I'm reading Qingliu's book Proposition 4.3.9.
Definition: A Dedekind scheme is an irreducible normal locally Noetherian scheme of dimension 0 or 1.
Proposition 4.3.9. Let $Y$ be a Dedekind Scheme, and $f:X\rightarrow Y$ be a morphism with $X$ reduced. Then $f$ is flat if and only if every irreducible component of $X$ dominates $Y$.
In the proof of the 'if' part, assume $y\in Y$ closed point, $f(x) = y$ and $\pi\in\mathcal O_{Y,y}$ be a uniformizing parameter for $\mathcal O_{Y,y}$. The author says the assumption implies $\pi$ does not belong to any minimal prime ideal of $\mathcal O_{X,x}$.
My questions:
Why $\pi$ does not belong to any minimal prime ideal of $\mathcal O_{X,x}$.
Is there any intuitive meaning of uniformizing parameter?
Thank you in advance!
If $y$ is a closed point of $Y$, then $\mathcal{O}_{Y,y}$ is a local ring that is not a field, and $\mathcal{O}_{Y,y}$ is a DVR (Proposition 3.1.12). This means $\mathcal{O}_{Y,y}$ is a local PID, i.e. the maximal ideal $\mathfrak{m}_y$ is principal. Its generator is called the uniformizing parameter for $\mathcal{O}_{Y,y}$. Of course it can't belong to any minimal prime of $\mathcal{O}_{Y,y}$ since $(\pi)$ is maximal (unless $\dim Y = 0$ in which case I'm concerned about the validity of Qing Liu's argument since all non-units of an Artinian ring are zero divisors, so it seems to be his argument is deficient. Perhaps he meant to assume $\dim Y >0$).
Regardless, if you are not comfortable with DVRs, chapter 4 of Qing Liu should probably wait until you have a stronger background in commutative algebra. This is a very challenging book (I have been struggling with it for over a year!) and having a really solid grasp of commutative algebra and basic scheme theory is essential for reading it. To learn about DVRs, I strongly recommend Atiyah-MacDonald Chapter 9 (make sure you understand integral extensions very well first), especially Proposition 9.2.