Etale is not Zariski + finite etale

Thanks to Bhargav Bhatt who sent me the latex
Slightly edited by me

Author: aisejohan

examples.tex

@@ -4179,6 +4179,90 @@ \section{Sheaf with quasi-compact flat covering which is not algebraic}
 
 
 
+\section{The \'etale topology vs Zariski and finite \'etale Covers}
+\label{section-etale-zariski-finite-etale}
+
+\noindent
+In this section we give an example, found by B.~Bhatt in response
+to a question by C.~Deninger, that shows the \'etale topology
+on algebraic varieties over $\mathbf{C}$ is strictly finer than the
+topology generated by open immersions and finite \'etale morphisms.
+Such examples are surely known to experts, perhaps going back all the way
+to Artin and Grothendieck\footnote{Anecdotally, Grothendieck first
+attempted to define the \'etale topology via Zariski open covers and
+finite \'etale covers, before Artin convinced him otherwise
+(presumably based on examples such as the one recorded here).}.
+If you know of a published example, please email
+\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}.
+
+\medskip\noindent
+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 \'etale surjective morphism, and
+$f^{-1}(x) = \{u\}$. 
+
+\begin{lemma}
+The \'etale cover $U \to X$ cannot be refined by any finite composition of open immersions and finite \'etale morphisms.
+\end{lemma}
+
+\begin{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 \'etale 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.
+
+\medskip\noindent
+For each $0 \le i \le m$, consider the fiber product $Z_i = Y \times_X W_i$.
+As $W_i \to X$ is \'etale 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 \'etale. 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 \'etale; this is
+impossible, so we obtain a contradiction.
+\end{proof}
+
+
+
+
+
+
+
+
+
 
 \section{Sheaves and specializations}
 \label{section-sheaves}