I am thinking about the following:
on a locally noetherian scheme X,$p$ and $q$ are generic points of two distinct irreducible components,do there exist two open sets $U,V$ of X with $p\in U,q\in V,U\cap V=\varnothing$?
when the irreducible components corresponding to p and q intersect at Q,we can just pick up a open affine near Q $Spec A$ where A is noetherian.
Spec A must contain p and q who correspond to minimal ideals in A.there are finitely many minimal prime ideal in A since A is noetherian.denote the intersection of minimal prime ideals except $p$ and $q$ as $I$,pick up a $f\in (p\cap I)\cap q^c$,$g\in (q\cap I)\cap p^c$(it’s easy to verify those sets are non-empty),consider $D(f),D(g)$ then it’s done.
So my questions are:
1.is my argument correct above?
2.how to deal with the situation when two irreducible components intersect not?
3.if it doesn’t hold,any simple counterexample?
Thanks for your help.