TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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The inhomogeneous one-cocycle model agrees with crossed homomorphisms in degree one

Statement

For an abelian $G$ -group $A$ , the quotient defined by crossed homomorphisms agrees canonically with the inhomogeneous degree-one cochain model:

$H^{1} (G, A) ≅ H_{i n h}^{1} (G, A) .$

Facts & Assumptions

Given: A group $G$ acting on an abelian group $A$ .

[L1]

The crossed-homomorphism model is $H^{1} (G, A) = Z^{1} (G, A) / B^{1} (G, A)$ (First cohomology via crossed homomorphisms).

[L2]

The inhomogeneous degree-one model is $\ker d^{1} / im d^{0}$ with $d^{1} (f) (g, h) = f (g) + g \cdot f (h) - f (g h), d^{0} (a) (g) = g \cdot a - a$ (First group cohomology via inhomogeneous one-cocycles).

Proof

technique · direct

1.1

A function $f : G \to A$ lies in $\ker d^{1}$ exactly when $0 = d^{1} (f) (g, h) = f (g) + g \cdot f (h) - f (g h)$ for all $g, h \in G$ , that is, exactly when $f (g h) = f (g) + g \cdot f (h)$ . So the one-cocycles in the inhomogeneous complex are exactly the crossed homomorphisms of [L1].

givenL1L2

2.1

A function lies in $im d^{0}$ exactly when it has the form $g \mapsto g \cdot a - a$ for some $a \in A$ . Those are precisely the principal crossed homomorphisms in [L1].

L1L2step 1.1

3.1

Steps 1.1 and 2.1 identify both the numerator and denominator of the two quotient constructions. Therefore the quotients themselves are canonically the same, giving the asserted isomorphism.

L1L2step 1.1step 2.1∎

Depends on

Used by

For a trivial action, first cohomology is Hom Corollary
FALSE: the cochain and crossed-homomorphism definitions of first cohomology agree automatically False statement
Inflation-restriction exact sequence in degree one Theorem

Dependency tree · two levels

6 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

Chaoli Li, Class field theory: proofs (standard reference, not scraped)
David A. Craven, Finite Group Theory (standard reference, not scraped)