Theorems · Definition · group theory

groupHomology.H1ToTensorOfIsTrivial

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

If a G-representation on A is trivial, this is the natural map H₁(G, A) → Gᵃᵇ ⊗[ℤ] A sending ⟦single g a⟧ to ⟦g⟧ ⊗ₜ a.

Defined in

Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Cited by

3 results in Mathlib

Foundations

Depth 117 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingGroupRep.IsTrivial