DefinitionDefinition: Literature-sourcedProof: AI-adaptedPipeline-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.
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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Crossed homomorphisms and first cohomology

Definition

For a left G-module A (written additively), put $Z_{c r}^{1} (G, A) = {d : G \to A : d (g h) = d (g) + g d (h)}, B_{c r}^{1} (G, A) = {δ a : g \mapsto g a - a} .$ Define $H_{c r}^{1} = Z_{c r}^{1} / B_{c r}^{1}$ . It agrees with normalized bar $H^{1} (G, A)$ ; comparison to derived cohomology retains the inherited DC and supplied-resolution convention. The crossed-homomorphism and bar quotient statements themselves require no choice axiom.

Facts & Assumptions

Given: G a group and A a left G-module.

[F1]

The inhomogeneous coboundary is the alternating multiplication formula (Inhomogeneous group cochains).

[F2]

Normalized inhomogeneous cochains compute group cohomology under its inherited conventions (Normalized cochains compute group cohomology).

Proof

1.1

In degrees zero and one the differential reads $(δ a) (g) = g a - a$ and $(δ d) (g, h) = g d (h) - d (g h) + d (g)$ . Thus $δ d = 0$ is exactly the crossed-homomorphism identity. Setting g=h=1 gives d(1)=0, so every crossed homomorphism is normalized. Also $(g h) a - a = (g a - a) + g (h a - a)$ , so every principal map is crossed. Both sets are additive groups and the principal maps form a subgroup.

F1givenalgebra

2.1

The cycles and boundaries of normalized degree one are therefore exactly the two groups in the Definition, so their quotient is bar H1 and hence, under the stated convention, the cohomology of F2. For trivial action the crossed identity is the homomorphism identity and every principal map is zero, giving $H^{1} (G, A) = Hom (G, A)$ . For G=1 or A=0 it is zero.

F2step 1.1algebra∎

Depends on

Used by

Degree-one restriction, inflation and quotient action Definition

Dependency tree · two levels

5 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)