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## Fundamenta Mathematicae

1993 | 143 | 2 | 183-190
Tytuł artykułu

### Almost split sequences for non-regular modules

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
183-190
Opis fizyczny
Daty
wydano
1993
otrzymano
1993-01-28
poprawiono
1993-04-30
Twórcy
autor
• Department of Mathematics, National University of Singapore, Singapore 0511, Republic of Singapore
Bibliografia
• [1] M. Auslander and I. Reiten, Representation theory of artin algebras III: Almost split sequences, Comm. Algebra 3 (1975), 239-294.
• [2] M. Auslander and I. Reiten, Representation theory of artin algebras IV: Invariants given by almost split sequences, ibid. 5 (1977), 443-518.
• [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.
• [4] R. Bautista and S. O. Smalø, Non-existent cycles, Comm. Algebra 11 (1983), 1755-1767.
• [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.
• [6] M. Harada and Y. Sai, On categories of indecomposable modules, Osaka J. Math. 7 (1970), 323-344.
• [7] S. Liu, Degrees of irreducible maps and the shapes of Auslander-Reiten quivers, J. London Math. Soc. (2) 45 (1992), 32-54.
• [8] S. Liu, Semi-stable components of an Auslander-Reiten quiver, ibid. 47 (1993), 405-416.
• [9] S. Liu, On short cycles in a module category, preprint.
• [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] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, Berlin, 1984.
• [12] Y. Zhang, The structure of stable components, Canad. J. Math. 43 (1991), 652-672.
Typ dokumentu
Bibliografia
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