The Stacks project

81.6 Coarse quotients

We only add this here so that we can later say that coarse quotients correspond to coarse moduli spaces (or moduli schemes).

Definition 81.6.1. Let $S$ be a scheme and $B$ an algebraic space over $S$. Let $j : R \to U \times _ B U$ be a pre-relation. A morphism $\phi : U \to X$ of algebraic spaces over $B$ is called a coarse quotient if

  1. $\phi $ is a categorical quotient, and

  2. $\phi $ is an orbit space.

If $S = B$, $U$, $R$ are all schemes, then we say a morphism of schemes $\phi : U \to X$ is a coarse quotient in schemes if

  1. $\phi $ is a categorical quotient in schemes, and

  2. $\phi $ is an orbit space.

In many situations the algebraic spaces $R$ and $U$ are locally of finite type over $B$ and the orbit space condition simply means that

\[ U(k)/\big (\text{equivalence relation generated by }j(R(k))\big ) \cong X(k) \]

for all algebraically closed fields $k$. See Lemma 81.5.20. If $j$ is also a (set-theoretic) pre-equivalence relation, then the condition is simply equivalent to $U(k)/j(R(k)) \to X(k)$ being bijective for all algebraically closed fields $k$.

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