Talk:Proper morphism

broken link to EGA —Preceding unsigned comment added by 129.175.50.70 (talk) 10:15, 19 March 2010 (UTC)

GAGA

I think the first statement in the section "Properties and characterizations of proper morphisms" (stating that f is a proper morphism iff f(C) is a proper map) is wrong, at least as it stands here. According to SGA 1, Exposé XII, Proposition 3.2 it is indeed true that f: X → Y is a proper morphism of schemes (of finite type over C, which isn't assumed here either) iff f(C): X(C) → Y(C) is a proper morphism of complex analytic spaces. BUT a proper morphism of complex analytic spaces as defined in loc. cit. is a morphism, which is a proper map of the underlying topological spaces in the sense of Bourbaki (where g is called proper iff for every topological space Z, the map g×1_Z is closed, which isn't Wikipedia's definition) and separated. TheLaeg (talk) 14:56, 11 January 2011 (UTC)

You're absolutely right. I have made correction accordingly. -- Taku (talk) 01:55, 18 August 2012 (UTC)

Formal schemes

Note that for a morphism of formal schemes, being of finite type has different meaning than for usual schemes. So the link to morphism of finite type is not appropriate. It would be simpler to define proper morphisms by the properties (i) f is adic (takes some ideal of definition of S to an ideal of definition of X) (ii) f₀ is proper in the usual sense. UL (talk) 15:01, 18 August 2012 (UTC)

I've just checked EGA, and yes you're right. I changed the definition (wasn't incorrect but misleading). morphism of finite type needs a definition for formal ones. -- Taku (talk) 15:59, 18 August 2012 (UTC) (except we actually don't have the article.)