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The Stacks project

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

  1. V_{i, j} \to U is a closed immersion passing through u,

  2. 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


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