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

Blockquote

> 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?

2 added 246 characters in body

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

Blockquote 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?

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

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

Blockquote 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?