Lemma 69.12.2. Let $f : X \to Y$ be a proper morphism of algebraic spaces over $\mathbf{Z}$ with $Y$ quasi-compact and quasi-separated. Then there exists a directed set $I$, an inverse system $(f_ i : X_ i \to Y_ i)$ of morphisms of algebraic spaces over $I$, such that the transition morphisms $X_ i \to X_{i'}$ and $Y_ i \to Y_{i'}$ are affine, such that $f_ i$ is proper and of finite presentation, such that $Y_ i$ is of finite presentation over $\mathbf{Z}$, and such that $(X \to Y) = \mathop{\mathrm{lim}}\nolimits (X_ i \to Y_ i)$.

Proof. By Lemma 69.12.1 we can write $X = \mathop{\mathrm{lim}}\nolimits _{k \in K} X_ k$ with $X_ k \to Y$ proper and of finite presentation. Next, by absolute Noetherian approximation (Proposition 69.8.1) we can write $Y = \mathop{\mathrm{lim}}\nolimits _{j \in J} Y_ j$ with $Y_ j$ of finite presentation over $\mathbf{Z}$. For each $k$ there exists a $j$ and a morphism $X_{k, j} \to Y_ j$ of finite presentation with $X_ k \cong Y \times _{Y_ j} X_{k, j}$ as algebraic spaces over $Y$, see Lemma 69.7.1. After increasing $j$ we may assume $X_{k, j} \to Y_ j$ is proper, see Lemma 69.6.13. The set $I$ will be consist of these pairs $(k, j)$ and the corresponding morphism is $X_{k, j} \to Y_ j$. For every $k' \geq k$ we can find a $j' \geq j$ and a morphism $X_{j', k'} \to X_{j, k}$ over $Y_{j'} \to Y_ j$ whose base change to $Y$ gives the morphism $X_{k'} \to X_ k$ (follows again from Lemma 69.7.1). These morphisms form the transition morphisms of the system. Some details omitted. $\square$

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