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Finite subintegral maps are cdh covers

PDF p. 5: HK18 Theorem 1.4

Statement

Let $p : T \to X$ be a finite subintegral morphism between separated finite-type schemes over a field. Then the singleton ${p}$ is a cdh covering family. Here finite means affine with finite-module coordinate rings; subintegral means integral, bijective on points and inducing isomorphisms on all residue fields. A cdh covering family generates a covering sieve for the topology of Completely decomposed proper covers and the cdh topology.

Facts & Assumptions

Given: A finite subintegral morphism $p : T \to X$ in the indicated scheme category.

Goal: Show that $p$ is a completely decomposed proper morphism and hence a cdh cover.

[F1]

A finite morphism is proper: it is separated, of finite type and universally closed. This follows affine-locally from integral maps being closed, a property preserved by base change. Completely decomposed proper covers and the cdh topology

[F2]

A completely decomposed proper morphism is a cdh covering singleton. Complete decomposition means that every target point has a preimage with isomorphic residue field. Completely decomposed proper covers and the cdh topology

Proof

technique · Apply the definitions

1.1F1given

By finiteness and [F1], $p$ is proper. By subintegrality, every $x \in X$ has a preimage $t$ and $κ (x) \to κ (t)$ is an isomorphism. Thus $p$ is completely decomposed.

2.1F2step 1.1∎

By [F2], the singleton ${p}$ is a cdh cover, as required.

Depends on

Used by · 2

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