Vanishing of the module-category surplus for string algebras

The literal Butler--Ringel count is stated for the quotient-category algebra carried by a string presentation. This file transports that count across the algebra equivalence bundled in an arbitrary string model and states the result for any complete duplicate-free right-module skeleton of the presented algebra.

theorem MagnitudeConjecture.RightModule.FiniteIndecomposableSkeleton.rightMiddleArity_le_two_of_admitsStringPresentation {k A : Type u} [Field k] [IsAlgClosed k] [Ring A] [Algebra k A] [FiniteDimensional k A] [IsNoetherianRing Aᵐᵒᵖ] (S : FiniteIndecomposableSkeleton k A) (hString : BoundQuiver.AdmitsStringPresentation k A) (i : Fin S.n) : FiniteTauMatrix.rightMiddleArity S.finiteTauCategoryData.toFiniteRightTauCategoryData i ≤ 2

Every right-mesh middle term of a representation-finite string algebra has total arity at most two, including the projective endpoints.

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

Every representation-finite algebra admitting a string presentation has beta ≤ 2: its nonprojective right-mesh middle terms have total arity at most two, and beta only counts their nonprojective summands.

theorem MagnitudeConjecture.RightModule.FiniteIndecomposableSkeleton.ambientARSurplus_eq_zero_of_admitsStringPresentation {k A : Type u} [Field k] [IsAlgClosed k] [Ring A] [Algebra k A] [FiniteDimensional k A] [IsNoetherianRing Aᵐᵒᵖ] (S : FiniteIndecomposableSkeleton k A) (hString : BoundQuiver.AdmitsStringPresentation k A) : S.ambientARSurplus = 0

Every representation-finite algebra admitting a string presentation has zero Auslander--Reiten surplus on any chosen indecomposable skeleton.