## 65.19 Points of the small étale site

This section is the analogue of Étale Cohomology, Section 59.29.

Definition 65.19.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.

A *geometric point* of $X$ is a morphism $\overline{x} : \mathop{\mathrm{Spec}}(k) \to X$, where $k$ is an algebraically closed field. We often abuse notation and write $\overline{x} = \mathop{\mathrm{Spec}}(k)$.

For every geometric point $\overline{x}$ we have the corresponding “image” point $x \in |X|$. We say that $\overline{x}$ is a *geometric point lying over $x$*.

It turns out that we can take stalks of sheaves on $X_{\acute{e}tale}$ at geometric points exactly in the same way as was done in the case of the small étale site of a scheme. In order to do this we define the notion of an étale neighbourhood as follows.

Definition 65.19.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$.

An *étale neighborhood* of $\overline{x}$ of $X$ is a commutative diagram

\[ \xymatrix{ & U \ar[d]^\varphi \\ {\bar x} \ar[r]^{\bar x} \ar[ur]^{\bar u} & X } \]

where $\varphi $ is an étale morphism of algebraic spaces over $S$. We will use the notation $\varphi : (U, \overline{u}) \to (X, \overline{x})$ to indicate this situation.

A *morphism of étale neighborhoods* $(U, \overline{u}) \to (U', \overline{u}')$ is an $X$-morphism $h : U \to U'$ such that $\overline{u}' = h \circ \overline{u}$.

Note that we allow $U$ to be an algebraic space. When we take stalks of a sheaf on $X_{\acute{e}tale}$ we have to restrict to those $U$ which are in $X_{\acute{e}tale}$, and so in this case we will only consider the case where $U$ is a scheme. Alternately we can work with the site $X_{space, {\acute{e}tale}}$ and consider all étale neighbourhoods. And there won't be any difference because of the last assertion in the following lemma.

Lemma 65.19.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. The category of étale neighborhoods is cofiltered. More precisely:

Let $(U_ i, \overline{u}_ i)_{i = 1, 2}$ be two étale neighborhoods of $\overline{x}$ in $X$. Then there exists a third étale neighborhood $(U, \overline{u})$ and morphisms $(U, \overline{u}) \to (U_ i, \overline{u}_ i)$, $i = 1, 2$.

Let $h_1, h_2: (U, \overline{u}) \to (U', \overline{u}')$ be two morphisms between étale neighborhoods of $\overline{s}$. Then there exist an étale neighborhood $(U'', \overline{u}'')$ and a morphism $h : (U'', \overline{u}'') \to (U, \overline{u})$ which equalizes $h_1$ and $h_2$, i.e., such that $h_1 \circ h = h_2 \circ h$.

Moreover, given any étale neighbourhood $(U, \overline{u}) \to (X, \overline{x})$ there exists a morphism of étale neighbourhoods $(U', \overline{u}') \to (U, \overline{u})$ where $U'$ is a scheme.

**Proof.**
For part (1), consider the fibre product $U = U_1 \times _ X U_2$. It is étale over both $U_1$ and $U_2$ because étale morphisms are preserved under base change and composition, see Lemmas 65.16.5 and 65.16.4. The map $\overline{u} \to U$ defined by $(\overline{u}_1, \overline{u}_2)$ gives it the structure of an étale neighborhood mapping to both $U_1$ and $U_2$.

For part (2), define $U''$ as the fibre product

\[ \xymatrix{ U'' \ar[r] \ar[d] & U \ar[d]^{(h_1, h_2)} \\ U' \ar[r]^-\Delta & U' \times _ X U'. } \]

Since $\overline{u}$ and $\overline{u}'$ agree over $X$ with $\overline{x}$, we see that $\overline{u}'' = (\overline{u}, \overline{u}')$ is a geometric point of $U''$. In particular $U'' \not= \emptyset $. Moreover, since $U'$ is étale over $X$, so is the fibre product $U'\times _ X U'$ (as seen above in the case of $U_1 \times _ X U_2$). Hence the vertical arrow $(h_1, h_2)$ is étale by Lemma 65.16.6. Therefore $U''$ is étale over $U'$ by base change, and hence also étale over $X$ (because compositions of étale morphisms are étale). Thus $(U'', \overline{u}'')$ is a solution to the problem posed by (2).

To see the final assertion, choose any surjective étale morphism $U' \to U$ where $U'$ is a scheme. Then $U' \times _ U \overline{u}$ is a scheme surjective and étale over $\overline{u} = \mathop{\mathrm{Spec}}(k)$ with $k$ algebraically closed. It follows (see Morphisms, Lemma 29.36.7) that $U' \times _ U \overline{u} \to \overline{u}$ has a section which gives us the desired $\overline{u}'$.
$\square$

Lemma 65.19.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x} : \mathop{\mathrm{Spec}}(k) \to X$ be a geometric point of $X$ lying over $x \in |X|$. Let $\varphi : U \to X$ be an étale morphism of algebraic spaces and let $u \in |U|$ with $\varphi (u) = x$. Then there exists a geometric point $\overline{u} : \mathop{\mathrm{Spec}}(k) \to U$ lying over $u$ with $\overline{x} = \varphi \circ \overline{u}$.

**Proof.**
Choose an affine scheme $U'$ with $u' \in U'$ and an étale morphism $U' \to U$ which maps $u'$ to $u$. If we can prove the lemma for $(U', u') \to (X, x)$ then the lemma follows. Hence we may assume that $U$ is a scheme, in particular that $U \to X$ is representable. Then look at the cartesian diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(k) \times _{\overline{x}, X, \varphi } U \ar[d]_{\text{pr}_1} \ar[r]_-{\text{pr}_2} & U \ar[d]^\varphi \\ \mathop{\mathrm{Spec}}(k) \ar[r]^-{\overline{x}} & X } \]

The projection $\text{pr}_1$ is the base change of an étale morphisms so it is étale, see Lemma 65.16.5. Therefore, the scheme $\mathop{\mathrm{Spec}}(k) \times _{\overline{x}, X, \varphi } U$ is a disjoint union of finite separable extensions of $k$, see Morphisms, Lemma 29.36.7. But $k$ is algebraically closed, so all these extensions are trivial, so $\mathop{\mathrm{Spec}}(k) \times _{\overline{x}, X, \varphi } U$ is a disjoint union of copies of $\mathop{\mathrm{Spec}}(k)$ and each of these corresponds to a geometric point $\overline{u}$ with $\varphi \circ \overline{u} = \overline{x}$. By Lemma 65.4.3 the map

\[ |\mathop{\mathrm{Spec}}(k) \times _{\overline{x}, X, \varphi } U| \longrightarrow |\mathop{\mathrm{Spec}}(k)| \times _{|X|} |U| \]

is surjective, hence we can pick $\overline{u}$ to lie over $u$.
$\square$

Lemma 65.19.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. Let $(U, \overline{u})$ an étale neighborhood of $\overline{x}$. Let $\{ \varphi _ i : U_ i \to U\} _{i \in I}$ be an étale covering in $X_{spaces, {\acute{e}tale}}$. Then there exist $i \in I$ and $\overline{u}_ i : \overline{x} \to U_ i$ such that $\varphi _ i : (U_ i, \overline{u}_ i) \to (U, \overline{u})$ is a morphism of étale neighborhoods.

**Proof.**
Let $u \in |U|$ be the image of $\overline{u}$. As $|U| = \bigcup _{i \in I} \varphi _ i(|U_ i|)$ there exists an $i$ and a point $u_ i \in U_ i$ mapping to $x$. Apply Lemma 65.19.4 to $(U_ i, u_ i) \to (U, u)$ and $\overline{u}$ to get the desired geometric point.
$\square$

Definition 65.19.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a presheaf on $X_{\acute{e}tale}$. Let $\overline{x}$ be a geometric point of $X$. The *stalk* of $\mathcal{F}$ at $\overline{x}$ is

\[ \mathcal{F}_{\bar x} = \mathop{\mathrm{colim}}\nolimits _{(U, \overline{u})} \mathcal{F}(U) \]

where $(U, \overline{u})$ runs over all étale neighborhoods of $\overline{x}$ in $X$ with $U \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$.

By Lemma 65.19.3, this colimit is over a filtered index category, namely the opposite of the category of étale neighborhoods in $X_{\acute{e}tale}$. More precisely Lemma 65.19.3 says the opposite of the category of all étale neighbourhoods is filtered, and the full subcategory of those which are in $X_{\acute{e}tale}$ is a cofinal subcategory hence also filtered.

This means an element of $\mathcal{F}_{\overline{x}}$ can be thought of as a triple $(U, \overline{u}, \sigma )$ where $U \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ and $\sigma \in \mathcal{F}(U)$. Two triples $(U, \overline{u}, \sigma )$, $(U', \overline{u}', \sigma ')$ define the same element of the stalk if there exists a third étale neighbourhood $(U'', \overline{u}'')$, $U'' \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ and morphisms of étale neighbourhoods $h : (U'', \overline{u}'') \to (U, \overline{u})$, $h' : (U'', \overline{u}'') \to (U', \overline{u}')$ such that $h^*\sigma = (h')^*\sigma '$ in $\mathcal{F}(U'')$. See Categories, Section 4.19.

This also implies that if $\mathcal{F}'$ is the sheaf on $X_{spaces, {\acute{e}tale}}$ corresponding to $\mathcal{F}$ on $X_{\acute{e}tale}$, then

65.19.6.1
\begin{equation} \label{spaces-properties-equation-stalk-spaces-etale} \mathcal{F}_{\overline{x}} = \mathop{\mathrm{colim}}\nolimits _{(U, \overline{u})} \mathcal{F}'(U) \end{equation}

where now the colimit is over all the étale neighbourhoods of $\overline{x}$. We will often jump between the point of view of using $X_{\acute{e}tale}$ and $X_{spaces, {\acute{e}tale}}$ without further mention.

In particular this means that if $\mathcal{F}$ is a presheaf of abelian groups, rings, etc then $\mathcal{F}_{\overline{x}}$ is an abelian group, ring, etc simply by the usual way of defining the group structure on a directed colimit of abelian groups, rings, etc.

slogan
Lemma 65.19.7. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. Consider the functor

\[ u : X_{\acute{e}tale}\longrightarrow \textit{Sets}, \quad U \longmapsto |U_{\overline{x}}| \]

Then $u$ defines a point $p$ of the site $X_{\acute{e}tale}$ (Sites, Definition 7.32.2) and its associated stalk functor $\mathcal{F} \mapsto \mathcal{F}_ p$ (Sites, Equation 7.32.1.1) is the functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ defined above.

**Proof.**
In the proof of Lemma 65.19.5 we have seen that the scheme $U_{\overline{x}}$ is a disjoint union of schemes isomorphic to $\overline{x}$. Thus we can also think of $|U_{\overline{x}}|$ as the set of geometric points of $U$ lying over $\overline{x}$, i.e., as the collection of morphisms $\overline{u} : \overline{x} \to U$ fitting into the diagram of Definition 65.19.2. From this it follows that $u(X)$ is a singleton, and that $u(U \times _ V W) = u(U) \times _{u(V)} u(W)$ whenever $U \to V$ and $W \to V$ are morphisms in $X_{\acute{e}tale}$. And, given a covering $\{ U_ i \to U\} _{i \in I}$ in $X_{\acute{e}tale}$ we see that $\coprod u(U_ i) \to u(U)$ is surjective by Lemma 65.19.5. Hence Sites, Proposition 7.33.3 applies, so $p$ is a point of the site $X_{\acute{e}tale}$. Finally, the our functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$ is given by exactly the same colimit as the functor $\mathcal{F} \mapsto \mathcal{F}_ p$ associated to $p$ in Sites, Equation 7.32.1.1 which proves the final assertion.
$\square$

Lemma 65.19.8. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$.

The stalk functor $\textit{PAb}(X_{\acute{e}tale}) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ is exact.

We have $(\mathcal{F}^\# )_{\overline{x}} = \mathcal{F}_{\overline{x}}$ for any presheaf of sets $\mathcal{F}$ on $X_{\acute{e}tale}$.

The functor $\textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ is exact.

Similarly the functors $\textit{PSh}(X_{\acute{e}tale}) \to \textit{Sets}$ and $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \textit{Sets}$ given by the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ are exact (see Categories, Definition 4.23.1) and commute with arbitrary colimits.

**Proof.**
This result follows from the general material in Modules on Sites, Section 18.36. This is true because $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ comes from a point of the small étale site of $X$, see Lemma 65.19.7. See the proof of Étale Cohomology, Lemma 59.29.9 for a direct proof of some of these statements in the setting of the small étale site of a scheme.
$\square$

We will see below that the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ is really the pullback along the morphism $\overline{x}$. In that sense the following lemma is a generalization of the lemma above.

Lemma 65.19.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

The functor $f_{small}^{-1} : \textit{Ab}(Y_{\acute{e}tale}) \to \textit{Ab}(X_{\acute{e}tale})$ is exact.

The functor $f_{small}^{-1} : \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ is exact, i.e., it commutes with finite limits and colimits, see Categories, Definition 4.23.1.

For any étale morphism $V \to Y$ of algebraic spaces we have $f_{small}^{-1}h_ V = h_{X \times _ Y V}$.

Let $\overline{x} \to X$ be a geometric point. Let $\mathcal{G}$ be a sheaf on $Y_{\acute{e}tale}$. Then there is a canonical identification

\[ (f_{small}^{-1}\mathcal{G})_{\overline{x}} = \mathcal{G}_{\overline{y}}. \]

where $\overline{y} = f \circ \overline{x}$.

**Proof.**
Recall that $f_{small}$ is defined via $f_{spaces, small}$ in Lemma 65.18.8. Parts (1), (2) and (3) are general consequences of the fact that $f_{spaces, {\acute{e}tale}} : X_{spaces, {\acute{e}tale}} \to Y_{spaces, {\acute{e}tale}}$ is a morphism of sites, see Sites, Definition 7.14.1 for (2), Modules on Sites, Lemma 18.31.2 for (1), and Sites, Lemma 7.13.5 for (3).

Proof of (4). This statement is a special case of Sites, Lemma 7.34.2 via Lemma 65.19.7. We also provide a direct proof. Note that by Lemma 65.19.8. taking stalks commutes with sheafification. Let $\mathcal{G}'$ be the sheaf on $Y_{spaces, {\acute{e}tale}}$ whose restriction to $Y_{\acute{e}tale}$ is $\mathcal{G}$. Recall that $f_{spaces, {\acute{e}tale}}^{-1}\mathcal{G}'$ is the sheaf associated to the presheaf

\[ U \longrightarrow \mathop{\mathrm{colim}}\nolimits _{U \to X \times _ Y V} \mathcal{G}'(V), \]

see Sites, Sections 7.13 and 7.5. Thus we have

\begin{align*} (f_{spaces, {\acute{e}tale}}^{-1}\mathcal{G}')_{\overline{x}} & = \mathop{\mathrm{colim}}\nolimits _{(U, \overline{u})} f_{spaces, {\acute{e}tale}}^{-1}\mathcal{G}'(U) \\ & = \mathop{\mathrm{colim}}\nolimits _{(U, \overline{u})} \mathop{\mathrm{colim}}\nolimits _{a : U \to X \times _ Y V} \mathcal{G}'(V) \\ & = \mathop{\mathrm{colim}}\nolimits _{(V, \overline{v})} \mathcal{G}'(V) \\ & = \mathcal{G}'_{\overline{y}} \end{align*}

in the third equality the pair $(U, \overline{u})$ and the map $a : U \to X \times _ Y V$ corresponds to the pair $(V, a \circ \overline{u})$. Since the stalk of $\mathcal{G}'$ (resp. $f_{spaces, {\acute{e}tale}}^{-1}\mathcal{G}'$) agrees with the stalk of $\mathcal{G}$ (resp. $f_{small}^{-1}\mathcal{G}$), see Equation (65.19.6.1) the result follows.
$\square$

The following theorem says that the small étale site of an algebraic space has enough points.

Theorem 65.19.12. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A map $a : \mathcal{F} \to \mathcal{G}$ of sheaves of sets is injective (resp. surjective) if and only if the map on stalks $a_{\overline{x}} : \mathcal{F}_{\overline{x}} \to \mathcal{G}_{\overline{x}}$ is injective (resp. surjective) for all geometric points of $X$. A sequence of abelian sheaves on $X_{\acute{e}tale}$ is exact if and only if it is exact on all stalks at geometric points of $S$.

**Proof.**
We know the theorem is true if $X$ is a scheme, see Étale Cohomology, Theorem 59.29.10. Choose a surjective étale morphism $f : U \to X$ where $U$ is a scheme. Since $\{ U \to X\} $ is a covering (in $X_{spaces, {\acute{e}tale}}$) we can check whether a map of sheaves is injective, or surjective by restricting to $U$. Now if $\overline{u} : \mathop{\mathrm{Spec}}(k) \to U$ is a geometric point of $U$, then $(\mathcal{F}|_ U)_{\overline{u}} = \mathcal{F}_{\overline{x}}$ where $\overline{x} = f \circ \overline{u}$. (This is clear from the colimits defining the stalks at $\overline{u}$ and $\overline{x}$, but it also follows from Lemma 65.19.9.) Hence the result for $U$ implies the result for $X$ and we win.
$\square$

The following lemma should be skipped on a first reading.

Lemma 65.19.13. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $p : \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ be a point of the small étale topos of $X$. Then there exists a geometric point $\overline{x}$ of $X$ such that the stalk functor $\mathcal{F} \mapsto \mathcal{F}_ p$ is isomorphic to the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$.

**Proof.**
By Sites, Lemma 7.32.7 there is a one to one correspondence between points of the site and points of the associated topos. Hence we may assume that $p$ is given by a functor $u : X_{\acute{e}tale}\to \textit{Sets}$ which defines a point of the site $X_{\acute{e}tale}$. Let $U \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ be an object whose structure morphism $j : U \to X$ is surjective. Note that $h_ U$ is a sheaf which surjects onto the final sheaf. Since taking stalks is exact we see that $(h_ U)_ p = u(U)$ is not empty (use Sites, Lemma 7.32.3). Pick $x \in u(U)$. By Sites, Lemma 7.35.1 we obtain a point $q : \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale})$ such that $p = j_{small} \circ q$, so that $\mathcal{F}_ p = (\mathcal{F}|_ U)_ q$ functorially. By Étale Cohomology, Lemma 59.29.12 there is a geometric point $\overline{u}$ of $U$ and a functorial isomorphism $\mathcal{G}_ q = \mathcal{G}_{\overline{u}}$ for $\mathcal{G} \in \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale})$. Set $\overline{x} = j \circ \overline{u}$. Then we see that $\mathcal{F}_{\overline{x}} \cong (\mathcal{F}|_ U)_{\overline{u}}$ functorially in $\mathcal{F}$ on $X_{\acute{e}tale}$ by Lemma 65.19.9 and we win.
$\square$

## Comments (4)

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