LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Divisibility survives the quotient and the support image

PDF p. 13: Proposition 6.1 support argument

Statement

Assume the Axiom of Choice and the fixed site and universe bounds of the motivic and support constructions. Let $k$ be an arbitrary algebraically closed field, let $X$ be an affine finite-type $k$ -scheme of Krull dimension three, put $U = X_{s m o o t h}$ , and let $T \subset X$ be a closed integral curve contained in $U$ . The group $H_{m o t, T}^{6, 3} (X, Z)$ is divisible, and its image under the actual support-to-global map into $H_{m o t}^{6, 3} (X, Z)$ is divisible. No injectivity of that map is required.

Facts & Assumptions

Given: The algebraically closed field $k$ , affine finite-type threefold $X$ , smooth locus $U$ , closed integral curve $T \subset U$ , Axiom of Choice, and fixed site and universe bounds in the statement.

Goal: Construct the stated actual maps and prove their consequence.

[F1]

The divisor presentation and actual Picard surjection supplies an actual surjection from a divisible group onto the supported group under exactly the stated geometry, field, Choice and size hypotheses. This includes the zero case on a lower-dimensional smooth stratum. Choice is inherited from its normalization and Picard constructions; The Axiom of Choice records that premise.

[F2]

Motivic and filtered K theory with support gives the actual support-to-global homomorphism under the stated Choice and size conventions.

Proof

1.1F1construct

For any surjection $q : D \to A$ with D divisible, fix $a \in A$ and positive n. Choose $d \in D$ with $q (d) = a$ , then $e \in D$ with $n e = d$ . Thus $n q (e) = a$ . Negative integers are handled by negation. Apply this to the actual surjection in the preceding lemma (or to its zero case). This proves divisibility of the supported group.

2.1F1F2step 1.1∎

Let $s : A \to G$ be the support-to-global homomorphism fixed by the completed support definition. For $s (a)$ and positive n, choose b in A with $n b = a$ from step 1.1. Then $n s (b) = s (a)$ , so every element of the image has an n-th root in that same image. Again handle negative n by negation. This proves the asserted image divisibility without identifying the support group with its image.

Depends on

Used by · 1

Divisibility of curve-supported top cohomology

Dependency tree · 3 levels

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