Lemma 70.11.3. 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 70.9.6 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 67.20.8. 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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