Let $R$ and $S$ be 2 associated, commutative, and unita $k$-algebras where $k$ is an algebraically closed field of characteristic $p$. We call these algebras inseparable isogeny or $F$-isomorphism if there is some $r>0$ such that $R^{p^r}\subseteq S\subseteq R$ where $R^{p^r}=\{x^{p^r}\in R~:~x\in R\}$. Is it true that if $R$ is Cohen-Macaulay then so is $S$, or vice versa? If not, what conditions give us this? I could not find any reference on this problem. I would greatly appreciate if anyone could give me some related book or paper. Thanks in advance.
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2$\begingroup$ In general this is false. For example, take $R=k[x,y], S=k[x^4,x^3y,xy^3,y^4]$, where $k$ has characteristic 2. $R$ is CM, $S$ is not CM and we have, $R^4\subset S\subset R$. $\endgroup$– MohanCommented Nov 14, 2013 at 16:02
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$\begingroup$ Thanks Mohan! Do you know any reference to learn about inseparable isogeny $k$-algebras? $\endgroup$– NN guestCommented Nov 14, 2013 at 20:13
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$\begingroup$ There are better experts in this forum. It all depends on what you want to know. $\endgroup$– MohanCommented Nov 14, 2013 at 20:20
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$\begingroup$ Dear Mohan, I am very interested in the relation between the depths of $R$ and $S$. Any suggestion please! $\endgroup$– NN guestCommented Nov 20, 2013 at 16:42
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$\begingroup$ There are no reasonable relation between the two. $\endgroup$– MohanCommented Nov 20, 2013 at 16:44
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