Almost split sequences for non-regular modules

S. Liu

Fundamenta Mathematicae (1993)

  • Volume: 143, Issue: 2, page 183-190
  • ISSN: 0016-2736

Abstract

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Let A be an Artin algebra and let 0 → X → ⊕ i = 1 r Y i → Z → 0 be an almost split sequence of A-modules with the Y i indecomposable. Suppose that X has a projective predecessor and Z has an injective successor in the Auslander-Reiten quiver Γ A of A. Then r ≤ 4, and r = 4 implies that one of the Y i is projective-injective. Moreover, if X → ⊕ j = 1 t Y j is a source map with the Y j indecomposable and X on an oriented cycle in Γ A , then t ≤ 4 and at most three of the Y j are not projective. The dual statement for a sink map holds. Finally, if an arrow X → Y in Γ A with valuation (d,d’) is on an oriented cycle, then dd’ ≤ 3.

How to cite

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Liu, S.. "Almost split sequences for non-regular modules." Fundamenta Mathematicae 143.2 (1993): 183-190. <http://eudml.org/doc/212001>.

@article{Liu1993,
abstract = {Let A be an Artin algebra and let $0 → X → ⊕_\{i = 1\}^rY_i → Z → 0$ be an almost split sequence of A-modules with the $Y_i$ indecomposable. Suppose that X has a projective predecessor and Z has an injective successor in the Auslander-Reiten quiver $Γ_A$ of A. Then r ≤ 4, and r = 4 implies that one of the $Y_i$ is projective-injective. Moreover, if $X → ⊕_\{j = 1\}^tY_j$ is a source map with the $Y_j$ indecomposable and X on an oriented cycle in $Γ_A$, then t ≤ 4 and at most three of the $Y_j$ are not projective. The dual statement for a sink map holds. Finally, if an arrow X → Y in $Γ_A$ with valuation (d,d’) is on an oriented cycle, then dd’ ≤ 3.},
author = {Liu, S.},
journal = {Fundamenta Mathematicae},
keywords = {Artin algebra; category of finitely generated left -modules; almost split sequence; connected components; Auslander-Reiten quiver; projective module; injective module; Bautista-Brenner theorem; finite representation type; indecomposable summands},
language = {eng},
number = {2},
pages = {183-190},
title = {Almost split sequences for non-regular modules},
url = {http://eudml.org/doc/212001},
volume = {143},
year = {1993},
}

TY - JOUR
AU - Liu, S.
TI - Almost split sequences for non-regular modules
JO - Fundamenta Mathematicae
PY - 1993
VL - 143
IS - 2
SP - 183
EP - 190
AB - Let A be an Artin algebra and let $0 → X → ⊕_{i = 1}^rY_i → Z → 0$ be an almost split sequence of A-modules with the $Y_i$ indecomposable. Suppose that X has a projective predecessor and Z has an injective successor in the Auslander-Reiten quiver $Γ_A$ of A. Then r ≤ 4, and r = 4 implies that one of the $Y_i$ is projective-injective. Moreover, if $X → ⊕_{j = 1}^tY_j$ is a source map with the $Y_j$ indecomposable and X on an oriented cycle in $Γ_A$, then t ≤ 4 and at most three of the $Y_j$ are not projective. The dual statement for a sink map holds. Finally, if an arrow X → Y in $Γ_A$ with valuation (d,d’) is on an oriented cycle, then dd’ ≤ 3.
LA - eng
KW - Artin algebra; category of finitely generated left -modules; almost split sequence; connected components; Auslander-Reiten quiver; projective module; injective module; Bautista-Brenner theorem; finite representation type; indecomposable summands
UR - http://eudml.org/doc/212001
ER -

References

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  8. [8] S. Liu, Semi-stable components of an Auslander-Reiten quiver, ibid. 47 (1993), 405-416. Zbl0818.16015
  9. [9] S. Liu, On short cycles in a module category, preprint. Zbl0824.16015
  10. [10] I. Reiten, The use of almost split sequences in the representation theory of artin algebras, in: Lecture Notes in Math. 944, Springer, Berlin, 1982, 29-104. 
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  12. [12] Y. Zhang, The structure of stable components, Canad. J. Math. 43 (1991), 652-672. Zbl0736.16007

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