A note on the cohomology of finite-dimensional cocommutative Hopf algebras
The author introduces the concept of a quasi-elementary sub-Hopf algebra of a finite-dimensional graded cocommutative Hopf algebra over a field of prime characteristic. Then he gives three cohomological results which show that in this context quasi-elementary Hopf algebras are the appropriate generalization of elementary abelian groups. The first result is an analogue of Carlson's generalization of the Quillen-Venkov detection theorem which reformulates and extends a result of \textit{C. Wilkerson} [Trans. Am. Math. Soc. 264, 137-150 (1981; Zbl 0465.55010)]; the second result is a related analogue of Chouinard's projectivity criterion. Finally, the third result is an analogue of Quillen's stratification theorem which is only proved in the connected case, but is conjectured by the author to be true in general. The first two results are proved by imitating the proofs in group theory. In contrast, the third result is deduced from the work of Hopkins and Smith on nilpotence and stable homotopy theory, and therefore its proof heavily uses the connectedness hypothesis. The paper also contains an appendix in which the author gives a slightly different characterization of connected quasi-elementary Hopf algebras, and (at least implicitly) examples of quasi-elementary sub-Hopf algebras in restricted universal enveloping algebras, mod \(p\) reduced polynomial algebras and mod \(p\) Steenrod algebras.
cohomology of cocommutative Hopf algebras
finite-dimensional graded cocommutative Hopf algebras
connected quasi-elementary Hopf algebras
restricted universal enveloping algebras
scientific article; zbMATH DE number 1013921
scientific article