The Stacks project

56.21 The étale topos

A topos is the category of sheaves of sets on a site, see Sites, Definition 7.15.1. Hence it is customary to refer to the use the phrase “étale topos of a scheme” to refer to the category of sheaves on the small étale site of a scheme. Here is the formal definition.

Definition 56.21.1. Let $S$ be a scheme.

  1. The étale topos, or the small étale topos of $S$ is the category $\mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale})$ of sheaves of sets on the small étale site of $S$.

  2. The Zariski topos, or the small Zariski topos of $S$ is the category $\mathop{\mathit{Sh}}\nolimits (S_{Zar})$ of sheaves of sets on the small Zariski site of $S$.

  3. For $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}, Zariski\} $ a big $\tau $-topos is the category of sheaves of set on a big $\tau $-topos of $S$.

Note that the small Zariski topos of $S$ is simply the category of sheaves of sets on the underlying topological space of $S$, see Topologies, Lemma 33.3.11. Whereas the small étale topos does not depend on the choices made in the construction of the small étale site, in general the big topoi do depend on those choices.

Here is a lemma, which is one of many possible lemmas expressing the fact that it doesn't matter too much which site we choose to define the small étale topos of a scheme.

Lemma 56.21.2. Let $S$ be a scheme. Let $S_{affine, {\acute{e}tale}}$ denote the full subcategory of $S_{\acute{e}tale}$ whose objects are those $U/S \in \mathop{\mathrm{Ob}}\nolimits (S_{\acute{e}tale})$ with $U$ affine. A covering of $S_{affine, {\acute{e}tale}}$ will be a standard étale covering, see Topologies, Definition 33.4.5. Then restriction

\[ \mathcal{F} \longmapsto \mathcal{F}|_{S_{affine, {\acute{e}tale}}} \]

defines an equivalence of topoi $\mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}) \cong \mathop{\mathit{Sh}}\nolimits (S_{affine, {\acute{e}tale}})$.

Proof. This you can show directly from the definitions, and is a good exercise. But it also follows immediately from Sites, Lemma 7.29.1 by checking that the inclusion functor $S_{affine, {\acute{e}tale}} \to S_{\acute{e}tale}$ is a special cocontinuous functor (see Sites, Definition 7.29.2). $\square$

Lemma 56.21.3. Let $S$ be a scheme. The étale topos of $S$ is independent (up to canonical equivalence) of the construction of the small étale site in Definition 56.20.4.

Proof. We have to show, given two big étale sites $\mathit{Sch}_{\acute{e}tale}$ and $\mathit{Sch}_{\acute{e}tale}'$ containing $S$, then $\mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}) \cong \mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}')$ with obvious notation. By Topologies, Lemma 33.12.1 we may assume $\mathit{Sch}_{\acute{e}tale}\subset \mathit{Sch}_{\acute{e}tale}'$. By Sets, Lemma 3.9.9 any affine scheme étale over $S$ is isomorphic to an object of both $\mathit{Sch}_{\acute{e}tale}$ and $\mathit{Sch}_{\acute{e}tale}'$. Thus the induced functor $S_{affine, {\acute{e}tale}} \to S_{affine, {\acute{e}tale}}'$ is an equivalence. Moreover, it is clear that both this functor and a quasi-inverse map transform standard étale coverings into standard étale coverings. Hence the result follows from Lemma 56.21.2. $\square$

Lemma 56.21.4. Let $S$ be a scheme. Let $S_{affine, Zar}$ denote the full subcategory of $S_{Zar}$ consisting of affine objects. A covering of $S_{affine, Zar}$ will be a standard Zariski covering, see Topologies, Definition 33.3.4. Then restriction

\[ \mathcal{F} \longmapsto \mathcal{F}|_{S_{affine, Zar}} \]

defines an equivalence of topoi $\mathop{\mathit{Sh}}\nolimits (S_{Zar}) \cong \mathop{\mathit{Sh}}\nolimits (S_{affine, Zar})$.

Proof. Please skip the proof of this lemma. It follows immediately from Sites, Lemma 7.29.1 by checking that the inclusion functor $S_{affine, Zar} \to S_{Zar}$ is a special cocontinuous functor (see Sites, Definition 7.29.2). $\square$


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