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Let $H$ be a finite dimensional Hopf algebra, $R$ be a $H$-module algebra and $V$ be a finite dimensional $H$-module such that $R\otimes_{k} V$ is a finitely generated $R$ module under the action: $r.(r'\otimes v)= rr'\otimes v$. Then, can we further show that $R\otimes_{k} V$ is also free as an $R$-module? If needed, please assume that $R$ is Noetherian.

P.S: I have posted this question in StackExchange but since I have not received any reply yet, I am posting it here.

Thanks in advance!

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    $\begingroup$ If one writes $V \cong \oplus k$ as $k$-vector spaces, then, isn't $R \otimes_k V \cong \oplus (R \otimes_k k) \cong \oplus R$ an isomorphism of $R$-modules under the given action of $R$ ? $\endgroup$
    – tj_
    Commented May 21, 2018 at 3:19
  • $\begingroup$ Yes, thanks! I got lost in the H- module stuff and missed this simple thing! $\endgroup$
    – Sam
    Commented May 21, 2018 at 14:55

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