LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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Free-presentation total homology and its degree-one edges

Statement

For the free-presentation bicomplex, $H_{n} (Tot D) = 0$ for $n > 1$ , $H_{1} (Tot D) = F_{a b}$ , $E_{01}^{2} = R / [F, R]$ , and $E_{p 0}^{2} = H_{p} (G; Z)$ . The degree-one maps are inclusion and quotient on abelianizations. This does not assert vanishing of every positive-degree E2 term. Retain DC and supplied-resolution conventions.

Facts & Assumptions

Given: The free presentation and bicomplex of the Definition, with the inherited homology conventions.

[F1]

Total homology is H*(F), E2 is H_p(G;H_q(R)), and the degree-one edges are identified (The free-presentation Lyndon bar bicomplex).

[F2]

A free group has a length-one free resolution (Generator differences form a basis of the free-group augmentation ideal).

[F3]

H1(R) conjugation coinvariants equal R/[F,R] (First integral homology and conjugation coinvariants).

Proof

1.1F1F2given

The total homology equals H*(F) by F1. The free resolution of F2 has no terms above degree one, so after tensoring it has zero homology there; in degree one it gives the free abelian group on the free generators, which is $F_{a b}$ . Hence the asserted total vanishing holds even for an infinite free generating set.

2.1F1F3step 1.1algebra∎

The zero homology of R with trivial integers is $Z$ : every vertex is identified, and each degree-one boundary is a difference of vertices. Conjugation fixes this generator. Consequently $E_{p 0}^{2} = H_{p} (G; Z)$ . Also $E_{01}^{2} = H_{0} (G; H_{1} (R; Z)) = R / [F, R]$ by F3. The edge calculations of F1 send r to its class in $F_{a b}$ and f to its image in $G_{a b}$ , with positive signs.

Depends on

Used by

The free-presentation homology five-term sequence Theorem

Dependency tree · two levels

13 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

Loh, Group Cohomology, Definitions 1.7.12–13 and Theorem 1.7.15 pp.63–64; Theorem 3.2.18 pp.129–132 (standard reference, not scraped)
Weibel, An Introduction to Homological Algebra, Chapter 6, Sections 6.4–6.8 (standard reference, not scraped)