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A flat finite-type equivalence relation has a generic scheme quotient

Statement

Assume the Axiom of Choice. Let $R ⇉ X$ be a flat finite-type equivalence-relation groupoid on a separated finite-type $k$ -scheme. There is a dense saturated open $W \subset X$ whose fppf quotient is a finite-type scheme $Y$ . The quotient $q : W \to Y$ is faithfully flat of finite presentation and $R_{W} ≅ W \times_{Y} W$ .

Facts & Assumptions

[F1]

Generic saturated quasi-sections with affine-contained finite subsets exist. (A flat finite-type equivalence relation has generic saturated quasi-sections)

[F2]

A quasi-section whose arrow map is finite locally free and whose finite-relation orbits lie in affine opens gives a scheme fppf quotient, a faithfully flat finite-presentation projection, and the prescribed kernel pair. (A flat equivalence relation with a suitable quasi-section has a scheme quotient)

Proof

Given: The schemes, maps, and hypotheses in the statement, and AC.

1.1F1F2givenconstruct

Apply [F1] and write its dense open as the finite disjoint union of saturated opens $W_{i}$ , with quasi-sections $U_{i}$ . The induced relation has finite locally free projections, so every orbit is finite; [F1] puts it in an affine open of $U_{i}$ . Thus every hypothesis of [F2] holds on each $W_{i}$ , and it supplies a finite-type scheme $Y_{i}$ representing $W_{i} / R_{W_{i}}$ , with the asserted projection and kernel pair.

2.1F1F2step 1.1construct∎

Set $Y = ∐_{i} Y_{i}$ . Since the $W_{i}$ are disjoint and saturated there are no relation arrows between different pieces, so this disjoint union represents the fppf quotient of $W = ∐_{i} W_{i}$ . Flatness, finite presentation, surjectivity and the kernel-pair identity hold piecewise and hence globally. AC is inherited from [F1]–[F2].

Depends on

Used by

Normal subgroup quotients of finite-type group schemes exist as fppf scheme quotients Theorem

Dependency tree · two levels

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Sources

SGA3, Expose V, Sections 7-8 (standard reference, not scraped)