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.

Induction and coinduction are the two adjoints

Statement

For $H \leq G$ , induction is left adjoint and coinduction is right adjoint to restriction.

Proof

Given: A left $H$ -module $M$ and a left $G$ -module $N$ .

1.1given

Evaluation at $1 \otimes m$ sends a $G$ -map $F : Z [G] \otimes_{Z [H]} M \to N$ to the $H$ -map $m \mapsto F (1 \otimes m)$ . Conversely, an $H$ -map $f : M \to Res N$ gives the well-defined $G$ -map $g \otimes m \mapsto g f (m)$ . These operations are inverse and natural, proving the induction adjunction for arbitrary group rings.

2.1step 1.1∎

The arbitrary-ring coextension adjunction identifies ${Hom}_{G} (N, {Hom}_{H} (Z [G], M))$ with ${Hom}_{H} (Res N, M)$ . Both identifications are natural, giving the two adjunctions.

Depends on

Used by

Dependency tree · two levels

8 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, §6.3 (standard reference, not scraped)