Splittings of bicommutative Hopf algebras.
Let \(A\to B\to C\) be a short exact sequence in the category of bicommutative biassociative Hopf algebras with an exhaustive primitive filtration. If \(A\) is cofree as a coalgebra and \(C\) is polynomial (free) as an algebra, then the sequence splits as Hopf algebras. This is the main result of this paper, and its proof relies heavily on the theory of Morava Dieudonné modules. A Morava Dieudonné module is a \(\mathbb{Z}/(p^n-1)\) graded group \(M_*\) with a \(\mathbb{Z}[V,F]/(VF-p)\)-module structure where \(F\colon M_t\to M_{pt}\) and \(V\colon M_t\to M_{t/p}\). A morphism \(M_*\to N_*\) is a graded module map, hence the set of Morava Dieudonné modules forms a category. The full subcategory consisting of modules with \(V\)-torsion is equivalent to the category of Morava (i.e., evenly graded) Hopf algebras via a functor \(m_*\colon(\text{Morava Hopf algebras})\to(\text{Morava}\) Dieudonné modules with \(V\)-torsion). The main result is thus proved by translating the problem into these Dieudonné modules. An application, in fact the motivation for this study, is found in the study of the fibration \(K(\mathbb{Z},3)\to BO\langle 8\rangle\to BSpin\), leading to the study of the short exact sequence obtained by applying \(K(2) _*\) to each part, where \(K(2)_*\) is the homology theory known as Morava \(K\)-theory. The paper concludes by showing that the group of homomorphisms \(m_*(K(2)_*BSpin)\to K\) is a nontrivial \(2\)-complete torsion-free \(\mathbb{Z}_2\)-module, and hence the space of splittings for the above short exact sequence is uncountable.
scientific article; zbMATH DE number 2116175