3 Added new (important) reference to example 2)

Let $R$ be an excellent Noetherian ring. A property $P$ is said to be open if the set $\{q \in \operatorname{Spec}(R) \ | \ R_q \ \text{satisfies} \ (P)\}$ is Zariski open. Examples of open properties include:

1) Regular (well-known), complete intersections, Gorenstein, Cohen-Macaulay, Serre's condition $(S_n)$ (Matsumura's book). These imply openness of other properties, for example normality, which means $(R_1)$ and $(S_2)$.

2) Factorial (for $R$ of characteristic $0$, since the proof uses resolution of singularities). UPDATE: in a recent very interesting preprint, the factorial and $\mathbb Q$-factorial property are proved to be open for varieties over any algebraically closed field.

3) $\mathbb Q$-Gorenstein, i.e. the canonical module is torsion in the class group (I don't know a convenient reference, please provide if you happen to know one).

My questions are:

Question 1: Do you know other interesting class of open properties?

Question 2: Are there good heuristic reasons for why a certain property should be open? Phrased a bit more ambitiously, are there common techniques for proving openness for certain class of properties?

Some comments: The excellent condition is quite mild but crucial. Question 2 was motivated by another question of mine, which I still don't know the answer to.

2 added 22 characters in body

Let $R$ be an excellent Noetherian ring. A property $P$ is said to be open if the set $\{q \in Spec(R) \operatorname{Spec}(R) \ | \ R_q \ \text{satisfies} \ (P)\}$ is Zariski open. Examples of open properties include:

1) Regular (well-known), complete intersections, Gorenstein, Cohen-Macaulay, Serre's condition $(S_n)$(Matsumura's (S_n)$(Matsumura's book). These imply openness of other properties, for example normality, which means$(R_1)$and$(S_2)$. 2) Factorial (for$R$of characteristic$0$, since the proof uses resolution of singularity)singularities). 3)$\mathbb Q$-Gorenstein, i.e. the canonical module is torsion in the class group (I don't know a convenient reference, please provide if you happen to know one). My questions are: Question 1: Do you know other interesting class of open properties? Question 2: Are there good heuristic reasons for why a certain property should be open? Phrased a bit more ambitiously, are there common techniques for proving openness for certain class of properties? Some comments: The excellent condition is quite mild but crucial. Question 2 was motivated by another question of mine, which I still don't know the answer to. 1 # What properties define open loci in excellent schemes? Let$R$be an excellent Noetherian ring. A property$P$is said to be open if the set $\{q \in Spec(R) | R_q \ \text{satisfies} \ (P)\}$ is Zariski open. Examples of open properties include: 1) Regular (well-known), complete intersections, Gorenstein, Cohen-Macaulay, Serre's condition$(S_n)$(Matsumura's book). These imply openness of other properties, for example normality, which means$(R_1)$and$(S_2)$. 2) Factorial (for$R$of characteristic$0$, since the proof uses resolution of singularity). 3)$\mathbb Q\$-Gorenstein, i.e. the canonical module is torsion in the class group (I don't know a convenient reference, please provide if you happen to know one).

My questions are:

Question 1: Do you know other interesting class of open properties?

Question 2: Are there good heuristic reasons for why a certain property should be open? Phrased a bit more ambitiously, are there common techniques for proving openness for certain class of properties?

Some comments: The excellent condition is quite mild but crucial. Question 2 was motivated by another question of mine, which I still don't know the answer to.