The Stacks project

Proof. Towards contradiction, assume there exists a tower of morphisms

where each map $W_ i \to W_{i - 1}$ is either an open immersion or a finite étale morphism, a lift $W_ m \to U$ of $W_ m \to X$ along $f : U \to X$, and a point $w \in W_ m$ mapping to $u \in U$ (and thus to $x \in X$). By replacing each $W_ k$ with the connected component containing the image $w_ k \in W_ k$ of $w$, we may assume that each $W_ k$ is smooth and connected.

For each $0 \le i \le m$, consider the fiber product $Z_ i = Y \times _ X W_ i$. As $W_ i \to X$ is étale and $Y$ is smooth, the curve $Z_ i$ is a disjoint union of smooth, connected curves and the maps $Z_ i \to W_ i$ are finite of degree $3$ by base change. Clearly, the fibre of $Z_ i \to W_ i$ over $w_ i$ consists of two points $z_ i, v_ i$ mapping to $y, u$ in $Y$. The morphism $Z_ i \to W_ i$ is unramified at $v_ i$ and ramified of degree $2$ at $z_ i$. Let $n_ i = \# \pi _0(Z_ i)$. Since the fibre of $Z_ i \to W_ i$ has $2$ points we see that $n_ i \in \{ 1, 2\} $. Moreover, $n_ i$ can only increase with $i$ as the transition maps $Z_ i \to Z_{i - 1}$ are dominant. We clearly have $n_0 = 1$. Moreover, we must have $n_ m \geq 2$: there is an $X$-morphism $W_ m \to U \subset Y$, so the base change $Z_ m \to W_ m$ has a section. There is then a minimal index $i \ge 1$ where $n_{i - 1} = 1$ and $n_ i = 2$. As the $n_ j$'s are determined by whether or not $Z_ i$ is irreducible, the map $W_ i \to W_{i - 1}$ cannot be an open immersion, so it must be finite étale. As $n_ i = 2$, the smooth curve $Z_ i$ has two connected components; as the map to $W_ i$ is finite, one of these components must then map isomorphically to $W_ i$, so we can write $Z_ i = W_ i \sqcup Z'_ i$, where $Z'_ i \to W_ i$ has degree $2$. The induced map $W_ i \subset Z_ i \to Z_{i - 1}$ is a dominant map between connected curves, both of which are finite over $W_{i - 1}$. Any such map must be surjective. Thus we may pick a point $t \in W_ i$ mapping to $z_{i - 1}$. We obtain extensions

of discrete valuation rings. Now the first one is ramified while the composite is unramified as $W_ i \to W_{i - 1}$ was étale; this is impossible, so we obtain a contradiction. $\square$