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I am actually going through "Twenty four hours of local cohomology". They discuss the Frobenius endomorphism in Chapter 21. Here is the Exercise 21.6 which I am finding hard to solve:

Find a ring $R$ of characteristic $p$ with the property that its Frobenius endomorphism $\varphi:R\rightarrow R$ is not flat but $R$ is flat over $\varphi(R)$.

The Frobenius endomorphism $\varphi:R\rightarrow R$ is the map $\varphi(r)=r^p$.

I am playing around with the ring $k[\![x^2,xy,y^2]\!]$ for which I know the frobenius endomorphism is not flat, but somehow i am not able to prove that $R$ is flat over $\varphi(R)$.

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    $\begingroup$ If $R$ has no nilpotent elements, then $\varphi$ is injective, and hence it is flat if and only if $R$ is flat over $\varphi(R)$. $\endgroup$
    – Piotr Achinger
    Commented Dec 6, 2021 at 7:25
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    $\begingroup$ Have you tried $R=k[x]/(x^2)$? $\endgroup$
    – Matthieu Romagny
    Commented Dec 6, 2021 at 7:49

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