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Let $(R,m)$ be Noetherian local ring which is an imagine of a Gorenstein ring $(S,n)$. Set $$ K_R:= Ext_S^{s-d}(R,S), $$ where $d=\dim R$, $s=\dim S$. If $K_R$ is Cohen-Macaulay (i.e. $R$ is a canonic …

Let $(R,\frak m)$ be a Noetherian local ring. I am trying to prove that if the natural map $\operatorname{Spec}(\widehat R)\rightarrow \operatorname{Spec}(R)$ is bijective then $\mathfrak{p}\widehat R …

Let $(R,m)$ be a Cohen-Macaulay ring and $M$ a maximal Cohen-Macaulay $R$-module. Assume there exists canonical module $K_R$. Set $L:=\{f(x)|x\in M, f\in Hom_R(M,K_R)\}$. Then $M$ is $R$-module. I …

Let $(R, \mathfrak m)$ be a Noetherian local ring. Let P be a property of $R$. Set $$ P(R) =\{\mathfrak p \in Spec(R)\,\,\, |\,\,\, R_{\mathfrak p}\, \, \mbox{is } P\},$$ $$ nP(R) =\{\mathfrak p \in S …

Please help me to find a Noetherian local ring $R$ such that: $R$ is universally catenary and all its formal fibers over minimal members of $Spec(R)$ are Cohen-Macaulay but $R$ has a nonCohen-Macaulay …