LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Changing the section changes the factor set by a coboundary

Statement

If $s$ and $s^{'}$ are normalized sections of the same abelian-kernel extension, then there is a normalized one-cochain $u : G \to M$ such that

$f_{s^{'}} = f_{s} + δ u .$

Facts & Assumptions

Given: Two normalized sections $s, s^{'} : G \to E$ of the same extension.

[F1]

Two-coboundaries have the form $(δ u) (g, h) = g \cdot u (h) - u (g h) + u (g)$ (Normalized two-cocycle and two-coboundary).

[F2]

Factor sets are defined by the section formula (Normalized set-theoretic section and factor set).

[L1]

Each factor set is a normalized two-cocycle (The factor set of a section is a normalized two-cocycle).

Proof

technique · direct

1.1

Since $π (s^{'} (g) s (g)^{- 1}) = 1$ , each quotient $s^{'} (g) s (g)^{- 1}$ lies in the kernel. So there is a unique $u (g) \in M$ with $s^{'} (g) = i (u (g)) s (g)$ . The normalization $s^{'} (1) = s (1) = 1$ gives $u (1) = 0$ .

F2givenchoosealgebra

2.1

Substitute $s^{'} (g) = i (u (g)) s (g)$ into the factor-set formula of [F2]. After moving kernel terms past lifts by the prescribed action, the result is $f_{s^{'}} (g, h) = u (g) + g \cdot u (h) - u (g h) + f_{s} (g, h) = f_{s} (g, h) + (δ u) (g, h) .$

F1F2step 1.1algebra

3.1

Thus the two factor sets differ by a coboundary; [L1] shows that both are indeed cocycles.

L1step 2.1∎

Depends on

Used by

An extension determines a well-defined H² class Corollary
Changing a section by a one-cochain Example
FALSE: the factor set is independent of the section as a function False statement
Cohomologous two-cocycles give equivalent extensions Lemma

Dependency tree · two levels

4 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

Clara Loh, Group Cohomology, SS 2019 (standard reference, not scraped)
Caroline Lassueur, Cohomology of Groups, SS 2021 (standard reference, not scraped)