The Stacks project

Lemma 66.18.5. In diagram (66.18.0.1) the set (66.18.0.2) is finite if $y$ can be represented by a monomorphism $\mathop{\mathrm{Spec}}(k) \to Y$ where $k$ is a field and $g$ is quasi-finite at $z$. (Special case: $Y$ is decent and $g$ is étale.)

Proof. By Lemma 66.18.3 applied twice we may replace $Z$ by $Z_ k = \mathop{\mathrm{Spec}}(k) \times _ Y Z$ and $X$ by $X_ k = \mathop{\mathrm{Spec}}(k) \times _ Y X$. We may and do replace $Y$ by $\mathop{\mathrm{Spec}}(k)$ as well. Note that $Z_ k \to \mathop{\mathrm{Spec}}(k)$ is quasi-finite at $z$ by Morphisms of Spaces, Lemma 65.27.2. Choose a scheme $V$, a point $v \in V$, and an étale morphism $V \to Z_ k$ mapping $v$ to $z$. Choose a scheme $U$, a point $u \in U$, and an étale morphism $U \to X_ k$ mapping $u$ to $x$. Again by Lemma 66.18.3 it suffices to show $F_{u, v}$ is finite for the diagram

\[ \xymatrix{ U \times _{\mathop{\mathrm{Spec}}(k)} V \ar[r] \ar[d] & V \ar[d] \\ U \ar[r] & \mathop{\mathrm{Spec}}(k) } \]

The morphism $V \to \mathop{\mathrm{Spec}}(k)$ is quasi-finite at $v$ (follows from the general discussion in Morphisms of Spaces, Section 65.22 and the definition of being quasi-finite at a point). At this point the finiteness follows from Example 66.18.1. The parenthetical remark of the statement of the lemma follows from the fact that on decent spaces points are represented by monomorphisms from fields and from the fact that an étale morphism of algebraic spaces is locally quasi-finite. $\square$


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