The Auslander-Reiten translation in submodule categories

Abstract

Let Λ \Lambda be an artin algebra or, more generally, a locally bounded associative algebra, and S ( Λ ) \mathcal {S}(\Lambda ) the category of all embeddings ( A ⊆ B ) (A\subseteq B) where B B is a finitely generated Λ \Lambda -module and A A is a submodule of B B . Then S ( Λ ) \mathcal {S}(\Lambda ) is an exact Krull-Schmidt category which has Auslander-Reiten sequences. In this manuscript we show that the Auslander-Reiten translation in S ( Λ ) \mathcal {S}(\Lambda ) can be computed within mod Λ \operatorname {mod}\,\Lambda by using our construction of minimal monomorphisms. If in addition Λ \Lambda is uniserial, then any indecomposable nonprojective object in S ( Λ ) \mathcal {S}(\Lambda ) is invariant under the sixth power of the Auslander-Reiten translation.