The Stacks project

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. $\square$

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].


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04HE. Beware of the difference between the letter 'O' and the digit '0'.