LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passverified 2026-09-26 (gpt-6-sol)

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.

Low-degree sequence of a free presentation

Statement

Assume the Axiom of Dependent Choice and supplied projective-resolution data for the group-homology construction. For an exact sequence $1 \to R \to F \to G \to 1$ with $F$ free, there is an exact sequence

$0 \to H_{2} (G; Z) \to R / [F, R] \to F_{a b} \to G_{a b} \to 0.$

Facts & Assumptions

Given: Dependent choice, the supplied group-homology resolution data, and a free presentation $1 \to R \to F \to G \to 1$ .

[L1]

Under the stated conventions, the free-presentation homology five-term theorem supplies the natural exact sequence $0 \to H_{2} (G; Z) \to R / [F, R] \to F_{a b} \to G_{a b} \to 0$ (The free-presentation homology five-term sequence).

Proof

technique · direct

1.1L1given

The hypotheses and group-homology conventions are exactly those of [L1]. Its first arrow is the free-presentation transgression, and its remaining arrows come from the inclusion $R ↪ F$ and quotient $F ↠ G$ .

2.1L1step 1.1∎

The exactness and initial injection asserted by [L1] therefore give the displayed sequence under the retained dependent-choice and resolution hypotheses.

Depends on

Used by

Hopf formula is presentation-independent Corollary
Kernel of the universal central extension Corollary
Hopf formula for the Schur multiplier Theorem

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Clara Löh, Group Cohomology (standard reference, not scraped)