106.12 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 83.1.
Definition 106.12.1. Let \mathcal{X} be an algebraic stack. Let f : \mathcal{X} \to Y be a morphism to an algebraic space Y.
We say f is a categorical moduli space if any morphism \mathcal{X} \to W to an algebraic space W factors uniquely through f.
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.
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.
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 106.12.2. 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 97.17.2 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 78.22.2) 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 78.23.2. You can also construct f from \phi using the explicit description of the quotient stack in Groupoids in Spaces, Lemma 78.24.1.
\square
Lemma 106.12.3. With assumption and notation as in Lemma 106.12.2. 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 106.12.2 that f is a categorical moduli space if and only if \phi is a categorical quotient (Quotients of Groupoids, Definition 83.4.1). 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 97.17.1). 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 94.16.3. 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 106.12.4. 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 65.9) 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.
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