# $H$-Hopf modules equal the tensor products of their coinvariants with H

In a comment for this old question, it was said that

> 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'm guessing that this means that given a (right) $H$-comodule $V$, for $H$ a Hopf algebra, equipped with a right $H$ action for which $\Delta_R(vh) = v_{(0)} g_{(0)} \otimes v_{(1)}g_{(1)}$, we have an isomorphism $$V \simeq V_{\text{inv}} \otimes H$$ Now it's easy to see that we have a surjective map $$V_{\text{inv}} \otimes H \to V, ~~~ v \otimes h \mapsto vh.$$ How does one show that this is an isomorphism?

EDIT: Wait, I think this is actually obvious: The map $$V \to V_{\text{inv}} \otimes H, ~~~~~ v \mapsto v_{(0)} S(v_{(1)}) \otimes v_{(2)},$$ seems to have the multiplication map as its inverse. So this gives us the isomorphism. Yes?

• A good reference is Sweedler's book on Hopf algebras. Jul 19 '12 at 20:55
• Havce you tried computing the two compositions of the maps you wrote in the question and seeing if they are identities? Jul 19 '12 at 20:57