Lemma 59.43.3. Let f : X \to Y be a finite morphism of schemes. Then property (B) holds.
Proof. Consider V \to Y étale, \{ U_ i \to X \times _ Y V\} an étale covering, and v \in V. We have to find a V' \to V and decomposition and maps as in Lemma 59.43.2. We may shrink V and Y, hence we may assume that V and Y are affine. Since X is finite over Y, this also implies that X is affine. During the proof we may (finitely often) replace (V, v) by an étale neighbourhood (V', v') and correspondingly the covering \{ U_ i \to X \times _ Y V\} by \{ V' \times _ V U_ i \to X \times _ Y V'\} .
Since X \times _ Y V \to V is finite there exist finitely many (pairwise distinct) points x_1, \ldots , x_ n \in X \times _ Y V mapping to v. We may apply More on Morphisms, Lemma 37.41.5 to X \times _ Y V \to V and the points x_1, \ldots , x_ n lying over v and find an étale neighbourhood (V', v') \to (V, v) such that
X \times _ Y V' = R \amalg \coprod T_ a
with T_ a \to V' finite with exactly one point p_ a lying over v' and moreover \kappa (v') \subset \kappa (p_ a) purely inseparable, and such that R \to V' has empty fibre over v'. Because X \to Y is finite, also R \to V' is finite. Hence after shrinking V' we may assume that R = \emptyset . Thus we may assume that X \times _ Y V = X_1 \amalg \ldots \amalg X_ n with exactly one point x_ l \in X_ l lying over v with moreover \kappa (v) \subset \kappa (x_ l) purely inseparable. Note that this property is preserved under refinement of the étale neighbourhood (V, v).
For each l choose an i_ l and a point u_ l \in U_{i_ l} mapping to x_ l. Now we apply property (A) for the finite morphism X \times _ Y V \to V and the étale morphisms U_{i_ l} \to X \times _ Y V and the points u_ l. This is permissible by Lemma 59.42.3 This gives produces an étale neighbourhood (V', v') \to (V, v) and decompositions
X \times _ Y V' = W_ l \amalg R_ l
and X-morphisms a_ l : W_ l \to U_{i_ l} whose image contains u_{i_ l}. Here is a picture:
\xymatrix{ & & & U_{i_ l} \ar[d] & \\ W_ l \ar[rrru] \ar[r] & W_ l \amalg R_ l \ar@{=}[r] & X \times _ Y V' \ar[r] \ar[d] & X \times _ Y V \ar[r] \ar[d] & X \ar[d] \\ & & V' \ar[r] & V \ar[r] & Y }
After replacing (V, v) by (V', v') we conclude that each x_ l is contained in an open and closed neighbourhood W_ l such that the inclusion morphism W_ l \to X \times _ Y V factors through U_ i \to X \times _ Y V for some i. Replacing W_ l by W_ l \cap X_ l we see that these open and closed sets are disjoint and moreover that \{ x_1, \ldots , x_ n\} \subset W_1 \cup \ldots \cup W_ n. Since X \times _ Y V \to V is finite we may shrink V and assume that X \times _ Y V = W_1 \amalg \ldots \amalg W_ n as desired. \square
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