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 37.41.3 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 37.41.5 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 37.41.1. 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].

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. $\square$

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