Theorems · Definition · group theory

groupHomology.cyclesMk

{k G : Type u} →
  [inst : CommRing k] →
    [inst_1 : Group G] →
      {A : Rep.{u, u, u} k G} →
        (m n : ℕ) →
          (ComplexShape.down ℕ).next m = n →
            (f : (Fin m → G) →₀ ↑A) →
              (CategoryTheory.ConcreteCategory.hom ((groupHomology.inhomogeneousChains A).d m n)) f = 0 →
                ↑(groupHomology.cycles A m)

When m = 0 this makes a term of cycles A 0 from any element of A (or more precisely any element in the kernel of d₀,₀ = 0). When m is positive, this makes a term of cycles A m from any element of the kernel of dₘ,ₘ₋₁.

Defined in

Mathlib.RepresentationTheory.Homological.GroupHomology.Basic

Cited by

6 results in Mathlib

Foundations

Depth 115 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingGroup