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.

Bockstein obstruction to prime torsion

PDF p. 5: Proposition 2.6: torsion freeness

Statement

For $C \in D (Z)$ and a prime $ℓ$ , multiplication by $ℓ$ on $H^{6} (C)$ is injective if and only if $H^{5} (C) \to H^{5} (C / ℓ)$ is surjective. Here $C / ℓ = C \otimes^{L} Z / ℓ$ and the map is the canonical reduction. In particular, $H^{5} (C / ℓ) = 0$ suffices.

Facts & Assumptions

[F1]

Functorial finite coefficients for derived complexes supplies the natural coefficient short exact sequence.

Proof

Given: $C$ and $ℓ$ as stated.

Goal: Prove the equivalence and sufficient vanishing.

1.1F1

Apply [F1] with $i = 5$ . The cokernel of the canonical reduction is $H^{6} (C) [ℓ]$ , the kernel of multiplication by $ℓ$ .

2.1step 1.1∎

That cokernel is zero precisely when reduction is surjective, equivalently when multiplication is injective. A zero target makes reduction surjective.

Depends on

Functorial finite coefficients for derived complexes

Used by · 1

Torsion freeness of top threefold motivic cohomology

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.