On the number of terms in the middle of almost split sequences over tame algebras

Abstract

Let A A be a finite dimensional tame algebra over an algebraically closed field k k . It has been conjectured that any almost split sequence 0 → X → ⊕ i = 1 n Y i → Z → 0 0 \to X \to \oplus _{i=1} ^n Y_i \to Z \to 0 with Y i Y_i indecomposable modules has n ≤ 5 n \le 5 and in case n = 5 n=5 , then exactly one of the Y i Y_i is a projective-injective module. In this work we show this conjecture in case all the Y i Y_i are directing modules, that is, there are no cycles of non-zero, non-iso maps Y i = M 1 → M 2 → ⋯ → M s = Y i Y_i =M_1 \to M_2 \to \cdots \to M_s=Y_i between indecomposable A A -modules. In case, Y 1 Y_1 and Y 2 Y_2 are isomorphic, we show that n ≤ 3 n \le 3 and give precise information on the structure of A A .