Lemma 76.33.1. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Let y \in |Y|. Let x_1, \ldots , x_ n \in |X| mapping to y. Assume that
f is locally of finite type,
f is separated,
f is quasi-finite at x_1, \ldots , x_ n, and
f is quasi-compact or Y is decent.
Then there exists an étale morphism (U, u) \to (Y, y) of pointed algebraic spaces and a decomposition
U \times _ Y X = W \amalg V
into open and closed subspaces such that the morphism V \to U is finite, every point of the fibre of |V| \to |U| over u maps to an x_ i, and the fibre of |W| \to |U| over u contains no point mapping to an x_ i.
Proof.
Let (U, u) \to (Y, y) be an étale morphism of algebraic spaces and consider the set of w \in |U \times _ Y X| mapping to u \in |U| and one of the x_ i \in |X|. By Decent Spaces, Lemma 68.18.4 (if f is of finite type) or Decent Spaces, Lemma 68.18.5 (if Y is decent) this set is finite. It follows that we may replace f by the base change U \times _ Y X \to U and x_1, \ldots , x_ n by the set of these w. In particular we may and do assume that Y is an affine scheme, whence X is a separated algebraic space.
Choose an affine scheme Z and an étale morphism Z \to X such that x_1, \ldots , x_ n are in the image of |Z| \to |X|. The fibres of |Z| \to |X| are finite, see Properties of Spaces, Lemma 66.6.7 (or the more general discussion in Decent Spaces, Section 68.6). Let \{ z_1, \ldots , z_ m\} \subset |Z| be the preimage of \{ x_1, \ldots , x_ n\} . By More on Morphisms, Lemma 37.41.4 there exists an étale morphism (U, u) \to (Y, y) such that U \times _ Y Z = Z_1 \amalg Z_2 with Z_1 \to U finite and (Z_1)_ y = \{ z_1, \ldots , z_ m\} . We may assume that U is affine and hence Z_1 is affine too.
Since f is separated, the image V of Z_1 \to X is both open and closed (Morphisms of Spaces, Lemma 67.40.6). Set W = X \setminus V to get a decomposition as in the lemma. To finish the proof we have to show that V \to U is finite. As Z_1 \to V is surjective and étale, V is the quotient of Z_1 by the étale equivalence relation R = Z_1 \times _ V Z_1, see Spaces, Lemma 65.9.1. Since f is separated, V \to U is separated and R is closed in Z_1 \times _ U Z_1. Since Z_1 \to U is finite, the projections s, t : R \to Z_1 are finite. Thus V is an affine scheme by Groupoids, Proposition 39.23.9. By Morphisms, Lemma 29.41.9 we conclude that V \to U is proper and by Morphisms, Lemma 29.44.11 we conclude that V \to U is finite, thereby finishing the proof.
\square
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