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Cdh sheaves invert finite subintegral maps

PDF p. 5: HK18 Theorem 1.4

Statement

If $p : T \to X$ is a finite subintegral morphism of separated finite-type schemes over a field and $F$ is a set-valued cdh sheaf, then restriction $F (X) \to F (T)$ is a bijection. In particular this applies to the finite seminormalization map when it belongs to this category.

Facts & Assumptions

Given: $p$ and $F$ as stated.

Goal: Prove bijectivity of restriction.

[F1]

The singleton ${p}$ is a cdh cover. Finite subintegral maps are cdh covers

[F2]

Every $s \in F (T)$ has equal pullbacks to $T \times_{X} T$ . Sections match across a finite subintegral cover

[F3]

A sheaf has unique amalgamations for covering sieves; restriction to a covering singleton is injective. Scheme sites and sheafification, A sheaf restricts injectively to a covering singleton

Proof

technique · Glue the matching section along the generated sieve

1.1F2construct

Given $s \in F (T)$ , assign to each arrow $v : V \to X$ in the sieve generated by $p$ the pullback $a^{*} s$ for any factorization $v = p a$ . This does not depend on the factorization: two such maps $a, b : V \to T$ define a map $V \to T \times_{X} T$ , and [F2] identifies $a^{*} s$ and $b^{*} s$ . These assignments are compatible with precomposition.

2.1F1F3step 1.1∎

By [F1] and [F3], this matching family has a unique amalgamation in $F (X)$ , whose restriction to $T$ is $s$ . Restriction is therefore surjective; [F3] also gives injectivity. Thus it is a bijection.

Depends on

Used by · 1

Representable cdh sheaves and seminormalization

Dependency tree · 3 levels

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