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

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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.

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The augmented bar complex is exact

Statement

The augmented complex $\dots \to B_{1} (G) \to B_{0} (G) \overset{ε}{\to} Z \to 0$ is exact as a complex of abelian groups.

Proof

Given: The augmented homogeneous bar complex.

1.1given

After forgetting the $G$ -action, set $s_{n} (g_{0}, \dots, g_{n}) = (1, g_{0}, \dots, g_{n})$ and $s_{- 1} (1) = (1)$ .

2.1step 1.1∎

For $n \geq 1$ , direct cancellation of the faces gives $d_{n + 1} s_{n} + s_{n - 1} d_{n} = 1_{B_{n}}$ . In degree zero it gives the correctly typed identity $d_{1} s_{0} + s_{- 1} ε = 1_{B_{0}}$ , and $ε s_{- 1} = 1_{Z}$ . Thus every cycle is a boundary and the augmentation is onto. The maps $s_{n}$ are only $Z$ -linear: left translation changes the inserted $1$ .

Depends on

Used by

The underlying bar contraction is not equivariant Counterexample
The bar contracting homotopy is group-equivariant False statement
The bar complex is a free resolution of the trivial module 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, Theorem 6.5.3 (standard reference, not scraped)