TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Shapiro lemma for group cohomology

Statement

Assume the Axiom of Choice (The Axiom of Choice).

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

Proof

Given: The Axiom of Choice and a projective $Z [G]$ -resolution $P \to Z$ .

1.1givenconstruct

Apply the Axiom of Choice to the family of nonempty right cosets $H \ G$ to obtain a transversal $T$ . The group-ring freeness supplier then gives $Z [G] ≅ ⨁_{t \in T} Z [H] t$ as left $Z [H]$ -modules. Restriction therefore takes the projective left $Z [G]$ -resolution $P$ to a projective left $Z [H]$ -resolution.

2.1step 1.1∎

Coinduction--restriction adjunction identifies ${Hom}_{G} (P, Coind M)$ with ${Hom}_{H} (Res P, M)$ as cochain complexes. Taking cohomology gives the claimed natural isomorphism.

Depends on

Used by

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

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, Shapiro Lemma 6.3.2 (standard reference, not scraped)