## 81.10 Geometric quotients

This is Mumford's definition of a geometric quotient (at least the definition from the first edition of GIT; as far as we can tell later editions changed “universally submersive” to “submersive”).

Definition 81.10.1. 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 *geometric quotient* if

$\phi $ is an orbit space,

condition (1) holds universally, i.e., $\phi $ is universally submersive, and

the functions on $X$ are the $R$-invariant functions on $U$.

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