Proposition 32.5.4. Let $S$ be a quasi-compact and quasi-separated scheme. There exist a directed set $I$ and an inverse system of schemes $(S_ i, f_{ii'})$ over $I$ such that

1. the transition morphisms $f_{ii'}$ are affine

2. each $S_ i$ is of finite type over $\mathbf{Z}$, and

3. $S = \mathop{\mathrm{lim}}\nolimits _ i S_ i$.

Proof. This is a special case of Lemma 32.5.3 with $V = \emptyset$. $\square$

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