The Stacks project

Lemma 70.10.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that

  1. $X$ is quasi-compact and quasi-separated, and

  2. $Y$ is quasi-separated.

Then $X = \mathop{\mathrm{lim}}\nolimits X_ i$ is a limit of a directed inverse system of algebraic spaces $X_ i$ of finite presentation over $Y$ with affine transition morphisms over $Y$.

Proof. Since $|f|(|X|)$ is quasi-compact we may replace $Y$ by a quasi-compact open subspace whose set of points contains $|f|(|X|)$. Hence we may assume $Y$ is quasi-compact as well. By Lemma 70.10.1 we can write $(X \to Y) = \mathop{\mathrm{lim}}\nolimits (X_ i \to Y_ i)$ for some directed inverse system of morphisms of finite type schemes over $\mathbf{Z}$ with affine transition morphisms. Since limits commute with limits (Categories, Lemma 4.14.10) we have $X = \mathop{\mathrm{lim}}\nolimits X_ i \times _{Y_ i} Y$. For $i \geq i'$ the transition morphism $X_ i \times _{Y_ i} Y \to X_{i'} \times _{Y_{i'}} Y$ is affine as the composition

\[ X_ i \times _{Y_ i} Y \to X_ i \times _{Y_{i'}} Y \to X_{i'} \times _{Y_{i'}} Y \]

where the first morphism is a closed immersion (by Morphisms of Spaces, Lemma 67.4.5) and the second is a base change of an affine morphism (Morphisms of Spaces, Lemma 67.20.5) and the composition of affine morphisms is affine (Morphisms of Spaces, Lemma 67.20.4). The morphisms $f_ i$ are of finite presentation (Morphisms of Spaces, Lemmas 67.28.7 and 67.28.9) and hence the base changes $X_ i \times _{f_ i, Y_ i} Y \to Y$ are of finite presentation (Morphisms of Spaces, Lemma 67.28.3). $\square$

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