The Stacks project

Lemma 68.11.3. Let $S$ be a scheme. Let $f : X \to Y$ be a finite morphism of algebraic spaces over $S$. Assume $Y$ quasi-compact and quasi-separated. Then $X$ can be written as a directed limit $X = \mathop{\mathrm{lim}}\nolimits X_ i$ where the transition maps are closed immersions and the objects $X_ i$ are finite and of finite presentation over $Y$.

Proof. Consider the finite quasi-coherent $\mathcal{O}_ Y$-module $\mathcal{A} = f_*\mathcal{O}_ X$. By Lemma 68.9.6 we can write $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ as a directed colimit of finite and finitely presented $\mathcal{O}_ Y$-algebras $\mathcal{A}_ i$ with surjective transition maps. Set $X_ i = \underline{\mathop{\mathrm{Spec}}}_ Y(\mathcal{A}_ i)$, see Morphisms of Spaces, Definition 65.20.8. By construction $X_ i \to Y$ is finite and of finite presentation, the transition maps are closed immersions, and $X = \mathop{\mathrm{lim}}\nolimits X_ i$. $\square$


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