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6) are self-hereditary. Note also that, given a hereditary stable tube T , the additive subcategory add T of mod A is hereditary in the sense that Ext2A (M, N ) = 0, for each pair of the objects M and N of add T . Obviously, every stable tube T of the Auslander–Reiten quiver Γ(mod A) of a hereditary algebra A is hereditary and, clearly, every hereditary stable tube T is self-hereditary, but the converse implication does not hold in general. We show in Section 4 that faithful standard stable tubes are hereditary.

Tubes k −2 and k in E, while the extreme terms are uniserial of E-length k −1 in E, the sequence (iii) does not split. In particular, Ei+1 [k −1] is not projective and hence τ Ei+1 [k −1] = 0. We claim that τ Ei+1 [k −1] ∼ = Ei [k −1]. By the induction hypothesis, we have an almost split sequence (corresponding to j = k − 1) pi,k−2 ui,k−1 [ ui+1,k−2 pi,k−1 ] 0 −→ Ei [k −2] −−−−−−→ Ei+1 [k −3]⊕Ei [k −1] −−−−−−−−−−−− → Ei+1 [k −2] −→ 0 for any i. Replacing i by i + 1, we get that the homomorphism ui+1,k−1 : Ei+1 [k − 2] −−−−→ Ei+1 [k − 1] is an irreducible morphism.

M−1}. Fix p ∈ {1, . . , m − 1} and choose a homomorphism h ∈ radpA (Ei , Ek ). We may assume that h is non-zero, p ≥ 2s (for i = k), and p ≥ 2r (for i = k). 1). 1), the non-zero homomorphism h has the form h = h1 + h2 + . . + hn , where the summand ht is the composite homomorphism htp h t t1 Ei = Xt1 −→X t2 −→ . . −→ Xtpt −→Xt,pt +1 = Ek , pt ≥ p and ht1 , . . , htpt are irreducible morphisms between indecomposable modules in mod A, for t ∈ {1, . . , n}. It follows that, for each t ∈ {1, .

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