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Auslander–Reiten translations in monomorphism categories

  • Bao-Lin Xiong EMAIL logo , Pu Zhang and Yue-Hui Zhang
Published/Copyright: February 21, 2012
Forum Mathematicum
From the journal Forum Mathematicum Volume 26 Issue 3

Abstract.

We generalize Ringel and Schmidmeier's theory on the Auslander–Reiten translation of the submodule category 𝒮2(A) to the monomorphism category 𝒮n(A); the category consists of all chains of (n-1) composable monomorphisms of A-modules. As in the case of n=2, 𝒮n(A) has Auslander–Reiten sequences, and the Auslander–Reiten translation τ𝒮 of 𝒮n(A) can be explicitly formulated via τ of A-mod. Furthermore, if A is a selfinjective algebra, we study the periodicity of τ𝒮 on the objects of 𝒮n(A) and of the Serre functor F𝒮 on the objects of the stable monomorphism category 𝒮n(A)̲. In particular, τ𝒮2m(n+1)X≅X for X∈𝒮n(Λ(m,t)), and F𝒮m(n+1)X≅X for X∈𝒮n(Λ(m,t))̲, where Λ(m,t), m≥1, t≥2, are the selfinjective Nakayama algebras.

Keywords: Monomorphism category; Auslander–Reiten translation; triangulated category; Serre functor
MSC: 16G10; 16G70; 18E30

Funding source: NSF of China

Award Identifier / Grant number: 10725104

Funding source: STCSM

Award Identifier / Grant number: 09XD1402500

The authors sincerely thank the anonymous referee for carefully reading the manuscript, and for the valuable comments and suggestions, which improve the presentation.

Received: 2011-1-26
Revised: 2012-1-28
Published Online: 2012-2-21
Published in Print: 2014-5-1

© 2014 by Walter de Gruyter Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Smooth global solutions for the two-dimensional Euler Poisson system
  3. Primitive ideals in quantum Schubert cells: Dimension of the strata
  4. Group valued null sequences and metrizable non-Mackey groups
  5. The splitting relation for Fréchet spaces over non-archimedean fields
  6. Bloch–Kato pro-p groups and locally powerful groups
  7. On Lp resolvent estimates for Laplace–Beltrami operators on compact manifolds
  8. The Ext functor and self-sums
  9. Auslander–Reiten translations in monomorphism categories
  10. An elementary proof of the vanishing of the second cohomology of the Witt and Virasoro algebra with values in the adjoint module
  11. Unitals over composition algebras
  12. Separation of bi-harmonic differential operators on Riemannian manifolds
https://doi.org/10.1515/forum-2011-0003
Keywords for this article
Monomorphism category; Auslander–Reiten translation; triangulated category; Serre functor

Articles in the same Issue

  1. Frontmatter
  2. Smooth global solutions for the two-dimensional Euler Poisson system
  3. Primitive ideals in quantum Schubert cells: Dimension of the strata
  4. Group valued null sequences and metrizable non-Mackey groups
  5. The splitting relation for Fréchet spaces over non-archimedean fields
  6. Bloch–Kato pro-p groups and locally powerful groups
  7. On Lp resolvent estimates for Laplace–Beltrami operators on compact manifolds
  8. The Ext functor and self-sums
  9. Auslander–Reiten translations in monomorphism categories
  10. An elementary proof of the vanishing of the second cohomology of the Witt and Virasoro algebra with values in the adjoint module
  11. Unitals over composition algebras
  12. Separation of bi-harmonic differential operators on Riemannian manifolds
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