Lemma 99.13.6 (0D1I)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 99: Quot and Hilbert Spaces
Section 99.13: The stack of algebraic spaces
Lemma 99.13.6 (cite)

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Lemma 99.13.6. The stack $p'_{fp, flat, proper} : \mathcal{S}\! \mathit{paces}'_{fp, flat, proper} \to \mathit{Sch}_{fppf}$ is limit preserving (Artin's Axioms, Definition 98.11.1).

Proof. Let $T = \mathop{\mathrm{lim}}\nolimits T_ i$ be the limits of a directed inverse system of affine schemes. By Limits of Spaces, Lemma 70.7.1 the category of algebraic spaces of finite presentation over $T$ is the colimit of the categories of algebraic spaces of finite presentation over $T_ i$ . To finish the proof use that flatness and properness descends through the limit, see Limits of Spaces, Lemmas 70.6.12 and 70.6.13. $\square$

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