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Let $R \subseteq S$ be local rings with maximal ideals $m_R$ and $m_S$. Assume that:

(1) $R$ and $S$ are (Noetherian) integral domains.

(2) $\dim(R)=\dim(S) < \infty$, where $\dim$ is the Krull dimension.

(3) $R$ is regular (hence a UFD).

(4) $S$ is Cohen-Macaulay.

(5) $R \subseteq S$ is simple, namely, $S=R[w]$ for some $w \in S$.

(6) $R \subseteq S$ is free.

(7) $R \subseteq S$ is integral, namely, every $s \in S$ satisfies a monic polynomial over $R$.

(8) $m_RS=m_S$, namely, the extension of $m_R$ to $S$ is $m_S$.

(9) It is not known whether the fields of fractions of $R$ and $S$, $Q(R)$ and $Q(S)$, are equal or not.

(10) It is not known if $R \subseteq S$ is separable or not.

Remark: It is known that if a (commutative) integral domains ring extension $A \subseteq B$ is integral+flat, then it is faithfully flat, and if also $Q(A)=Q(B)$, then $A=B$. This is why I did not want to assume that $Q(R)=Q(S)$, since in this case $R=S$ immediately.

Question: Is it true that, assuming (1)-(10) imply that $S$ is regular or $R=S$?

Example: $R=\mathbb{C}[x(x-1)]_{x(x-1)}$ and $S=\mathbb{C}[x]_{(x)}$, with $R \neq S$ and $S$ is regular.

Non-example: $R=\mathbb{C}[x^2]_{(x^2)}$ and $S=\mathbb{C}[x^2,x^3]_{(x^2,x^3)}$, but condition (8) is not satisfied.

Relevant questions, for example: a, b, c, d.

Thank you very much! I have asked the above question here, with no comments (yet).

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Condition (8) implies that the map is unramified, and since it is also flat by (6) (which by the way implies $S$ is CM by Miracle Flatness), it is etale, and so $S$ is regular. Note you don't need (5) here.

Without condition (8) you can take $R=\mathbf{C}[[x,y]]$ and $S=R[w]/(w^2 - xy)$.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – Stefan Kohl
    Apr 22, 2021 at 7:59
  • $\begingroup$ With the help of the following resources, things are more clear for me now, so there is no need to further discuss my above question, unless you wish to add something, then you are welcome; thank you for your help! stacks.math.columbia.edu/tag/024L jstor.org/stable/2372926?seq=1 $\endgroup$
    – user237522
    Apr 23, 2021 at 2:14

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