TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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Homogeneous and inhomogeneous cochains agree

Statement

The complex ${Hom}_{Z [G]} (B_{∙} (G), M)$ is naturally isomorphic to $(C^{∙} (G, M), d)$ .

Proof

Given: Homogeneous bars and the inhomogeneous differential.

1.1

Send a homogeneous equivariant cochain $F$ to $f (g_{1}, \dots, g_{n}) = F (1, g_{1}, g_{1} g_{2}, \dots, g_{1} \dots g_{n})$ .

given

2.1

Its inverse sends $f$ to $F (g_{0}, \dots, g_{n}) = g_{0} f (g_{0}^{- 1} g_{1}, \dots, g_{n - 1}^{- 1} g_{n})$ . The formulas are inverse and $G$ -equivariance is immediate; applying alternating face deletion yields exactly the displayed formula for $d$ .

step 1.1∎

Depends on

Used by

The bar cochain complex computes group cohomology Theorem

Dependency tree · two levels

4 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

Weibel, Appendix 6.5.5 (standard reference, not scraped)