Theorems · Definition · group theory

Rep.FiniteCyclicGroup.homResolutionIso

{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) →
              (Rep.FiniteCyclicGroup.resolution k g hg).complex.linearYonedaObj k A ≅
                Rep.FiniteCyclicGroup.moduleCatCochainComplex A g

Given a finite cyclic group G generated by g : G and a k-linear G-representation A, the periodic cochain complex 0 ⟶ Hom(k[G], A) --(- ∘ (ρ(g) - 𝟙))--> Hom(k[G], A) --(- ∘ N)--> Hom(k[G], A) ⟶ ... is isomorphic as a complex in ModuleCat k to 0 ⟶ A --(ρ(g) - 𝟙)--> A --N--> A --(ρ(g) - 𝟙)--> A --N--> A ⟶ ....

Defined in

Mathlib.RepresentationTheory.Homological.GroupCohomology.FiniteCyclic

Cited by

2 results in Mathlib

Foundations

Depth 112 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommGroupFintype