Let $A$ be a normal Noetherian ring. Then will the singular locus of $A$, i.e., $\{\mathcal{p}\in Spec(A)|A_{\mathcal{p}}\ \text{is not a regular ring}\} $, be closed in $Spec(A)$? If so, what will be the ideal defining it and what can be said about the height of that ideal?
Thanks in advance!
I think it is worthwhile to point out that the singular locus is not always closed, even for normal domains:
Theorem [Abhyankar and Heinzer 1988, Thm. 6.3]. There exists a two-dimensional normal noetherian domain, all of whose localizations at prime ideals are excellent, but whose singular locus is not closed.
There are also local examples. Nishimura attributes the following to Brodmann and Rotthaus.
Theorem [Nishimura 2012, Ex. 2.11]. There exists a three-dimensional normal noetherian local domain, which is a complete intersection and universally Japanese, but whose singular locus is not closed.
The condition that the singular locus is closed is called J-1. Affine algebras over a field are excellent, hence J-1 (pretty much by definition), so there is no conflict between these examples and Mohan's comment.