Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
You are of course right about the containment! I probably want the embedding $k(p(x)) \hookrightarrow k(x)$ to factor through an algebraic closure of $k(p(x))$. Is this possible?
A delicate aspect. I know that as a Grothendieck topology I would have to consider jointly surjective flat morphisms (+ some properties perhaps). But in "Going-down implies generalized going-down" by Dobbs--Hetzel (Lemma 2.1) there is a topology (in the "classical" sense) on Spec(R) defined (but this should have been considered before), which I thought is also referred to as the "flat topology". I clearly don't know how this is connected to the flat topology in the Grothendieck sense.
This is about the fact that $Spec(R_P)$ is homeomorphic to the "generization" of $P$ in $Spec(R)$? Does this perhaps imply an answer about my neighborhood question?
Well, V(P) contains all $Q \supset P$, so its complement is the set of all $Q \not\supset P$. But $Spec(R_P)$ consists of all $Q$ with $Q \cap (R \setminus P) = \emptyset$, so $Q \subset P$. This is not the complement of $V(P)$. Am I wrong? This would be great. :-)
