For $H$ an Hopf algebra, and $V$ a right $H$-comodule with coaction $\Delta_R$, for which there exists a vector space decomposition $V=V_+ \oplus V_-$. Are the following two statements equivalent?

**Statement I**: For any element of $V$, with decomposition $v = v_+ + v_-$, we have $v$ coinvariant, if and only if both $v_{+}$ and $v_{-}$ are both coinvariant, ie
$$
\Delta_R(v) = v \otimes 1 \Longleftrightarrow \Delta_R(v_{\pm}) = v_{\pm} \otimes 1
$$

and

**Statement II**: The decomposition $v = V_+ \oplus V_-$ is a decomposition of comodules, ie
$$
\Delta_R(V_+) \subseteq V_+ \otimes H, ~~~~~~ \Delta_R(V_-) \subseteq V_- \otimes H
$$

That II implies I is easy, but I can't see what's happening in the other direction.

shouldbe equivalent -- not every comodule needs to have coinvariants! Maybe your $V$ is aHopfmodule? – darij grinberg Mar 24 '12 at 16:10