Auslander–Reiten translations in monomorphism categories
Abstract. We generalize Ringel and Schmidmeier's theory on the Auslander–Reiten translation of the submodule category 𝒮 2 (A)$\mathcal {S}_2(A)$ to the monomorphism category 𝒮 n (A)$\mathcal {S}_n(A)$ ; the category consists of all chains of (n-1)$(n-1)$ composable monomorphisms of A-modules. As in the case of n=2$n=2$ , 𝒮 n (A)$\mathcal {S}_n(A)$ has Auslander–Reiten sequences, and the Auslander–Reiten translation τ 𝒮 $\tau _{\mathcal {S}}$ of 𝒮 n (A)$\mathcal {S}_n(A)$ can be explicitly formulated via τ of A-mod. Furthermore, if A is a selfinjective algebra, we study the periodicity of τ 𝒮 $\tau _{\mathcal {S}}$ on the objects of 𝒮 n (A)$\mathcal {S}_n(A)$ and of the Serre functor F 𝒮 $F_{\mathcal {S}}$ on the objects of the stable monomorphism category 𝒮 n (A) ̲$\underline{\mathcal {S}_n(A)}$ . In particular, τ 𝒮 2m(n+1) X≅X${\tau _{\mathcal {S}}^{2m(n+1)}X\cong X}$ for X∈𝒮 n (Λ(m,t))${X\in \mathcal {S}_n(\Lambda (m, t))}$ , and F 𝒮 m(n+1) X≅X${F_{\mathcal {S}}^{m(n+1)}X\cong X}$ for X∈𝒮 n (Λ(m,t)) ̲${X\in \underline{\mathcal {S}_n(\Lambda (m, t))}}$ , where Λ(m,t)$\Lambda (m, t)$ , m≥1$m\ge 1$ , t≥2$t\ge 2$ , are the selfinjective Nakayama algebras.