A note on the cohomology of finite-dimensional cocommutative Hopf algebras
type: Item
instance of: scholarly article
title: A note on the cohomology of finite-dimensional cocommutative Hopf algebras
zbMATH Open document ID: 0873.16005
author: John H. Palmieri
published in: Journal of Algebra
publication date: 1997-10-27T00:00:00Z
review text: 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.
reviewed by: Jörg Feldvoss
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 57T05
Mathematics Subject Classification ID: 55S10
zbMATH DE Number: 1013921
zbMATH Keywords: cohomology of cocommutative Hopf algebras
zbMATH Keywords: finite-dimensional graded cocommutative Hopf algebras
zbMATH Keywords: Quillen-Venkov detection theorem
zbMATH Keywords: Chouinard's projectivity criterion
zbMATH Keywords: Quillen's stratification theorem
zbMATH Keywords: connected quasi-elementary Hopf algebras
zbMATH Keywords: restricted universal enveloping algebras
zbMATH Keywords: polynomial algebras
zbMATH Keywords: Steenrod algebras
MaRDI profile type: Publication
full work available at URL: jabr.1996.6812
OpenAlex ID: W2078219639
cites work: Modules over the Steenrod algebra
cites work: Q3997902
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DOI: 10.1006/JABR.1996.6812
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description: scientific article; zbMATH DE number 1013921