Homological algebra MOC

Group cohomology

Cochain complex

To a group $𝐺$ and a $ℤ 𝐺$ -module $𝑀$ we associate the cochain complex $(𝐶 ∙ (𝐺, 𝑀), 𝑑 ∙)$ where the $𝑘$ -cochains

𝐶 𝑘 (𝐺, 𝑀) = 𝖲 𝖾 𝗍 (𝐺 𝑘, 𝑀)

are functions $𝛼 : 𝐺 𝑘 \to 𝑀$ . For each $𝑘 \in ℕ$ , the coboundary operator

d 𝑘 + 1 : 𝐶 𝑘 (𝐺, 𝑀) \to 𝐶 𝑘 + 1 (𝐺, 𝑀)

acts as follows: Given a $𝑘$ -cochain $𝛼$ , the $(𝑘 + 1)$ -cochain $d 𝛼$ takes $(𝑔 1, \dots, 𝑔 𝑘 + 1)$ to the alternating sum

𝑔 1 𝛼 (𝑔 2, \dots, 𝑔 𝑘 + 1) + 𝑘 \sum 𝑖 = 1 (- 1) 𝑘 𝛼 (𝑔 1, \dots, 𝑔 𝑖 𝑔 𝑖 + 1, \dots, 𝑔 𝑘 + 1) + (- 1) 𝑘 𝛼 (𝑔 1, \dots, 𝑔 𝑘) .

Special cases

Second cohomology of a group with coëfficients in an abelian group

See also

Central group extension

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