cocycles₂ — Mathlib

English

The 2-cocycles Z^2(G,A) are defined as the kernel of the differential d23: Fun(G × G, A) → Fun(G × G × G, A).

Русский

2-цикл Z^2(G,A) определяется как ядро дифференциала d23: Fun(G × G, A) → Fun(G × G × G, A).

LaTeX

$$Z^2(G,A) = Ker(d_{23} A)$$

Lean4

/-- The 2-cocycles `Z²(G, A)` of `A : Rep k G`, defined as the kernel of the map
`Fun(G × G, A) → Fun(G × G × G, A)` sending
`(f, (g₁, g₂, g₃)) ↦ ρ_A(g₁)(f(g₂, g₃)) - f(g₁g₂, g₃) + f(g₁, g₂g₃) - f(g₁, g₂).` -/
def cocycles₂ : Submodule k (G × G → A) :=
  LinearMap.ker (d₂₃ A).hom

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