The Stacks project

59.27 Étale coverings

We recall the definition.

Definition 59.27.1. An étale covering of a scheme $U$ is a family of morphisms of schemes $\{ \varphi _ i : U_ i \to U\} _{i \in I}$ such that

  1. each $\varphi _ i$ is an étale morphism,

  2. the $U_ i$ cover $U$, i.e., $U = \bigcup _{i\in I}\varphi _ i(U_ i)$.

Proof. (See also Topologies, Lemma 34.9.7.) Let $\{ \varphi _ i : U_ i \to U\} _{i \in I}$ be an étale covering. Since an étale morphism is flat, and the elements of the covering should cover its target, the property fp (faithfully flat) is satisfied. To check the property qc (quasi-compact), let $V \subset U$ be an affine open, and write $\varphi _ i^{-1}(V) = \bigcup _{j \in J_ i} V_{ij}$ for some affine opens $V_{ij} \subset U_ i$. Since $\varphi _ i$ is open (as étale morphisms are open), we see that $V = \bigcup _{i\in I} \bigcup _{j \in J_ i} \varphi _ i(V_{ij})$ is an open covering of $V$. Further, since $V$ is quasi-compact, this covering has a finite refinement. $\square$

So any statement which is true for fpqc coverings remains true a fortiori for étale coverings. For instance, the étale site is subcanonical.

Definition 59.27.3. (For more details see Section 59.20, or Topologies, Section 34.4.) Let $S$ be a scheme. The big étale site over $S$ is the site $(\mathit{Sch}/S)_{\acute{e}tale}$, see Definition 59.20.2. The small étale site over $S$ is the site $S_{\acute{e}tale}$, see Definition 59.20.2. We define similarly the big and small Zariski sites on $S$, denoted $(\mathit{Sch}/S)_{Zar}$ and $S_{Zar}$.

Loosely speaking the big étale site of $S$ is made up out of schemes over $S$ and coverings the étale coverings. The small étale site of $S$ is made up out of schemes étale over $S$ with coverings the étale coverings. Actually any morphism between objects of $S_{\acute{e}tale}$ is étale, in virtue of Proposition 59.26.2, hence to check that $\{ U_ i \to U\} _{i \in I}$ in $S_{\acute{e}tale}$ is a covering it suffices to check that $\coprod U_ i \to U$ is surjective.

The small étale site has fewer objects than the big étale site, it contains only the “opens” of the étale topology on $S$. It is a full subcategory of the big étale site, and its topology is induced from the topology on the big site. Hence it is true that the restriction functor from the big étale site to the small one is exact and maps injectives to injectives. This has the following consequence.

Proposition 59.27.4. Let $S$ be a scheme and $\mathcal{F}$ an abelian sheaf on $(\mathit{Sch}/S)_{\acute{e}tale}$. Then $\mathcal{F}|_{S_{\acute{e}tale}}$ is a sheaf on $S_{\acute{e}tale}$ and

\[ H^ p_{\acute{e}tale}(S, \mathcal{F}|_{S_{\acute{e}tale}}) = H^ p_{\acute{e}tale}(S, \mathcal{F}) \]

for all $p \geq 0$.

Proof. This is a special case of Lemma 59.20.3. $\square$

In accordance with the general notation introduced in Section 59.20 we write $H_{\acute{e}tale}^ p(S, \mathcal{F})$ for the above cohomology group.

Comments (2)

Comment #1340 by yogesh more on

minor typo:03PF penultimate sentence of the proof of lemma 45.27.2, line 2708 of the code: " an open covering of " (instead of ).

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