Magnitude conjecture

MagnitudeConjecture.Algebra.RightModuleStandardFormARIncomingFactors

Factorization through the recovered incoming map #

source
@[instance_reducible]
def MagnitudeConjecture.RightModule.FiniteIndecomposableSkeleton.standardFormARIncomingFactorsQuiver {k A : Type u} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A] [IsNoetherianRing Aᵐᵒᵖ] (S : FiniteIndecomposableSkeleton k A) :
Quiver (Fin S.n)
Instances For
    source
    @[instance_reducible]
    noncomputable def MagnitudeConjecture.RightModule.FiniteIndecomposableSkeleton.standardFormARIncomingFactorsArrowFintype {k A : Type u} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A] [IsNoetherianRing Aᵐᵒᵖ] (S : FiniteIndecomposableSkeleton k A) (x y : Fin S.n) :
    Fintype (x ⟶ y)
    Instances For
      source
      theorem MagnitudeConjecture.RightModule.FiniteIndecomposableSkeleton.standardFormRecoveredIncomingMap_factors_obj {k A : Type u} [Field k] [IsAlgClosed k] [Ring A] [Algebra k A] [FiniteDimensional k A] [IsNoetherianRing Aᵐᵒᵖ] (S : FiniteIndecomposableSkeleton k A) (x z : Fin S.n) (q : S.standardFormProjectiveVertexModuleIndecomposableSkeleton.obj x ⟶ S.standardFormProjectiveVertexModuleIndecomposableSkeleton.obj z) (hq : ¬CategoryTheory.IsSplitEpi q) :
      ∃ (factor : S.standardFormProjectiveVertexModuleIndecomposableSkeleton.obj x ⟶ S.standardFormAdditiveRestrictedYonedaFunctor.obj (S.standardFormRightMeshData.additiveIncomingObj z)), CategoryTheory.CategoryStruct.comp factor (S.standardFormRecoveredIncomingMap z) = q

      Every nonsplit morphism between recovered skeleton objects factors through the complete recovered incoming map.