Covariant stable representables detect uniserial modules #
This file completes Auslander--Reiten, Proposition 1.1(a), in the finite right-module skeleton: a nonzero indecomposable nonprojective module whose projective-stable covariant representable is uniserial is itself uniserial.
theorem
MagnitudeConjecture.RightModule.FiniteIndecomposableSkeleton.isUniserialModule_of_projectiveStableCovariantRepresentable
{k A : Type u}
[Field k]
[Ring A]
[Algebra k A]
[FiniteDimensional k A]
[IsNoetherianRing Aᵐᵒᵖ]
(S : FiniteIndecomposableSkeleton k A)
{X : FinitelyGeneratedCategory A}
(hX : CategoryTheory.Indecomposable X)
(hXnonprojective : ¬CategoryTheory.Projective X)
(hstable : IsUniserialObject (S.finiteProjectiveStableCovariantRepresentable X))
:
IsUniserialModule Aᵐᵒᵖ ↑X
Auslander--Reiten Proposition 1.1(a), in the exact finite-skeleton interface used by the magnitude campaign.