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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.