Auslander–Reiten translations in monomorphism categories
Auslander–Reiten translations in monomorphism categories
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DOI：
10.1515/forum-2011-0003
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发表时间：
2011-01
影响因子：
0.8
通讯作者：
Bao-Lin Xiong;Pu Zhang;Yuehui Zhang
中科院分区：
文献类型：
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作者：
Bao-Lin Xiong;Pu Zhang;Yuehui Zhang
摘要。我们推广了Ringel和Schmidmeier关于子模范畴的Auslander-Reiten翻译理论𝒮2 (A)$\mathcal {S}_2(A)$ 到单态范畴𝒮n (A)$\mathcal {S}_n(A)$ ；范畴由（n-1）的所有链组成$(n-1)$ a -模的可组合单态。在n=2的情况下$n=2$ ，𝒮n (A)$\mathcal {S}_n(A)$ 有Auslander-Reiten序列，而Auslander-Reiten翻译τ𝒮 $\tau _{\mathcal {S}}$ 的𝒮n (A)$\mathcal {S}_n(A)$ 可以通过τ (A-mod)显式表示。进一步，如果A是自射代数，我们研究了τ𝒮的周期性 $\tau _{\mathcal {S}}$ 关于𝒮n (A)的对象$\mathcal {S}_n(A)$ 和服务函子F𝒮 $F_{\mathcal {S}}$ 关于稳定单态范畴𝒮n (A) ＿的对象$\underline{\mathcal {S}_n(A)}$ ． 特别地，τ𝒮2m(n+1) X × X${\tau _{\mathcal {S}}^{2m(n+1)}X\cong X}$ 对于X∈𝒮n （Λ（m,t））${X\in \mathcal {S}_n(\Lambda (m, t))}$ ， F𝒮m(n+1) X × X${F_{\mathcal {S}}^{m(n+1)}X\cong X}$ 对于X∈𝒮n (Λ（m,t）${X\in \underline{\mathcal {S}_n(\Lambda (m, t))}}$ ，其中Λ（m,t）$\Lambda (m, t)$ ， m≥1$m\ge 1$ ， t≥2$t\ge 2$ ，是自射的中山代数。
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.