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.

Shapiro lemma for group homology

Statement

For $H \leq G$ , a left $H$ -module $M$ , and $n \geq 0$ , there is a natural isomorphism $H_{n} (G; {Ind}_{H}^{G} M) ≅ H_{n} (H; M)$ .

Proof

Given: A projective right $Z [G]$ -resolution $P \to Z$ .

1.1

Restriction of $P$ is a projective right $Z [H]$ -resolution of the right trivial module: a left-coset transversal gives $Z [G] ≅ ⨁_{t \in T} t Z [H]$ as right $Z [H]$ -modules, so restriction sends free, hence projective, right $Z [G]$ -modules to projective right $Z [H]$ -modules. Exactness and the augmentation are unchanged on restriction.

given

2.1

Tensor associativity gives an isomorphism of chain complexes $P \otimes_{Z [G]} (Z [G] \otimes_{Z [H]} M) ≅ (Res P) \otimes_{Z [H]} M$ . The left and right sides compute respectively $H_{n} (G; {Ind}_{H}^{G} M)$ and $H_{n} (H; M)$ , so their homology gives the asserted natural isomorphism.

step 1.1∎

Depends on

Used by

Shapiro lemma for the trivial subgroup Example
Shapiro lemma needs no induction/coinduction distinction False statement

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, Shapiro Lemma 6.3.2 (standard reference, not scraped)