Special-biserial presentations of ambient algebras

The category algebra of covariant representables on selected right projectives is the opposite ambient algebra. Combining that calculation with the ordinary-arrow presentation first gives a special-biserial presentation of the opposite algebra. Applying the same construction to the opposite primitive-projective presentation removes this variance and gives a presentation of the original ambient algebra.

theorem MagnitudeConjecture.RightModule.FiniteIndecomposableSkeleton.PrimitiveProjectivePresentation.ambientOpposite_admitsSpecialBiserialPresentation_of_isBiserial {k A : Type u} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A] [IsNoetherianRing A] [IsNoetherianRing Aᵐᵒᵖ] [IsAlgClosed k] {S : FiniteIndecomposableSkeleton k A} (P : S.PrimitiveProjectivePresentation) (hP : P.IsBiserial) : BoundQuiver.AdmitsSpecialBiserialPresentation k Aᵐᵒᵖ

A biserial primitive-projective presentation gives a literal special- biserial bound-quiver presentation of the opposite ambient algebra. This is the variance-correct form of the category-algebra calculation.

theorem MagnitudeConjecture.RightModule.FiniteIndecomposableSkeleton.PrimitiveProjectivePresentation.ambient_admitsSpecialBiserialPresentation_of_isBiserial {k A : Type u} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A] [IsNoetherianRing A] [IsNoetherianRing Aᵐᵒᵖ] [IsAlgClosed k] {S : FiniteIndecomposableSkeleton k A} (P : S.PrimitiveProjectivePresentation) (hP : P.IsBiserial) : BoundQuiver.AdmitsSpecialBiserialPresentation k A

A complete biserial primitive-projective presentation gives a literal special-biserial bound-quiver presentation of its ambient algebra. Apply the variance-correct category-algebra calculation to the opposite presentation and then remove the double opposite.