Theorems · Definition · group theory

groupCohomologyIso

{k G : Type u} →
  [inst : CommRing k] →
    [inst_1 : Group G] →
      (A : Rep.{u, u, u} k G) →
        (n : ℕ) →
          (P : CategoryTheory.ProjectiveResolution (Rep.trivial k G k)) →
            groupCohomology A n ≅ HomologicalComplex.homology (P.complex.linearYonedaObj k A) n

The nth group cohomology of a k-linear G-representation A is isomorphic to Hⁿ(Hom(P, A)), where P is any projective resolution of k as a trivial k-linear G-representation.

Defined in

Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic

Cited by

0 results in Mathlib

Foundations

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

Assumes

CommRingGroup