Proceedings of the 44th Symposium on Ring Theory and ...

Pro<strong>of</strong>. Observe first that for each indecomposable non projective R-module M, we have
l(Ω 2 M) = l(τM). Now, applying <strong>the</strong> previous Lemma 3.3, we obtain for each i ≥ 0
that l(Ω i B) ≤ l(Ω i C) + α/2, and l(Ω i+2 C) ≤ l(Ω i B) + α/2 for some positive number α.
Hence, for each i ≥ 0 we have l(Ω i+2 C) − l(Ω i C) ≤ α. In particular, for each n ≥ 0 we
have l(Ω 2n C) − l(C) ≤ nα. This means that <strong>the</strong> complexity <strong>of</strong> C is bounded by 2. If
cx C = 1, <strong>the</strong>n, since it is not τ periodic, <strong>the</strong> module C must lie in a ZA ∞ -component
by [17], but for <strong>the</strong>se components every irreducible map is eventually Ω-perfect. Hence
cx C = 2.
□
We obtain <strong>the</strong> following immediate consequence: assume that we have a component C,
whose stable part C s is <strong>of</strong> <strong>the</strong> form ZA ∞ ∞, and assume also that <strong>the</strong>re exists an Auslander-
Reiten sequence 0 → τC → E ⊕ F → C → 0 where E and F are indecomposable, and
nei<strong>the</strong>r E → C nor F → C is eventually Ω-perfect. Observe also that in this case, no
irreducible map in C s between indecomposable modules is eventually Ω-perfect. It follows
immediately from <strong>the</strong> previous proposition that every non projective module in C has
complexity 2. This situation can actually occur. The following example is due to Ringel.
Example 18. Let R be <strong>the</strong> finite dimensional selfinjective string algebra given by <strong>the</strong>
quiver
α
β
1 2 3
γ
modulo <strong>the</strong> relations αβ = 0, δγ = 0, γαγα = βδβδ and αγαγα = δβδβδ = 0. There
exists a ZA ∞ ∞ component where none <strong>of</strong> <strong>the</strong> irreducible maps between <strong>the</strong> indecomposable
modules is eventually Ω-perfect, (or even τ-perfect). For instance, consider <strong>the</strong> string
module M = r 3 P 2 . It is easy to see that M is not eventually Ω-perfect, that α(M) = 2,
and that no irreducible map from an indecomposable module to M is eventually Ω-perfect.
Moreover, by [6], M lies in a component consisting entirely <strong>of</strong> string modules. But <strong>the</strong>
only string modules lying on <strong>the</strong> boundary <strong>of</strong> an Auslander-Reiten component can lie on
tubes (see [12], II.6.4), so this module belongs to a ZA ∞ ∞ component. Note also that <strong>the</strong>
simple modules S 1 and S 3 are Ω-periodic <strong>of</strong> period 6, and that <strong>the</strong>y both lie on tubes <strong>of</strong>
rank 3.
Example 19. Following Erdmann [12], for each positive integer m, we denote by Λ m <strong>the</strong>
local symmetric string algebra over a field K,
Λ m = K〈x, y〉/〈x 2 , (xy) m+1 − (yx) m+1 , x 2 − (yx) m y, x 3 〉
If <strong>the</strong> characteristic <strong>of</strong> K is 2, and m+1 = 2 n ≥ 4, <strong>the</strong>n <strong>the</strong> algebra Λ m modulo its socle is
isomorphic to <strong>the</strong> group algebra <strong>of</strong> <strong>the</strong> semidihedral group <strong>of</strong> order 2 n+2 modulo its socle.
Motivated by this fact, Erdmann calls this algebra semidihedral. She proves that Λ m has
infinitely many stable components <strong>of</strong> type ZA ∞ ∞ and ZD ∞ ([12], Propositions II,10.1 and
II,10.2), and that <strong>the</strong> o<strong>the</strong>r stable components are tubes <strong>of</strong> rank 1 and 2. Moreover, she
shows that <strong>the</strong> unique simple module lies in a component <strong>of</strong> type ZD ∞ so it is not periodic.
Therefore, every indecomposable non projective Λ m -module is eventually Ω-perfect by
[18]. Note that in <strong>the</strong> same book, Erdmann generalizes <strong>the</strong> notion <strong>of</strong> semidihedral algebra
to that <strong>of</strong> algebras <strong>of</strong> semidihedral type and one also obtains interesting examples for <strong>the</strong>
non local case ([12], Lemma VIII. 2.1.).
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