Proposition 12.5 for the algebra A₀ of a representation-finite algebra #
deformation-refinements.tex (prop:full-hochschild-comparison). For the mesh
category C = k(Γ_A) of a basic representation-finite algebra A, the mesh presentations are
resolutions (they come from the almost split sequences; the paragraph after Lemma 12.3), so
Ψ : HH²(A₀)_q → coker D_{q+1} is an isomorphism in every internal degree q ≥ 1
(MeshWalk.fullHochschildComparison).
theorem
MultiplicativeBasisDeformations.IndSkeleton.hres_ar
{k A : Type w}
[Field k]
[IsAlgClosed k]
[Ring A]
[Algebra k A]
[FiniteDimensional k A]
[IsNoetherianRing Aᵐᵒᵖ]
(S : IndSkeleton k A)
(b : S.ArrowBases)
(w x : S.ARVertex)
:
Function.Injective
⇑(FlatPert.relMap (FlatPert.meshPres (S.arCoefficients b))
(FlatPert.idealC (FlatPert.meshPres (S.arCoefficients b))) (FlatPert.meshPres (S.arCoefficients b)).r w x) ∧ (FlatPert.alpha (FlatPert.idealC (FlatPert.meshPres (S.arCoefficients b))) w x).ker ≤ (FlatPert.relMap (FlatPert.meshPres (S.arCoefficients b)) (FlatPert.idealC (FlatPert.meshPres (S.arCoefficients b)))
(FlatPert.meshPres (S.arCoefficients b)).r w x).range
The mesh presentations of k(Γ_A) are resolutions: the representables applied to the
almost split sequences (deformation-refinements.tex).
theorem
MultiplicativeBasisDeformations.IndSkeleton.fullHochschildComparison_ar
{k A : Type w}
[Field k]
[IsAlgClosed k]
[Ring A]
[Algebra k A]
[FiniteDimensional k A]
[IsNoetherianRing Aᵐᵒᵖ]
(S : IndSkeleton k A)
(b : S.ArrowBases)
{q : ℕ}
(hq : 1 ≤ q)
:
(∀ (z : ScalarMeshCategory.MeshFamily (S.arCoefficients b)),
(∀ (x : ↑S.arTranslationQuiver.nonprojective),
z x ∈ ScalarMeshCategory.degreePart (S.arCoefficients b) (q + 2) ↑(S.arTranslationQuiver.tau x) ↑x) →
∃ (μ :
(CatAlg.Alg (S.arMeshData b).F0)ᵐᵒᵖ →ₗ[k] (CatAlg.Alg (S.arMeshData b).F0)ᵐᵒᵖ →ₗ[k] (CatAlg.Alg (S.arMeshData b).F0)ᵐᵒᵖ)
(_ : GExt.IsHCocycle μ) (_ : GExt.HDeg (MeshWalk.galgA0 (S.arMeshData b) ⋯) μ ↑q) (Ld :
MeshWalk.LiftData (S.arMeshData b) ⋯ q μ), MeshWalk.Ψc Ld = Submodule.Quotient.mk z) ∧ ∀
(μ :
(CatAlg.Alg (S.arMeshData b).F0)ᵐᵒᵖ →ₗ[k] (CatAlg.Alg (S.arMeshData b).F0)ᵐᵒᵖ →ₗ[k] (CatAlg.Alg (S.arMeshData b).F0)ᵐᵒᵖ),
GExt.IsHCocycle μ →
GExt.HDeg (MeshWalk.galgA0 (S.arMeshData b) ⋯) μ ↑q →
∀ (Ld : MeshWalk.LiftData (S.arMeshData b) ⋯ q μ),
MeshWalk.Ψc Ld = 0 →
∃ (H : (CatAlg.Alg (S.arMeshData b).F0)ᵐᵒᵖ →ₗ[k] (CatAlg.Alg (S.arMeshData b).F0)ᵐᵒᵖ),
(∀ (p : ℤ) (a : (CatAlg.Alg (S.arMeshData b).F0)ᵐᵒᵖ),
(MeshWalk.galgA0 (S.arMeshData b) ⋯).gr.IsHom p a →
(MeshWalk.galgA0 (S.arMeshData b) ⋯).gr.IsHom (p + ↑q) (H a)) ∧ μ = GExt.hcobdry H
Proposition 12.5 for A₀ = k(Γ_A)-projective corner: Ψ is an isomorphism in every
internal degree q ≥ 1.