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A Noetherian local ring $R$ admits a small Cohen-Macaulay (CM) algebra if there is an injective map of rings $R\hookrightarrow S$ such that every system of parameters of $R$ becomes a regular sequence in $S$ and $S$ is a finite $R$-module.

If $R$ is a non CM normal domain containing the rationals, then $R$ cannot admit a small CM algebra. This is because there exists a retraction from $S\rightarrow R$ using the trace map corresponding to the fraction fields.

My question: Is an example of the failure of this non existence known in mixed characteristic $p>0$? More precisely, is there a concrete example of a Noetherian local non CM normal domain $R$ of mixed characteristic $p>0$ such that $R$ admits a small CM algebra? Thank you.

Please note: I am a graduate student and would like to know if such examples are already known. I believe I am able to construct examples but I am not sure whether this is new.

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    $\begingroup$ If you take some mixed characteristic domain $R$ such that $R[1/p]$ is a non-CM normal domain, then it seems there cannot be a small CM $R$-algebra simply because there isn't one after inverting $p$. $\endgroup$
    – Anonymous
    Dec 15 '20 at 0:29
  • $\begingroup$ True, in such a case $R$ does not admit a small CM algebra. However, $R[1/p]$ could be CM : would you happen to know if examples in this situation exist? Thanks for taking the time. $\endgroup$
    – Phoenix
    Dec 15 '20 at 1:38
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    $\begingroup$ You may already know this: There is a paper by Bhatt which gives examples not admitting a small CM algebra in equal characteristic $p$: arxiv.org/abs/1207.5413. $\endgroup$ Dec 15 '20 at 12:02
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    $\begingroup$ Also, there are some positive results in equal characteristic $p$: e.g. arxiv.org/abs/1911.05335. I'm not aware of results in this direction in mixed characteristic off the top of my head. $\endgroup$ Dec 15 '20 at 12:28
  • $\begingroup$ @AxelStäbler Thanks for your comments. Yes, Bhatt's paper shows it goes both ways in equal characteristic $p$. I think the paper from Schoutens seems to show the existence of small CM modules (not algebras) in certain cases of equal char $p$. I was thinking about the existence of small CM modules in a certain situation in mixed characteristic and could answer it in the positive. in some cases it turns out it even admits a small CM algebra. $\endgroup$
    – Phoenix
    Dec 15 '20 at 17:21
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Example 3.2 in this paper provides an example of a non Cohen-Macaulay normal domain of mixed characteristic that admits a small Cohen-Macaulay algebra.

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