Transposes and translates detect indecomposable isomorphism classes

For non-projective finite-length indecomposable modules, an isomorphism between transposes of finite projective presentations implies an isomorphism of the original modules. When the transposes are reflexive over the scalar ring, the same holds for their scalar duals, the Auslander--Reiten translates. For minimal presentations this becomes an iff: the translate detects isomorphism classes on the non-projective indecomposables.

Minimality is needed only for the forward implication. Arbitrary finite projective presentations suffice for reflection. The scalar base need not be a field; reflexivity is automatic for finite-dimensional transposes over a field.

The proofs use the full and faithful stable transpose, the projective stable quotient's detection of non-projective indecomposable isomorphism classes, and scalar double duality.

References

• M. Auslander, I. Reiten, S. O. Smalø, Representation Theory of Artin Algebras, Cambridge University Press (1995), Section IV.1.

theorem TauCeti.FiniteProjectivePresentation.nonempty_linearEquiv_of_auslanderReitenTranspose {A : Type u} [Ring A] [Small.{v, u} A] {M N : ModuleCat A} (P : FiniteProjectivePresentation M) (Q : FiniteProjectivePresentation N) (hM : IsFiniteLength A ↑M) (hN : IsFiniteLength A ↑N) (hiM : IsIndecomposableModule A ↑M) (hiN : IsIndecomposableModule A ↑N) (hpM : ¬Module.Projective A ↑M) (e : AuslanderReitenTranspose P.p ≃ₗ[Aᵐᵒᵖ] AuslanderReitenTranspose Q.p) : Nonempty (↑M ≃ₗ[A] ↑N)

Isomorphic transposes of finite projective presentations detect isomorphism of finite-length indecomposable modules, provided the source module is not projective. The presentations need not be minimal.

theorem TauCeti.FiniteProjectivePresentation.nonempty_linearEquiv_of_auslanderReitenTranslate {A : Type u} [Ring A] [Small.{v, u} A] {M N : ModuleCat A} {k : Type u_1} [CommSemiring k] [Algebra k A] (P : FiniteProjectivePresentation M) (Q : FiniteProjectivePresentation N) [Module.IsReflexive k (AuslanderReitenTranspose P.p)] [Module.IsReflexive k (AuslanderReitenTranspose Q.p)] (hM : IsFiniteLength A ↑M) (hN : IsFiniteLength A ↑N) (hiM : IsIndecomposableModule A ↑M) (hiN : IsIndecomposableModule A ↑N) (hpM : ¬Module.Projective A ↑M) (e : AuslanderReitenTranslate k P.p ≃ₗ[A] AuslanderReitenTranslate k Q.p) : Nonempty (↑M ≃ₗ[A] ↑N)

Isomorphic Auslander--Reiten translates detect isomorphism of non-projective, finite-length indecomposable modules when both transposes are scalar-reflexive. No minimality of either finite projective presentation is needed.

theorem TauCeti.FiniteProjectivePresentation.nonempty_linearEquiv_auslanderReitenTranslate_iff {A : Type u} [Ring A] [Small.{v, u} A] {M N : ModuleCat A} {k : Type u_1} [CommSemiring k] [Algebra k A] (P : FiniteProjectivePresentation M) (Q : FiniteProjectivePresentation N) [Module.IsReflexive k (AuslanderReitenTranspose P.p)] [Module.IsReflexive k (AuslanderReitenTranspose Q.p)] (hP : IsMinimalProjectivePresentation P.p P.π) (hQ : IsMinimalProjectivePresentation Q.p Q.π) (hM : IsFiniteLength A ↑M) (hN : IsFiniteLength A ↑N) (hiM : IsIndecomposableModule A ↑M) (hiN : IsIndecomposableModule A ↑N) (hpM : ¬Module.Projective A ↑M) : Nonempty (AuslanderReitenTranslate k P.p ≃ₗ[A] AuslanderReitenTranslate k Q.p) ↔ Nonempty (↑M ≃ₗ[A] ↑N)

On non-projective finite-length indecomposables, the Auslander--Reiten translate computed from finite minimal projective presentations detects exactly the module isomorphism classes, when the transposes are scalar-reflexive. Only the source needs an explicit non-projectivity hypothesis.