If $H$ is a finite-dimensional Hopf algebra, then the antipode is bijective?

Question

If I’m given a finite-dimensional Hopf algebra $H$, how do I show the antipode is bijective? It's obvious that if we prove either injective or surjective, we get the other one for free since $H$ is finite-dimensional. Can someone nudge me in the right direction?

Answer 1

This is a basic fact in the theory of Hopf algebras. It is a corollary of the fundamental theorem of Hopf modules. Reference : Montgomery Susan, Hopf algebras and their actions on rings, Theorem 2.1.3.