10-04957-3Search in Google Scholar
[7] M. Dokuchaev and M. Khrypchenko, Partial cohomology of groups, J. Algebra 427 (2015), 142–182. 10.1016/j.jalgebra.2014.11.030Search in Google Scholar
[8] M. Dokuchaev and M. Khrypchenko, Twisted partial actions and extensions of semilattices of groups by groups, Internat. J. Algebra Comput. 27 (2017), no. 7, 887–933. 10.1142/S0218196717500424Search in Google Scholar
[9] M. Dokuchaev and M. Khrypchenko, Partial cohomology of groups and extensions of semilattices of abelian groups, J. Pure Appl. Algebra 222 (2018), no. 10, 2897–2930. 10.1016/j.jpaa.2017.11.005Search in Google Scholar
[10] M. Dokuchaev and B. Novikov, Partial projective representations and partial actions, J. Pure Appl. Algebra 214 (2010), no. 3, 251–268. 10.1016/j.jpaa.2009.05.001Search in Google Scholar
[11] M. Dokuchaev, A. Paques and H. Pinedo, Partial Galois cohomology and related homomorphisms, Q. J. Math. 70 (2019), no. 2, 737–766. 10.1093/qmath/hay062Search in Google Scholar
[12] M. Dokuchaev, A. Paques, H. Pinedo and I. Rocha, Partial generalized crossed products and a seven-term exact sequence, preprint (2019), https://arxiv.org/abs/1908.05820. 10.1016/j.jalgebra.2020.12.014Search in Google Scholar
[13] M. Dokuchaev and N. Sambonet, Schur’s theory for partial projective representations, Israel J. Math. 232 (2019), no. 1, 373–399. 10.1007/s11856-019-1876-4Search in Google Scholar
[14]
R. Exel,
Twisted partial actions: a classification of regular
[15] M. Kennedy and C. Schafhauser, Noncommutative boundaries and the ideal structure of reduced crossed products, Duke Math. J. 168 (2019), no. 17, 3215–3260. 10.1215/00127094-2019-0032Search in Google Scholar
[16] H. Lausch, Cohomology of inverse semigroups, J. Algebra 35 (1975), 273–303. 10.1016/0021-8693(75)90051-4Search in Google Scholar
[17] M. V. Lawson, Inverse Semigroups. The Theory of Partial Symmetries, World Scientific, River Edge, 1998. 10.1142/3645Search in Google Scholar
[18] S. MacLane, Homology, Springer, Berlin, 1963. 10.1007/978-3-642-62029-4Search in Google Scholar
[19] M. Petrich, Inverse Semigroups, Pure Appl. Math. (N. Y.), John Wiley & Sons, New York, 1984. Search in Google Scholar
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