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.

The coinvariants functor

Definition

For a left $G$ -module $M$ , its coinvariants are $M_{G} = M / ⟨ g m - m : g \in G, m \in M ⟩$ . If $f : M \to N$ is $G$ -linear, define $f_{G} : M_{G} \to N_{G}$ by $f_{G} ([m]) = [f (m)]$ . Equivariance makes this well defined, and identities and composites are preserved, so $M \mapsto M_{G}$ is a functor. The map $1 \otimes m \mapsto [m]$ naturally identifies $Z \otimes_{Z [G]} M$ (with right trivial $Z$ ) with $M_{G}$ .

Depends on

Used by

The coinvariants functor is right exact Proposition

Dependency tree · two levels

10 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, §6.1 (standard reference, not scraped)