Surjectivity of the Auslander–Reiten translate

Every finite-dimensional non-injective indecomposable module over a finite-dimensional algebra is the Auslander–Reiten translate of a finite-dimensional non-projective indecomposable. The preimage is the right transpose of a minimal presentation of the scalar dual. A minimal presentation of this preimage then computes its translate.

The result concerns actual modules, rather than objects modulo injectives or projectives. It supplies the surjectivity half of the correspondence on indecomposable isomorphism classes. The field is arbitrary and the algebra may be noncommutative.

References

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

theorem ModuleCat.exists_auslanderReitenTranslate {k : Type u} {A : Type v} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A] (M : ModuleCat A) [Module.Finite k ↑M] (hiM : CategoryTheory.Indecomposable M) (hinj : ¬CategoryTheory.Injective M) : ∃ (L : ModuleCat A) (_ : Module.Finite k ↑L), CategoryTheory.Indecomposable L ∧ ¬CategoryTheory.Projective L ∧ ∃ (P : TauCeti.FiniteProjectivePresentation L), TauCeti.IsMinimalProjectivePresentation P.p P.π ∧ Nonempty (TauCeti.AuslanderReitenTranslate k P.p ≃ₗ[A] ↑M)

Every finite-dimensional non-injective indecomposable is D Tr of a finite minimal presentation of a finite-dimensional non-projective indecomposable.