Auslander-Reiten translations in monomorphism categories

Abstract

We generalize Ringel and Schmidmeier's theory on the Auslander-Reiten translation of the submodule category ${\mathcal{S}}_2(A)$ to the monomorphism category ${\mathcal{S}}_n(A)$. As in the case of $n=2$, ${\mathcal{S}}_n(A)$ has Auslander-Reiten sequences, and the Auslander-Reiten translation ${\tau }_{\mathcal{S}}$ of ${\mathcal{S}}_n(A)$ can be explicitly formulated via $\tau$ of $A$-mod. Furthermore, if $A$ is a selfinjective algebra, we study the periodicity of ${\tau }_{\mathcal{S}}$ on the objects of ${\mathcal{S}}_n(A)$, and of the Serre functor $F_{\mathcal{S}}$ on the objects of the stable monomorphism category $\underline{{\mathcal{S}}_n(A)}$. In particular, ${\tau }_{\mathcal{S}}^{2m(n+1)}X\cong X$ for $X\in\mathcal{S}_n(\A(m, t))$; and $F_{\mathcal{S}}^{m(n+1)}X\cong X$ for $X\in\underline{\mathcal{S}_n(\A(m, t))}$, where $\A(m, t), \ m\ge1, \ t\ge2,$ are the selfinjective Nakayama algebras.

33 pages, 1 figures