Lemma 99.15.3 (0D52)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 99: Quot and Hilbert Spaces
Section 99.15: The stack of curves
Lemma 99.15.3 (cite)

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Lemma 99.15.3. The category $\mathcal{C}\! \mathit{urves}$ is a stack in groupoids over $\mathit{Sch}_{fppf}$ .

Proof. Using the embedding (99.15.1.1), the description of the image, and the corresponding fact for $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ (Lemma 99.13.3) this reduces to the following statement: Given an object $X \to S$ of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ and an fppf covering $\{ S_ i \to S\} _{i \in I}$ the following are equivalent:

$X \to S$ has relative dimension $\leq 1$ , and
for each $i$ the base change $X_ i \to S_ i$ has relative dimension $\leq 1$ .

This follows from Morphisms of Spaces, Lemma 67.34.3. $\square$

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