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

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The bar complex is a free resolution of the trivial module

Statement

The augmented homogeneous bar complex is a free $Z [G]$ -resolution of the trivial module $Z$ .

Proof

Given: The augmented bar complex.

1.1

Every diagonal orbit in $G^{n + 1}$ has the unique representative $(1, g_{0}^{- 1} g_{1}, \dots, g_{0}^{- 1} g_{n})$ ; hence $B_{n} (G)$ is free over $Z [G]$ on these representatives.

given

2.1

The differential and augmentation are $G$ -linear, and the underlying augmented complex is exact. Thus it is a free $Z [G]$ -resolution.

step 1.1∎

Depends on

Used by

The bar contracting homotopy is group-equivariant False statement
The bar cochain complex computes group cohomology Theorem

Dependency tree · two levels

5 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, Theorem 6.5.3 (standard reference, not scraped)