##
Favorite open or closed conditions?

Posted by ravivakil under

Big lists
[9] 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 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?

June 10, 2011 at 8:39 am

If is a morphism, and is a quasicoherent sheaf on . The locus (on both and ) where is flat over 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 to be locally of finite presentation, and 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 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 is flat, under reasonable circumstances. If 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 is reduced. It can’t be extended farther, as when is never flat at any nonreduced points of .

June 10, 2011 at 8:30 am

The reduced locus of a scheme is open.

June 10, 2011 at 8:31 am

The locus where a morphism is smooth of relative dimension (e.g. where is etale) is open on and on . (I should later find the necessarily hypotheses.)

June 10, 2011 at 8:33 am

Given a finite type quasicoherent sheaf on a scheme , the locus of points on where the stalk is generated by global sections is open on . (Exercise 16.3.C(c))

June 10, 2011 at 8:37 am

If is a proper morphism, and is a line bundle on , then the locus on where the 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.

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.

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 ) the locus on where the fiber of is Cohen-Maculay of a given dimension 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 are open.

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

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 […]

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).