Criterion for open morphisms without constructible sets? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T01:54:11Z http://mathoverflow.net/feeds/question/38991 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38991/criterion-for-open-morphisms-without-constructible-sets Criterion for open morphisms without constructible sets? Martin Brandenburg 2010-09-16T16:36:13Z 2010-09-16T20:32:33Z <p>The following theorem is proved in EGA IV 2.4.6:</p> <p>Every morphism of schemes, which is flat and locally of finite presentation, is open.</p> <p>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.</p> <p>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?</p> http://mathoverflow.net/questions/38991/criterion-for-open-morphisms-without-constructible-sets/39020#39020 Answer by Matthieu Romagny for Criterion for open morphisms without constructible sets? Matthieu Romagny 2010-09-16T20:32:33Z 2010-09-16T20:32:33Z <p>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).</p> <p>More precisely :</p> <p>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.</p> <p>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 <em>flat</em> 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 <em>flat</em> 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.</p> <p>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).</p>