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.