Auslander-Reiten translations in monomorphism categories
Bao-Lin Xiong, Pu Zhang, Yue-Hui Zhang

TL;DR
This paper extends the theory of Auslander-Reiten translations from submodule categories to monomorphism categories, providing explicit formulations and studying periodicity properties for selfinjective algebras.
Contribution
It generalizes the Auslander-Reiten translation theory to monomorphism categories $ ext{S}_n(A)$ and analyzes periodicity in the selfinjective case.
Findings
Explicit formulation of $ au_{ ext{S}}$ via $ au$ of $A$-mod.
Periodicity results for $ au_{ ext{S}}$ and Serre functor $F_{ ext{S}}$ in selfinjective Nakayama algebras.
Extension of Auslander-Reiten theory to higher monomorphism categories.
Abstract
We generalize Ringel and Schmidmeier's theory on the Auslander-Reiten translation of the submodule category to the monomorphism category . As in the case of , has Auslander-Reiten sequences, and the Auslander-Reiten translation of can be explicitly formulated via of -mod. Furthermore, if is a selfinjective algebra, we study the periodicity of on the objects of , and of the Serre functor on the objects of the stable monomorphism category . In particular, for ; and for , where are the selfinjective Nakayama algebras.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology
