83.4 Categorical quotients
This is the most basic kind of quotient one can consider.
Definition 83.4.1. Let S be a scheme, and let B be an algebraic space over S. Let j = (t, s) : R \to U \times _ B U be pre-relation in algebraic spaces over B.
We say a morphism \phi : U \to X of algebraic spaces over B is a categorical quotient if it is R-invariant, and for every R-invariant morphism \psi : U \to Y of algebraic spaces over B there exists a unique morphism \chi : X \to Y such that \psi = \phi \circ \chi .
Let \mathcal{C} be a full subcategory of the category of algebraic spaces over B. Assume U, R are objects of \mathcal{C}. In this situation we say a morphism \phi : U \to X of algebraic spaces over B is a categorical quotient in \mathcal{C} if X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}), and \phi is R-invariant, and for every R-invariant morphism \psi : U \to Y with Y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) there exists a unique morphism \chi : X \to Y such that \psi = \phi \circ \chi .
If B = S and \mathcal{C} is the category of schemes over S, then we say U \to X is a categorical quotient in the category of schemes, or simply a categorical quotient in schemes.
We often single out a category \mathcal{C} of algebraic spaces over B by some separation axiom, see Example 83.4.3 for some standard cases. Note that \phi : U \to X is a categorical quotient if and only if U \to X is a coequalizer for the morphisms t, s : R \to U in the category. Hence we immediately deduce the following lemma.
Lemma 83.4.2. Let S be a scheme, and let B be an algebraic space over S. Let j : R \to U \times _ B U be a pre-relation in algebraic spaces over B. If a categorical quotient in the category of algebraic spaces over B exists, then it is unique up to unique isomorphism. Similarly for categorical quotients in full subcategories of \textit{Spaces}/B.
Proof.
See Categories, Section 4.11.
\square
Example 83.4.3. Let S be a scheme, and let B be an algebraic space over S. Here are some standard examples of categories \mathcal{C} that we often come up when applying Definition 83.4.1:
\mathcal{C} is the category of all algebraic spaces over B,
B is separated and \mathcal{C} is the category of all separated algebraic spaces over B,
B is quasi-separated and \mathcal{C} is the category of all quasi-separated algebraic spaces over B,
B is locally separated and \mathcal{C} is the category of all locally separated algebraic spaces over B,
B is decent and \mathcal{C} is the category of all decent algebraic spaces over B, and
S = B and \mathcal{C} is the category of schemes over S.
In this case, if \phi : U \to X is a categorical quotient then we say U \to X is (1) a categorical quotient, (2) a categorical quotient in separated algebraic spaces, (3) a categorical quotient in quasi-separated algebraic spaces, (4) a categorical quotient in locally separated algebraic spaces, (5) a categorical quotient in decent algebraic spaces, (6) a categorical quotient in schemes.
Definition 83.4.4. Let S be a scheme, and let B be an algebraic space over S. Let \mathcal{C} be a full subcategory of the category of algebraic spaces over B closed under fibre products. Let j = (t, s) : R \to U \times _ B U be pre-relation in \mathcal{C}, and let U \to X be an R-invariant morphism with X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}).
We say U \to X is a universal categorical quotient in \mathcal{C} if for every morphism X' \to X in \mathcal{C} the morphism U' = X' \times _ X U \to X' is the categorical quotient in \mathcal{C} of the base change j' : R' \to U' of j.
We say U \to X is a uniform categorical quotient in \mathcal{C} if for every flat morphism X' \to X in \mathcal{C} the morphism U' = X' \times _ X U \to X' is the categorical quotient in \mathcal{C} of the base change j' : R' \to U' of j.
Lemma 83.4.5. In the situation of Definition 83.4.1. If \phi : U \to X is a categorical quotient and U is reduced, then X is reduced. The same holds for categorical quotients in a category of spaces \mathcal{C} listed in Example 83.4.3.
Proof.
Let X_{red} be the reduction of the algebraic space X. Since U is reduced the morphism \phi : U \to X factors through i : X_{red} \to X (Properties of Spaces, Lemma 66.12.4). Denote this morphism by \phi _{red} : U \to X_{red}. Since \phi \circ s = \phi \circ t we see that also \phi _{red} \circ s = \phi _{red} \circ t (as i : X_{red} \to X is a monomorphism). Hence by the universal property of \phi there exists a morphism \chi : X \to X_{red} such that \phi _{red} = \phi \circ \chi . By uniqueness we see that i \circ \chi = \text{id}_ X and \chi \circ i = \text{id}_{X_{red}}. Hence i is an isomorphism and X is reduced.
To show that this argument works in a category \mathcal{C} one just needs to show that the reduction of an object of \mathcal{C} is an object of \mathcal{C}. We omit the verification that this holds for each of the standard examples.
\square
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