Almost split sequences for non-regular modules

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Department of Mathematics, National University of Singapore, Singapore 0511, Republic of Singapore

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

Γ_{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

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.

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