Hochschild cohomology ring of the modular group

Alekhin, A.; Volkov, Yu.; Generalov, A.

doi:10.1090/S1061-0022-2014-01328-3

Hochschild cohomology ring of the modular group
HTML articles powered by AMS MathViewer

by 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

PDF | Request permission

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