Hochschild cohomology ring of the modular group
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A. P. Alekhin, Yu. V. Volkov and A. I. Generalov
Translated by: A. I. Generalov - St. Petersburg Math. J. 26 (2015), 1-25
- DOI: https://doi.org/10.1090/S1061-0022-2014-01328-3
- Published electronically: November 21, 2014
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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
- 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 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 19 (2008), no. 5, 723–763. MR 2381942, DOI 10.1090/S1061-0022-08-01018-2
- A. I. Generalov, Hochschild cohomology for the integer group ring of the semidihedral group, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 388 (2011), no. Voprosy Teorii Predstavleniĭ Algebr i Grupp. 21, 119–151, 310–311 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 183 (2012), no. 5, 640–657. MR 2822518, DOI 10.1007/s10958-012-0829-9
- Takao 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
- Takao Hayami, On Hochschild cohomology ring of the integral group ring of the quaternion group, Tsukuba J. Math. 29 (2005), no. 2, 363–387. MR 2177018, DOI 10.21099/tkbjm/1496164962
- Takao Hayami, Hochschild cohomology ring of the integral group ring of dihedral groups, Tsukuba J. Math. 31 (2007), no. 1, 99–127. MR 2337122, DOI 10.21099/tkbjm/1496165117
- Takao Hayami, Hochschild cohomology ring of the integral group ring of the semidihedral 2-group, Algebra Colloq. 18 (2011), no. 2, 241–258. MR 2785780, DOI 10.1142/S1005386711000150
- Stephen F. Siegel and Sarah J. Witherspoon, The Hochschild cohomology ring of a group algebra, Proc. London Math. Soc. (3) 79 (1999), no. 1, 131–157. MR 1687539, DOI 10.1112/S0024611599011958
- C. T. C. Wall, Resolutions for extensions of groups, Proc. Cambridge Philos. Soc. 57 (1961), 251–255. MR 178046, DOI 10.1017/s0305004100035155
- 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), no. Voprosy Teorii Predstavleniĭ Algebr i Grupp. 18, 143–150, 264 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 161 (2009), no. 4, 537–541. MR 2749138, DOI 10.1007/s10958-009-9590-0
- I. J. Leary, The cohomology of certain groups, PhD Thesis, Cambridge, 1992.
- C. B. Thomas, Chern classes and metacyclic $p$-groups, Mathematika 18 (1971), 196–200. MR 299699, DOI 10.1112/S0025579300005453
- D. J. Benson, Representations and cohomology. II, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, Cambridge, 1998. Cohomology of groups and modules. MR 1634407
- Samuel Eilenberg and Saunders MacLane, Cohomology theory in abstract groups. I, Ann. of Math. (2) 48 (1947), 51–78. MR 19092, DOI 10.2307/1969215
- Saunders Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften, Band 114, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156879
- Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
Bibliographic 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
- Received by editor(s): March 10, 2013
- Published electronically: November 21, 2014
- © Copyright 2014 American Mathematical Society
- Journal: St. Petersburg Math. J. 26 (2015), 1-25
- MSC (2010): Primary 16E40
- DOI: https://doi.org/10.1090/S1061-0022-2014-01328-3
- MathSciNet review: 3234804