I am studying Kunz theorem using the notes by Karen smith, in the forward part regularity implies flatness, i am having difficulty verifying that $K[[x_1^{1/p},…,x_n^{1/p}]]$ is free over $K[[x_1,…,x_n]]$. Here $R$ is a complete local ring of characteristic $p$ and $R = K[[x_1,…,x_n]]$. How to see this naturally?
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The $\prod_{i=1}^n x_i^{a_i/p}$ with $a_i\in 0\ldots p-1$ are a free $R$-module basis (rank $p^n$). – reuns Mar 18 '23 at 23:18
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Can u elaborate on that, because i tried to prove that this set is linearly independent over $K[[x_1,...,x_n]]$ and i am getting a messy equation. @reuns – Yossarain Mar 18 '23 at 23:44
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You shouldn't get anything messy, for $a'\in {0\ldots p-1}^n$ the sub-module generated by the $x^a, a\ne a'$ has all its $x^{a'+p b}$ coefficients $=0$. – reuns Mar 19 '23 at 00:58