Theorems · Definition · group theory

groupHomology.H1AddEquivOfIsTrivial

{k G : Type u} →
  [inst : CommRing k] →
    [inst_1 : Group G] →
      (A : Rep.{u, u, u} k G) →
        [A.IsTrivial] → ↑(groupHomology.H1 A) ≃+ TensorProduct ℤ (Additive (Abelianization G)) ↑A

If a G-representation on A is trivial, this is the group isomorphism between H₁(G, A) ≃+ Gᵃᵇ ⊗[ℤ] A defined by ⟦single g a⟧ ↦ ⟦g⟧ ⊗ a.

Defined in

Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Cited by

4 results in Mathlib

Foundations

Depth 125 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingGroupRep.IsTrivial