Homological algebra MOC

Group cohomology

To a group 𝐺 and a [[Module|ℤ[𝐺]-module]] we associate the cochain complex (𝐶∙(𝐺,𝑀),𝑑∙) homology where the 𝑘-cochains

𝐶𝑘(𝐺,𝑀)={(}𝐺𝑘,𝑀)

are functions 𝛼 :𝐺𝑘 →𝑀 and the coboundary operators

𝑑𝑘+1:𝐶𝑘(𝐺,𝑀)→𝐶𝑘+1(𝐺,𝑀)

are defined by the rather unwieldy formula

𝑑𝑘+1𝛼(𝑔1,⋮,𝑔𝑘+1)=𝑔1𝛼(𝑔1,⋮,𝑔𝑘+1)+𝑘+1∑𝑖=1(−1)𝑖𝛼(𝑔1,⋮,𝑔𝑖−1,𝑔𝑖𝑔𝑖+1,𝑔𝑖+2,⋮,𝑔𝑘+1)

where we interpret 𝑔𝑘+2 =𝑒.

See also

• Central group extension

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