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
such that each V_ j \to U is 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
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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