Lemma 109.70.1. The stack in groupoids

whose category of sections over a scheme $S$ is the category of flat, proper, finitely presented algebraic spaces over $S$ (see Quot, Section 98.13) is not an algebraic stack.

Lemma 109.70.1. 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 98.13) is not an algebraic stack.

**Proof.**
If it was an algebraic stack, then every formal object would be effective, see Artin's Axioms, Lemma 97.9.5. 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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