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:
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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 $(S_2)$.
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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.
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$\mathbb Q$-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.
Best Answer
to give a shamelessly trivial answer to your question 2, I would say that the point is usually that the failure of these properties is closed, often "obviously". I would also add one more auxiliary property, that acts as a meta property for some of these:
A coherent sheaf being locally free is an open property. This follows from Nakayama's lemma.
So, here is your list. The properties on the left fail along the loci on the right. [Caveat: I did not include conditions that are sometimes necessary, but I figured that this is a philosophical question and so the answer does not have to be stated in the most precise way.]
regular/smooth -------------- zero set of the Jacobian ideal, also the locus where $\Omega_X$ is not locally free
Cohen-Macaulay, $S_n$ ----- support of appropriate Ext sheaves
Gorenstein ------------------ {not CM} $\bigcup$ {CM but $\omega_X$ is not a line bundle}
$\mathbb Q$-Gorenstein --------------- $\bigcap_m$ {$\omega_X^{[m]}$ is not a line bundle}
rational singularity --------- (in char $0$) $\bigcup_{i=1}^{\dim X}{\rm supp\,}R^i\phi_*\mathcal O_{\widetilde X}$ where $\phi:\widetilde X\to X$ is a resolution.
klt singularity ---------------- zero set of the multiplier ideal.
Du Bois singularity --------- $\bigcup_{i\neq 0} {\rm supp\,} h^i(\underline\Omega_X^0)\bigcup \,{\rm supp\,}{\rm coker\,}[\mathcal O_X\to h^0(\underline\Omega_X^0)]$
(semi-)normality ------------ ${\rm supp\,}{\rm coker\,}[\mathcal O_X\to \pi_*\mathcal O_{Y}]$, where $\pi:Y\to X$ is the (semi-)normalization.
etcetera...