Lemma 102.5.1. Let $I$ be a directed set. Let $(X_ i, f_{ii'})$ be an inverse system of algebraic spaces over $I$. Assume
the morphisms $f_{ii'} : X_ i \to X_{i'}$ are affine,
the spaces $X_ i$ are quasi-compact and quasi-separated.
102.5 Descending relative objects
This section is the analogue of Limits of Spaces, Section 70.7.
Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$. If $\mathcal{X}$ is an algebraic stack of finite presentation over $X$, then there exists an $i \in I$ and an algebraic stack $\mathcal{X}_ i$ of finite presentation over $X_ i$ with $\mathcal{X} \cong \mathcal{X}_ i \times _{X_ i} X$ as algebraic stacks over $X$.
Proof. By Morphisms of Stacks, Definition 101.27.1 the morphism $\mathcal{X} \to X$ is quasi-compact, locally of finite presentation, and quasi-separated. Since $X$ is quasi-compact and $\mathcal{X} \to X$ is quasi-compact, we see that $\mathcal{X}$ is quasi-compact (Morphisms of Stacks, Definition 101.7.2). Hence we can find an affine scheme $U$ and a surjective smooth morphism $U \to \mathcal{X}$ (Properties of Stacks, Lemma 100.6.2). Set $R = U \times _\mathcal {X} U$. We obtain a smooth groupoid in algebraic spaces $(U, R, s, t, c)$ over $X$ such that $\mathcal{X} = [U/R]$, see Algebraic Stacks, Lemma 94.16.2. Since $\mathcal{X} \to X$ is quasi-separated and $X$ is quasi-separated we see that $\mathcal{X}$ is quasi-separated (Morphisms of Stacks, Lemma 101.4.10). Thus $R \to U \times U$ is quasi-compact and quasi-separated (Morphisms of Stacks, Lemma 101.4.7) and hence $R$ is a quasi-separated and quasi-compact algebraic space. On the other hand $U \to X$ is locally of finite presentation and hence also $R \to X$ is locally of finite presentation (because $s : R \to U$ is smooth hence locally of finite presentation). Thus $(U, R, s, t, c)$ is a groupoid object in the category of algebraic spaces which are of finite presentation over $X$. By Limits of Spaces, Lemma 70.7.1 there exists an $i$ and a groupoid in algebraic spaces $(U_ i, R_ i, s_ i, t_ i, c_ i)$ over $X_ i$ whose pullback to $X$ is isomorphic to $(U, R, s, t, c)$. After increasing $i$ we may assume that $s_ i$ and $t_ i$ are smooth, see Limits of Spaces, Lemma 70.6.3. The quotient stack $\mathcal{X}_ i = [U_ i/R_ i]$ is an algebraic stack (Algebraic Stacks, Theorem 94.17.3).
There is a morphism $[U/R] \to [U_ i/R_ i]$, see Groupoids in Spaces, Lemma 78.21.1. We claim that combined with the morphisms $[U/R] \to X$ and $[U_ i/R_ i] \to X_ i$ (Groupoids in Spaces, Lemma 78.20.2) we obtain an isomorphism (i.e., equivalence)
\[ [U/R] \longrightarrow [U_ i/R_ i] \times _{X_ i} X \]
The corresponding map
\[ [U/_{\! p}R] \longrightarrow [U_ i/_{\! p}R_ i] \times _{X_ i} X \]
on the level of “presheaves of groupoids” as in Groupoids in Spaces, Equation (78.20.0.1) is an isomorphism. Thus the claim follows from the fact that stackification commutes with fibre products, see Stacks, Lemma 8.8.4. $\square$
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