TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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The inflation–restriction–transgression five-term sequence

Statement

Assume AC. For every group extension $1 \to N \to G \to Q \to 1$ and left G-module A, the sequence $0 ⟶ H^{1} (Q, A^{N}) \overset{\inf}{\to} H^{1} (G, A) \overset{res}{\to} H^{1} (N, A)^{Q} \overset{Tra}{\to} H^{2} (Q, A^{N}) \overset{\inf}{\to} H^{2} (G, A)$ is exact at every term having a following displayed arrow. Tra uses the normalizer quotient and its printed DHW cocycle sign; degree-two inflation includes coefficient inclusion. Cohomology is normalized bar cohomology, with its inherited derived interpretation. No exactness assertion at the last term is intended.

Facts & Assumptions

Given: AC, the extension and module A.

[F1]

Inflation is injective and its image is the kernel of restriction (Degree-one inflation–restriction is exact).

[F2]

The kernel of Tra is the restriction image (The kernel of transgression is the image of restriction).

[F3]

The kernel of degree-two inflation is the image of Tra (The kernel of degree-two inflation is the transgression image).

Proof

1.1

Use the degree-one inflation and restriction formulas with coefficients $A^{N} \subseteq A$ . F1 gives injectivity of the first arrow and exactness at $H^{1} (G, A)$ . Its codomain for restriction is precisely $H^{1} (N, A)^{Q}$ , the domain of Tra in F2.

F1F2given

2.1

F2 gives exactness at $H^{1} (N, A)^{Q}$ . F3 applies to the same normalizer transgression and the pullback/coefficient-inclusion inflation and gives exactness at $H^{2} (Q, A^{N})$ . These cover every asserted location, including zero composites. No result about the cokernel of the final map is used or claimed. For N=1 the two inflation maps are identities and H1(N,A)=0; for Q=1 the restriction map is the identity and both positive-degree Q groups vanish, so the degenerate sequences agree.

F2F3step 1.1algebra∎

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Dependency tree · two levels

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Sources

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)
Weibel, An Introduction to Homological Algebra, Chapter 6, Sections 6.4–6.8 (standard reference, not scraped)