Injectivity of the Auslander–Reiten translate

Over a finite-dimensional algebra, the translate of a finite minimal projective presentation is injective exactly when the presented module is projective. In that case the translate is zero. Thus D Tr takes non-projective modules to non-injective modules, as required for the Auslander–Reiten correspondence.

Linear duality exchanges injectivity and projectivity. The transpose of a minimal presentation has no nonzero projective summands, so an injective translate must vanish. These results do not require indecomposability or algebraic closedness of the ground field.

References

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

theorem TauCeti.IsMinimalProjectivePresentation.moduleInjective_auslanderReitenTranslate_iff_projective {k : Type u_1} {A : Type u_2} {P₀ : Type u_3} {P₁ : Type u_4} {M : Type u_5} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A] [AddCommGroup P₀] [Module A P₀] [Module.Finite A P₀] [AddCommGroup P₁] [Module A P₁] [Module.Finite A P₁] [AddCommGroup M] [Module A M] {p₁ : P₁ →ₗ[A] P₀} {p₀ : P₀ →ₗ[A] M} (h : IsMinimalProjectivePresentation p₁ p₀) : Module.Injective A (AuslanderReitenTranslate k p₁) ↔ Module.Projective A M

The translate of a finite minimal projective presentation is injective exactly when the presented module is projective. In particular a non-projective module has non-injective translate, even without an indecomposability hypothesis.