DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

Restriction, induction, and coinduction

Definition

For $H \leq G$ , restriction forgets from $Z [G]$ to $Z [H]$ . For a left $H$ -module $M$ , set ${Ind}_{H}^{G} M = Z [G] \otimes_{Z [H]} M$ and ${Coind}_{H}^{G} M = {Hom}_{Z [H]} (Z [G], M)$ , with $G$ acting on the latter by $(g φ) (x) = φ (x g)$ .

Depends on

Used by

Restriction and corestriction in group cohomology Definition
The group ring is free over a subgroup ring Lemma
Induction and coinduction are the two adjoints Theorem

Dependency tree · two levels

14 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, Definition 6.3.1 (standard reference, not scraped)