Degenerations in the additive categories of almost cyclic coherent Auslander-Reiten components Piotr Malicki (ICTP, Trieste, February 2010) • A – finite-dimensional algebra over a fixed algebraically closed field k . • mod A ( d ) – affine variety of d -dimensional A -modules. • Gl d ( k ) – acts on mod A ( d ) by conjugation. • O ( M ) – Gl d ( k )-orbit of a module M in mod A ( d ). • For M, N ∈ mod A ( d ) , N is called degener- ation of M if N ∈ O ( M ) ( M ≤ deg N ). 1
Fact. ≤ deg is a partial order on mod A ( d ). Remarks. 1. Riedtmann has proved that if M, N, Z are modules in mod A such that there is an exact sequence in mod A one of the forms 0 → N → M ⊕ Z → Z → 0 or 0 → Z → Z ⊕ M → N → 0 then M ≤ deg N . 2. Zwara has proved that the converse impli- cation is also true. 2
• Γ A – Auslander-Reiten quiver of A . • C – connected component of Γ A . • C is said to be generalized standard if rad ∞ ( X, Y ) = 0 for all modules X , Y in C . • C is said to be almost cyclic if all but finitely many modules of C lie on oriented cycles contained entirely in C . • C is said to be coherent if the following two conditions are satisfied: (C1) For each projective module P in C there is an infinite sectional path P = X 1 → X 2 → · · · → X i → X i +1 → · · · in C , (C2) For each injective module I in C there is an infinite sectional path · · · → Y j +1 → Y j → · · · → Y 2 → Y 1 = I in C . 3
� � � � � � � � � � � � � � � � � � � � • A proper subtube of Γ A is a full transla- tion subquiver T ( X, p, q ), p, q ≥ 1, obtained from the translation quiver T ( X ) of the form X � � � � � � � � � � ϕX ψX � � � � � � � � � � � � � ϕ 2 X ψ 2 X ϕψX � � � � � � � � � � � � � � � � � � ϕ 3 X ϕ 2 ψX ϕψ 2 X ψ 3 X � � � � � � � � � � � � � � � � � � � � � � with the set of vertices T ( X ) 0 = { ϕ i ψ j X ; i, j ≥ 0 } , and the set of arrows ϕ i +1 ψ j X → ϕ i ψ j X, ϕ i ψ j X → ϕ i ψ j +1 X, where τ ( ϕ i ψ j +1 X ) = ϕ i +1 ψ j X , for all i, j ≥ 0, by identifying the vertices ϕ i + p ψ j X with ϕ i ψ j + q X for all pairs i, j ≥ 0. We set ϕ i ψ 0 X = ϕ i X, ϕ 0 ψ j X = ψ j X, ϕ 0 ψ 0 X = X. 4
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • A full translation subquiver of Γ A of the form Y 1 Y t � � � � � � � � � � � � Y 2 ◦ ◦ ◦ � � � � � � � � � � � � � � � � � � Y 2 ◦ ◦ ◦ � � � � � � � � � � � � � � � � � � Y 1 Y t ◦ ◦ � � � � � � � � � � � � � � � � � � � � � � � � U t ◦ ◦ � � � � � � � � � � � � � � � � � � ◦ ◦ � � � � � � � � � � � � ◦ ◦ ◦ � � � � � � � � � � � � � � � � � � ◦ ◦ ◦ ◦ � � � � � � � � � � � � ◦ ◦ ◦ � � � � � � ◦ ◦ � � � � � � D t where t ≥ 2, is said to be a M¨ obius con- figuration . 5
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • A full translation subquiver of Γ A of the form A 1 A t A 1 A t ◦ ◦ ◦ ◦ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ◦ � ◦ � � � � � ◦ � ◦ � ◦ � ◦ � ◦ � ◦ B 1 B t B 1 B t ◦ ◦ ◦ ◦ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ � � � � � � � � � � � � � � � � � � � � � � � � U t � � � � � � � � � � � � ◦ ◦ � � � � � � � � � � � � ◦ ◦ ◦ � � � � � � � � � � � � � � � � � � ◦ ◦ ◦ ◦ � � � � � � � � � � � � ◦ ◦ ◦ � � � � � � ◦ ◦ � � � � � � D t where t ≥ 2, is said to be a coil configu- ration . 6
• For a module M ∈ mod A , we shall denote by [ M ] the image of M in the Grothendieck group K 0 ( A ) of A . • Thus [ M ] = [ N ] if and only if M and N have the same simple composition factors including the multiplicities. Proposition. Let A be an algebra and C a generalized standard component in Γ A which contains a M¨ obius configuration or a coil con- figuration. Then there exist indecomposable modules M and N in C such that M < deg N . Proof. We need the following fact. 7
Lemma. Let A be an algebra and [ f 1 ,u 1 ] t [ u 2 ,f 2 ] 0 → M 1 − − − − → N 1 ⊕ M 2 − − − − → N 2 → 0 [ f 2 ,v 1 ] t [ v 2 ,f 3 ] 0 → M 2 − − − − → N 2 ⊕ M 3 − − − − → N 3 → 0 be short exact sequences in mod A . Then the sequence [ f 1 ,v 1 u 1 ] t [ − v 2 u 2 ,f 3 ] 0 → M 1 − − − − − − → N 1 ⊕ M 3 − − − − − − → N 3 → 0 is exact. 8
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Assume first that C admits a M¨ obius configu- ration. Y t Y 1 � � � � � � � � � � � � ◦ ◦ ◦ Y 2 � � � � � � � � � � � � � � � � � � Y 2 ◦ ◦ ◦ � � � � � � � � � � � � � � � � � � Y 1 ◦ ◦ Y t � � � � � � � � � � � � � � � � � � � � � � � � ◦ N ◦ � � � � � � � � � � � � � � � � � � ◦ ◦ � � � � � � � � � � � � ◦ ◦ ◦ � � � � � � � � � � � � � � � � � � ◦ ◦ ◦ Z � � � � � � � � � � � � ◦ ◦ ◦ � � � � � � ◦ ◦ � � � � � � M Applying Lemma to the short exact sequences given by the meshes of the above translation quiver we get exact sequences 0 → N → Y 1 ⊕ Z → Y t → 0 and 0 → Y 1 → Y t ⊕ M → Z → 0 . Applying Lemma again to the above two se- quences we obtain an exact sequence 0 → N → M ⊕ Z → Z → 0 . 9
Finally, by Riedtmann’s result we infer that M ≤ deg N . Then M < deg N , since M �≃ N . If C admits a coil configuration the proof is similar. Examples 1 and 2. Theorem A. Let A be an algebra and Γ a gen- eralized standard almost cyclic coherent com- ponent of Γ A . Let M and N be A -modules such that M ∈ add(Γ) , N ∈ Γ , [ M ] = [ N ] . The following conditions are equivalent: (i) M < deg N . (ii) There exist a M¨ obius configuration or a coil configuration in Γ and a number k ≥ 0 such that M = ϕ kp D t = ψ kq D t and N = ϕ kp U t = ψ kq U t , where t ≥ 2 , U t , D t are modules lying in some proper subtube of Γ having p rays and q corays. 12
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Example 3. Consider the algebra A given by the quiver � 25 23 24 ϕ � 26 σ � θ � 21 35 22 � ����������� χ ̟ 37 36 ψ ϑ � 31 1 5 34 30 ν ι α � 4 π � ω � 28 � 27 2 6 33 29 β 3 8 7 32 µ γ ζ 9 10 20 δ 11 15 η ̺ ξ λ κ 12 13 14 17 18 19 ε 16 bound by αβ = 0, γδ = 0, ηε = 0, κλ̺ = 0, ζγ = 0, ξκλ = 0, µζ = 0, νπ = 0, πω = 0, σθ = 0, ϕψ = 0, ϑι = 0, χ̟ = 0. 13
Following Abeasis and del Fra: M ≤ ext N ⇔ there are modules M i , U i , V i and short exact sequences 0 → U i → M i → V i → 0 in mod A such that M = M 1 , M i +1 = U i ⊕ V i , 1 ≤ i ≤ s , and N = M s +1 for some 1 ≤ s ∈ N . Facts. 1. ≤ ext is a partial order on mod A . 2. For all modules M, N ∈ mod A ( d ), we have M ≤ ext N = ⇒ M ≤ deg N. Remark. The converse implication is not true in general even for very simple representation-finite alge- bras as in the following Riedtmann’s example. 15
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