LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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

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: Let $1 \to R \to F \to G \to 1$ be a free presentation.

[L1]

The Lyndon--Hochschild--Serre low-degree homology sequence for this extension is $H_{2} (F; Z) \to H_{2} (G; Z) \to R / [F, R] \to F_{a b} \to G_{a b} \to 0$ .

Proof

technique · direct

1.1

The Lyndon--Hochschild--Serre low-degree homology sequence for this extension is $H_{2} (F; Z) \to H_{2} (G; Z) \to R / [F, R] \to F_{a b} \to G_{a b} \to 0$ . This is the five-term sequence recorded in the cited source.

L1given

2.1

A free group has zero second integral homology, so the first map has zero source and exactness gives the displayed sequence beginning with $0$ .

step 1.1algebra∎

Depends on

Used by

Hopf formula for the Schur multiplier Theorem

Dependency tree · two levels

8 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)