The Auslander-Reiten translation in submodule categories

Abstract:

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