TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge 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.

Finite integral cohomological dimension implies torsion-free

Statement

If ${cd}_{Z} G < \infty$ , then $G$ is torsion-free.

Proof

Given: A finite projective resolution of the trivial $Z [G]$ -module.

1.1given

If $g \in G$ has finite order $m > 1$ , restrict the resolution to $C = ⟨ g ⟩$ ; subgroup freeness preserves projectives, so ${cd}_{Z} C < \infty$ .

2.1step 1.1∎

In $Z [C]$ , the alternating maps $g - 1$ and $N = 1 + g + \dots + g^{m - 1}$ form a periodic free resolution: $(g - 1) N = N (g - 1) = 0$ , and coefficient comparison gives the two kernels as the two images. With trivial coefficients $Z / m$ , its Hom complex has zero differentials, so $H^{q} (C; Z / m) \neq 0$ for arbitrarily large $q$ . This contradicts finite cohomological dimension.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Löh, Corollary 1.7.3 (standard reference, not scraped)