The Auslander-Reiten Translation in Submodule Categories

Abstract

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

Dedicated to Idun Reiten

References in corpus (1)

• Invariant Subspaces of Nilpotent Linear Operators. I

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