Let $Y$ be an irreducible locally noetherian, universally catenary scheme. Let $y\in Y$. Then
does there exists an open affine neighborhood $\operatorname{Spec}B$ such that
- $B$ is a discrete valuation ring.
- $\operatorname{Spec}B $ contains the $y$ as the speical point.
?
This question originates from trial to show that
"Let $Y$ be an irreducible locally noetherian, universally catenary scheme. If $f:X→Y$ is dominant and locally of finte type, then each fiber of $f$ is equidimensional? "
(c.f. See : Why $\operatorname{dim}\mathcal{O}_{X,x} \otimes_{\mathcal{O}_{Y,y}} \kappa(y)=0$ in certain situation? , answer of Sisi (Edit). There, he uses https://stacks.math.columbia.edu/tag/00QK.
If needed, I will upload my own more detailed argument or progress that I made. )
Can anyone help?
Best Answer
Let us follow the convention of the stacks project: that $\dim_y Y := \operatorname{min}(\dim U)$ where $U$ ranges over the open subsets of $Y$ containing $y$. If what you said was true, then $\dim_y Y \leq 1$ for all $y \in Y$. Therefore, $$\dim Y = \sup_{y \in Y} \dim_y Y \leq 1$$ where the equality is lemma 28.10.2 in the stacks project. As such, this couldn't work for schemes $Y$ of dimension $\geq 2$.