The cohomology algebras of finite-dimensional Hopf algebras

Wilkerson, Clarence

doi:10.1090/S0002-9947-1981-0597872-X

The cohomology algebras of finite-dimensional Hopf algebras
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by Clarence Wilkerson

Trans. Amer. Math. Soc. 264 (1981), 137-150

DOI: https://doi.org/10.1090/S0002-9947-1981-0597872-X

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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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