I am working on some algebras over complete regular local algebras. But I am not sure whether such rings are worth to study. I am looking for some examples of these algebras. Let $(R,\mathfrak{m})$ be a complete regular local ring. Is there any (nontrivial) example of an $R$-algebra $\Lambda$ which is finitely generated and maximal Cohen-Macaulay as an $R$-module, with the property that every $\Lambda$-module which is maximal Cohen-Macaulay as an $\widehat{R}$-module is locally projective on the puctured spectrum of $R$? More gemerally, Let $R$ be a Henselian regular local ring. Let $\Lambda$ be an $R$-algebra with infinite global dimension which is finitely generated and CM as an $R$-module. I am looking for an example of such algebra with the property that every $\widehat{\Lambda}$-module which is $MCM$ as an $\widehat{R}$-module is locally projective on the punctured spedtrum of $R$.Is there any axample of such algebra where $R$ is not complete?
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2$\begingroup$ Are you considering some version of "complete Cohen-Macaulay" that applies in the non-Noetherian case? If your local ring is Noetherian of finite global dimension, then it is regular. Is there a reason that you do not just say that $R$ is regular? $\endgroup$– Jason StarrCommented Jul 30, 2017 at 19:38
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$\begingroup$ You are right. I added the global dimension condition in the final step and I didn't notice that this ring is a regular local ring and o it is CM. But can we give an example of an algebra as above with infinite global dimension? I mean, I am looking for an algebra with the above properties with infinite global dimension over a complete regular local ring. $\endgroup$– Homa81Commented Jul 31, 2017 at 5:56
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