The Stacks project

Example 35.32.2. Consider the property $\mathcal{P}$ of morphisms of schemes defined by the rule $\mathcal{P}(f : X \to Y) =$“for every $y \in Y$ which is a specialization of some $f(x)$, $x \in X$ the local ring $\mathcal{O}_{Y, y}$ is Noetherian”. Let us verify that this is étale local on the source and étale local on the target. We will freely use Schemes, Lemma 26.13.2.

Local on the target: Let $\{ g_ i : Y_ i \to Y\}$ be an étale covering. Let $f_ i : X_ i \to Y_ i$ be the base change of $f$, and denote $h_ i : X_ i \to X$ the projection. Assume $\mathcal{P}(f)$. Let $f(x_ i) \leadsto y_ i$ be a specialization. Then $f(h_ i(x_ i)) \leadsto g_ i(y_ i)$ so $\mathcal{P}(f)$ implies $\mathcal{O}_{Y, g_ i(y_ i)}$ is Noetherian. Also $\mathcal{O}_{Y, g_ i(y_ i)} \to \mathcal{O}_{Y_ i, y_ i}$ is a localization of an étale ring map. Hence $\mathcal{O}_{Y_ i, y_ i}$ is Noetherian by Algebra, Lemma 10.31.1. Conversely, assume $\mathcal{P}(f_ i)$ for all $i$. Let $f(x) \leadsto y$ be a specialization. Choose an $i$ and $y_ i \in Y_ i$ mapping to $y$. Since $x$ can be viewed as a point of $\mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}) \times _ Y X$ and $\mathcal{O}_{Y, y} \to \mathcal{O}_{Y_ i, y_ i}$ is faithfully flat, there exists a point $x_ i \in \mathop{\mathrm{Spec}}(\mathcal{O}_{Y_ i, y_ i}) \times _ Y X$ mapping to $x$. Then $x_ i \in X_ i$, and $f_ i(x_ i)$ specializes to $y_ i$. Thus we see that $\mathcal{O}_{Y_ i, y_ i}$ is Noetherian by $\mathcal{P}(f_ i)$ which implies that $\mathcal{O}_{Y, y}$ is Noetherian by Algebra, Lemma 10.164.1.

Local on the source: Let $\{ h_ i : X_ i \to X\}$ be an étale covering. Let $f_ i : X_ i \to Y$ be the composition $f \circ h_ i$. Assume $\mathcal{P}(f)$. Let $f(x_ i) \leadsto y$ be a specialization. Then $f(h_ i(x_ i)) \leadsto y$ so $\mathcal{P}(f)$ implies $\mathcal{O}_{Y, y}$ is Noetherian. Thus $\mathcal{P}(f_ i)$ holds. Conversely, assume $\mathcal{P}(f_ i)$ for all $i$. Let $f(x) \leadsto y$ be a specialization. Choose an $i$ and $x_ i \in X_ i$ mapping to $x$. Then $y$ is a specialization of $f_ i(x_ i) = f(x)$. Hence $\mathcal{P}(f_ i)$ implies $\mathcal{O}_{Y, y}$ is Noetherian as desired.

We claim that there exists a commutative diagram

with surjective étale vertical arrows, such that $h$ has $\mathcal{P}$ and $f$ does not have $\mathcal{P}$. Namely, let

and let $X \subset Y$ be the open subscheme which is the complement of the point all of whose coordinates $x_ n = 0$. Let $U = X$, let $V = X \amalg Y$, let $a, b$ the obvious map, and let $h : U \to V$ be the inclusion of $U = X$ into the first summand of $V$. The claim above holds because $U$ is locally Noetherian, but $Y$ is not.