We only add this here so that we can later say that coarse quotients correspond to coarse moduli spaces (or moduli schemes).
Definition 83.6.1. 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
$\phi $ is a categorical quotient, and
$\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
$\phi $ is a categorical quotient in schemes, and
$\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
for all algebraically closed fields $k$. See Lemma 83.5.20. 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 $k$.
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)