DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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.

Degree-one restriction, inflation and quotient action

Definition

For $1 \to N \to G \overset{π}{\to} Q \to 1$ and a left G-module A, let $A^{N} = {a : n a = a for all n \in N}$ , with action $q a = g a$ for any lift g of q. Define $res (d) = d ∣_{N}, \inf (c) (g) = c (π (g)), (g \cdot d) (n) = g d (g^{- 1} n g) .$ Inflation starts with crossed maps $Q \to A^{N}$ . The last formula is a G-action on crossed maps $N \to A$ ; it is its induced action on H1 that factors through Q. All H1 symbols first denote the explicit bar quotient; the derived interpretation uses its inherited conventions. The next lemma supplies well-definedness on the quotient.

Depends on

Crossed homomorphisms and first cohomology

Used by

Degree-one maps and the quotient action are well-defined Lemma

Dependency tree · two levels

3 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, An Introduction to Homological Algebra, Chapter 6, Sections 6.4–6.8 (standard reference, not scraped)
Dekimpe–Hartl–Wauters, A seven-term exact sequence for the cohomology of a group extension, Sections 2–5 pp.2–11 and Section 10.2 p.21 (standard reference, not scraped)