Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The cohomology algebras of finite-dimensional Hopf algebras


Author: Clarence Wilkerson
Journal: Trans. Amer. Math. Soc. 264 (1981), 137-150
MSC: Primary 16A61; Secondary 16A24, 57T05
DOI: https://doi.org/10.1090/S0002-9947-1981-0597872-X
MathSciNet review: 597872
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The cohomology algebra of a finite dimensional graded connected cocommutative biassociative Hopf algebra over a field $ K$ is shown to be a finitely generated $ K$-algebra. Counterexamples to the analogue of a result of Quillen (that nonnilpotent cohomology classes should have nonzero restriction to some abelian sub-Hopf algebra) are constructed, but an elementary proof of the validity of this "detection principle" for the special case of finite sub-Hopf algebras of the $ \operatorname{mod} 2$ Steenrod algebra is given. As an application, an explicit formula for the Krull dimension of the cohomology algebras of the finite skeletons of the $ \operatorname{mod} 2$ Steenrod algebra is given.


References [Enhancements On Off] (What's this?)

  • [1] J. F. Adams, On the nonexistence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20-104. MR 0141119 (25:4530)
  • [2] J. F. Adams and H. R. Margolis, Sub-Hopf-algebras of the Steenrod algebra, Proc. Cambridge Philos. Soc. 76 (1974), 45-52. MR 0341487 (49:6238)
  • [3] D. Anderson and D. Davis, A vanishing theorem in homological algebra, Comment. Math. Helv. 48 (1973), 318-327. MR 0334207 (48:12526)
  • [4] N. Bourbaki, Commutative algebra, Elements of Mathematics, Addison-Wesley, Reading, Mass., and Hermann, Paris, 1972.
  • [5] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1956. MR 0077480 (17:1040e)
  • [6] L. Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961), 224-239. MR 0137742 (25:1191)
  • [7] E. Golod, The cohomology ring on a finite $ p$-group, Dokl. Akad. Nauk SSSR 125 (1959), 703-706. (Russian) MR 0104720 (21:3473)
  • [8] D. Kraines, Massey higher products, Trans. Amer. Math. Soc. 124 (1966), 431-449. MR 0202136 (34:2010)
  • [9] W. H. Lin, Cohomology of sub-Hopf-algebras of the Steenrod algebra. I; II, J. Pure Appl. Algebra (2) 10 (1977), 101-114; 11 (1977), 105-110. MR 0454975 (56:13217)
  • [10] A. Liulevicius, The factorization of cyclic reduced powers by secondary operations, Mem. Amer. Math. Soc. No. 42 (1962). MR 0182001 (31:6226)
  • [11] -, Notes on homotopy of Thom spectra, Amer. J. Math. (1) 86 (1964), 1-16. MR 0166787 (29:4060)
  • [12] C. Löfwall, Une algèbre nilpotente dont la série de Poincaré est non rationelle, C. R. Acad. Sci. Paris Sér. A (5) 288 (1979), 327-330. MR 526127 (80e:16016)
  • [13] H. Matsumura, Commutative algebra, Benjamin, New York, 1970. MR 0266911 (42:1813)
  • [14] J. P. May, A general algebraic approach to Steenrod operations, Lecture Notes in Math., vol. 168, Springer-Verlag, Berlin and New York, 1970, pp. 153-231. MR 0281196 (43:6915)
  • [15] J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211-264. MR 0174052 (30:4259)
  • [16] D. Quillen, The spectrum of an equivariant cohomology ring. I; II, Ann. of Math. (2) 94 (1971), 549-572; 573-602. MR 0298694 (45:7743)
  • [17] D. Quillen and B. B. Venkov, Cohomology of finite groups and elementary abelian subgroups, Topology 11 (1972), 317-318. MR 0294506 (45:3576)
  • [18] J. P. Serre, Sur la dimension cohomologique des groupes profinis, Topology 3 (1965), 413-420. MR 0180619 (31:4853)
  • [19] J. Stallings, Homology and central series of groups, J. Algebra 2 (1965), 170-181. MR 0175956 (31:232)
  • [20] R. G. Swan, The nontriviality of the restriction map in the cohomology ring of groups, Proc. Amer. Math. Soc. 11 (1960), 885-887. MR 0124050 (23:A1370)
  • [21] B. B. Venkov, Cohomology algebras for some classifying spaces, Dokl. Akad. Nauk SSSR 127 (1959), 943-944. (Russian) MR 0108788 (21:7500)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16A61, 16A24, 57T05

Retrieve articles in all journals with MSC: 16A61, 16A24, 57T05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0597872-X
Keywords: Cohomology of Hopf algebras, Steenrod operations, Steenrod algebra, spectral sequences, cohomology of classifying spaces
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society