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 with $h_ t(t) = u$.
\[ \xymatrix{ T \ar[d]^\pi & W_ t \ar[l] \ar[rr]_{h_ t} \ar[rd] & & U \ar[dl] \\ V \ar[rr] & & S } \]
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: where the south-west arrow is a Zariski-covering.
\[ \xymatrix{ & \coprod T_ i \ar[rd] \ar[ld] & \\ T \ar[rd]^\pi & & U \ar[ld]_ f \\ & S & } \]
Proof. This is a restatement of Algebra, Lemma 10.144.6. $\square$
Remark 37.40.3. In terms of topologies Lemmas 37.40.1 and 37.40.2 mean the following. Let $S$ be any scheme. Let $\{ f_ i : U_ i \to S\} $ be an étale covering of $S$. There exists a Zariski open covering $S = \bigcup V_ j$, for each $j$ a finite locally free, surjective morphism $W_ j \to V_ j$, and for each $j$ a Zariski open covering $\{ W_{j, k} \to W_ j\} $ such that the family $\{ W_{j, k} \to S\} $ refines the given étale covering $\{ f_ i : U_ i \to S\} $. What does this mean in practice? Well, for example, suppose we have a descent problem which we know how to solve for Zariski coverings and for fppf coverings of the form $\{ \pi : T \to S\} $ with $\pi $ finite locally free and surjective. Then this descent problem has an affirmative answer for étale coverings as well. This trick was used by Gabber in his proof that $\text{Br}(X) = \text{Br}'(X)$ for an affine scheme $X$, see [Hoobler].
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