# On some loci of rings

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!

• I don't know an algorithm for 1, if you are finite type over a perfect field you can use the Jacobian, or see the recent work of Hochster-Jeffries. For 2, there are options. First, a ring is Gorenstein if and only if it is Cohen-Macaulay with locally free canonical module. The non-Cohen-Macaulay locus can be expressed as the vanishing of the annihilator of certain Ext groups. The locus where the canonical module $\omega_R$ is not locally free can be computed in various ways (from homological, to in the case of a reduced ring, via a map $\omega_R \cdot \omega_R^{-1} \to R$). Sep 15 at 14:49
• Thank @KarlSchwede very much for your comment. I understand the second part now.
– TNAn
Sep 16 at 21:55