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David Lampert
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Let $S$ be the local ring of nodal curve, $R$ = inverse limit $Frob: S \to S$. For example:

  • $k$ a perfect field,
  • $f(x,y)=y^{p+1}-x^{p+1}(1+x)$,
  • $R=k[x^{1/p^{\infty}},y^{1/p^{\infty}}]/(f^{1/p^{\infty}})$.

Here's a complete local example:

  • $k$ a perfect field,
  • $f(x,y)=y^{p}-x^{p}y-x^{p+1}$,
  • $R=k[[x^{1/p^{\infty}},y^{1/p^{\infty}}]]/(f^{1/p^{\infty}})$.

In each example $(y/x)$ is integral over $R$.

David Lampert
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