Remark 11.5: the indicator cocycles span HH²(A₀)_{>0}

rem:classical-mixtures (binary-families.tex): if V is the span of the indicator cocycles of Proposition 11.2, Corollary 10.5 gives the exact sequence 0 → V ∩ B²(A₀) → V → HH²(A₀)_{>0} → 0.

The left part is the definition of the map V → HH². The content is surjectivity: every Hochschild two-cocycle of A₀ of positive internal degree is a combination of indicator cocycles plus a coboundary (classicalMixtures), for a basic representation-finite algebra in characteristic two. Here HH² is ordinary Hochschild cohomology (all cocycles and coboundaries of the given internal degree; RelativeHochschild.lean), and Corollary 10.5 is the proved cor105Hyp.

theorem MultiplicativeBasisDeformations.IndSkeleton.classicalMixtures {k A : Type u} [Field k] [IsAlgClosed k] [Ring A] [Algebra k A] [FiniteDimensional k A] [IsNoetherianRing Aᵐᵒᵖ] (S : IndSkeleton k A) (b : S.ArrowBases) [CharP k 2] (hbasic : IsBasicAlgebra A) (hA : IsRepresentationFinite k A) (v₀ : S.ARVertex) : let D := S.arMeshData b; have G := MeshWalk.lenA0 D ⋯; ∃ (hF : IsRepresentationFinite k (CatAlg.Alg D.F0)ᵐᵒᵖ) (ι : Type) (x : Fintype ι) (x_1 : DecidableEq ι) (Wd : ι → MeshWalk.MeshWt S.arTranslationQuiver ℤ) (n : ℕ) (E : Fin n → Quotient (MeshWalk.SysD D v₀ hF Wd).pairSetoid) (qd : Fin n → ℕ), (∀ (j : Fin n), 0 < qd j ∧ LenGrading.IsRel₂ (MeshWalk.VxA0 D v₀ hF Wd) ((MeshWalk.SysD D v₀ hF Wd).transp k (MeshWalk.ΘD D v₀ hF Wd) ((MeshWalk.SysD D v₀ hF Wd).muE k (E j))) ∧ LenGrading.IsCocycle₂ ((MeshWalk.SysD D v₀ hF Wd).transp k (MeshWalk.ΘD D v₀ hF Wd) ((MeshWalk.SysD D v₀ hF Wd).muE k (E j))) ∧ G.HasDeg₂ (qd j) ((MeshWalk.SysD D v₀ hF Wd).transp k (MeshWalk.ΘD D v₀ hF Wd) ((MeshWalk.SysD D v₀ hF Wd).muE k (E j)))) ∧ ∀ (q : ℕ), 0 < q → ∀ (φ : (CatAlg.Alg D.F0)ᵐᵒᵖ →ₗ[k] (CatAlg.Alg D.F0)ᵐᵒᵖ →ₗ[k] (CatAlg.Alg D.F0)ᵐᵒᵖ), G.HasDeg₂ q φ → LenGrading.IsCocycle₂ φ → ∃ (cf : Fin n → k) (f : (CatAlg.Alg D.F0)ᵐᵒᵖ →ₗ[k] (CatAlg.Alg D.F0)ᵐᵒᵖ), G.HasDeg₁ q f ∧ φ = ∑ j : Fin n with qd j = q, cf j • (MeshWalk.SysD D v₀ hF Wd).transp k (MeshWalk.ΘD D v₀ hF Wd) ((MeshWalk.SysD D v₀ hF Wd).muE k (E j)) + LenGrading.cobdry k f

Remark 11.5 (rem:classical-mixtures): for a basic representation-finite algebra in characteristic two, the transported indicator cocycles μ_E of Proposition 11.2 (normalized relative cocycles of positive degree) span HH²(A₀)_{>0}: every Hochschild two-cocycle φ of A₀ of internal degree q > 0 is Σ c_j μ_{E_j} + d f with f of degree q.