Group G
G-additive group M
Natural number n
Gn = { σ1, … , σn | σi ∈ G }
Cn = ( G, M ) = { φ : Gn → M | φ is map as set }
Co = ( G, M ) = M
Against Cn = ( G, M )
( φ + ψ ) ( σ ) = φ ( σ ) + φ( σ ) φ , ψ ∈ Cn σ∈ Cn
Element of Cn = ( G, M ) n- Cochain
Homomorphism dn = Cn ( G, M ) → Cn+1 ( G, M ) n ≥ 0
d n+1. dn = 0
Zn ( G, M ) = Ker ( dn ) n ≥ 0
Bn ( G, M ) = Im ( dn-1 ) n ≥ 1
Element of Zn ( G, M ) is n-cosylcle.
Element of Bn ( G, M ) is n-coboundary.
Bn ( G, M ) ⊂ Zn ( G, M )
Cohomology group of M is below.
Hn ( G, M ) = Zn ( G, M ) / Bn ( G, M )
H0 ( G, M ) = Z0 ( G, M )
[Note]
Cohomology group may be helpful to the meaning of words and their variations.
[References]
Tokyo July 29 2007
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