Lemma 66.13.1. Let $S$ be a scheme. Let $Z$ be an algebraic space over $S$. Let $k$ be a field and let $\mathop{\mathrm{Spec}}(k) \to Z$ be surjective and flat. Then any morphism $\mathop{\mathrm{Spec}}(k') \to Z$ where $k'$ is a field is surjective and flat.

## 66.13 Reduced singleton spaces

A *singleton* space is an algebraic space $X$ such that $|X|$ is a singleton. It turns out that these can be more interesting than just being the spectrum of a field, see Spaces, Example 63.14.7. We develop a tiny bit of machinery to be able to talk about these.

**Proof.**
Consider the fibre square

Note that $T \to \mathop{\mathrm{Spec}}(k')$ is flat and surjective hence $T$ is not empty. On the other hand $T \to \mathop{\mathrm{Spec}}(k)$ is flat as $k$ is a field. Hence $T \to Z$ is flat and surjective. It follows from Morphisms of Spaces, Lemma 65.31.5 that $\mathop{\mathrm{Spec}}(k') \to Z$ is flat. It is surjective as by assumption $|Z|$ is a singleton. $\square$

Lemma 66.13.2. Let $S$ be a scheme. Let $Z$ be an algebraic space over $S$. The following are equivalent

$Z$ is reduced and $|Z|$ is a singleton,

there exists a surjective flat morphism $\mathop{\mathrm{Spec}}(k) \to Z$ where $k$ is a field, and

there exists a locally of finite type, surjective, flat morphism $\mathop{\mathrm{Spec}}(k) \to Z$ where $k$ is a field.

**Proof.**
Assume (1). Let $W$ be a scheme and let $W \to Z$ be a surjective étale morphism. Then $W$ is a reduced scheme. Let $\eta \in W$ be a generic point of an irreducible component of $W$. Since $W$ is reduced we have $\mathcal{O}_{W, \eta } = \kappa (\eta )$. It follows that the canonical morphism $\eta = \mathop{\mathrm{Spec}}(\kappa (\eta )) \to W$ is flat. We see that the composition $\eta \to Z$ is flat (see Morphisms of Spaces, Lemma 65.30.3). It is also surjective as $|Z|$ is a singleton. In other words (2) holds.

Assume (2). Let $W$ be a scheme and let $W \to Z$ be a surjective étale morphism. Choose a field $k$ and a surjective flat morphism $\mathop{\mathrm{Spec}}(k) \to Z$. Then $W \times _ Z \mathop{\mathrm{Spec}}(k)$ is a scheme étale over $k$. Hence $W \times _ Z \mathop{\mathrm{Spec}}(k)$ is a disjoint union of spectra of fields (see Remark 66.4.1), in particular reduced. Since $W \times _ Z \mathop{\mathrm{Spec}}(k) \to W$ is surjective and flat we conclude that $W$ is reduced (Descent, Lemma 35.16.1). In other words (1) holds.

It is clear that (3) implies (2). Finally, assume (2). Pick a nonempty affine scheme $W$ and an étale morphism $W \to Z$. Pick a closed point $w \in W$ and set $k = \kappa (w)$. The composition

is locally of finite type by Morphisms of Spaces, Lemmas 65.23.2 and 65.39.9. It is also flat and surjective by Lemma 66.13.1. Hence (3) holds. $\square$

The following lemma singles out a slightly better class of singleton algebraic spaces than the preceding lemma.

Lemma 66.13.3. Let $S$ be a scheme. Let $Z$ be an algebraic space over $S$. The following are equivalent

$Z$ is reduced, locally Noetherian, and $|Z|$ is a singleton, and

there exists a locally finitely presented, surjective, flat morphism $\mathop{\mathrm{Spec}}(k) \to Z$ where $k$ is a field.

**Proof.**
Assume (2) holds. By Lemma 66.13.2 we see that $Z$ is reduced and $|Z|$ is a singleton. Let $W$ be a scheme and let $W \to Z$ be a surjective étale morphism. Choose a field $k$ and a locally finitely presented, surjective, flat morphism $\mathop{\mathrm{Spec}}(k) \to Z$. Then $W \times _ Z \mathop{\mathrm{Spec}}(k)$ is a scheme étale over $k$, hence a disjoint union of spectra of fields (see Remark 66.4.1), hence locally Noetherian. Since $W \times _ Z \mathop{\mathrm{Spec}}(k) \to W$ is flat, surjective, and locally of finite presentation, we see that $\{ W \times _ Z \mathop{\mathrm{Spec}}(k) \to W\} $ is an fppf covering and we conclude that $W$ is locally Noetherian (Descent, Lemma 35.13.1). In other words (1) holds.

Assume (1). Pick a nonempty affine scheme $W$ and an étale morphism $W \to Z$. Pick a closed point $w \in W$ and set $k = \kappa (w)$. Because $W$ is locally Noetherian the morphism $w : \mathop{\mathrm{Spec}}(k) \to W$ is of finite presentation, see Morphisms, Lemma 29.21.7. Hence the composition

is locally of finite presentation by Morphisms of Spaces, Lemmas 65.28.2 and 65.39.8. It is also flat and surjective by Lemma 66.13.1. Hence (2) holds. $\square$

Lemma 66.13.4. Let $S$ be a scheme. Let $Z' \to Z$ be a monomorphism of algebraic spaces over $S$. Assume there exists a field $k$ and a locally finitely presented, surjective, flat morphism $\mathop{\mathrm{Spec}}(k) \to Z$. Then either $Z'$ is empty or $Z' = Z$.

**Proof.**
We may assume that $Z'$ is nonempty. In this case the fibre product $T = Z' \times _ Z \mathop{\mathrm{Spec}}(k)$ is nonempty, see Properties of Spaces, Lemma 64.4.3. Now $T$ is an algebraic space and the projection $T \to \mathop{\mathrm{Spec}}(k)$ is a monomorphism. Hence $T = \mathop{\mathrm{Spec}}(k)$, see Morphisms of Spaces, Lemma 65.10.8. We conclude that $\mathop{\mathrm{Spec}}(k) \to Z$ factors through $Z'$. But as $\mathop{\mathrm{Spec}}(k) \to Z$ is surjective, flat and locally of finite presentation, we see that $\mathop{\mathrm{Spec}}(k) \to Z$ is surjective as a map of sheaves on $(\mathit{Sch}/S)_{fppf}$ (see Spaces, Remark 63.5.2) and we conclude that $Z' = Z$.
$\square$

The following lemma says that to each point of an algebraic space we can associate a canonical reduced, locally Noetherian singleton algebraic space.

Lemma 66.13.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. Then there exists a unique monomorphism $Z \to X$ of algebraic spaces over $S$ such that $Z$ is an algebraic space which satisfies the equivalent conditions of Lemma 66.13.3 and such that the image of $|Z| \to |X|$ is $\{ x\} $.

**Proof.**
Choose a scheme $U$ and a surjective étale morphism $U \to X$. Set $R = U \times _ X U$ so that $X = U/R$ is a presentation (see Spaces, Section 63.9). Set

The canonical morphism $U' \to U$ is a monomorphism. Let

Because $U' \to U$ is a monomorphism we see that the projections $s', t' : R' \to U'$ factor as a monomorphism followed by an étale morphism. Hence, as $U'$ is a disjoint union of spectra of fields, using Remark 66.4.1, and using Schemes, Lemma 26.23.11 we conclude that $R'$ is a disjoint union of spectra of fields and that the morphisms $s', t' : R' \to U'$ are étale. Hence $Z = U'/R'$ is an algebraic space by Spaces, Theorem 63.10.5. As $R'$ is the restriction of $R$ by $U' \to U$ we see $Z \to X$ is a monomorphism by Groupoids, Lemma 39.20.6. Since $Z \to X$ is a monomorphism we see that $|Z| \to |X|$ is injective, see Morphisms of Spaces, Lemma 65.10.9. By Properties of Spaces, Lemma 64.4.3 we see that

is surjective which implies (by our choice of $U'$) that $|Z| \to |X|$ has image $\{ x\} $. We conclude that $|Z|$ is a singleton. Finally, by construction $U'$ is locally Noetherian and reduced, i.e., we see that $Z$ satisfies the equivalent conditions of Lemma 66.13.3.

Let us prove uniqueness of $Z \to X$. Suppose that $Z' \to X$ is a second such monomorphism of algebraic spaces. Then the projections

are monomorphisms. The algebraic space in the middle is nonempty by Properties of Spaces, Lemma 64.4.3. Hence the two projections are isomorphisms by Lemma 66.13.4 and we win. $\square$

We introduce the following terminology which foreshadows the residual gerbes we will introduce later, see Properties of Stacks, Definition 98.11.8.

Definition 66.13.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. The *residual space of $X$ at $x$*^{1} is the monomorphism $Z_ x \to X$ constructed in Lemma 66.13.5.

In particular we know that $Z_ x$ is a locally Noetherian, reduced, singleton algebraic space and that there exists a field and a surjective, flat, locally finitely presented morphism

It turns out that $Z_ x$ is a regular algebraic space as follows from the following lemma.

Lemma 66.13.7. A reduced, locally Noetherian singleton algebraic space $Z$ is regular.

**Proof.**
Let $Z$ be a reduced, locally Noetherian singleton algebraic space over a scheme $S$. Let $W \to Z$ be a surjective étale morphism where $W$ is a scheme. Let $k$ be a field and let $\mathop{\mathrm{Spec}}(k) \to Z$ be surjective, flat, and locally of finite presentation (see Lemma 66.13.3). The scheme $T = W \times _ Z \mathop{\mathrm{Spec}}(k)$ is étale over $k$ in particular regular, see Remark 66.4.1. Since $T \to W$ is locally of finite presentation, flat, and surjective it follows that $W$ is regular, see Descent, Lemma 35.16.2. By definition this means that $Z$ is regular.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)