Becker-Gottlieb transfer for Hochschild cohomology
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- by Fei Xu
- Proc. Amer. Math. Soc. 142 (2014), 2593-2608
- DOI: https://doi.org/10.1090/S0002-9939-2014-12013-2
- Published electronically: April 15, 2014
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Abstract:
Let $G$ be a finite group. Over any finite $G$-poset $\mathcal {P}$ we may define a transporter category $G\propto \mathcal {P}$ as the corresponding Grothendieck construction. There exists a Becker-Gottlieb transfer from the ordinary cohomology of $G\propto \mathcal {P}$ to that of $G$. We shall construct it using module-theoretic methods and then extend it to a transfer from the Hochschild cohomology of $k(G\propto \mathcal {P})$ to that of $kG$, where $k$ is a base field.References
- J. C. Becker and D. H. Gottlieb, Transfer maps for fibrations and duality, Compositio Math. 33 (1976), no. 2, 107–133. MR 436137
- D. J. Benson, Representations and cohomology. I, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1998. Basic representation theory of finite groups and associative algebras. MR 1644252
- 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
- W. G. Dwyer, Homology decompositions for classifying spaces of finite groups, Topology 36 (1997), no. 4, 783–804. MR 1432421, DOI 10.1016/S0040-9383(96)00031-6
- William G. Dwyer and Hans-Werner Henn, Homotopy theoretic methods in group cohomology, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2001. MR 1926776, DOI 10.1007/978-3-0348-8356-6
- W. G. Dwyer and C. W. Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Ann. of Math. (2) 139 (1994), no. 2, 395–442. MR 1274096, DOI 10.2307/2946585
- P. J. Hilton and U. Stammbach, A course in homological algebra, 2nd ed., Graduate Texts in Mathematics, vol. 4, Springer-Verlag, New York, 1997. MR 1438546, DOI 10.1007/978-1-4419-8566-8
- Markus Linckelmann, Transfer in Hochschild cohomology of blocks of finite groups, Algebr. Represent. Theory 2 (1999), no. 2, 107–135. MR 1702272, DOI 10.1023/A:1009979222100
- Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
- Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129
- Mark A. Ronan and Stephen D. Smith, Sheaves on buildings and modular representations of Chevalley groups, J. Algebra 96 (1985), no. 2, 319–346. MR 810532, DOI 10.1016/0021-8693(85)90013-4
- Daniel E. Swenson, The Steinberg complex of an arbitrary finite group in arbitrary positive characteristic, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–University of Minnesota. MR 2713697
- Peter Webb, An introduction to the representations and cohomology of categories, Group representation theory, EPFL Press, Lausanne, 2007, pp. 149–173. MR 2336640
- Fei Xu, Hochschild and ordinary cohomology rings of small categories, Adv. Math. 219 (2008), no. 6, 1872–1893. MR 2455628, DOI 10.1016/j.aim.2008.07.014
- Fei Xu, Tensor structure on $k\scr C$-mod and cohomology, Proc. Edinb. Math. Soc. (2) 56 (2013), no. 1, 349–370. MR 3021416, DOI 10.1017/S0013091512000107
- Fei Xu, On local categories of finite groups, Math. Z. 272 (2012), no. 3-4, 1023–1036. MR 2995153, DOI 10.1007/s00209-011-0971-y
Bibliographic Information
- Fei Xu
- Affiliation: Department of Mathematics, Shantou University, Shantou, Guangdong 515063, People’s Republic of China
- Email: fxu@stu.edu.cn
- Received by editor(s): April 3, 2012
- Received by editor(s) in revised form: August 11, 2012
- Published electronically: April 15, 2014
- Additional Notes: The author \CJK*{UTF8} \CJKtilde\CJKfamily{gbsn}(å¾ æ) \endCJK* was supported in part by a Beatriu de Pinós research fellowship from the government of Catalonia of Spain.
- Communicated by: Pham Huu Tiep
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 2593-2608
- MSC (2010): Primary 20C05; Secondary 20J99
- DOI: https://doi.org/10.1090/S0002-9939-2014-12013-2
- MathSciNet review: 3209315