Indecomposability of minimal transposes

The transpose of a finite minimal projective presentation of a finite-length indecomposable module is indecomposable exactly when the original module is not projective. For arbitrary finite projective presentations, the corresponding assertion holds in the projective stable category. Minimality excludes the projective retracts that this category forgets, so the conclusion then holds for actual modules.

Together with TauCeti.AuslanderReitenTranslate.isIndecomposableModule_iff, the transpose characterization gives the corresponding result for the Auslander–Reiten translate when the transpose is reflexive over the base. In particular, this applies to finite-dimensional modules over finite-dimensional algebras over a field, and supplies the preservation of indecomposability needed to classify modules by their translates.

References

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

@[simp]

theorem TauCeti.FiniteProjectivePresentation.indecomposable_stableTransposeObj_iff {A : Type u} [Ring A] {M : ModuleCat A} [Small.{v, u} A] (P : FiniteProjectivePresentation M) (hM : IsFiniteLength A ↑M) (hiM : IsIndecomposableModule A ↑M) : CategoryTheory.Indecomposable ((ExactStructure.abelian (ModuleCat Aᵐᵒᵖ)).projectiveStableFunctor.obj ↧(AuslanderReitenTranspose P.p)) ↔ ¬Module.Projective A ↑M

The stable transpose of a finite projective presentation of a finite-length indecomposable module is indecomposable exactly when the module is not projective. Minimality is not needed.

@[simp]

theorem TauCeti.FiniteProjectivePresentation.isIndecomposableModule_auslanderReitenTranspose_iff {A : Type u} [Ring A] {M : ModuleCat A} [Small.{v, u} A] (P : FiniteProjectivePresentation M) (hP : IsMinimalProjectivePresentation P.p P.π) (hM : IsFiniteLength A ↑M) (hiM : IsIndecomposableModule A ↑M) : IsIndecomposableModule Aᵐᵒᵖ (AuslanderReitenTranspose P.p) ↔ ¬Module.Projective A ↑M

The transpose of a finite minimal presentation of a finite-length indecomposable module is indecomposable exactly when the original module is not projective.