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The diagonal of a finite subintegral map is a cdh cover

PDF p. 5: HK18 Theorem 1.4

Statement

For a finite subintegral morphism $p : T \to X$ of separated finite-type schemes over a field, its diagonal $Δ : T \to T \times_{X} T$ is a finite subintegral morphism and a cdh covering singleton. The diagonal is the unique map whose composites with both projections are the identity of $T$ .

Facts & Assumptions

Given: Such a morphism $p$ .

Goal: Prove the assertion about $Δ$ .

[F1]

A subintegral morphism is a universal homeomorphism and induces residue-field isomorphisms. Seminormality and seminormalization

[F2]

A finite morphism is separated, so its diagonal is a closed immersion; a closed immersion is finite and induces isomorphisms on residue fields at its image points.

[F3]

A finite subintegral morphism in this category is a cdh covering singleton. Finite subintegral maps are cdh covers

Proof

technique · Use the base change of the universal homeomorphism

1.1F1given

The first projection $q : T \times_{X} T \to T$ is a base change of $p$ and is therefore a homeomorphism on underlying spaces by [F1]. Since $q Δ = {id}_{T}$ , the diagonal is bijective on points.

2.1F2step 1.1

By [F2], $Δ$ is finite and induces residue-field isomorphisms at image points. Together with step 1.1, it is subintegral.

3.1F3step 2.1∎

The fibre product is again separated and of finite type over the field. Apply [F3] to $Δ$ to obtain the stated cdh cover.

Depends on

Used by · 1

Sections match across a finite subintegral cover

Dependency tree · 3 levels

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