## 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 $k$.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04A1. Beware of the difference between the letter 'O' and the digit '0'.