Question

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?

Answer 1

You can of course assume that the base scheme is affine. Then it works in two steps : morally the result is really a result for morphisms locally of finite type (or finite presentation, as you wish) between locally noetherian schemes ; but a morphism locally of finite presentation comes by base change from a noetherian base scheme (remember we have an affine base).

More precisely :

1) There is a nice proof without constructibility in Milne's Etale Cohomoloy, theorem 2.12, for morphisms locally of finite type. That does the job in case the base scheme is locally noetherian.

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.

I am preparing notes for a course on schemes with this theorem included. If you wish, I can send you the relevant pages (it's in french).