Degenerations in the additive categories of almost cyclic coherent - - PDF document

Degenerations in the additive categories

• f almost cyclic coherent

Auslander-Reiten components Piotr Malicki (ICTP, Trieste, February 2010)

• A – finite-dimensional algebra over a fixed

algebraically closed field k.

• modA(d) – affine variety of d-dimensional

A-modules.

• Gld(k) – acts on modA(d) by conjugation.
• O(M) – Gld(k)-orbit of a module M in modA(d).
• For M, N ∈ modA(d), N is called degener-

ation of M if N ∈ O(M) (M ≤deg N).

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Fact. ≤deg is a partial order on modA(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

• r

0 → Z → Z ⊕ M → N → 0 then M ≤deg N.

• 2. Zwara has proved that the converse impli-

cation is also true.

2

• ΓA – Auslander-Reiten quiver of A.
• C – connected component of ΓA.
• 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 = X1 → X2 → · · · → Xi → Xi+1 → · · · in C, (C2) For each injective module I in C there is an infinite sectional path · · · → Yj+1 → Yj → · · · → Y2 → Y1 = I in C.

3

• A proper subtube of ΓA 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
• ϕ2X
• ϕψX
• ψ2X
• ϕ3X
• ϕ2ψX
• ϕψ2X
• ψ3X
• with the set of vertices

T (X)0 = {ϕiψjX; i, j ≥ 0}, and the set of arrows ϕi+1ψjX → ϕiψjX, ϕiψjX → ϕiψj+1X, where τ(ϕiψj+1X) = ϕi+1ψjX, for all i, j ≥ 0, by identifying the vertices ϕi+pψjX with ϕiψj+qX for all pairs i, j ≥ 0. We set ϕiψ0X = ϕiX, ϕ0ψjX = ψjX, ϕ0ψ0X = X.

4

• A full translation subquiver of ΓA of the

form

Yt

• Y1
• Y2

Y2

• Y1
• Yt
• Ut
• Dt
• where t ≥ 2, is said to be a M¨
• bius con-

figuration.

5

• A full translation subquiver of ΓA of the

form

A1

• At
• A1
• At
• B1
• Bt
• B1
• Bt
• Ut
• Dt
• where t ≥ 2, is said to be a coil configu-

ration.

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• For a module M ∈ mod A, we shall denote

by [M] the image of M in the Grothendieck group K0(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 ΓA which contains a M¨

• bius 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.

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• Lemma. Let A be an algebra and

0 → M1

[f1,u1]t

− − − − → N1 ⊕ M2

[u2,f2]

− − − − → N2 → 0 0 → M2

[f2,v1]t

− − − − → N2 ⊕ M3

[v2,f3]

− − − − → N3 → 0 be short exact sequences in mod A. Then the sequence 0 → M1

[f1,v1u1]t

− − − − − − → N1 ⊕ M3

[−v2u2,f3]

− − − − − − → N3 → 0 is exact.

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Assume first that C admits a M¨

• bius configu-

ration.

Yt

• Y1
• Y2

Y2

• Y1
• Yt
• 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 → Y1 ⊕ Z → Yt → 0 and 0 → Y1 → Yt ⊕ M → Z → 0. Applying Lemma again to the above two se- quences we obtain an exact sequence 0 → N → M ⊕ Z → Z → 0.

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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 ΓA. 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¨

• bius configuration or a

coil configuration in Γ and a number k ≥ 0 such that M = ϕkpDt = ψkqDt and N = ϕkpUt = ψkqUt, where t ≥ 2, Ut, Dt are modules lying in some proper subtube of Γ having p rays and q corays.

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Example 3. Consider the algebra A given by the quiver

23

• 24

25

35 σ ̟

• 22

θ 21 ϕ 26 ψ

• 37

ϑ

• 36

χ

• 1
• 5

34 ν

• 30

ι

• 31

2 α 4

• β
• 6
• 33

π

• 29

ω 28

27

3

• 8
• 7

32 µ

• 9

δ

• 10

γ

• 20
• ζ
• 11
• 15

η

• 12

13 ̺

• 14

λ

• ε
• 17

κ

• 18

ξ

• 19
• 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.

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Following Abeasis and del Fra: M ≤ext N ⇔ there are modules Mi, Ui, Vi and short exact sequences 0 → Ui → Mi → Vi → 0 in mod A such that M = M1, Mi+1 = Ui ⊕ Vi, 1 ≤ i ≤ s, and N = Ms+1 for some 1 ≤ s ∈ N. Facts.

• 1. ≤ext is a partial order on mod A.
• 2. For all modules M, N ∈ modA(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.

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Example 4. Let A = kQ/I, where Q : 1 β 2

• α
• ,

I = α2. Let M and N be two A-modules given by the following representations: M : k

• 1
• k2
• 0 0

1 0

• N :

k

• 1
• k2
• 0 0

1 0

• Then kQ/I is an algebra of finite type and

M, N are nonisomorphic and indecomposable. Moreover, M <deg N, but M <ext N. (M <ext N = ⇒ N − decomposable)

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Remark. Zwara has proved that the orders ≤deg and ≤ext are equivalent for all modules in mod A if and only if for any modules M, N in mod A with M <deg N, the module N is decomposable. Theorem B. Let A be an algebra and C a gen- eralized standard almost cyclic coherent com- ponent of ΓA. The following conditions are equivalent: (i) C contains neither a M¨

• bius configuration

nor a coil configuration. (ii) The partial orders ≤deg and ≤ext coincide

• n add(C).

Remark. In the proofs of Theorems A and B the follow- ing characterization of almost cyclic coherent component of ΓA is essentially applied: Theorem. [Malicki-Skowro´ nski] Let C be a connected component of ΓA. Then C is co- herent and almost cyclic if and only if C is a generalized multicoil.

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Let A be an algebra. A family C = (Ci)i∈I of components of ΓA is said to be separating in mod A if the modules in ind A split into three disjoint classes PA, CA = C and QA such that: (S1) CA is a sincere generalized standard family of components; (S2) HomA(QA, PA) = 0, HomA(QA, CA) = 0, HomA(CA, PA) = 0; (S3) any morphism from PA to QA factors through add(CA).

• Note that then PA and QA are uniquely

determined by CA.

• Recall that CA is called sincere if every sim-

ple A-module occurs as a composition fac- tor of a module in CA.

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Corollary A. Let A be an algebra with a sep- arating family of almost cyclic coherent com- ponents in ΓA. The following conditions are equivalent: (i) A is tame. (ii) If M, M′, N are A-modules such that M <deg N, M′ <deg N and N is indecomposable, then M ≃ M′ and is indecomposable. (iii) There exists an integer t such that for any sequence Mr <deg . . . <deg M2 <deg M1 with M1, . . . , Mr indecomposable A-modules, the inequality r ≤ t holds. Corollary B. Let A be an algebra with a sepa- rating family CA of almost cyclic coherent com- ponents in ΓA. Then the orders ≤deg and ≤ext coincide on mod A if and only if A is tame and CA contains neither a M¨

• bius configuration nor

a coil configuration.

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MAIN REFERENCES

• P. Malicki, Degenerations in the module

varieties of almost cyclic coherent Auslander- Reiten components, Colloq. Math. 114, 253– 276 (2009).

• P. Malicki, Degenerations for indecomposable

modules in almost cyclic coherent Auslander- Reiten components, J. Pure Appl. Algebra, in press.

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