LemmaStatement: 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.

The group ring is free over a subgroup ring

Statement

If $H \leq G$ and a left (respectively right) coset transversal is supplied, then $Z [G]$ is free as a right (respectively left) $Z [H]$ -module on it.

Proof

Given: A supplied left coset transversal $T$ .

1.1

Every $g \in G$ has a unique form $t h$ with $t \in T, h \in H$ .

given

2.1

Group-ring basis expansion therefore gives $Z [G] = ⨁_{t \in T} t Z [H]$ as right modules. The opposite-side assertion follows from a right transversal. For arbitrary cosets, choosing $T$ is the only choice input.

step 1.1∎

Depends on

Used by

Restriction and corestriction in group cohomology Definition
Finite integral cohomological dimension implies torsion-free Theorem
Shapiro lemma for group cohomology Theorem
Shapiro lemma for group homology Theorem

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, Proposition 1.7.2 (standard reference, not scraped)