The beta characterization of representation-finite special-biserial algebras

The forward implication transports to a literal special-biserial Morita representative, applies the proved projective-injective socle reduction, and uses the string-algebra middle-term bound. The converse is developed below through the representation-finite Auslander--Reiten structure theorem.

theorem MagnitudeConjecture.specialBiserial_beta_le_two {k A : Type u} [Field k] [IsAlgClosed k] [Ring A] [Algebra k A] [FiniteDimensional k A] [IsNoetherianRing Aᵐᵒᵖ] (S : RightModule.FiniteIndecomposableSkeleton k A) (hSpecial : BoundQuiver.IsSpecialBiserial k A) : FiniteTauMatrix.beta S.finiteTauCategoryData.toFiniteRightTauCategoryData ≤ 2

A representation-finite special-biserial algebra has at most two nonprojective occurrences in every almost-split middle term.

theorem MagnitudeConjecture.representationFinite_isSpecialBiserial_iff_beta_le_two {k A : Type u} [Field k] [IsAlgClosed k] [Ring A] [Algebra k A] [FiniteDimensional k A] [IsNoetherianRing Aᵐᵒᵖ] (S : RightModule.FiniteIndecomposableSkeleton k A) : BoundQuiver.IsSpecialBiserial k A ↔ FiniteTauMatrix.beta S.finiteTauCategoryData.toFiniteRightTauCategoryData ≤ 2

For a finite-dimensional representation-finite algebra over an algebraically closed field, special-biseriality is equivalent to the bound β ≤ 2 on nonprojective occurrences in almost-split middle terms.

theorem MagnitudeConjecture.representationFinite_isSpecialBiserial_iff_beta_le_two_of_presentation {k A : Type u} [Field k] [IsAlgClosed k] [Ring A] [Algebra k A] [FiniteDimensional k A] [IsNoetherianRing Aᵐᵒᵖ] [IsNoetherianRing A] [IsNoetherianRing Aᵐᵒᵖᵐᵒᵖ] (S : RightModule.FiniteIndecomposableSkeleton k A) (_P : S.PrimitiveProjectivePresentation) : BoundQuiver.IsSpecialBiserial k A ↔ FiniteTauMatrix.beta S.finiteTauCategoryData.toFiniteRightTauCategoryData ≤ 2

The beta characterization specialized to an algebra carrying a complete duplicate-free primitive-projective presentation.