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.

Corestriction after restriction multiplies by the index

Statement

If $H \leq G$ has finite index, then ${cor}_{H}^{G} {res}_{H}^{G} = [G : H]$ on $H^{n} (G; M)$ for all $n \geq 0$ .

Proof

Given: A finite-index subgroup $H \leq G$ and a left $G$ -module $M$ .

1.1given

In degree zero, if $m \in M^{G}$ , restriction regards $m$ as $H$ -fixed and the norm sends it to $\sum_{x H \in G / H} x m = [G : H] m$ . Thus ${cor}_{H}^{G} {res}_{H}^{G}$ and multiplication by $[G : H]$ have the same degree-zero component.

2.1step 1.1∎

Both are morphisms from the universal cohomological delta functor $H^{*} (G; -)$ to itself. By A morphism between universal delta functors is determined in degree zero, equality in degree zero forces equality in every degree. Hence the composite is multiplication by $[G : H]$ on $H^{n} (G; M)$ for all $n \geq 0$ .

Depends on

Used by

Finite groups annihilate positive cohomology by their order Proposition

Dependency tree · two levels

12 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, Lemma 6.7.17 (standard reference, not scraped)