# Almost split sequences for non-regular modules

Fundamenta Mathematicae (1993)

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

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## Abstract

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Let A be an Artin algebra and let $0\to X\to {\oplus }_{i=1}^{r}{Y}_{i}\to Z\to 0$ 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 ${\Gamma }_{A}$ of A. Then r ≤ 4, and r = 4 implies that one of the ${Y}_{i}$ is projective-injective. Moreover, if $X\to {\oplus }_{j=1}^{t}{Y}_{j}$ is a source map with the ${Y}_{j}$ indecomposable and X on an oriented cycle in ${\Gamma }_{A}$, 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 ${\Gamma }_{A}$ 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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1. [1] M. Auslander and I. Reiten, Representation theory of artin algebras III: Almost split sequences, Comm. Algebra 3 (1975), 239-294. Zbl0331.16027
2. [2] M. Auslander and I. Reiten, Representation theory of artin algebras IV: Invariants given by almost split sequences, ibid. 5 (1977), 443-518. Zbl0396.16007
3. [3] R. Bautista and S. Brenner, On the number of terms in the middle of an almost split sequence, in: Lecture Notes in Math. 903, Springer, Berlin, 1981, 1-8. Zbl0483.16030
4. [4] R. Bautista and S. O. Smalø, Non-existent cycles, Comm. Algebra 11 (1983), 1755-1767. Zbl0515.16013
5. [5] D. Happel, U. Preiser and C. M. Ringel, Vinberg's characterization of Dynkin diagrams using subadditive functions with applications to DTr-periodic modules, in: Lecture Notes in Math. 832, Springer, Berlin, 1980, 280-294. Zbl0446.16032
6. [6] M. Harada and Y. Sai, On categories of indecomposable modules, Osaka J. Math. 7 (1970), 323-344. Zbl0248.18018
7. [7] S. Liu, Degrees of irreducible maps and the shapes of Auslander-Reiten quivers, J. London Math. Soc. (2) 45 (1992), 32-54. Zbl0703.16010
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.
11. [11] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, Berlin, 1984.
12. [12] Y. Zhang, The structure of stable components, Canad. J. Math. 43 (1991), 652-672. Zbl0736.16007

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