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AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Sections match across a finite subintegral cover

PDF p. 5: HK18 Theorem 1.4

Statement

Let $p : T \to X$ be a finite subintegral morphism of separated finite-type schemes over a field, and let $F$ be a set-valued cdh sheaf. For every $s \in F (T)$ , its pullbacks along the two projections $T \times_{X} T ⇉ T$ agree. The fibre product represents pairs of maps to $T$ with the same composite to $X$ .

Facts & Assumptions

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

Goal: Prove equality of the two pullbacks.

[F1]

The diagonal $Δ : T \to T \times_{X} T$ is a cdh covering singleton. The diagonal of a finite subintegral map is a cdh cover

[F2]

Restriction of a sheaf along a covering singleton is injective. A sheaf restricts injectively to a covering singleton

Proof

technique · Compare after restriction to the diagonal

1.1given

Each projection composed with $Δ$ is ${id}_{T}$ . Consequently the two pullbacks of $s$ restrict along $Δ$ to the same element $s$ .

2.1F1F2step 1.1∎

By [F1] and [F2], restriction along $Δ$ is injective. The two pullbacks are therefore equal.

Depends on

Used by · 1

Cdh sheaves invert finite subintegral maps

Dependency tree · 3 levels

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