Lemma 110.57.1. The étale cover $U \to X$ cannot be refined by any finite composition of open immersions and finite étale morphisms.
110.57 The étale topology vs Zariski and finite étale Covers
In this section we give an example, found by B. Bhatt in response to a question by C. Deninger, that shows the étale topology on algebraic varieties over $\mathbf{C}$ is strictly finer than the topology generated by open immersions and finite étale morphisms. Such examples are surely known to experts, perhaps going back all the way to Artin and Grothendieck1. If you know of a published example, please email stacks.project@gmail.com.
Let $X$ be a smooth, connected affine curve over $\mathbf{C}$. Choose a finite morphism $g: Y \to X$ of degree $3$ from a smooth connected curve $Y$ such that there is a point $x \in X$ whose fiber $g^{-1}(x) = \{ u, y\} $ consists of an unramified point $u$ (degree 1) and a ramified point $y$ (degree 2). By replacing $X$ with a Zariski open neighborhood of $x$, we may assume $g$ is unramified everywhere except at $y$. Let $U = Y \setminus \{ y\} $. Then $f=g|_ U: U \to X$ is an étale surjective morphism, and $f^{-1}(x) = \{ u\} $.
Proof. Towards contradiction, assume there exists a tower of morphisms
\[ W_ m \to W_{m-1} \to \dots \to W_1 \to W_0 = X \]
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
\[ \mathcal{O}_{W_{i - 1}, w_{i - 1}} \to \mathcal{O}_{Z_ i, z_ i} \to \mathcal{O}_{W_ i, z} \]
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. $\square$
[1] Anecdotally, Grothendieck first attempted to define the étale topology via Zariski open covers and finite étale covers, before Artin convinced him otherwise (presumably based on examples such as the one recorded here).
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