Alphabeta Math
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≤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.

1.1

Evaluation at 1⊗m sends a G-map F:Z[G]⊗Z[H]M→N to the H-map m↦F(1⊗m). Conversely, an H-map f:M→Res⁡N gives the well-defined G-map g⊗m↦gf(m). These operations are inverse and natural, proving the induction adjunction for arbitrary group rings.

given
2.1

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.

step 1.1∎

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