is the kernel of homorphism of finite group schemes once again finite?
(i.e. if you have $G = Spec A, H = Spec B$ over a field $k$, i.e. $A,B$ finite dimensional $k$-vector spaces, is ker$\varphi$ once again finite?)
In my opinion, this should be equivalent to asking whether the $k$-algebra given by $A/I_BA$, where $I_B$ is the augmentation ideal of $B$, is finite dimensional as a $k$-vector space. Is this true? I don't seem to get anywhere with this :)