69.16 Finite cover by a scheme

As an application of the limit results of this chapter, we prove that given any quasi-compact and quasi-separated algebraic space $X$, there is a scheme $Y$ and a surjective, finite morphism $Y \to X$. We will rely on the already proven result that we can find a finite integral cover by a scheme, which was proved in Decent Spaces, Section 67.9.

Proposition 69.16.1. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.

1. There exists a surjective finite morphism $Y \to X$ of finite presentation where $Y$ is a scheme,

2. given a surjective étale morphism $U \to X$ we may choose $Y \to X$ such that for every $y \in Y$ there is an open neighbourhood $V \subset Y$ such that $V \to X$ factors through $U$.

Proof. Part (1) is the special case of (2) with $U = X$. Let $Y \to X$ be as in Decent Spaces, Lemma 67.9.2. Choose a finite affine open covering $Y = \bigcup V_ j$ such that $V_ j \to X$ factors through $U$. We can write $Y = \mathop{\mathrm{lim}}\nolimits Y_ i$ with $Y_ i \to X$ finite and of finite presentation, see Lemma 69.11.2. For large enough $i$ the algebraic space $Y_ i$ is a scheme, see Lemma 69.5.11. For large enough $i$ we can find affine opens $V_{i, j} \subset Y_ i$ whose inverse image in $Y$ recovers $V_ j$, see Lemma 69.5.7. For even larger $i$ the morphisms $V_ j \to U$ over $X$ come from morphisms $V_{i, j} \to U$ over $X$, see Proposition 69.3.10. This finishes the proof. $\square$

Lemma 69.16.2. Let $S$ be a scheme. Let $f : X \to Y$ be an integral morphism of algebraic spaces over $S$. Assume $Y$ quasi-compact and quasi-separated. Let $V \subset Y$ be a quasi-compact open subspace such that $f^{-1}(V) \to V$ is finite and of finite presentation. Then $X$ can be written as a directed limit $X = \mathop{\mathrm{lim}}\nolimits X_ i$ where $f_ i : X_ i \to Y$ are finite and of finite presentation such that $f^{-1}(V) \to f_ i^{-1}(V)$ is an isomorphism for all $i$.

Proof. This lemma is a slight refinement of Proposition 69.16.1. Consider the integral quasi-coherent $\mathcal{O}_ Y$-algebra $\mathcal{A} = f_*\mathcal{O}_ X$. In the next paragraph, we will write $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ as a directed colimit of finite and finitely presented $\mathcal{O}_ Y$-algebras $\mathcal{A}_ i$ such that $\mathcal{A}_ i|_ V = \mathcal{A}|_ V$. Having done this we set $X_ i = \underline{\mathop{\mathrm{Spec}}}_ Y(\mathcal{A}_ i)$, see Morphisms of Spaces, Definition 66.20.8. By construction $X_ i \to Y$ is finite and of finite presentation, $X = \mathop{\mathrm{lim}}\nolimits X_ i$, and $f_ i^{-1}(V) = f^{-1}(V)$.

The proof of the assertion on algebras is similar to the proof of part (2) of Lemma 69.9.7. First, write $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ as a colimit of finitely presented $\mathcal{O}_ Y$-modules using Lemma 69.9.1. Since $\mathcal{A}|_ V$ is a finite type $\mathcal{O}_ V$-module we may and do assume that $\mathcal{F}_ i|_ V \to \mathcal{A}|_ V$ is surjective for all $i$. For each $i$, let $\mathcal{J}_ i$ be the kernel of the map

$\text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_ i) \longrightarrow \mathcal{A}$

For $i' \geq i$ there is an induced map $\mathcal{J}_ i \to \mathcal{J}_{i'}$. We have $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_ i)/\mathcal{J}_ i$. Moreover, the quasi-coherent $\mathcal{O}_ X$-algebras $\text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_ i)/\mathcal{J}_ i$ are finite (as finite type quasi-coherent subalgebras of the integral quasi-coherent $\mathcal{O}_ Y$-algebra $\mathcal{A}$ over $\mathcal{O}_ X$). The restriction of $\text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_ i)/\mathcal{J}_ i$ to $V$ is $\mathcal{A}|_ V$ by the surjectivity above. Hence $\mathcal{J}_ i|_ V$ is finitely generated as an ideal sheaf of $\text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_ i)|_ V$ due to the fact that $\mathcal{A}|_ V$ is finitely presented as an $\mathcal{O}_ Y$-algebra. Write $\mathcal{J}_ i = \mathop{\mathrm{colim}}\nolimits \mathcal{E}_{ik}$ as a colimit of finitely presented $\mathcal{O}_ X$-modules. We may and do assume that $\mathcal{E}_{ik}|_ V$ generates $\mathcal{J}_ i|_ V$ as a sheaf of ideal of $\text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_ i)|_ V$ by the statement on finite generation above. Given $i' \geq i$ and $k$ there exists a $k'$ such that we have a map $\mathcal{E}_{ik} \to \mathcal{E}_{i'k'}$ making

$\xymatrix{ \mathcal{J}_ i \ar[r] & \mathcal{J}_{i'} \\ \mathcal{E}_{ik} \ar[u] \ar[r] & \mathcal{E}_{i'k'} \ar[u] }$

commute. This follows from Cohomology of Spaces, Lemma 68.5.3. This induces a map

$\mathcal{A}_{ik} = \text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_ i)/(\mathcal{E}_{ik}) \longrightarrow \text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_{i'})/(\mathcal{E}_{i'k'}) = \mathcal{A}_{i'k'}$

where $(\mathcal{E}_{ik})$ denotes the ideal generated by $\mathcal{E}_{ik}$. The quasi-coherent $\mathcal{O}_ X$-algebras $\mathcal{A}_{ki}$ are of finite presentation and finite for $k$ large enough (see proof of Lemma 69.9.6). Moreover we have $\mathcal{A}_{ik}|_ V = \mathcal{A}|_ V$ by construction. Finally, we have

$\mathop{\mathrm{colim}}\nolimits \mathcal{A}_{ik} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i = \mathcal{A}$

Namely, the first equality was shown in the proof of Lemma 69.9.6 and the second equality because $\mathcal{A}$ is the colimit of the modules $\mathcal{F}_ i$. $\square$

Lemma 69.16.3. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$ such that $|X|$ has finitely many irreducible components.

1. There exists a surjective finite morphism $f : Y \to X$ of finite presentation where $Y$ is a scheme such that $f$ is finite étale over a quasi-compact dense open $U \subset X$,

2. given a surjective étale morphism $V \to X$ we may choose $Y \to X$ such that for every $y \in Y$ there is an open neighbourhood $W \subset Y$ such that $W \to X$ factors through $V$.

Proof. Part (1) is the special case of (2) with $V = X$.

Proof of (2). Let $\pi : Y \to X$ be as in Decent Spaces, Lemma 67.9.3 and let $U \subset X$ be a quasi-compact dense open such that $\pi ^{-1}(U) \to U$ is finite étale. Choose a finite affine open covering $Y = \bigcup W_ j$ such that $W_ j \to X$ factors through $V$. We can write $Y = \mathop{\mathrm{lim}}\nolimits Y_ i$ with $\pi _ i : Y_ i \to X$ finite and of finite presentation such that $\pi ^{-1}(U) \to \pi _ i^{-1}(U)$ is an isomorphism, see Lemma 69.16.2. For large enough $i$ the algebraic space $Y_ i$ is a scheme, see Lemma 69.5.11. For large enough $i$ we can find affine opens $W_{i, j} \subset Y_ i$ whose inverse image in $Y$ recovers $W_ j$, see Lemma 69.5.7. For even larger $i$ the morphisms $W_ j \to V$ over $X$ come from morphisms $W_{i, j} \to U$ over $X$, see Proposition 69.3.10. This finishes the proof. $\square$

Lemma 69.16.4. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. There exists a $t \geq 0$ and closed subspaces

$X \supset Z_0 \supset Z_1 \supset \ldots \supset Z_ t = \emptyset$

such that $Z_ i \to X$ is of finite presentation, $Z_0 \subset X$ is a thickening, and for each $i = 0, \ldots t - 1$ there exists a scheme $Y_ i$, a surjective, finite, and finitely presented morphism $Y_ i \to Z_ i$ which is finite étale over $Z_ i \setminus Z_{i + 1}$.

Proof. We may view $X$ as an algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Spaces, Definition 64.16.2 and Properties of Spaces, Definition 65.3.1. Thus we may apply Proposition 69.8.1. It follows that we can find an affine morphism $X \to X_0$ with $X_0$ of finite presentation over $\mathbf{Z}$. If we can prove the lemma for $X_0$, then we can pull back the stratification and the morphisms to $X$ and get the result for $X$; some details omitted. This reduces us to the case discussed in the next paragraph.

Assume $X$ is of finite presentation over $\mathbf{Z}$. Then $X$ is Noetherian and $|X|$ is a Noetherian topological space (with finitely many irreducible components) of finite dimension. Hence we may use induction on $\dim (|X|)$. Any finite morphism towards $X$ is of finite presentation, so we can ignore that requirement in the rest of the proof. By Lemma 69.16.3 there exists a surjective finite morphism $Y \to X$ which is finite étale over a dense open $U \subset X$. Set $Z_0 = X$ and let $Z_1 \subset X$ be the reduced closed subspace with $|Z_1| = |X| \setminus |U|$. By induction we find an integer $t \geq 0$ and a filtration

$Z_1 \supset Z_{1, 0} \supset Z_{1, 1} \supset \ldots \supset Z_{1, t} = \emptyset$

by closed subspaces, where $Z_{1, 0} \to Z_1$ is a thickening and there exist finite surjective morphisms $Y_{1, i} \to Z_{1, i}$ which are finite étale over $Z_{1, i} \setminus Z_{1, i + 1}$. Since $Z_1$ is reduced, we have $Z_1 = Z_{1, 0}$. Hence we can set $Z_ i = Z_{1, i - 1}$ and $Y_ i = Y_{1, i - 1}$ for $i \geq 1$ and the lemma is proved. $\square$

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