Geodesic
Almost split sequences for non-regular modules
Fundamenta Mathematicae, Tome 143 (1993) no. 2, pp. 183-190 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/fm-143-2-183-190

S. Liu 1

1 
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S. Liu. Almost split sequences for non-regular modules. Fundamenta Mathematicae, Tome 143 (1993) no. 2, pp. 183-190. doi: 10.4064/fm-143-2-183-190

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