Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Hi,

I would like to understand and prove the following two "well-known" facts:

1)

If $B$ is a scheme and $P$ a property for which I know:

i) if $B=Spec(V)$ where $V$ is a complete DVR, if $P$ holds on the special point then it holds on the generic point of $B$

ii) $P$ is true on a constructible set of $B$

then $P$ holds on an open subset of $B$.

2)

Assume $B$ integral. If $P$ true on the generic fiber implies that there exists an open dense of $B$ where $P$ holds then $P$ holds on a constructible set.

share|improve this question
    
What do you mean by a property here and is there a difference between "P holds on U" and "U has P"? –  Martin Brandenburg Nov 29 '11 at 17:35
    
Sorry, by "propriety" I meant "property". In my case I have a family $X\rightarrow B$ having a property $P$ satisfying i) and ii) –  uuuk Nov 29 '11 at 18:25
    
The spectrum of a complete DVR has two points, the closed one and the generic one, so I don't understand (i). –  Graham Leuschke Nov 29 '11 at 18:48
    
Sorry, now it is fixed. –  uuuk Nov 30 '11 at 15:40
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.