The Stacks project

Lemma 99.15.6. The stack $\mathcal{C}\! \mathit{urves}\to \mathit{Sch}_{fppf}$ is limit preserving (Artin's Axioms, Definition 98.11.1).

Proof. Using the embedding (, the description of the image, and the corresponding fact for $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ (Lemma 99.13.6) this reduces to the following statement: Let $T = \mathop{\mathrm{lim}}\nolimits T_ i$ be the limits of a directed inverse system of affine schemes. Let $i \in I$ and let $X_ i \to T_ i$ be an object of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ over $T_ i$. Assume that $T \times _{T_ i} X_ i \to T$ has relative dimension $\leq 1$. Then for some $i' \geq i$ the morphism $T_{i'} \times _{T_ i} X_ i \to T_ i$ has relative dimension $\leq 1$. This follows from Limits of Spaces, Lemma 70.6.14. $\square$

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