Let $(R, \mathfrak m)$ be a Noetherian local ring. Let P be a property of $R$. Set $$ P(R) =\{\mathfrak p \in Spec(R)\,\,\, |\,\,\, R_{\mathfrak p}\, \, \mbox{is } P\},$$ $$ nP(R) =\{\mathfrak p \in Spec(R)\,\,\, |\,\,\, R_{\mathfrak p}\, \, \mbox{is not} P\}.$$ I knew that:
(1) If $R$ is excellent ring then $P(R)$ is open, where P is "regular";
(2) If $R$ is homomorphic image of a Gorenstein ring then $P(R)$ is open, where P is "Gorenstein".
The methods: Using Topological Nagata criterion and Ring theoretic Nagata criterion. In these cases $nP(R)$ is closed.
My question: Find $I$, $J$ so that $nP(R)=Var(I)$ in Case 1 and $nP(R)=Var(J)$ in Case 2. Thank you for your comment or answer for me!
