The Stacks project

82.9 Good quotients

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

Definition 82.9.1. 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_quotients] 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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