Group cohomology

To a group $𝐺$ and a $ℤ [𝐺]$ -module we associate the cochain complex $(𝐶 ∙ (𝐺, 𝑀), 𝑑 ∙)$ #m/def/homology where the $𝑘$ -cochains

𝐶 𝑘 (𝐺, 𝑀) = 𝖲 𝖾 𝗍 (𝐺 𝑘, 𝑀)

are functions $𝛼 : 𝐺 𝑘 \to 𝑀$ and the coboundary operators

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

are defined by the rather unwieldy formula

𝑑 𝑘 + 1 𝛼 (𝑔 1, ⋮, 𝑔 𝑘 + 1) = 𝑔 1 𝛼 (𝑔 1, ⋮, 𝑔 𝑘 + 1) + 𝑘 + 1 \sum 𝑖 = 1 (- 1) 𝑖 𝛼 (𝑔 1, ⋮, 𝑔 𝑖 - 1, 𝑔 𝑖 𝑔 𝑖 + 1, 𝑔 𝑖 + 2, ⋮, 𝑔 𝑘 + 1)

where we interpret $𝑔 𝑘 + 2 = 𝑒$ .

See also