## 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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