A Note on the Cohomology of Finite-Dimensional Cocommutative Hopf Algebras

In the context of finite-dimensional cocommutative Hopf algebras, we prove versions of various group cohomology results: the Quillen–Venkov theorem on detecting nilpotence in group cohomology, Chouinard's theorem on determining whether akG-module is projective by restricting to elementary abelianp-subgroups ofG, and Quillen's theorem which identifies the cohomology ofG, “modulo nilpotent elements.” This last result is only proved for graded connected Hopf algebras.