41.17 Étale local structure of unramified morphisms
In the chapter More on Morphisms, Section 37.41 the reader can find some results on the étale local structure of quasi-finite morphisms. In this section we want to combine this with the topological properties of unramified morphisms we have seen in this chapter. The basic overall picture to keep in mind is
\[ \xymatrix{ V \ar[r] \ar[dr] & X_ U \ar[d] \ar[r] & X \ar[d]^ f \\ & U \ar[r] & S } \]
see More on Morphisms, Equation (37.41.0.1). We start with a very general case.
Lemma 41.17.1. Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots , x_ n \in X$ be points having the same image $s$ in $S$. Assume $f$ is unramified at each $x_ i$. Then there exists an étale neighbourhood $(U, u) \to (S, s)$ and opens $V_{i, j} \subset X_ U$, $i = 1, \ldots , n$, $j = 1, \ldots , m_ i$ such that
$V_{i, j} \to U$ is a closed immersion passing through $u$,
$u$ is not in the image of $V_{i, j} \cap V_{i', j'}$ unless $i = i'$ and $j = j'$, and
any point of $(X_ U)_ u$ mapping to $x_ i$ is in some $V_{i, j}$.
Proof.
By Morphisms, Definition 29.35.1 there exists an open neighbourhood of each $x_ i$ which is locally of finite type over $S$. Replacing $X$ by an open neighbourhood of $\{ x_1, \ldots , x_ n\} $ we may assume $f$ is locally of finite type. Apply More on Morphisms, Lemma 37.41.3 to get the étale neighbourhood $(U, u)$ and the opens $V_{i, j}$ finite over $U$. By Lemma 41.7.3 after possibly shrinking $U$ we get that $V_{i, j} \to U$ is a closed immersion.
$\square$
Lemma 41.17.2. Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots , x_ n \in X$ be points having the same image $s$ in $S$. Assume $f$ is separated and $f$ is unramified at each $x_ i$. Then there exists an étale neighbourhood $(U, u) \to (S, s)$ and a disjoint union decomposition
\[ X_ U = W \amalg \coprod \nolimits _{i, j} V_{i, j} \]
such that
$V_{i, j} \to U$ is a closed immersion passing through $u$,
the fibre $W_ u$ contains no point mapping to any $x_ i$.
In particular, if $f^{-1}(\{ s\} ) = \{ x_1, \ldots , x_ n\} $, then the fibre $W_ u$ is empty.
Proof.
Apply Lemma 41.17.1. We may assume $U$ is affine, so $X_ U$ is separated. Then $V_{i, j} \to X_ U$ is a closed map, see Morphisms, Lemma 29.41.7. Suppose $(i, j) \not= (i', j')$. Then $V_{i, j} \cap V_{i', j'}$ is closed in $V_{i, j}$ and its image in $U$ does not contain $u$. Hence after shrinking $U$ we may assume that $V_{i, j} \cap V_{i', j'} = \emptyset $. Moreover, $\bigcup V_{i, j}$ is a closed and open subscheme of $X_ U$ and hence has an open and closed complement $W$. This finishes the proof.
$\square$
The following lemma is in some sense much weaker than the preceding one but it may be useful to state it explicitly here. It says that a finite unramified morphism is étale locally on the base a closed immersion.
Lemma 41.17.3. Let $f : X \to S$ be a finite unramified morphism of schemes. Let $s \in S$. There exists an étale neighbourhood $(U, u) \to (S, s)$ and a finite disjoint union decomposition
\[ X_ U = \coprod \nolimits _ j V_ j \]
such that each $V_ j \to U$ is a closed immersion.
Proof.
Since $X \to S$ is finite the fibre over $s$ is a finite set $\{ x_1, \ldots , x_ n\} $ of points of $X$. Apply Lemma 41.17.2 to this set (a finite morphism is separated, see Morphisms, Section 29.44). The image of $W$ in $U$ is a closed subset (as $X_ U \to U$ is finite, hence proper) which does not contain $u$. After removing this from $U$ we see that $W = \emptyset $ as desired.
$\square$
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