# Hopf Comodule Decompositions and Coinvariant Elements

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.

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I don't see much of a reason why they should be equivalent -- not every comodule needs to have coinvariants! Maybe your $V$ is a Hopf module? –  darij grinberg Mar 24 '12 at 16:10
If that helps, then yes assume it a Hopf module. –  Abtan Massini Mar 24 '12 at 16:35
There is a theorem that Hopf modules are (up to isomorphism) the tensor products of their coinvariants with $H$. (This theorem is usually worded in terms of a category equivalence. I don't know a good reference.) I guess it will help here. –  darij grinberg Mar 24 '12 at 16:57
I still don't believe Statements I and II are equivalent. Just take $V^+$ to be the subspace of covariants, and $V^-$ to be an arbitrary vector-space complement to $V^+$. Then, I holds but II does not (in general.) Please be more open about the context where your question comes from. –  darij grinberg Mar 24 '12 at 22:46