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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is shown to be a finitely generated <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mod 2"> <mml:semantics> <mml:mrow> <mml:mi>mod</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {mod} 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Steenrod algebra is given. As an application, an explicit formula for the Krull dimension of the cohomology algebras of the finite skeletons of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mod 2"> <mml:semantics> <mml:mrow> <mml:mi>mod</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {mod} 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Steenrod algebra is given.