## When is a Surjective Comodule Endomorphism an Automorphism?

Given a Hopf algebra $H$, a left $H$-comodule $V$, and a surjective comodule endomorphism $f: V \to V$. Can somebody give:

(i) a set of neccessary, or sufficient, or both neccessary and sufficient, conditions for $f$ to have zero kernel?

(ii) an example of such a comodule map with non-zero kernel?

For (ii), take $H=k$, your base field, $V$ any infinite dimensional $k$-vector space, with its obvious $H$-comodule structure, and $f$ any non-injective surjection – Mariano Suárez-Alvarez Aug 22 at 17:35