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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