The Stacks project

66.4 Finiteness conditions and points

In this section we elaborate on the question of when points can be represented by monomorphisms from spectra of fields into the space.

Remark 66.4.1. Before we give the proof of the next lemma let us recall some facts about étale morphisms of schemes:

  1. An étale morphism is flat and hence generalizations lift along an étale morphism (Morphisms, Lemmas 29.35.12 and 29.25.9).

  2. An étale morphism is unramified, an unramified morphism is locally quasi-finite, hence fibres are discrete (Morphisms, Lemmas 29.35.16, 29.34.10, and 29.20.6).

  3. A quasi-compact étale morphism is quasi-finite and in particular has finite fibres (Morphisms, Lemmas 29.20.9 and 29.20.10).

  4. An étale scheme over a field $k$ is a disjoint union of spectra of finite separable field extension of $k$ (Morphisms, Lemma 29.35.7).

For a general discussion of étale morphisms, please see Étale Morphisms, Section 41.11.

Lemma 66.4.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. The following are equivalent:

  1. there exists a family of schemes $U_ i$ and étale morphisms $\varphi _ i : U_ i \to X$ such that $\coprod \varphi _ i : \coprod U_ i \to X$ is surjective, and such that for each $i$ the fibre of $|U_ i| \to |X|$ over $x$ is finite, and

  2. for every affine scheme $U$ and étale morphism $\varphi : U \to X$ the fibre of $|U| \to |X|$ over $x$ is finite.

Proof. The implication (2) $\Rightarrow $ (1) is trivial. Let $\varphi _ i : U_ i \to X$ be a family of étale morphisms as in (1). Let $\varphi : U \to X$ be an étale morphism from an affine scheme towards $X$. Consider the fibre product diagrams

\[ \xymatrix{ U \times _ X U_ i \ar[r]_-{p_ i} \ar[d]_{q_ i} & U_ i \ar[d]^{\varphi _ i} \\ U \ar[r]^\varphi & X } \quad \quad \xymatrix{ \coprod U \times _ X U_ i \ar[r]_-{\coprod p_ i} \ar[d]_{\coprod q_ i} & \coprod U_ i \ar[d]^{\coprod \varphi _ i} \\ U \ar[r]^\varphi & X } \]

Since $q_ i$ is étale it is open (see Remark 66.4.1). Moreover, the morphism $\coprod q_ i$ is surjective. Hence there exist finitely many indices $i_1, \ldots , i_ n$ and a quasi-compact opens $W_{i_ j} \subset U \times _ X U_{i_ j}$ which surject onto $U$. The morphism $p_ i$ is étale, hence locally quasi-finite (see remark on étale morphisms above). Thus we may apply Morphisms, Lemma 29.55.10 to see the fibres of $p_{i_ j}|_{W_{i_ j}} : W_{i_ j} \to U_ i$ are finite. Hence by Properties of Spaces, Lemma 64.4.3 and the assumption on $\varphi _ i$ we conclude that the fibre of $\varphi $ over $x$ is finite. In other words (2) holds. $\square$

Lemma 66.4.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. The following are equivalent:

  1. there exists a scheme $U$, an étale morphism $\varphi : U \to X$, and points $u, u' \in U$ mapping to $x$ such that setting $R = U \times _ X U$ the fibre of

    \[ |R| \to |U| \times _{|X|} |U| \]

    over $(u, u')$ is finite,

  2. for every scheme $U$, étale morphism $\varphi : U \to X$ and any points $u, u' \in U$ mapping to $x$ setting $R = U \times _ X U$ the fibre of

    \[ |R| \to |U| \times _{|X|} |U| \]

    over $(u, u')$ is finite,

  3. there exists a morphism $\mathop{\mathrm{Spec}}(k) \to X$ with $k$ a field in the equivalence class of $x$ such that the projections $\mathop{\mathrm{Spec}}(k) \times _ X \mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(k)$ are étale and quasi-compact, and

  4. there exists a monomorphism $\mathop{\mathrm{Spec}}(k) \to X$ with $k$ a field in the equivalence class of $x$.

Proof. Assume (1), i.e., let $\varphi : U \to X$ be an étale morphism from a scheme towards $X$, and let $u, u'$ be points of $U$ lying over $x$ such that the fibre of $|R| \to |U| \times _{|X|} |U|$ over $(u, u')$ is a finite set. In this proof we think of a point $u = \mathop{\mathrm{Spec}}(\kappa (u))$ as a scheme. Note that $u \to U$, $u' \to U$ are monomorphisms (see Schemes, Lemma 26.23.7), hence $u \times _ X u' \to R = U \times _ X U$ is a monomorphism. In this language the assumption really means that $u \times _ X u'$ is a scheme whose underlying topological space has finitely many points. Let $\psi : W \to X$ be an étale morphism from a scheme towards $X$. Let $w, w' \in W$ be points of $W$ mapping to $x$. We have to show that $w \times _ X w'$ is a scheme whose underlying topological space has finitely many points. Consider the fibre product diagram

\[ \xymatrix{ W \times _ X U \ar[r]_ p \ar[d]_ q & U \ar[d]^\varphi \\ W \ar[r]^\psi & X } \]

As $x$ is the image of $u$ and $u'$ we may pick points $\tilde w, \tilde w'$ in $W \times _ X U$ with $q(\tilde w) = w$, $q(\tilde w') = w'$, $u = p(\tilde w)$ and $u' = p(\tilde w')$, see Properties of Spaces, Lemma 64.4.3. As $p$, $q$ are étale the field extensions $\kappa (w) \subset \kappa (\tilde w) \supset \kappa (u)$ and $\kappa (w') \subset \kappa (\tilde w') \supset \kappa (u')$ are finite separable, see Remark 66.4.1. Then we get a commutative diagram

\[ \xymatrix{ w \times _ X w' \ar[d] & \tilde w \times _ X \tilde w' \ar[l] \ar[d] \ar[r] & u \times _ X u' \ar[d] \\ w \times _ X w' & \tilde w \times _ S \tilde w' \ar[l] \ar[r] & u \times _ S u' } \]

where the squares are fibre product squares. The lower horizontal morphisms are étale and quasi-compact, as any scheme of the form $\mathop{\mathrm{Spec}}(k) \times _ S \mathop{\mathrm{Spec}}(k')$ is affine, and by our observations about the field extensions above. Thus we see that the top horizontal arrows are étale and quasi-compact and hence have finite fibres. We have seen above that $|u \times _ X u'|$ is finite, so we conclude that $|w \times _ X w'|$ is finite. In other words, (2) holds.

Assume (2). Let $U \to X$ be an étale morphism from a scheme $U$ such that $x$ is in the image of $|U| \to |X|$. Let $u \in U$ be a point mapping to $x$. Then we have seen in the previous paragraph that $u = \mathop{\mathrm{Spec}}(\kappa (u)) \to X$ has the property that $u \times _ X u$ has a finite underlying topological space. On the other hand, the projection maps $u \times _ X u \to u$ are the composition

\[ u \times _ X u \longrightarrow u \times _ X U \longrightarrow u \times _ X X = u, \]

i.e., the composition of a monomorphism (the base change of the monomorphism $u \to U$) by an étale morphism (the base change of the étale morphism $U \to X$). Hence $u \times _ X U$ is a disjoint union of spectra of fields finite separable over $\kappa (u)$ (see Remark 66.4.1). Since $u \times _ X u$ is finite the image of it in $u \times _ X U$ is a finite disjoint union of spectra of fields finite separable over $\kappa (u)$. By Schemes, Lemma 26.23.11 we conclude that $u \times _ X u$ is a finite disjoint union of spectra of fields finite separable over $\kappa (u)$. In other words, we see that $u \times _ X u \to u$ is quasi-compact and étale. This means that (3) holds.

Let us prove that (3) implies (4). Let $\mathop{\mathrm{Spec}}(k) \to X$ be a morphism from the spectrum of a field into $X$, in the equivalence class of $x$ such that the two projections $t, s : R = \mathop{\mathrm{Spec}}(k) \times _ X \mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(k)$ are quasi-compact and étale. This means in particular that $R$ is an étale equivalence relation on $\mathop{\mathrm{Spec}}(k)$. By Spaces, Theorem 63.10.5 we know that the quotient sheaf $X' = \mathop{\mathrm{Spec}}(k)/R$ is an algebraic space. By Groupoids, Lemma 39.20.6 the map $X' \to X$ is a monomorphism. Since $s, t$ are quasi-compact, we see that $R$ is quasi-compact and hence Properties of Spaces, Lemma 64.15.3 applies to $X'$, and we see that $X' = \mathop{\mathrm{Spec}}(k')$ for some field $k'$. Hence we get a factorization

\[ \mathop{\mathrm{Spec}}(k) \longrightarrow \mathop{\mathrm{Spec}}(k') \longrightarrow X \]

which shows that $\mathop{\mathrm{Spec}}(k') \to X$ is a monomorphism mapping to $x \in |X|$. In other words (4) holds.

Finally, we prove that (4) implies (1). Let $\mathop{\mathrm{Spec}}(k) \to X$ be a monomorphism with $k$ a field in the equivalence class of $x$. Let $U \to X$ be a surjective étale morphism from a scheme $U$ to $X$. Let $u \in U$ be a point over $x$. Since $\mathop{\mathrm{Spec}}(k) \times _ X u$ is nonempty, and since $\mathop{\mathrm{Spec}}(k) \times _ X u \to u$ is a monomorphism we conclude that $\mathop{\mathrm{Spec}}(k) \times _ X u = u$ (see Schemes, Lemma 26.23.11). Hence $u \to U \to X$ factors through $\mathop{\mathrm{Spec}}(k) \to X$, here is a picture

\[ \xymatrix{ u \ar[r] \ar[d] & U \ar[d] \\ \mathop{\mathrm{Spec}}(k) \ar[r] & X } \]

Since the right vertical arrow is étale this implies that $k \subset \kappa (u)$ is a finite separable extension. Hence we conclude that

\[ u \times _ X u = u \times _{\mathop{\mathrm{Spec}}(k)} u \]

is a finite scheme, and we win by the discussion of the meaning of property (1) in the first paragraph of this proof. $\square$

Lemma 66.4.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. Let $U$ be a scheme and let $\varphi : U \to X$ be an étale morphism. The following are equivalent:

  1. $x$ is in the image of $|U| \to |X|$, and setting $R = U \times _ X U$ the fibres of both

    \[ |U| \longrightarrow |X| \quad \text{and}\quad |R| \longrightarrow |X| \]

    over $x$ are finite,

  2. there exists a monomorphism $\mathop{\mathrm{Spec}}(k) \to X$ with $k$ a field in the equivalence class of $x$, and the fibre product $\mathop{\mathrm{Spec}}(k) \times _ X U$ is a finite nonempty scheme over $k$.

Proof. Assume (1). This clearly implies the first condition of Lemma 66.4.3 and hence we obtain a monomorphism $\mathop{\mathrm{Spec}}(k) \to X$ in the class of $x$. Taking the fibre product we see that $\mathop{\mathrm{Spec}}(k) \times _ X U \to \mathop{\mathrm{Spec}}(k)$ is a scheme étale over $\mathop{\mathrm{Spec}}(k)$ with finitely many points, hence a finite nonempty scheme over $k$, i.e., (2) holds.

Assume (2). By assumption $x$ is in the image of $|U| \to |X|$. The finiteness of the fibre of $|U| \to |X|$ over $x$ is clear since this fibre is equal to $|\mathop{\mathrm{Spec}}(k) \times _ X U|$ by Properties of Spaces, Lemma 64.4.3. The finiteness of the fibre of $|R| \to |X|$ above $x$ is also clear since it is equal to the set underlying the scheme

\[ (\mathop{\mathrm{Spec}}(k) \times _ X U) \times _{\mathop{\mathrm{Spec}}(k)} (\mathop{\mathrm{Spec}}(k) \times _ X U) \]

which is finite over $k$. Thus (1) holds. $\square$

Lemma 66.4.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. The following are equivalent:

  1. for every affine scheme $U$, any étale morphism $\varphi : U \to X$ setting $R = U \times _ X U$ the fibres of both

    \[ |U| \longrightarrow |X| \quad \text{and}\quad |R| \longrightarrow |X| \]

    over $x$ are finite,

  2. there exist schemes $U_ i$ and étale morphisms $U_ i \to X$ such that $\coprod U_ i \to X$ is surjective and for each $i$, setting $R_ i = U_ i \times _ X U_ i$ the fibres of both

    \[ |U_ i| \longrightarrow |X| \quad \text{and}\quad |R_ i| \longrightarrow |X| \]

    over $x$ are finite,

  3. there exists a monomorphism $\mathop{\mathrm{Spec}}(k) \to X$ with $k$ a field in the equivalence class of $x$, and for any affine scheme $U$ and étale morphism $U \to X$ the fibre product $\mathop{\mathrm{Spec}}(k) \times _ X U$ is a finite scheme over $k$,

  4. there exists a quasi-compact monomorphism $\mathop{\mathrm{Spec}}(k) \to X$ with $k$ a field in the equivalence class of $x$, and

  5. there exists a quasi-compact morphism $\mathop{\mathrm{Spec}}(k) \to X$ with $k$ a field in the equivalence class of $x$.

Proof. The equivalence of (1) and (3) follows on applying Lemma 66.4.4 to every étale morphism $U \to X$ with $U$ affine. It is clear that (3) implies (2). Assume $U_ i \to X$ and $R_ i$ are as in (2). We conclude from Lemma 66.4.2 that for any affine scheme $U$ and étale morphism $U \to X$ the fibre of $|U| \to |X|$ over $x$ is finite. Say this fibre is $\{ u_1, \ldots , u_ n\} $. Then, as Lemma 66.4.3 (1) applies to $U_ i \to X$ for some $i$ such that $x$ is in the image of $|U_ i| \to |X|$, we see that the fibre of $|R = U \times _ X U| \to |U| \times _{|X|} |U|$ is finite over $(u_ a, u_ b)$, $a, b \in \{ 1, \ldots , n\} $. Hence the fibre of $|R| \to |X|$ over $x$ is finite. In this way we see that (1) holds. At this point we know that (1), (2), and (3) are equivalent.

If (4) holds, then for any affine scheme $U$ and étale morphism $U \to X$ the scheme $\mathop{\mathrm{Spec}}(k) \times _ X U$ is on the one hand étale over $k$ (hence a disjoint union of spectra of finite separable extensions of $k$ by Remark 66.4.1) and on the other hand quasi-compact over $U$ (hence quasi-compact). Thus we see that (3) holds. Conversely, if $U_ i \to X$ is as in (2) and $\mathop{\mathrm{Spec}}(k) \to X$ is a monomorphism as in (3), then

\[ \coprod \mathop{\mathrm{Spec}}(k) \times _ X U_ i \longrightarrow \coprod U_ i \]

is quasi-compact (because over each $U_ i$ we see that $\mathop{\mathrm{Spec}}(k) \times _ X U_ i$ is a finite disjoint union spectra of fields). Thus $\mathop{\mathrm{Spec}}(k) \to X$ is quasi-compact by Morphisms of Spaces, Lemma 65.8.8.

It is immediate that (4) implies (5). Conversely, let $\mathop{\mathrm{Spec}}(k) \to X$ be a quasi-compact morphism in the equivalence class of $x$. Let $U \to X$ be an étale morphism with $U$ affine. Consider the fibre product

\[ \xymatrix{ F \ar[r] \ar[d] & U \ar[d] \\ \mathop{\mathrm{Spec}}(k) \ar[r] & X } \]

Then $F \to U$ is quasi-compact, hence $F$ is quasi-compact. On the other hand, $F \to \mathop{\mathrm{Spec}}(k)$ is étale, hence $F$ is a finite disjoint union of spectra of finite separable extensions of $k$ (Remark 66.4.1). Since the image of $|F| \to |U|$ is the fibre of $|U| \to |X|$ over $x$ (Properties of Spaces, Lemma 64.4.3), we conclude that the fibre of $|U| \to |X|$ over $x$ is finite. The scheme $F \times _{\mathop{\mathrm{Spec}}(k)} F$ is also a finite union of spectra of fields because it is also quasi-compact and étale over $\mathop{\mathrm{Spec}}(k)$. There is a monomorphism $F \times _ X F \to F \times _{\mathop{\mathrm{Spec}}(k)} F$, hence $F \times _ X F$ is a finite disjoint union of spectra of fields (Schemes, Lemma 26.23.11). Thus the image of $F \times _ X F \to U \times _ X U = R$ is finite. Since this image is the fibre of $|R| \to |X|$ over $x$ by Properties of Spaces, Lemma 64.4.3 we conclude that (1) holds. $\square$

Lemma 66.4.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent:

  1. there exist schemes $U_ i$ and étale morphisms $U_ i \to X$ such that $\coprod U_ i \to X$ is surjective and each $U_ i \to X$ has universally bounded fibres, and

  2. for every affine scheme $U$ and étale morphism $\varphi : U \to X$ the fibres of $U \to X$ are universally bounded.

Proof. The implication (2) $\Rightarrow $ (1) is trivial. Assume (1). Let $(\varphi _ i : U_ i \to X)_{i \in I}$ be a collection of étale morphisms from schemes towards $X$, covering $X$, such that each $\varphi _ i$ has universally bounded fibres. Let $\psi : U \to X$ be an étale morphism from an affine scheme towards $X$. For each $i$ consider the fibre product diagram

\[ \xymatrix{ U \times _ X U_ i \ar[r]_{p_ i} \ar[d]_{q_ i} & U_ i \ar[d]^{\varphi _ i} \\ U \ar[r]^\psi & X } \]

Since $q_ i$ is étale it is open (see Remark 66.4.1). Moreover, we have $U = \bigcup \mathop{\mathrm{Im}}(q_ i)$, since the family $(\varphi _ i)_{i \in I}$ is surjective. Since $U$ is affine, hence quasi-compact we can finite finitely many $i_1, \ldots , i_ n \in I$ and quasi-compact opens $W_ j \subset U \times _ X U_{i_ j}$ such that $U = \bigcup p_{i_ j}(W_ j)$. The morphism $p_{i_ j}$ is étale, hence locally quasi-finite (see remark on étale morphisms above). Thus we may apply Morphisms, Lemma 29.55.10 to see the fibres of $p_{i_ j}|_{W_ j} : W_ j \to U_{i_ j}$ are universally bounded. Hence by Lemma 66.3.2 we see that the fibres of $W_ j \to X$ are universally bounded. Thus also $\coprod _{j = 1, \ldots , n} W_ j \to X$ has universally bounded fibres. Since $\coprod _{j = 1, \ldots , n} W_ j \to X$ factors through the surjective étale map $\coprod q_{i_ j}|_{W_ j} : \coprod _{j = 1, \ldots , n} W_ j \to U$ we see that the fibres of $U \to X$ are universally bounded by Lemma 66.3.5. In other words (2) holds. $\square$

Lemma 66.4.7. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent:

  1. there exists a Zariski covering $X = \bigcup X_ i$ and for each $i$ a scheme $U_ i$ and a quasi-compact surjective étale morphism $U_ i \to X_ i$, and

  2. there exist schemes $U_ i$ and étale morphisms $U_ i \to X$ such that the projections $U_ i \times _ X U_ i \to U_ i$ are quasi-compact and $\coprod U_ i \to X$ is surjective.

Proof. If (1) holds then the morphisms $U_ i \to X_ i \to X$ are étale (combine Morphisms, Lemma 29.35.3 and Spaces, Lemmas 63.5.4 and 63.5.3 ). Moreover, as $U_ i \times _ X U_ i = U_ i \times _{X_ i} U_ i$, both projections $U_ i \times _ X U_ i \to U_ i$ are quasi-compact.

If (2) holds then let $X_ i \subset X$ be the open subspace corresponding to the image of the open map $|U_ i| \to |X|$, see Properties of Spaces, Lemma 64.4.10. The morphisms $U_ i \to X_ i$ are surjective. Hence $U_ i \to X_ i$ is surjective étale, and the projections $U_ i \times _{X_ i} U_ i \to U_ i$ are quasi-compact, because $U_ i \times _{X_ i} U_ i = U_ i \times _ X U_ i$. Thus by Spaces, Lemma 63.11.4 the morphisms $U_ i \to X_ i$ are quasi-compact. $\square$


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