Take the 2-minute tour ×  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 $$ \Delta_R(v) = v \otimes 1 \Longleftrightarrow
\Delta_R(v_{\pm}) = v_{\pm} \otimes 1 $$   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.
share|improve this question 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