The Stacks project

66.8 Stratifying algebraic spaces by schemes

In this section we prove that a quasi-compact and quasi-separated algebraic space has a finite stratification by locally closed subspaces each of which is a scheme and such that the glueing of the parts is by elementary distinguished squares. We first prove a slightly weaker result for reasonable algebraic spaces.

Lemma 66.8.1. Let $S$ be a scheme. Let $W \to X$ be a morphism of a scheme $W$ to an algebraic space $X$ which is flat, locally of finite presentation, separated, locally quasi-finite with universally bounded fibres. There exist reduced closed subspaces

\[ \emptyset = Z_{-1} \subset Z_0 \subset Z_1 \subset Z_2 \subset \ldots \subset Z_ n = X \]

such that with $X_ r = Z_ r \setminus Z_{r - 1}$ the stratification $X = \coprod _{r = 0, \ldots , n} X_ r$ is characterized by the following universal property: Given $g : T \to X$ the projection $W \times _ X T \to T$ is finite locally free of degree $r$ if and only if $g(|T|) \subset |X_ r|$.

Proof. Let $n$ be an integer bounding the degrees of the fibres of $W \to X$. Choose a scheme $U$ and a surjective étale morphism $U \to X$. Apply More on Morphisms, Lemma 37.40.3 to $W \times _ X U \to U$. We obtain closed subsets

\[ \emptyset = Y_{-1} \subset Y_0 \subset Y_1 \subset Y_2 \subset \ldots \subset Y_ n = U \]

characterized by the property stated in the lemma for the morphism $W \times _ X U \to U$. Clearly, the formation of these closed subsets commutes with base change. Setting $R = U \times _ X U$ with projection maps $s, t : R \to U$ we conclude that

\[ s^{-1}(Y_ r) = t^{-1}(Y_ r) \]

as closed subsets of $R$. In other words the closed subsets $Y_ r \subset U$ are $R$-invariant. This means that $|Y_ r|$ is the inverse image of a closed subset $Z_ r \subset |X|$. Denote $Z_ r \subset X$ also the reduced induced algebraic space structure, see Properties of Spaces, Definition 64.12.5.

Let $g : T \to X$ be a morphism of algebraic spaces. Choose a scheme $V$ and a surjective étale morphism $V \to T$. To prove the final assertion of the lemma it suffices to prove the assertion for the composition $V \to X$ (by our definition of finite locally free morphisms, see Morphisms of Spaces, Section 65.46). Similarly, the morphism of schemes $W \times _ X V \to V$ is finite locally free of degree $r$ if and only if the morphism of schemes

\[ W \times _ X (U \times _ X V) \longrightarrow U \times _ X V \]

is finite locally free of degree $r$ (see Descent, Lemma 35.20.30). By construction this happens if and only if $|U \times _ X V| \to |U|$ maps into $|Y_ r|$, which is true if and only if $|V| \to |X|$ maps into $|Z_ r|$. $\square$

Lemma 66.8.2. Let $S$ be a scheme. Let $W \to X$ be a morphism of a scheme $W$ to an algebraic space $X$ which is flat, locally of finite presentation, separated, and locally quasi-finite. Then there exist open subspaces

\[ X = X_0 \supset X_1 \supset X_2 \supset \ldots \]

such that a morphism $\mathop{\mathrm{Spec}}(k) \to X$ factors through $X_ d$ if and only if $W \times _ X \mathop{\mathrm{Spec}}(k)$ has degree $\geq d$ over $k$.

Proof. Choose a scheme $U$ and a surjective étale morphism $U \to X$. Apply More on Morphisms, Lemma 37.40.5 to $W \times _ X U \to U$. We obtain open subschemes

\[ U = U_0 \supset U_1 \supset U_2 \supset \ldots \]

characterized by the property stated in the lemma for the morphism $W \times _ X U \to U$. Clearly, the formation of these closed subsets commutes with base change. Setting $R = U \times _ X U$ with projection maps $s, t : R \to U$ we conclude that

\[ s^{-1}(U_ d) = t^{-1}(U_ d) \]

as open subschemes of $R$. In other words the open subschemes $U_ d \subset U$ are $R$-invariant. This means that $U_ d$ is the inverse image of an open subspace $X_ d \subset X$ (Properties of Spaces, Lemma 64.12.2). $\square$

Lemma 66.8.3. Let $S$ be a scheme. Let $X$ be a quasi-compact algebraic space over $S$. There exist open subspaces

\[ \ldots \subset U_4 \subset U_3 \subset U_2 \subset U_1 = X \]

with the following properties:

  1. setting $T_ p = U_ p \setminus U_{p + 1}$ (with reduced induced subspace structure) there exists a separated scheme $V_ p$ and a surjective étale morphism $f_ p : V_ p \to U_ p$ such that $f_ p^{-1}(T_ p) \to T_ p$ is an isomorphism,

  2. if $x \in |X|$ can be represented by a quasi-compact morphism $\mathop{\mathrm{Spec}}(k) \to X$ from a field, then $x \in T_ p$ for some $p$.

Proof. By Properties of Spaces, Lemma 64.6.3 we can choose an affine scheme $U$ and a surjective étale morphism $U \to X$. For $p \geq 0$ set

\[ W_ p = U \times _ X \ldots \times _ X U \setminus \text{all diagonals} \]

where the fibre product has $p$ factors. Since $U$ is separated, the morphism $U \to X$ is separated and all fibre products $U \times _ X \ldots \times _ X U$ are separated schemes. Since $U \to X$ is separated the diagonal $U \to U \times _ X U$ is a closed immersion. Since $U \to X$ is étale the diagonal $U \to U \times _ X U$ is an open immersion, see Morphisms of Spaces, Lemmas 65.39.10 and 65.38.9. Similarly, all the diagonal morphisms are open and closed immersions and $W_ p$ is an open and closed subscheme of $U \times _ X \ldots \times _ X U$. Moreover, the morphism

\[ U \times _ X \ldots \times _ X U \longrightarrow U \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} \ldots \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} U \]

is locally quasi-finite and separated (Morphisms of Spaces, Lemma 65.4.5) and its target is an affine scheme. Hence every finite set of points of $U \times _ X \ldots \times _ X U$ is contained in an affine open, see More on Morphisms, Lemma 37.40.1. Therefore, the same is true for $W_ p$. There is a free action of the symmetric group $S_ p$ on $W_ p$ over $X$ (because we threw out the fix point locus from $U \times _ X \ldots \times _ X U$). By the above and Properties of Spaces, Proposition 64.14.1 the quotient $V_ p = W_ p/S_ p$ is a scheme. Since the action of $S_ p$ on $W_ p$ was over $X$, there is a morphism $V_ p \to X$. Since $W_ p \to X$ is étale and since $W_ p \to V_ p$ is surjective étale, it follows that also $V_ p \to X$ is étale, see Properties of Spaces, Lemma 64.16.3. Observe that $V_ p$ is a separated scheme by Properties of Spaces, Lemma 64.14.3.

We let $U_ p \subset X$ be the open subspace which is the image of $V_ p \to X$. By construction a morphism $\mathop{\mathrm{Spec}}(k) \to X$ with $k$ algebraically closed, factors through $U_ p$ if and only if $U \times _ X \mathop{\mathrm{Spec}}(k)$ has $\geq p$ points; as usual observe that $U \times _ X \mathop{\mathrm{Spec}}(k)$ is scheme theoretically a disjoint union of (possibly infinitely many) copies of $\mathop{\mathrm{Spec}}(k)$, see Remark 66.4.1. It follows that the $U_ p$ give a filtration of $X$ as stated in the lemma. Moreover, our morphism $\mathop{\mathrm{Spec}}(k) \to X$ factors through $T_ p$ if and only if $U \times _ X \mathop{\mathrm{Spec}}(k)$ has exactly $p$ points. In this case we see that $V_ p \times _ X \mathop{\mathrm{Spec}}(k)$ has exactly one point. Set $Z_ p = f_ p^{-1}(T_ p) \subset V_ p$. This is a closed subscheme of $V_ p$. Then $Z_ p \to T_ p$ is an étale morphism between algebraic spaces which induces a bijection on $k$-valued points for any algebraically closed field $k$. To be sure this implies that $Z_ p \to T_ p$ is universally injective, whence an open immersion by Morphisms of Spaces, Lemma 65.51.2 hence an isomorphism and (1) has been proved.

Let $x : \mathop{\mathrm{Spec}}(k) \to X$ be a quasi-compact morphism where $k$ is a field. Then the composition $\mathop{\mathrm{Spec}}(\overline{k}) \to \mathop{\mathrm{Spec}}(k) \to X$ is quasi-compact as well (Morphisms of Spaces, Lemma 65.8.5). In this case the scheme $U \times _ X \mathop{\mathrm{Spec}}(\overline{k})$ is quasi-compact. In view of the fact (seen above) that it is a disjoint union of copies of $\mathop{\mathrm{Spec}}(\overline{k})$ we find that it has finitely many points. If the number of points is $p$, then we see that indeed $x \in T_ p$ and the proof is finished. $\square$

Lemma 66.8.4. Let $S$ be a scheme. Let $X$ be a quasi-compact, reasonable algebraic space over $S$. There exist an integer $n$ and open subspaces

\[ \emptyset = U_{n + 1} \subset U_ n \subset U_{n - 1} \subset \ldots \subset U_1 = X \]

with the following property: setting $T_ p = U_ p \setminus U_{p + 1}$ (with reduced induced subspace structure) there exists a separated scheme $V_ p$ and a surjective étale morphism $f_ p : V_ p \to U_ p$ such that $f_ p^{-1}(T_ p) \to T_ p$ is an isomorphism.

Proof. The proof of this lemma is identical to the proof of Lemma 66.8.3. Let $n$ be an integer bounding the degrees of the fibres of $U \to X$ which exists as $X$ is reasonable, see Definition 66.6.1. Then we see that $U_{n + 1} = \emptyset $ and the proof is complete. $\square$

Lemma 66.8.5. Let $S$ be a scheme. Let $X$ be a quasi-compact, reasonable algebraic space over $S$. There exist an integer $n$ and open subspaces

\[ \emptyset = U_{n + 1} \subset U_ n \subset U_{n - 1} \subset \ldots \subset U_1 = X \]

such that each $T_ p = U_ p \setminus U_{p + 1}$ (with reduced induced subspace structure) is a scheme.

Proof. Immediate consequence of Lemma 66.8.4. $\square$

The following result is almost identical to [Proposition 5.7.8, GruRay].

reference

Lemma 66.8.6. Let $X$ be a quasi-compact and quasi-separated algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. There exist an integer $n$ and open subspaces

\[ \emptyset = U_{n + 1} \subset U_ n \subset U_{n - 1} \subset \ldots \subset U_1 = X \]

with the following property: setting $T_ p = U_ p \setminus U_{p + 1}$ (with reduced induced subspace structure) there exists a quasi-compact separated scheme $V_ p$ and a surjective étale morphism $f_ p : V_ p \to U_ p$ such that $f_ p^{-1}(T_ p) \to T_ p$ is an isomorphism.

Proof. The proof of this lemma is identical to the proof of Lemma 66.8.3. Observe that a quasi-separated space is reasonable, see Lemma 66.5.1 and Definition 66.6.1. Hence we find that $U_{n + 1} = \emptyset $ as in Lemma 66.8.4. At the end of the argument we add that since $X$ is quasi-separated the schemes $U \times _ X \ldots \times _ X U$ are all quasi-compact. Hence the schemes $W_ p$ are quasi-compact. Hence the quotients $V_ p = W_ p/S_ p$ by the symmetric group $S_ p$ are quasi-compact schemes. $\square$

The following lemma probably belongs somewhere else.

Lemma 66.8.7. Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$. Let $E \subset |X|$ be a subset. Then $E$ is étale locally constructible (Properties of Spaces, Definition 64.8.2) if and only if $E$ is a locally constructible subset of the topological space $|X|$ (Topology, Definition 5.15.1).

Proof. Assume $E \subset |X|$ is a locally constructible subset of the topological space $|X|$. Let $f : U \to X$ be an étale morphism where $U$ is a scheme. We have to show that $f^{-1}(E)$ is locally constructible in $U$. The question is local on $U$ and $X$, hence we may assume that $X$ is quasi-compact, $E \subset |X|$ is constructible, and $U$ is affine. In this case $U \to X$ is quasi-compact, hence $f : |U| \to |X|$ is quasi-compact. Observe that retrocompact opens of $|X|$, resp. $U$ are the same thing as quasi-compact opens of $|X|$, resp. $U$, see Topology, Lemma 5.27.1. Thus $f^{-1}(E)$ is constructible by Topology, Lemma 5.15.3.

Conversely, assume $E$ is étale locally constructible. We want to show that $E$ is locally constructible in the topological space $|X|$. The question is local on $X$, hence we may assume that $X$ is quasi-compact as well as quasi-separated. We will show that in this case $E$ is constructible in $|X|$. Choose open subspaces

\[ \emptyset = U_{n + 1} \subset U_ n \subset U_{n - 1} \subset \ldots \subset U_1 = X \]

and surjective étale morphisms $f_ p : V_ p \to U_ p$ inducing isomorphisms $f_ p^{-1}(T_ p) \to T_ p = U_ p \setminus U_{p + 1}$ where $V_ p$ is a quasi-compact separated scheme as in Lemma 66.8.6. By definition the inverse image $E_ p \subset V_ p$ of $E$ is locally constructible in $V_ p$. Then $E_ p$ is constructible in $V_ p$ by Properties, Lemma 28.2.5. Thus $E_ p \cap |f_ p^{-1}(T_ p)| = E \cap |T_ p|$ is constructible in $|T_ p|$ by Topology, Lemma 5.15.7 (observe that $V_ p \setminus f_ p^{-1}(T_ p)$ is quasi-compact as it is the inverse image of the quasi-compact space $U_{p + 1}$ by the quasi-compact morphism $f_ p$). Thus

\[ E = (|T_ n| \cap E) \cup (|T_{n - 1}| \cap E) \cup \ldots \cup (|T_1| \cap E) \]

is constructible by Topology, Lemma 5.15.14. Here we use that $|T_ p|$ is constructible in $|X|$ which is clear from what was said above. $\square$


Comments (0)


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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0A4I. Beware of the difference between the letter 'O' and the digit '0'.