The cohomology algebras of finite-dimensional Hopf algebras

Abstract

The cohomology algebra of a finite dimensional graded connected cocommutative biassociative Hopf algebra over a field K K is shown to be a finitely generated K 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 mod ⁡ 2 \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 mod ⁡ 2 \operatorname {mod} 2 Steenrod algebra is given.