Research proof index

Assume that we have a short exact sequence 0 → A → B → C → 0 in Rep R G and that the sequence of H- invariants is also a short exact in Rep R (G ⧸ H) : 0 → Aᴴ → Bᴴ → Cᴴ → 0. Then we have a commuting square Hⁿ(G ⧸ H, Cᴴ) ⟶ H^{n+1}(G ⧸ H, Aᴴ) | | ↓ ↓ Hⁿ(G , C) ⟶ H^{n+1}(G,A) where the horizontal maps are connecting homomorphisms an...

Given any short exact sewuence 0 → A → B → C → 0 in Rep R G and any subgroup H of G, the following diagram is commutative Hⁿ(G,C) ⟶ H^{n+1}(G A) | | ↓ ↓ Hⁿ(H,C) ⟶ H^{n+1}(G A). The vertical arrows are restriction and the horizontals are connecting homomorphisms. For this, it would be sensible to define restriction as a natural transformation, so t...

If H²ⁿ⁺²(H,M) and H²ᵐ⁺¹(H,M) are both zero for every subgroup H of G then M is acyclic. -/ theorem groupCohomology.trivialCohomology_of_even_of_odd_of_solvable [Finite G] [Group.IsSolvable G] (M : Rep R G) (n m : ℕ) -- todo: don't quantify over all types (h_even : ∀ (H : Type) [Group H] {φ : H →* G} (_ : Function.Injective φ), IsZero (groupCohom...

If H²ⁿ⁺²(H,M) and H²ᵐ⁺¹(H,M) are both zero for every subgroup H of G then M is acyclic.

Let G be a finite group acting on S, and R be the fixed subring. If Q is a prime of S with finite residue field, then there exists a Frobenius element σ : G at Q.

If φ : 𝔸_K^f ⊗ V → 𝔸_K^f ⊗ V is 𝔸_K^f-linear and φₚ is its local component at a place p then for all x : 𝔸_K^f ⊗ V we have (evalₚ ⊗ id_V) (φ x) = φₚ ((evalₚ ⊗ id_V) x), or, more colloquiually, (φ x)ₚ = φₚ (xₚ).