The Stacks project

Lemma 32.7.2. Let $f : X \to S$ be a morphism of schemes. Assume that

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

  2. $S$ is quasi-separated.

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

Proof. Since $f(X)$ is quasi-compact we may replace $S$ by a quasi-compact open containing $f(X)$. Hence we may assume $S$ is quasi-compact. By Lemma 32.7.1 we can write $(X \to S) = \mathop{\mathrm{lim}}\nolimits (X_ i \to S_ i)$ 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 $X = \mathop{\mathrm{lim}}\nolimits X_ i \times _{S_ i} S$. Let $i \geq i'$ in $I$. The morphism $X_ i \times _{S_ i} S \to X_{i'} \times _{S_{i'}} S$ is affine as the composition

\[ X_ i \times _{S_ i} S \to X_ i \times _{S_{i'}} S \to X_{i'} \times _{S_{i'}} S \]

where the first morphism is a closed immersion (by Schemes, Lemma 26.21.9) and the second is a base change of an affine morphism (Morphisms, Lemma 29.11.8) and the composition of affine morphisms is affine (Morphisms, Lemma 29.11.7). The morphisms $f_ i$ are of finite presentation (Morphisms, Lemmas 29.21.9 and 29.21.11) and hence the base changes $X_ i \times _{f_ i, S_ i} S \to S$ are of finite presentation (Morphisms, Lemma 29.21.4). $\square$

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