# Example of existence of small CM algebra in mixed characteristic

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.

• 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 $p$. Dec 15 '20 at 0:29
• 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. Dec 15 '20 at 1:38
• 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. Dec 15 '20 at 12:02
• 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. Dec 15 '20 at 12:28
• @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. Dec 15 '20 at 17:21