The Stacks project

Lemma 32.13.3. Let $f : X \to S$ be a proper morphism with $S$ quasi-compact and quasi-separated. Then there exists a directed set $I$, an inverse system $(f_ i : X_ i \to S_ i)$ of morphisms of schemes over $I$, such that the transition morphisms $X_ i \to X_{i'}$ and $S_ i \to S_{i'}$ are affine, such that $f_ i$ is proper, such that $S_ i$ is of finite type over $\mathbf{Z}$, and such that $(X \to S) = \mathop{\mathrm{lim}}\nolimits (X_ i \to S_ i)$.

Proof. By Lemma 32.13.2 we can write $X = \mathop{\mathrm{lim}}\nolimits _{k \in K} X_ k$ with $X_ k \to S$ proper and of finite presentation. Next, by absolute Noetherian approximation (Proposition 32.5.4) we can write $S = \mathop{\mathrm{lim}}\nolimits _{j \in J} S_ j$ with $S_ j$ of finite type over $\mathbf{Z}$. For each $k$ there exists a $j$ and a morphism $X_{k, j} \to S_ j$ of finite presentation with $X_ k \cong S \times _{S_ j} X_{k, j}$ as schemes over $S$, see Lemma 32.10.1. After increasing $j$ we may assume $X_{k, j} \to S_ j$ is proper, see Lemma 32.13.1. The set $I$ will be consist of these pairs $(k, j)$ and the corresponding morphism is $X_{k, j} \to S_ j$. For every $k' \geq k$ we can find a $j' \geq j$ and a morphism $X_{j', k'} \to X_{j, k}$ over $S_{j'} \to S_ j$ whose base change to $S$ gives the morphism $X_{k'} \to X_ k$ (follows again from Lemma 32.10.1). These morphisms form the transition morphisms of the system. Some details omitted. $\square$

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