Favorite open or closed conditions?

June 10, 2011

Favorite open or closed conditions?

Posted by ravivakil under Big lists
[11] Comments

What are your favorite open and closed and locally closed conditions (under reasonable hypotheses). (My motivation for doing all three of these at once: to keep people saying “the reduced locus is open” and “the nonreduced locus is closed”.)

For simplicity, please put one per comment, so people can respond.

Related question: I was intending to have one list on properties that can be checked at closed points, but I realized that all examples I had in mind were of two sorts, which I am asking separately. (i) Some are true because they are open conditions, and thus for a quasicompact scheme it suffices to check them at closed points (see the comment after Important Exercise 5.1.E [used to be 6.1.E]). (ii) Some are true because they a variety-specific result, usually involving the Nullstellensatz. Are there any things you like to check at closed points that aren’t of either of these two sorts?

11 Responses to “Favorite open or closed conditions?”

ravivakil Says:

June 10, 2011 at 8:39 am
If $\pi: X \rightarrow Y$ is a morphism, and $\mathcal{F}$ is a quasicoherent sheaf on $X$ . The locus (on both $X$ and $Y$ ) where $\mathcal{F}$ is flat over $Y$ is open. I should think about this, and get hypotheses right (certainly locally Noetherian and finite type is needed), and give references!

Update July 19 and 20, 2011: for openness on the source: we need $\pi$ to be locally of finite presentation, and $\mathcal{F}$ to be finitely presented. This is EGA IV_3.11.3.1. The much easier locally Noetherian case (to which the general case is reduced to) is EGA IV_3.11.1.1, and also Matsumura’s Commutative Ring Theory Theorem 24.3 (p. 187). Update December 14, 2012: see stacks tag 00RC.

I see no great version of opennness on the target, other than the case when $\pi$ is proper. (Am I missing something obvious or important?) A cheaper (and very useful) variant of this is that there should exist a dense open subset of the target over which $\pi$ is flat, under reasonable circumstances. If $Y$ is integral (and Noetherian), this can be shown using Grothendieck’s “generic freeness [or flatness]” theorem. It can be extended easily to the case where $Y$ is reduced. It can’t be extended farther, as when $Y^{red} \rightarrow Y$ is never flat at any nonreduced points of $Y$ .

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ravivakil Says:

June 10, 2011 at 8:30 am
The reduced locus of a scheme is open.

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ravivakil Says:

June 10, 2011 at 8:31 am
The locus where a morphism $\pi: X \rightarrow Y$ is smooth of relative dimension $n$ (e.g. where $\pi$ is etale) is open on $X$ and on $Y$ . (I should later find the necessarily hypotheses.)

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ravivakil Says:

June 10, 2011 at 8:33 am
Given a finite type quasicoherent sheaf $\mathcal{F}$ on a scheme $X$ , the locus of points $x$ on $X$ where the stalk $\mathcal{F}_x$ is generated by global sections is open on $X$ . (Exercise 16.3.C(c))

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ravivakil Says:

June 10, 2011 at 8:37 am
If $\pi: X \rightarrow Y$ is a proper morphism, and $\mathcal{L}$ is a line bundle on $X$ , then the locus on $Y$ where the $\mathcal{L}$ is ample (when restricted to the fiber) is an open condition. I intend to add a reference later.

I’ve never used this, but it makes me feel good.

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ravivakil Says:

June 10, 2011 at 8:42 am
Nonsingularity on a locally Noetherian scheme is an open condition. Reference: Matsumura’s Commutative Ring Theory Thm 24.4 (p. 187), attributed there to Nagata.

This is a stronger condition than Fact 13.3.8 that localizations of regular local rings are regular local rings (Eisenbud Cor. 19.14; Matsumura Commutative Ring Theory Thm. 19.3).

I’ve never used this, but it makes me feel good.

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ravivakil Says:

June 10, 2011 at 8:45 am
The Cohen-Macaulay locus on a scheme (under some hypotheses) is an open condition. Update June 17 2011: from the stacks project, tag 00RF, if S is a finite type k-scheme, the Cohen-Macaulay locus is open. More generally (Stacks Project tag 00RH) if A is a finitely presented flat R-algebra (so think geometrically $\pi: \text{Spec} \; A \rightarrow \text{Spec} \; R$ ) the locus on $\text{Spec} \; A$ where the fiber of $\pi$ is Cohen-Maculay of a given dimension $d$ is open. Update June 21 2011: the Cohen-Macaulay locus on a locally Noetherian scheme is open (Matsumura Commutative Ring Theory Theorem 24.5, p. 188). Related fact: on a locally Noetherian scheme, the Gorenstein locus is open (Matsumura Thm 24.6).

Boissiere, Gabber, and Serman have a recent cool paper showing that the factorial locus and the Q-factorial locus of varieties over any field but $\overline{\mathbb{F}}_p$ are open.

I’ve never used these, but they make me feel good.

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Favorite properties of varieties (finite type k-schemes) checkable at closed points? « Math 216: Foundations of Algebraic Geometry Says:

June 13, 2011 at 9:35 am
[…] open condition will work, but please list those here. I’m looking for some “variety-specific” facts. There seem to be remarkably […]

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ravivakil Says:

July 20, 2011 at 4:28 am
Local completion intersection locus. Matsumura’s Commutative Ring Theory p. 189 attributes this to Greco and Marinari [1] (along with the argument for the opennness of the Gorenstein locus).

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Xingbang Hao Says:

March 31, 2019 at 6:52 pm
Why could we can only check fiber’s property to check the whole one? And where is the Important Exercise 6.1.E?

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1. ravivakil Says:
  
  April 25, 2019 at 2:25 pm
  That’s now become 5.1.E since that was originally written — sorry!
  
  Reply