Proposition 2.10(2): A₀ is basic #
prop:projective-evaluation (2): "The algebra A₀ is basic and representation-finite." The
vertex idempotents of A₀ are primitive (FpA0); a corner element e_i x e_j with i ≠ j has
no degree-zero part (degree zero is spanned by the vertex idempotents, VxG), so an inverse pair
x ∈ e_iA₀e_j, y ∈ e_jA₀e_i would put e_i = xy in positive degree (isBasic_A0).
theorem
MultiplicativeBasisDeformations.MeshWalk.isBasic_A0
{k : Type u}
[Field k]
{V : Type}
[Quiver V]
[Fintype V]
[DecidableEq V]
{Q : TranslationQuiver V}
[(x : V) → Fintype ((y : V) × (y ⟶ x))]
{c : MeshCoefficients k Q}
{L : CatAlg.LocalFamily (meshFam Q c)}
(D : CatAlg.MeshData (meshFam Q c) L)
[IsAlgClosed k]
[hone : ∀ (x y : V), Subsingleton (x ⟶ y)]
[(x y : V) → Fintype (x ⟶ y)]
[hfd : ∀ (x y : V), FiniteDimensional k (meshFam Q c x ⟶ meshFam Q c y)]
(v₀ : V)
(hF : IsRepresentationFinite k (CatAlg.Alg D.F0)ᵐᵒᵖ)
:
IsBasicAlgebra (CatAlg.Alg D.F0)ᵐᵒᵖ
A₀ is basic (Proposition 2.10(2)).
theorem
MultiplicativeBasisDeformations.IndSkeleton.projectiveAlgebra_basic_repFinite
{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)
(v₀ : S.ARVertex)
:
IsBasicAlgebra (CatAlg.Alg (S.arMeshData b).F0)ᵐᵒᵖ ∧ IsRepresentationFinite k (CatAlg.Alg (S.arMeshData b).F0)ᵐᵒᵖ
Proposition 2.10(2) for a basic representation-finite algebra A: A₀ is basic and
representation-finite.