Coconnected Hopf algebras give fixed vectors in every nonzero comodule

For $r\ge 0$ put $V_r=\{v\in V:\rho (v)\in V{\otimes }_k{C}_r\}$; these are $k$-linear subspaces of $V$ with $V_r\subseteq V_{r+1}$ and, by [F1] and [F2], ${\bigcup }_r{V}_r=V$. The space $V_0$ consists exactly of the fixed vectors: if $\rho (v)\in V\otimes C_0=V\otimes k\cdot 1$ then $\rho (v)=w\otimes 1$ for some $w$, and applying $\mathrm{id}\otimes \epsilon$ gives $v=w$ by the counit identity, so $\rho (v)=v\otimes 1$; conversely a fixed vector lies in $V_0$.

I claim that $V_r=0$ implies $V_{r+1}=0$ whenever $r\ge 0$. Let $\pi :A\to A\mathrm{/}{C}_r$ be the quotient map. If $v\in V_{r+1}$, then $\rho (v)\in V\otimes C_{r+1}$, and by [F1] every element of $C_{r+1}$ maps to zero under $\pi \otimes \pi$ applied to $\mathrm{\Delta }$, because $\mathrm{\Delta }(C_{r+1})\subseteq C_r\otimes A+A\otimes C_r$; hence $(\mathrm{id}\otimes \pi \otimes \pi )(\mathrm{id}\otimes \mathrm{\Delta })\rho (v)=0$. By coassociativity [F2] this is $(\mathrm{id}\otimes \pi \otimes \pi )(\rho \otimes \mathrm{id})\rho (v)=0$. Choose a finite expansion $(\mathrm{id}\otimes \pi )\rho (v)={\sum }_i{v}_i\otimes {\bar{a}}_i$ with the $v_i$ linearly independent, by taking a finite basis of the span of the first factors of any tensor expansion; then the last identity reads ${\sum }_i(\mathrm{id}\otimes \pi )\rho (v_i)\otimes {\bar{a}}_i=0$. The map $(\mathrm{id}\otimes \pi )\rho$ is injective on $V$ when $V_r=0$, since its kernel is exactly $V_r$; therefore the elements $(\mathrm{id}\otimes \pi )\rho (v_i)$ are linearly independent, so each ${\bar{a}}_i=0$, that is, $(\mathrm{id}\otimes \pi )\rho (v)=0$ and hence $\rho (v)\in V\otimes C_r$, i.e. $v\in V_r=0$. Thus $V_{r+1}=0$.

Since $V\mathrm{\ne }0$, [F2] gives an element $v\mathrm{\ne }0$ with $\rho (v)\in V\otimes C_r$ for some $r$, so $V_r\mathrm{\ne }0$. Iterating [step 2.1] downwards, $V_0\mathrm{\ne }0$: if $V_0=0$ then $V_1=0$, then $V_2=0$, and by induction $V_r=0$ for all $r$, contradicting $V_r\mathrm{\ne }0$. By [step 1.1] any nonzero element of $V_0$ is a nonzero fixed vector, which proves the first assertion. The group-theoretic form follows from the comodule dictionary [F3], since for $G$ with $O(G)=A$ the rational representations of $G$ are exactly the $A$-comodules and fixed vectors are the elements with coaction $v\otimes 1$.