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


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 $B$.


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.

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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
A similar question was asked on Stackexchange here:math.stackexchange.com/questions/903891/… Does anybody know the answer to this? –  YangMills Aug 21 '14 at 10:13

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