The Stacks project

64.19 Points of the small étale site

This section is the analogue of Étale Cohomology, Section 58.29.

Definition 64.19.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.

  1. 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 $\overline{x} = \mathop{\mathrm{Spec}}(k)$.

  2. 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 point 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 64.19.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$.

  1. 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.

  2. 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 64.19.3. 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:

  1. 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 $(U, \overline{u}) \to (U_ i, \overline{u}_ i)$, $i = 1, 2$.

  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 64.16.5 and 64.16.4. 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 64.16.6. 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 29.34.7) that $U' \times _ U \overline{u} \to \overline{u}$ has a section which gives us the desired $\overline{u}'$. $\square$

Lemma 64.19.4. 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 $\overline{x} = f \circ \overline{u}$.

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 64.16.5. 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 29.34.7. 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 $f \circ \overline{u} = \overline{x}$. By Lemma 64.4.3 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 64.19.5. 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 64.19.4 to $(U_ i, u_ i) \to (U, u)$ and $\overline{u}$ to get the desired geometric point. $\square$

Definition 64.19.6. 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 64.19.3, 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 64.19.3 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

64.19.6.1
\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 64.19.7. 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 64.19.5 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 64.19.1. 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 64.19.5. 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 64.19.8. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$.

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

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

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

  4. 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 64.19.7. See the proof of Étale Cohomology, Lemma 58.29.9 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 64.19.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

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

  2. 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.

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

  4. 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 64.18.7. 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 64.19.7. We also provide a direct proof. Note that by Lemma 64.19.8. 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 (64.19.6.1) the result follows. $\square$

Remark 64.19.10. This remark is the analogue of Étale Cohomology, Remark 58.55.6. 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 $X$. By Étale Cohomology, Theorem 58.55.3 the category of sheaves on $\mathop{\mathrm{Spec}}(k)_{\acute{e}tale}$ is equivalent to the category of sets (by taking a sheaf to its global sections). Hence it follows from Lemma 64.19.9 part (4) applied to the morphism $\overline{x}$ that the functor

\[ \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \longrightarrow \textit{Sets}, \quad \mathcal{F} \longmapsto \mathcal{F}_{\overline{x}} \]

is isomorphic to the functor

\[ \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathop{\mathrm{Spec}}(k)_{\acute{e}tale}) = \textit{Sets}, \quad \mathcal{F} \longmapsto \overline{x}^*\mathcal{F} \]

Hence we may view the stalk functors as pullback functors along geometric morphisms (and not just some abstract morphisms of topoi as in the result of Lemma 64.19.7).

Remark 64.19.11. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. We claim that for any pair of geometric points $\overline{x}$ and $\overline{x}'$ lying over $x$ the stalk functors are isomorphic. By definition of $|X|$ we can find a third geometric point $\overline{x}''$ so that there exists a commutative diagram

\[ \xymatrix{ \overline{x}'' \ar[r] \ar[d] \ar[rd]^{\overline{x}''} & \overline{x}' \ar[d]^{\overline{x}'} \\ \overline{x} \ar[r]^{\overline{x}} & X. } \]

Since the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ is given by pullback along the morphism $\overline{x}$ (and similarly for the others) we conclude by functoriality of pullbacks.

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

Theorem 64.19.12. 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 58.29.10. 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 64.19.9.) 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 64.19.13. 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 58.29.12 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 64.19.9 and we win. $\square$


Comments (1)

Comment #5098 by Klaus Mattis on

In Lemma 05VN and its proof, the morphism \phi is called f multiple times.


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