BeckerGottlieb transfer for Hochschild cohomology
Author:
Fei Xu
Journal:
Proc. Amer. Math. Soc. 142 (2014), 25932608
MSC (2010):
Primary 20C05; Secondary 20J99
Published electronically:
April 15, 2014
MathSciNet review:
3209315
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Abstract: Let be a finite group. Over any finite poset we may define a transporter category as the corresponding Grothendieck construction. There exists a BeckerGottlieb transfer from the ordinary cohomology of to that of . We shall construct it using moduletheoretic methods and then extend it to a transfer from the Hochschild cohomology of to that of , where is a base field.
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 [1]
 J. C. Becker and D. H. Gottlieb, Transfer maps for fibrations and duality, Compositio Math. 33 (1976), no. 2, 107133. MR 0436137 (55 #9087)
 [2]
 D. J. Benson, Representations and cohomology. I. Basic representation theory of finite groups and associative algebras, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1998. MR 1644252 (99f:20001a)
 [3]
 D. J. Benson, Representations and cohomology. II. Cohomology of groups and modules, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, Cambridge, 1998. MR 1634407 (99f:20001b)
 [4]
 W. G. Dwyer, Homology decompositions for classifying spaces of finite groups, Topology 36 (1997), no. 4, 783804. MR 1432421 (97m:55016), http://dx.doi.org/10.1016/S00409383(96)000316
 [5]
 William G. Dwyer and HansWerner Henn, Homotopy theoretic methods in group cohomology, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2001. MR 1926776 (2003h:20093)
 [6]
 W. G. Dwyer and C. W. Wilkerson, Homotopy fixedpoint methods for Lie groups and finite loop spaces, Ann. of Math. (2) 139 (1994), no. 2, 395442. MR 1274096 (95e:55019), http://dx.doi.org/10.2307/2946585
 [7]
 P. J. Hilton and U. Stammbach, A course in homological algebra, 2nd ed., Graduate Texts in Mathematics, vol. 4, SpringerVerlag, New York, 1997. MR 1438546 (97k:18001)
 [8]
 Markus Linckelmann, Transfer in Hochschild cohomology of blocks of finite groups, Algebr. Represent. Theory 2 (1999), no. 2, 107135. MR 1702272 (2000h:20024), http://dx.doi.org/10.1023/A:1009979222100
 [9]
 Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, SpringerVerlag, New York, 1998. MR 1712872 (2001j:18001)
 [10]
 Daniel Quillen, Higher algebraic theory. I, Algebraic theory, I: Higher theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85147. MR 0338129 (49 #2895)
 [11]
 Mark A. Ronan and Stephen D. Smith, Sheaves on buildings and modular representations of Chevalley groups, J. Algebra 96 (1985), no. 2, 319346. MR 810532 (87h:20087), http://dx.doi.org/10.1016/00218693(85)900134
 [12]
 Daniel E. Swenson, The Steinberg complex of an arbitrary finite group in arbitrary positive characteristic, Thesis (Ph.D.)University of Minnesota, 2009, ProQuest LLC, Ann Arbor, MI. MR 2713697
 [13]
 Peter Webb, An introduction to the representations and cohomology of categories, Group representation theory, EPFL Press, Lausanne, 2007, pp. 149173. MR 2336640 (2008f:18013)
 [14]
 Fei Xu, Hochschild and ordinary cohomology rings of small categories, Adv. Math. 219 (2008), no. 6, 18721893. MR 2455628 (2009h:18025), http://dx.doi.org/10.1016/j.aim.2008.07.014
 [15]
 F. Xu, Tensor structure on mod and cohomology, Proc. Edinb. Math. Soc. (2) 56 (2013), no. 1, 349370. MR 3021416
 [16]
 Fei Xu, On local categories of finite groups, Math. Z. 272 (2012), no. 34, 10231036. MR 2995153, http://dx.doi.org/10.1007/s002090110971y
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Additional Information
Fei Xu
Affiliation:
Department of Mathematics, Shantou University, Shantou, Guangdong 515063, People’s Republic of China
Email:
fxu@stu.edu.cn
DOI:
http://dx.doi.org/10.1090/S000299392014120132
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 \begin{CJK*}UTF8 \CJKtilde\CJKfamily{gbsn}(徐 斐) \end{CJK*} was supported in part by a Beatriu de Pinós research fellowship from the government of Catalonia of Spain.
Communicated by:
Pham Huu Tiep
Article copyright:
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