## 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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