0
$\begingroup$

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.

Improve this question
$\endgroup$
5
  • $\begingroup$ What do you mean by a property here and is there a difference between "P holds on U" and "U has P"? $\endgroup$
    – Martin Brandenburg
    Commented Nov 29, 2011 at 17:35
  • $\begingroup$ Sorry, by "propriety" I meant "property". In my case I have a family $X\rightarrow B$ having a property $P$ satisfying i) and ii) $\endgroup$
    – uuuk
    Commented Nov 29, 2011 at 18:25
  • $\begingroup$ The spectrum of a complete DVR has two points, the closed one and the generic one, so I don't understand (i). $\endgroup$
    – Graham Leuschke
    Commented Nov 29, 2011 at 18:48
  • $\begingroup$ Sorry, now it is fixed. $\endgroup$
    – uuuk
    Commented Nov 30, 2011 at 15:40
  • $\begingroup$ A similar question was asked on Stackexchange here:math.stackexchange.com/questions/903891/… Does anybody know the answer to this? $\endgroup$
    – YangMills
    Commented Aug 21, 2014 at 10:13

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .