## 105.12 Moduli spaces

This section discusses morphisms $f : \mathcal{X} \to Y$ from algebraic stacks to algebraic spaces. Under suitable hypotheses $Y$ is called a moduli space for $\mathcal{X}$. If $\mathcal{X} = [U/R]$ is a presentation, then we obtain an $R$-invariant morphism $U \to Y$ and (under suitable hypotheses) $Y$ is a quotient of the groupoid $(U, R, s, t, c)$. A discussion of the different types of quotients can be found starting with Quotients of Groupoids, Section 82.1.

Definition 105.12.1. Let $\mathcal{X}$ be an algebraic stack. Let $f : \mathcal{X} \to Y$ be a morphism to an algebraic space $Y$.

1. We say $f$ is a categorical moduli space if any morphism $\mathcal{X} \to W$ to an algebraic space $W$ factors uniquely through $f$.

2. We say $f$ is a uniform categorical moduli space if for any flat morphism $Y' \to Y$ of algebraic spaces the base change $f' : Y' \times _ Y \mathcal{X} \to Y'$ is a categorical moduli space.

Let $\mathcal{C}$ be a full subcategory of the category of algebraic spaces.

1. We say $f$ is a categorical moduli space in $\mathcal{C}$ if $Y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and any morphism $\mathcal{X} \to W$ with $W \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ factors uniquely through $f$.

2. We say is a uniform categorical moduli space in $\mathcal{C}$ if $Y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and for every flat morphism $Y' \to Y$ in $\mathcal{C}$ the base change $f' : Y' \times _ Y \mathcal{X} \to Y'$ is a categorical moduli space in $\mathcal{C}$.

By the Yoneda lemma a categorical moduli space, if it exists, is unique. Let us match this with the language introduced for quotients.

Lemma 105.12.2. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces with $s, t : R \to U$ flat and locally of finite presentation. Consider the algebraic stack $\mathcal{X} = [U/R]$. Given an algebraic space $Y$ there is a $1$-to-$1$ correspondence between morphisms $f : \mathcal{X} \to Y$ and $R$-invariant morphisms $\phi : U \to Y$.

Proof. Criteria for Representability, Theorem 96.17.2 tells us $\mathcal{X}$ is an algebraic stack. Given a morphism $f : \mathcal{X} \to Y$ we let $\phi : U \to Y$ be the composition $U \to \mathcal{X} \to Y$. Since $R = U \times _\mathcal {X} U$ (Groupoids in Spaces, Lemma 77.22.2) it is immediate that $\phi$ is $R$-invariant. Conversely, if $\phi : U \to Y$ is an $R$-invariant morphism towards an algebraic space, we obtain a morphism $f : \mathcal{X} \to Y$ by Groupoids in Spaces, Lemma 77.23.2. You can also construct $f$ from $\phi$ using the explicit description of the quotient stack in Groupoids in Spaces, Lemma 77.24.1. $\square$

Lemma 105.12.3. With assumption and notation as in Lemma 105.12.2. Then $f$ is a (uniform) categorical moduli space if and only if $\phi$ is a (uniform) categorical quotient. Similarly for moduli spaces in a full subcategory.

Proof. It is immediate from the $1$-to-$1$ correspondence established in Lemma 105.12.2 that $f$ is a categorical moduli space if and only if $\phi$ is a categorical quotient (Quotients of Groupoids, Definition 82.4.1). If $Y' \to Y$ is a morphism, then $U' = Y' \times _ Y U \to Y' \times _ Y \mathcal{X} = \mathcal{X}'$ is a surjective, flat, locally finitely presented morphism as a base change of $U \to \mathcal{X}$ (Criteria for Representability, Lemma 96.17.1). And $R' = Y' \times _ Y R$ is equal to $U' \times _{\mathcal{X}'} U'$ by transitivity of fibre products. Hence $\mathcal{X}' = [U'/R']$, see Algebraic Stacks, Remark 93.16.3. Thus the base change of our situation to $Y'$ is another situation as in the statement of the lemma. From this it immediately follows that $f$ is a uniform categorical moduli space if and only if $\phi$ is a uniform categorical quotient. $\square$

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 étale 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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