Talk:Flat morphism

Openness

"Flat morphisms, which are locally of finite type are open" -- I'm sure this should say one of:

• Flat morphisms, which are locally of finite type, are open;
• Flat morphisms which are locally of finite type are open.

These mean quite different things; does anyone know which is correct? Jowa fan (talk) 08:49, 17 December 2009 (UTC)

The second sentence is the correct one. Liu (talk) 00:16, 30 December 2010 (UTC)

Generic

What do you mean by generic dimension? The dimension of f^(-1)(t), where t is the generic point? If yes (or if it has other meaning) it should be explained 86.71.242.159 (talk) 17:54, 17 September 2010 (UTC)

I don't think "generic" has a role in the article. Look at EGA IV.2 Corollaire 6.1.4 ...it follows that every fiber has the same dimension...not only the generic one...so...what did you mean by generic? May be "general"...anyway...I suggest to delete this word. 86.71.242.159 (talk) 23:15, 17 September 2010 (UTC)

This is not what says EGA. For the fibers to have the same dimension dimension equal to dim X - dim Y, the target scheme should be universally catenary. Liu (talk) 00:16, 30 December 2010 (UTC)

Descent properties

I believe some of the converses to the descent properties are wrong.

For example, the statement "Conversely, if f is also of finite presentation and f⁻¹(y) is reduced or normal, respectively, at x, then X is reduced or normal, respectively, at x." is false. If X = Y is a non-reduced scheme and f is the identity, then the fibers of f are spectra of fields and therefore reduced, whereas X is not.

This would be fixed by assuming moreover that Y is reduced, an assumption that matches the one made in the cited proposition. — Preceding unsigned comment added by Svdlugt (talk • contribs) 13:23, 30 March 2017 (UTC)

References