# Criterion for open morphisms without constructible sets?

The following theorem is proved in EGA IV 2.4.6:

Every morphism of schemes, which is flat and locally of finite presentation, is open.

I've already seen some applications of this theorem, so I want to understand the proof. But it is based on the whole theory of (ind)constructible sets, whose development seems to be quite long and (sorry!) uninteresting in EGA.

So I want to know: Is it possible to give a direct proof? We may reduce to the affine case, so perhaps it's an observation from commutative algebra?

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See Proposition 28.8 in the Commutative Algebra chapter of the stacks project, math.columbia.edu/algebraic_geometry/stacks-git/browse.html. – mdeland Sep 16 '10 at 16:54
This also uses constructible sets. But I think it is more concise as EGA. – Martin Brandenburg Sep 16 '10 at 19:27

2) In the general case, there is a technical result in EGA (see in particular EGA IV, corollaire 11.2.6.1 and proposition 11.3.9) that says if $f:X\to Spec(A)$ is locally of finite presentation and flat then there exists a finitely generated sub-$\mathbb{Z}$-algebra $A_0\subset A$ (hence a noetherian ring) and an $A_0$-scheme $X_0$ which is locally of finite presentation (or finite type, as you wish) and flat such that $X\simeq X_0\otimes_{A_0} A$. Then you are in case 1). I understand that you want to avoid too much technical stuff, but I think that this very point can not be avoided, unless you are OK with locally noetherian schemes which after all is quite reasonable.