# 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?

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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 '12 at 17:35
For (i), at the level of generality with which you wrote the question it is difficult to say something useful. One nice criterion for injectivity is that it is enough to check injectivity of the restriction of the map to the socle. –  Mariano Suárez-Alvarez Aug 22 '12 at 17:46
How do you define the scole of a comodule, as The sum of the minimal nonzero sub-comodules? –  Dyke Acland Aug 22 '12 at 18:22