The third author is a researcher from CONICET, Argentina. LEMMA 2.1 Let A be an artin algebra, and M be an indecomposable A-module such that there exists a path M 0 Then there exists an indecomposable A-module L with pd A L ≥ 2 and Hom A (L, M ) = 0. Proof. Assume that this is not the case, that is, there exist M ∈ ind A and a path Proof. Assume M to be indecomposable and Ext-injective in add L A . Then τ −1 A M ∈ L A . By Theorem 1.1, there exists an indecomposable A-module L such that pd A L

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... 2 and Hom A (L, τ −1 A M ) = 0. Hence, there exists an indecomposable injective I such that Hom A (I, τ A L) = 0. So, either Hom A (τ A L, M ) = 0 and the path I → τ A L → M yields M ∈ E 1 , or else Hom A (τ A L, M ) = 0 in which case the Auslander-Reiten formula gives Hom A (L, τ −1 A M ) Hom A (τ A L, M ) = 0. Since Hom A (L, τ −1 A M ) = 0, there exists an indecomposable projective P and a path L → P → τ −1 A M . Since pd A L ≥ 2, we have L ∈ L A , so P ∈ L A and consequently, M ∈ E 2 . Conversely, let M ∈ E 1 ∪ E 2 . Clearly, if M ∈ E 2 , then M ∈ L A but τ −1 A M ∈ L A (because τ −1 A M succedes a projective not in L A ) so that M is Ext-injective in add L A . If M ∈ E 1 , then either M is injective, or else there exists an indecomposable injective I and a path This establishes the first assertion. The second follows immediately. The following corollary gives equivalent characterisations of the sets E 1 and E 2 . COROLLARY 3.2 Let M ∈ L A . Then: (a) The following conditions are equivalent: