The Hochschild cohomology group for the integral group ring of the semidihedral 2-group SD2k is computed. In calculations, the free bimodule resolution of the corresponding group ring is used. In turn, to construct it the usual resolution of the trivial SD2k-module \(\mathbb{Z}\) was first built. Bibliography: 22 titles.
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References
S. Eilenberg and S. MacLane, “Cohomology theory in abstract groups. I,” Ann. Math., 48 51–78 (1947).
A. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton (1956).
M. Gerstenhaber, “The cohomology structure of an associative ring,” Ann. Math., 78, 267–288 (1963).
S. F. Siegel and S. J. Witherspoon, “The Hochschild cohomology ring of a group algebra,” Proc. London Math. Soc., 79, 131–157 (1999).
K. Erdmann and Th. Holm, “Twisted bimodules and Hochschild cohomology for self-injective algebras of class An,” Forum Math., 11, 177–201 (1999).
A. I. Generalov, “Hochschild cohomology of algebras of dihedral type. I: the family \(D\left( {3{\mathcal K}} \right)\) in characteristic 2,” Algebra Analiz, 16, No. 6, 53–122 (2004).
K. Erdmann, “Blocks of tame representation type and related algebras,” Lecture Notes Math., 1428, Berlin, Heidelberg (1990).
A. I. Generalov, “Hochschild cohomology of algebras of quaternion type. I: generalized quaternion groups,” Algebra Analiz, 18, No. 1, 55–107 (2006).
A. I. Generalov, A. A. Ivanov, and S. O. Ivanov, “Hochschild cohomology of algebras of quaternion type. II: The family \(Q{\left( {2{\mathcal B}} \right)_1}\) in characteristic 2,” Zap. Nauchn. Semin. POMI, 349, 53–134 (2007).
A. I. Generalov, “Hochschild cohomology of algebras of quaternion type. III: Algebras with a small parameter,” Zap. Nauchn. Semin. POMI, 356, 46–84 (2008).
A. I. Generalov and N. Yu. Kossovskaya, “Hochschild cohomology of the Liu-Schulz algebras,” Algebra Analiz, 18, No. 4, 39–82 (2006).
A. I. Generalov, “Hochschild cohomology of the integral group algebra of the dihedral group. I. Even case,” Algebra Analiz, 19, No. 5, 70–123 (2007).
T. Hayami, “Hochschild cohomology ring of the integral group algebra of dihedral groups,” Tsukuba J. Math., 31, 99–127 (2007).
T. Hayami, “Hochschild cohomlogy ring of the integral group ring of the generalized quaternion group,” SUT J. Math., 38, No. 1, 83–126 (2002).
T. Hayami, “On Hochschild cohomology ring of the integral group ring of the quaternion group,” Tsukuba J. Math., 29, No. 2, 363–387 (2005).
T. Holm, “Hochschild cohomology of tame blocks,” J. Algebra, 271, 798–826 (2002).
K. Erdmann. Th. Holm, and N. Snashall, “Twisted bimodules and Hochschild cohomology for self-injective algebras of class A n . II,” Algebras and Repr. Theory, 5, 457–482 (2002).
M. A. Kachalova, “Hochschild cohomology for Möbius algebras,” Zap. Nauchn. Semin. POMI, 330, 173–200 (2006).
A. A. Ivanov, “Hochschild cohomology of algebras of quaternion type: the family \(Q{\left( {2{\mathcal B}} \right)_1}\left( {k,s,a,c} \right)\) over a field with characteristic different from 2,” Vestn. Peterburg Univ.. Ser.1, Math. Mekh., Astr., No. 1 63–72 (2010).
A. I. Generalov and S. O. Ivanov, “Bimodule resolution of a group algebra,” Zap. Nauchn. Semin. POMI, 365, 143–150 (2009).
C. T. C. Wall, “Resolutions for extensions of groups,” Proc. Cambridge Phil. Soc., 57, No. 2, 251–255 (1961).
T. Hayami, “Hochschild cohomology ring of the integral group ring of the semidihedral 2-groups,” Algebra Colloq., 18, No. 2, 241–258 (2011).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 388, 2011, pp. 119–151.
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Generalov, A.I. Hochschild cohomology of the integral group ring of the semidihedral group. J Math Sci 183, 640–657 (2012). https://doi.org/10.1007/s10958-012-0829-9
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DOI: https://doi.org/10.1007/s10958-012-0829-9