The Stacks project

Lemma 110.71.1. The stack in groupoids

\[ p'_{fp, flat, proper} : \mathcal{S}\! \mathit{paces}'_{fp, flat, proper} \longrightarrow \mathit{Sch}_{fppf} \]

whose category of sections over a scheme $S$ is the category of flat, proper, finitely presented algebraic spaces over $S$ (see Quot, Section 99.13) is not an algebraic stack.

Proof. If it was an algebraic stack, then every formal object would be effective, see Artin's Axioms, Lemma 98.9.5. The discussion above show this is not the case after base change to $\mathop{\mathrm{Spec}}(\mathbf{C})$. Hence the conclusion. $\square$

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