Research proof index

Maddy's Lemma : Density implies continuity.

Andrew's Lemma : Density for algebraic extensions.

Given map f: M ⟶ N and q : ℕ, if H^{q+1}(M) ⟶ H^{q+1}(N) is surjective, then any z : Z^{q+1}(N) can be written as f(z') + d(y) for some z' : Z^{q+1}(M) and y : C^q(M). Note that d is spelled as toCocycles.

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...