37.40 Finite free locally dominates étale
In this section we explain a result that roughly states that étale coverings of a scheme S can be refined by Zariski coverings of finite locally free covers of S.
Lemma 37.40.1. Let S be a scheme. Let s \in S. Let f : (U, u) \to (S, s) be an étale neighbourhood. There exists an affine open neighbourhood s \in V \subset S and a surjective, finite locally free morphism \pi : T \to V such that for every t \in \pi ^{-1}(s) there exists an open neighbourhood t \in W_ t \subset T and a commutative diagram
\xymatrix{ T \ar[d]^\pi & W_ t \ar[l] \ar[rr]_{h_ t} \ar[rd] & & U \ar[dl] \\ V \ar[rr] & & S }
with h_ t(t) = u.
Proof.
The problem is local on S hence we may replace S by any open neighbourhood of s. We may also replace U by an open neighbourhood of u. Hence, by Morphisms, Lemma 29.36.14 we may assume that U \to S is a standard étale morphism of affine schemes. In this case the lemma (with V = S) follows from Algebra, Lemma 10.144.5.
\square
Lemma 37.40.2. Let f : U \to S be a surjective étale morphism of affine schemes. There exists a surjective, finite locally free morphism \pi : T \to S and a finite open covering T = T_1 \cup \ldots \cup T_ n such that each T_ i \to S factors through U \to S. Diagram:
\xymatrix{ & \coprod T_ i \ar[rd] \ar[ld] & \\ T \ar[rd]^\pi & & U \ar[ld]_ f \\ & S & }
where the south-west arrow is a Zariski-covering.
Proof.
This is a restatement of Algebra, Lemma 10.144.6.
\square
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