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Transactions of the American Mathematical Society
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
MathSciNet review: 597872
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0597872-X
PII: S 0002-9947(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