The Stacks project

Lemma 105.12.4. Let $f : \mathcal{X} \to Y$ be a morphism from an algebraic stack to an algebraic space. If for every affine scheme $Y'$ and flat morphism $Y' \to Y$ the base change $f' : Y' \times _ Y \mathcal{X} \to Y'$ is a categorical moduli space, then $f$ is a uniform categorical moduli space.

Proof. Choose an ├ętale covering $\{ Y_ i \to Y\} $ where $Y_ i$ is an affine scheme. For each $i$ and $j$ choose a affine open covering $Y_ i \times _ Y Y_ j = \bigcup Y_{ijk}$. Set $\mathcal{X}_ i = Y_ i \times _ Y \mathcal{X}$ and $\mathcal{X}_{ijk} = Y_{ijk} \times _ Y \mathcal{X}$. Let $g : \mathcal{X} \to W$ be a morphism towards an algebraic space. Then we consider the diagram

\[ \xymatrix{ \mathcal{X}_ i \ar[r] \ar[d] & \mathcal{X} \ar[d] \ar[r]_ g & W \\ Y_ i \ar[r] \ar@{..>}[rru] & Y } \]

The assumption that $\mathcal{X}_ i \to Y_ i$ is a categorical moduli space, produces a unique dotted arrow $h_ i : Y_ i \to W$. The assumption that $\mathcal{X}_{ijk} \to Y_{ijk}$ is a categorical moduli space, implies the restriction of $h_ i$ and $h_ j$ to $Y_{ijk}$ are equal. Hence $h_ i$ and $h_ j$ agree on $Y_ i \times _ Y Y_ j$. Since $Y = \coprod Y_ i / \coprod Y_ i \times _ Y Y_ j$ (by Spaces, Section 64.9) we conclude that there is a unique morphism $Y \to W$ through which $g$ factors. Thus $f$ is a categorical moduli space. The same argument applies after a flat base change, hence $f$ is a uniform categorical moduli space. $\square$

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