On the square of the antipode in a connected filtered Hopf algebra (Q6585124)

\textit{M. Aguiar} and \textit{A. Lauve} [Algebra Number Theory 9, No. 3, 547--583 (2015; Zbl 1330.16020)] proved that in a graded an connected Hopf algebra \(H\) of antipode \(S\), for any \(n\geq 1\),\N\[\N(S^2-Id_H)^n(H_n)=(0).\N\]\NThis result is here improved in several directions: firstly, by replacing the base field by a commutative ring; secondly, by weakening the coassociativity condition; then, by replacing \(S\) and \(Id_H\) be other coalgebra morphisms. It is then proved that the exponent \(n\) can be lowered to \(n-1\) for any \(n\geq 2\) if and only if it can be done for \(n=2\), which happens for example if the elements of \(H_1\) commute.