Second cohomology of a group with coëfficients in an abelian group

Given a group $𝐺$ and an abelian group $𝐵$ , we may consider $𝐵$ as a $ℤ 𝐺$ -module with $𝐺$ acting trivially, and therefore the cohomology of $𝐺$ with coëfficents in $𝐵$ . In the resulting second chain complex, the 2-cochains $𝐶 2 (𝐴, 𝐵)$ are maps¹

𝜀 0 : 𝐴 \times 𝐴 \to 𝐵

and the 2-cocycles $𝑍 2 (𝐺, 𝐵)$ are 2-cochains such that

𝜀 0 (𝑎, 𝑏) + 𝜀 0 (𝑎 𝑏, 𝑐) = 𝜀 0 (𝑏, 𝑐) + 𝜀 0 (𝑎, 𝑏 𝑐) \forall 𝑎, 𝑏, 𝑐 \in 𝐺

and the 2-coboundaries $𝐵 2 (𝐺, 𝐵)$ are 2-cochains such that

𝜀 0 (𝑎 𝑏) = 𝜂 (𝑎 𝑏) - 𝜂 (𝑎) - 𝜂 (𝑏) \forall 𝑎, 𝑏 \in 𝐺

for some 1-cochain $𝜂 : 𝐴 \to 𝐵$ . Thus, in particular, $ℤ$ -bilinear maps $𝐴 \times 𝐴 \to 𝐵$ are 2-cocycles. Thus

𝐻 2 (𝐴, 𝐵) = 𝑍 2 (𝐴, 𝐵) / 𝐵 2 (𝐴, 𝐵) .

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1988. Vertex operator algebras and the Monster, §5.1, p. 103 ↩

Second cohomology of a group with coëfficients in an abelian group

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