## 81.9 Good quotients

Especially when taking quotients by group actions the following definition is useful.

Definition 81.9.1. 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 good quotient if

1. $\phi$ is invariant,

2. $\phi$ is affine,

3. $\phi$ is surjective,

4. condition (3) holds universally, and

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

In Seshadri gives almost the same definition, except that instead of (4) he simply requires the condition (3) to hold – he does not require it to hold universally.

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