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.

Integral group modules and the trivial module

Definition

Throughout this page a $G$ -module is a left $Z [G]$ -module. The abelian group $Z$ is a left trivial module by $g \cdot z = z$ , equivalently through the augmentation $Z [G] \to Z$ . When it occurs on the left of $\otimes_{Z [G]}$ , the same underlying group is instead the right trivial module $z \cdot g = z$ .

Depends on

Used by

Bar augmentation Definition
Inhomogeneous group cochains Definition
Integral cohomological dimension Definition
Restriction, induction, and coinduction Definition
The augmented unnormalized homogeneous bar complex Definition
The coinvariants functor Definition
The invariants functor Definition
A periodic resolution for a finite cyclic group Example
Invariants are Hom from the trivial module Theorem

Dependency tree · two levels

13 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, Group Homology and Cohomology, §6.1 (standard reference, not scraped)