Commutative Hopf algebras and cocommutative Hopf algebras in positive characteristic

Abstract

Let A be a commutative Hopf algebra over a field k of characteristic p > 0. Let ϑ: C → B be a surjective map of commutative algebras such that x p = 0 for any x in Ker(ϑ), so that the map F c:k 1 p ⊗ C → C, λ ⊗ a ↦ λ pa p factors through k 1 p ⊗ ϑ , yielding a map F C B: k 1 p ⊗ B → C . A map of algebras f: A → B can be lifted to an algebra map \\ ̄ tf: A → C such that f = ϑ ∘ \\ ̄ tf if and only if Ker(F A)⊂ Ker(F C B ∘ (k 1 p ⊗ f)) . In particular, if F A is injective, any algebra map A → B can be lifted to A → C. The dual results will be given for cocommutative Hopf algebras and coalgebra maps.