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Hochschild cohomology ring of the modular group


Authors: A. P. Alekhin, Yu. V. Volkov and A. I. Generalov
Translated by: A. I. Generalov
Original publication: Algebra i Analiz, tom 26 (2014), nomer 1.
Journal: St. Petersburg Math. J. 26 (2015), 1-25
MSC (2010): Primary 16E40
DOI: https://doi.org/10.1090/S1061-0022-2014-01328-3
Published electronically: November 21, 2014
MathSciNet review: 3234804
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Abstract: A description in terms of generators and relations is given for the cohomology ring and the Hochschild cohomology ring of the group algebra for the even modular group over the ring of integers. The free resolution of the trivial module described by Wall is used for that. Moreover, the bimodule resolution of the group algebra in question is described.


References [Enhancements On Off] (What's this?)

  • 1. A. I. Generalov, The Hochschild cohomology of the integer group ring of a dihedral group. I, The even case, Algebra i Analiz 19 (2007), no. 5, 70-123; English transl., St. Petersburg Math. J. 19 (2008), no. 5, 723-763. MR 2381942 (2009e:16015)
  • 2. -, Hochschild cohomology for the integer group ring of the semidihedral group, Zap. Nauchn. Sem. S.-Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 388 (2011), 119-151; English transl., J. Math. Sci. (N.Y.) 183 (2012), no. 5, 640-657. MR 2822518 (2012e:16024)
  • 3. T. Hayami, Hochschild cohomology ring of the integral group ring of the generalized quaternion group, SUT J. Math. 38 (2002), no. 1, 83-126. MR 1919950 (2003f:16014)
  • 4. -, On Hochschild cohomology ring of the integral group ring of the quaternion group, Tsukuba J. Math. 29 (2005), no. 2, 363-387. MR 2177018 (2006j:16016)
  • 5. -, Hochschild cohomology ring of the integral group ring of dihedral groups, Tsukuba J. Math. 31 (2007), no. 1, 99-127. MR 2337122 (2008d:16016)
  • 6. -, Hochschild cohomology ring of the integral group ring of the semidehidral $ 2$-group, Algebra Colloq. 18 (2011), no. 2, 241-258. MR 2785780 (2012c:16031)
  • 7. S. F. Siegel and S. J. Witherspoon, The Hochschild cohomology ring of a group algebra, Proc. London Math. Soc.(3) 79 (1999), no. 1, 131-157. MR 1687539 (2000b:16016)
  • 8. C. T. C. Wall, Resolutions for extensions of groups, Proc. Cambridge Philos. Soc. 57 (1961), 251-255. MR 0178046 (31:2304)
  • 9. A. I. Generalov and S. O. Ivanov, Bimodule resolution of a group algebra, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 365 (2009), 143-150; English transl., J. Math. Sci. (N.Y.) 161 (2009), no. 4, 537-541. MR 2749138 (2012b:16016)
  • 10. I. J. Leary, The cohomology of certain groups, PhD Thesis, Cambridge, 1992.
  • 11. C. B. Thomas, Chern classes and metacyclic $ p$-groups, Mathematika 18 (1971), 196-200. MR 0299699 (45:8747)
  • 12. D. J. Benson, Representations and cohomology. II, Cohomology of groups and modules, Cambridge Stud. Adv. Math. vol. 30, Cambridge Univ. Press, Cambridge, 1998. MR 1634407 (99f:20001b)
  • 13. S. Eilenberg and S. MacLane, Cohomology theory in abstract groups. I, Ann. of Math. (2) 48 (1947), 51-78. MR 0019092 (8:367f)
  • 14. S. MacLane, Homology, Grundlehren Math. Wiss., Bd. 114, Springer-Verlag, Berlin, 1963. MR 0156879 (28:122)
  • 15. K. Brown, Cohomology of groups, Grad. Texts in Math., vol. 87, Springer, Berlin, 1982. MR 672956 (83k:20002)

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Additional Information

A. P. Alekhin
Affiliation: Research Group, Era7 Bioinformatics, Plaza Campo Verde 3 Atico, Granada 18001, Spain
Email: aalekhin@ohnosequences.com

Yu. V. Volkov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskii pr., 28, Staryi Peterhof, St. Petersburg 198504, Russia
Email: wolf86_666@list.ru

A. I. Generalov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskii pr., 28, Staryi Peterhof, St. Petersburg 198504, Russia
Email: ageneralov@gmail.com

DOI: https://doi.org/10.1090/S1061-0022-2014-01328-3
Keywords: Hochschild cohomology, group algebras, modular $2$-groups
Received by editor(s): March 10, 2013
Published electronically: November 21, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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