Tilted Special Biserial Algberas François Huard and Shiping Liu ...

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an indecomposable automorphism of a k-vector space V . One defines a band module N = N(w, φ) determined by w and φ as follows: For 1 ≤ i ≤ n, define V i = V . For 1 ≤ i
there is no clockwise cycle in (Q, Ī). Clearly a reduced cycyle of Q is clockwise in (Q, Ī) if and only if it is clockwise in (Q, I). Therefore it suffices to show that A is of directed representation type if and only if so is Ā. FirstthatĀ is not of directed representation type implies trivially that A is not either. Assume now that A is not of directed representation type. If A is of infinite representation type, then so is Ā. ThusĀ is not of directed representation type [15]. Suppose that A is of finite representation type. Then there is no band module in mod A and there is a cycle f 1 f r (∗) M 0 −→ M1 →···→M r−1 −→ Mr = M 0 in mod A, wheretheM i are indecomposable modules and the f i are irreducible maps. Then r>1andM i−1 ≁ = Mi for all 1 ≤ i ≤ r. Note that each M i with 0 ≤ i ≤ r is either a string module or a projective-injective module corresponding to a binomial relation of (Q, I). Let s with 0 ≤ s ≤ r be such that M s is not a string module. We may assume that 0
Page 1 and 2: Tilted Special Biserial Algberas Fr
Page 3: 1.2. Definition. Let A = kQ/I be a
Page 7 and 8: e a string in (Q, I), wheren ≥ 1,
Page 9 and 10: P (s(v n+1 )) since (PD2) does not
Page 11 and 12: Write p 1 = α 1 u 1 , q 1 = β 1 v
Page 13 and 14: 4. Main result In this final sectio
Page 15 and 16: is a zero-relation, where γ 1 is t
Page 17 and 18: the contrary that (Q, I) admits a s
Page 19 and 20: [2]M.C.R.ButlerandC.M.Ringel,Auslan

Tilted Special Biserial Algberas François Huard and Shiping Liu ...

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