Theorems · Theorem · group theory

Rep.FiniteCyclicGroup.coinvariantsTensorResolutionIso_hom_f_hom_apply

∀ {k G : Type u} [inst : CommRing k] [inst_1 : CommGroup G] [inst_2 : Fintype G] (A : Rep.{u, u, u} k G) (g : G)
  (hg : ∀ (x : G), x ∈ Subgroup.zpowers g) (i : ℕ)
  (a :
    TensorProduct k (↑A) (MonoidAlgebra k G) ⧸
      (Representation.Coinvariants.ker (A.ρ.tprod (Representation.leftRegular k G))).toAddSubgroup),
  (ModuleCat.Hom.hom ((Rep.FiniteCyclicGroup.coinvariantsTensorResolutionIso A g hg).hom.f i)) a =
    (QuotientAddGroup.lift (Representation.Coinvariants.ker (A.ρ.tprod (Representation.leftRegular k G))).toAddSubgroup
        (TensorProduct.lift ((Finsupp.linearCombination k fun g => A.ρ g⁻¹) ∘ₗ ↑(MonoidAlgebra.coeffLinearEquiv k)) ∘ₗ
            ↑(TensorProduct.comm k (↑A) (MonoidAlgebra k G))).toAddMonoidHom
        ⋯)
      a

Defined in

Mathlib.RepresentationTheory.Homological.GroupHomology.FiniteCyclic

Cited by

0 results in Mathlib

Foundations

Depth 114 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommGroupFintype