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An extension determines a well-defined H^2 class

Statement

An extension of $G$ by the abelian $G$ -module $M$ , together with a normalized section, determines a cohomology class $[f_{s}] \in H^{2} (G, M)$ that is independent of the chosen normalized section.

Facts & Assumptions

Given: An extension $1 \to M \to E \to G \to 1$ inducing the given action, and a normalized section $s$ .

[F1]

The second cohomology group is the quotient of cocycles by coboundaries (Second cohomology by factor sets).

[L1]

Replacing the section changes the factor set by a coboundary (Changing the section changes the factor set by a coboundary).

Proof

technique · direct

1.1

Because $s (1) = 1$ , the factor-set formula gives $f_{s} (1, g) = f_{s} (g, 1) = 0$ for every $g \in G$ .

givenalgebra

2.1

Comparing $(s (g) s (h)) s (k)$ with $s (g) (s (h) s (k))$ and translating the conjugation term through the prescribed $G$ -action yields $g \cdot f_{s} (h, k) - f_{s} (g h, k) + f_{s} (g, h k) - f_{s} (g, h) = 0.$ So $f_{s}$ is a normalized two-cocycle.

step 1.1givenalgebra

3.1

Steps 1.1 and 2.1 show that $f_{s} \in Z^{2} (G, M)$ , so [F1] defines a class $[f_{s}] \in H^{2} (G, M)$ .

F1step 1.1step 2.1

4.1

If $s^{'}$ is another normalized section, then [L1] gives $f_{s^{'}} = f_{s} + δ u$ for some one-cochain $u$ . Thus $f_{s}$ and $f_{s^{'}}$ define the same coset in the quotient [F1].

F1L1step 3.1

5.1

Therefore the extension determines a well-defined class in $H^{2} (G, M)$ independent of the chosen normalized section.

step 4.1∎

Depends on

Used by

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

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