The Stacks project

Our proof follows closely the proof given in [Theorem 1.2.2, CLO].

Proposition 70.8.1. Let $X$ be a quasi-compact and quasi-separated algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. There exist a directed set $I$ and an inverse system of algebraic spaces $(X_ i, f_{ii'})$ over $I$ such that

  1. the transition morphisms $f_{ii'}$ are affine

  2. each $X_ i$ is quasi-separated and of finite type over $\mathbf{Z}$, and

  3. $X = \mathop{\mathrm{lim}}\nolimits X_ i$.

Proof. We apply Decent Spaces, Lemma 68.8.6 to get open subspaces $U_ p \subset X$, schemes $V_ p$, and morphisms $f_ p : V_ p \to U_ p$ with properties as stated. Note that $f_ n : V_ n \to U_ n$ is an étale morphism of algebraic spaces whose restriction to the inverse image of $T_ n = (V_ n)_{red}$ is an isomorphism. Hence $f_ n$ is an isomorphism, for example by Morphisms of Spaces, Lemma 67.51.2. In particular $U_ n$ is a quasi-compact and separated scheme. Thus we can write $U_ n = \mathop{\mathrm{lim}}\nolimits U_{n, i}$ as a directed limit of schemes of finite type over $\mathbf{Z}$ with affine transition morphisms, see Limits, Proposition 32.5.4. Thus, applying descending induction on $p$, we see that we have reduced to the problem posed in the following paragraph.

Here we have $U \subset X$, $U = \mathop{\mathrm{lim}}\nolimits U_ i$, $Z \subset X$, and $f : V \to X$ with the following properties

  1. $X$ is a quasi-compact and quasi-separated algebraic space,

  2. $V$ is a quasi-compact and separated scheme,

  3. $U \subset X$ is a quasi-compact open subspace,

  4. $(U_ i, g_{ii'})$ is a directed inverse system of quasi-separated algebraic spaces of finite type over $\mathbf{Z}$ with affine transition morphisms whose limit is $U$,

  5. $Z \subset X$ is a closed subspace such that $|X| = |U| \amalg |Z|$,

  6. $f : V \to X$ is a surjective étale morphism such that $f^{-1}(Z) \to Z$ is an isomorphism.

Problem: Show that the conclusion of the proposition holds for $X$.

Note that $W = f^{-1}(U) \subset V$ is a quasi-compact open subscheme étale over $U$. Hence we may apply Lemmas 70.7.1 and 70.6.2 to find an index $0 \in I$ and an étale morphism $W_0 \to U_0$ of finite presentation whose base change to $U$ produces $W$. Setting $W_ i = W_0 \times _{U_0} U_ i$ we see that $W = \mathop{\mathrm{lim}}\nolimits _{i \geq 0} W_ i$. After increasing $0$ we may assume the $W_ i$ are schemes, see Lemma 70.5.11. Moreover, $W_ i$ is of finite type over $\mathbf{Z}$.

Apply Limits, Lemma 32.5.3 to $W = \mathop{\mathrm{lim}}\nolimits _{i \geq 0} W_ i$ and the inclusion $W \subset V$. Replace $I$ by the directed set $J$ found in that lemma. This allows us to write $V$ as a directed limit $V = \mathop{\mathrm{lim}}\nolimits V_ i$ of finite type schemes over $\mathbf{Z}$ with affine transition maps such that each $V_ i$ contains $W_ i$ as an open subscheme (compatible with transition morphisms). For each $i$ we can form the push out

\[ \xymatrix{ W_ i \ar[r] \ar[d]_\Delta & V_ i \ar[d] \\ W_ i \times _{U_ i} W_ i \ar[r] & R_ i } \]

in the category of schemes. Namely, the left vertical and upper horizontal arrows are open immersions of schemes. In other words, we can construct $R_ i$ as the glueing of $V_ i$ and $W_ i \times _{U_ i} W_ i$ along the common open $W_ i$ (see Schemes, Section 26.14). Note that the étale projection maps $W_ i \times _{U_ i} W_ i \to W_ i$ extend to étale morphisms $s_ i, t_ i : R_ i \to V_ i$. It is clear that the morphism $j_ i = (t_ i, s_ i) : R_ i \to V_ i \times V_ i$ is an étale equivalence relation on $V_ i$. Note that $W_ i \times _{U_ i} W_ i$ is quasi-compact (as $U_ i$ is quasi-separated and $W_ i$ quasi-compact) and $V_ i$ is quasi-compact, hence $R_ i$ is quasi-compact. For $i \geq i'$ the diagram
\begin{equation} \label{spaces-limits-equation-cartesian} \vcenter { \xymatrix{ R_ i \ar[r] \ar[d]_{s_ i} & R_{i'} \ar[d]^{s_{i'}} \\ V_ i \ar[r] & V_{i'} } } \end{equation}

is cartesian because

\[ (W_{i'} \times _{U_{i'}} W_{i'}) \times _{U_{i'}} U_ i = W_{i'} \times _{U_{i'}} U_ i \times _{U_ i} U_ i \times _{U_{i'}} W_{i'} = W_ i \times _{U_ i} W_ i. \]

Consider the algebraic space $X_ i = V_ i/R_ i$ (see Spaces, Theorem 65.10.5). As $V_ i$ is of finite type over $\mathbf{Z}$ and $R_ i$ is quasi-compact we see that $X_ i$ is quasi-separated and of finite type over $\mathbf{Z}$ (see Properties of Spaces, Lemma 66.6.5 and Morphisms of Spaces, Lemmas 67.8.6 and 67.23.4). As the construction of $R_ i$ above is compatible with transition morphisms, we obtain morphisms of algebraic spaces $X_ i \to X_{i'}$ for $i \geq i'$. The commutative diagrams

\[ \xymatrix{ V_ i \ar[r] \ar[d] & V_{i'} \ar[d] \\ X_ i \ar[r] & X_{i'} } \]

are cartesian as ( is cartesian, see Groupoids, Lemma 39.20.7. Since $V_ i \to V_{i'}$ is affine, this implies that $X_ i \to X_{i'}$ is affine, see Morphisms of Spaces, Lemma 67.20.3. Thus we can form the limit $X' = \mathop{\mathrm{lim}}\nolimits X_ i$ by Lemma 70.4.1. We claim that $X \cong X'$ which finishes the proof of the proposition.

Proof of the claim. Set $R = \mathop{\mathrm{lim}}\nolimits R_ i$. By construction the algebraic space $X'$ comes equipped with a surjective étale morphism $V \to X'$ such that

\[ V \times _{X'} V \cong R \]

(use Lemma 70.4.1). By construction $\mathop{\mathrm{lim}}\nolimits W_ i \times _{U_ i} W_ i = W \times _ U W$ and $V = \mathop{\mathrm{lim}}\nolimits V_ i$ so that $R$ is the union of $W \times _ U W$ and $V$ glued along $W$. Property (6) implies the projections $V \times _ X V \to V$ are isomorphisms over $f^{-1}(Z) \subset V$. Hence the scheme $V \times _ X V$ is the union of the opens $\Delta _{V/X}(V)$ and $W \times _ U W$ which intersect along $\Delta _{W/X}(W)$. We conclude that there exists a unique isomorphism $R \cong V \times _ X V$ compatible with the projections to $V$. Since $V \to X$ and $V \to X'$ are surjective étale we see that

\[ X = V/ V \times _ X V = V/R = V/V \times _{X'} V = X' \]

by Spaces, Lemma 65.9.1 and we win. $\square$

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