A Wild Bocs Having Strong Homogeneous Property^{1}

Raymundo Bautista, William Crawley-Boevey, Tiangang Lei, Yingbo Zhang

Abstract

In this paper we give a bocs A of wild representation type over an algebraically closed field k, and R(A), the category of finite-dimensional representations of A. For any indecomposable object M in R(A) there is an almost split sequence starting and ending at M. And each connected component of Auslander-Reiten quiver of R(A) is a homogeneous tube. This is a counter-example for the converse of Crawley-Boevey’s theorem [CB1] for bocses.

1. Introduction.

1.1. Bocs, which is the abbreviated form of bimodule over a category with coalgebra structure, was introduced by Roiter and Kleiner in 1975 [RK], developed by Drozed in 1977 [D], and reformulated by Crawley-Boevey in 1987 [CB1]. Let k be an algebraically closed field, Λ a finite-dimensional k-algebra. Then there exists a bocs B over k associated to Λ [D], [CB1]. From this relation Drozd proved one of the most important theorems in representation theory of algebras, namely, a finitedimentsional k-algebra is either of representation tame type or of representation wild type, but not both [D]. After 10 years, Crawley-Boevey proved the following theorem: if Λ is of representation tame type, then for each dimension d, almost all modules with dim M ≤ d have the property DTr M ≃ M [CB1]. It is natural to ask the converse of the theorem: if for each dimension d, almost all modules M with dim M ≤ d have the property DTr M ≃ M, is Λ of representation tame type? Or equivalently, if Λ is of representation wild type, do exist infinitely many modules M with dim M ≤ d for some positive integer d, such that DTr M ≄ M ?

Since every finite-dimensional k-algebra Λ is associated to a finite-dimensional layered triangular bocs B. And any finite-dimensional layered triangular bocs has almost split sequences [BK], we may consider the above problem first in the case of bocses.

We will introduce in the following a special bocs denoted by A.

1.2. A = (A, V) is a bocs defined over an algebraically closed field k, with layer L = (A′; ω; x, a; v), where the set of indecomposables in A′ consists of only one object X with A′(X, X) = k. The differentials are given by δ(x) = 0, δ(a) = xv − vx, δ(v) = 0. By [CB1,3.10] A is wild. The bigraph of A is Denote by R(A) the category of finite-dimensional representations of A. An object M in R(A) is given by a triple M = (M_{X}, M(x), M(a)) where M_{X} is a finitedimensional k-vector space and M(x), M(a) are two linear maps on M_{X}. If N = (N_{X}, N(x), N(a)) is also an object in R(A), a morphism φ : M → N in R(A) is given by two linear transformations: φ_{X}, φ_{v} : M_{X} → N_{X}, satisfying the following conditions: N(x)φ_{X} − φ_{X}M(x) = 0 and N(a)φ_{X} − φ_{X}M(a) = N(x)φ_{v} − φ_{v}M(x). If ψ : N → L is another morphism in R(A), then the composition ψφ : M → L consists of (ψφ)_{X} = ψ_{X}φ_{X} and (ψφ)_{v} = ψ_{X}φ_{v} + ψ_{v}φ_{X}. The identity morphism id : M → M consists of (id)_{X} = id_{M_{X}} and (id)_{v} = 0.

1Project supported by National Science Foundation of China.

Since V is a left triangular tensor coalgebra over A [BK,5.8]. Therefore the idempotents split in R(A) [BK,5.13,5.1(e)]. Since R(A) is a finite-dimensional kadditive category with idempotents split, then R(A) is a Krull-Schmidt category.

1.3. Definitions.

Let B = (B, W) be a bocs with layer L = (B′; ω; a_{1} ⋯ a_{n}; v_{1} ⋯ v_{m}), denote by R(B) the category of finite-dimensional representations of B.

(1) Recall that a morphism ι : M → E is said to be left almost split, if (a) ι is not a split monomorphism, and (b) for any morphism φ : M → N in R(B), which is not a split monomorphism, there exists φ′ : E → N, such that φ′ · ι = φ. If in addition for any η : E → E with η · ι = ι, η is an isomorphism, then ι is called a minimal left almost split map. Dually we have the definition of right almost split map and minimal right almost split map [AR2, p. 454].

(2) A sequence 0 → M ⟶^{ι} E ⟶^{π} N → 0 in R(B) ia called an almost split sequence, if M, N are indecomposable; ι is a minimal left almost split map and π is a minimal right almost split map; and πι = 0.

(3) A morphism φ : M → N is called a proper epimorphism (monomorphism) if (i, ω)^{∗}M ⟶^{(i,ω)^{∗}φ} (i, ω)^{∗}N is an epimorphism (monomorphism) in R(B′) [CB1, 3.2].

(4) A sequence ⋯ ⟶ M ⟶^{ι} E ⟶^{π} N ⟶ ⋯ is called a proper exact sequence if

⋯ ⟶ (i, ω)^{∗}M ⟶^{(i,ω)^{∗}ι} (i, ω)^{∗}E ⟶^{(i,ω)^{∗}π} (i, ω)^{∗}N ⟶ ⋯

is an exact sequence in R(B′).

(5) An object N in R(B) is called proper projective, if any proper epimorphism φ : M → N splits. Dually an object M in R(B) is called proper injective if any proper monomorphism ψ : M → N in R(B) splits.

(6) An almost split sequence 0 → M ⟶^{ι} E ⟶^{π} N → 0 is called a proper almost split sequence if it is proper exact.

(7) R(B) is said to have almost split sequences if (a) any indecomposable object M of R(B) has left and right almost split maps; (b) for any non proper injective indecomposable object M in R(B), there exists a proper almost split sequence 0 → M → E → N → 0 in R(B); (c) for any non proper projective indecomposable object N in R(B), there exists a proper almost split sequence 0 → M → E → N → 0 in R(B).

If R(B) has almost split sequences, then R(B) has an Auslander-Reiten quiver as in the case of Artin algebras. The vertices are isomorphism classes of indecomposable objects [M], and we put an arrow [M] → [N] if there is an irreducible map φ : M → N.

1.4. Homogeneous property.

Let B = (B, W) be a layered bocs.

(1) An indecomposable object M in R(B) is called homogeneous, if there is a proper almost split sequence

0 → M ⟶^{ι} E ⟶^{π} M → 0

in R(B), the sequence is also called homogeneous.

(2) R(B) is said to have the homogeneous property if R(B) has almost split sequences and for each dimension d, almost all indecomposable modules M with dimM ≤ d are homogeneous.

(3) R(B) is said to have strong homogeneous property if (a) R(B) has almost split sequences; (b) there exist neither proper-projectives, nor proper-injectives in R(B); and (c) for any indecomposable M in R(B), M is homogeneous.

1.5. Theorem. Let A = (A, V) be the wild bocs defined in 1.2. Then R(A) has strong homogeneous property. Moreover each connected component of the Auslander-Reiten quiver of R(A) is a homogeneous tube.

2. Proof of strong homogeneous property.

2.1. Take Λ = k[ε]/(ε^{2}) and R = Λ[y], the polynomial algebra in y over Λ, R is a commutative ring. Denote by Mod R the module category over R, and by mod R the category of finite-dimensional (over k) R-modules. Let C be the full subcategory of mod R whose objects are the objects of mod R which are free as Λ-modules. Define a functor F : R(A) ⟶ C, sending a representation M = (M_{X}, M(x), M(a)) to the R-module M_{X} ⊕ M_{X}, with ε and y acting as

( \begin{matrix}0 & I \\ 0 & 0 \end{matrix} ) and ( \begin{matrix}M(x) & M(a) \\ 0 & M(x) \end{matrix} )

respectively, where I is the identity matrix of size dim M_{X}, and sending a morphism φ = (φ_{X}, φ_{v}) : M → N to the R-morphism M_{X} ⊕ M_{X} → N_{X} ⊕ N_{X} with matrix

( \begin{matrix}φ_{X} & φ_{v} \\ 0 & φ_{X} \end{matrix} ).

It is easy to check that εF (φ) = F (φ)ε and yF (φ) = F (φ)y by the conditions on φ_{X} and φ_{v} given in 1.2. And F (id_{M}) = id_{F (M)}, F (ψφ) = F (ψ)F (φ) by the composition rule of two morphisms in R(A) given in 1.2.

2.2 Lemma. F : R(A) ⟶ C is an equivalence functor.

Proof. We first prove that F is a dense functor. Any object in C is of the form Λ⊗_{k}M_{X} with M_{X} finite-dimensional k-vector space. Then Λ⊗_{k}M_{X} = (ε⊗M_{X})⊕(1⊗

M_{X}). If we identify M_{X} with ε ⊗ M_{X} and with 1 ⊗ M_{X} via the maps sending m to ε ⊗ m and 1 ⊗ m respectively, then in terms of the above decomposition of Λ ⊗_{k} M_{X},

the action of ε is given by ( \begin{matrix}0 & I \\ 0 & 0 \end{matrix} ), the action of y by ( \begin{matrix}M_{11} & M_{12} \\ 0 & M_{22} \end{matrix} ) but both

actions commute, then M_{11} = M_{22}. Take M = (M_{X}, M(x) = M_{11}, M(a) = M_{12}), then F (M) ≃ Λ ⊗_{k} M_{X}. Now it is easy to see that F is full and faithful, thus F is an equivalence.

2.3 Lemma. If M is in C, then TrM ≃ DM with DM = Hom_{k}(M, k).

Proof. Consider the following sequence of maps in Mod R:

0 ⟶ R ⊗_{Λ} M ⟶^{f} R ⊗_{Λ} M ⟶^{g} M ⟶ 0

with f(r ⊗ m) = ry ⊗ m − r ⊗ ym, g(r ⊗ m) = rm. This sequence is exact. In fact, clearly g is epimorphism and gf = 0. Any element m in R ⊗_{Λ} M has

a unique expression as m = ∑_{i=0}^{l} y^{i} ⊗ m_{i}, m_{i} ∈ M. If f( ∑_{i=0}^{l} y^{i} ⊗ m_{i}) = 0, then

∑_{i=0}^{l} y^{i+1} ⊗ m_{i} − ∑_{i=0}^{l} y^{i} ⊗ ym_{i} = 0. Thus y^{l+1} ⊗ m_{l} = 0, y^{i} ⊗ (m_{i−1} − ym_{i}) = 0, i =

l, l−1, ⋯, 1. Therefore m_{l} = 0, m_{i−1} = ym_{i} = 0, i = l, l−1, ⋯, 1 by induction, thus

∑_{i=0}^{l} y^{i} ⊗ m_{i} = 0, f is a monomorphism. Now if g( ∑_{i=0}^{l} y^{i} ⊗ m_{i}) = 0, then ∑_{i=0}^{l} y^{i}m_{i} = 0.

Therefore

∑_{i=0}^{l} y^{i} ⊗ m_{i} =y^{l} ⊗ m_{l} − y^{l−1} ⊗ ym_{l} + y^{l−1} ⊗ (m_{l−1} + ym_{l})
− y^{l−2} ⊗ y(m_{l−1} + ym_{l}) + ⋯ + y ⊗ (m_{1} + ym_{2} + ⋯ + y^{l−1}m_{l})
− 1 ⊗ y(m_{1} + ym_{2} + ⋯ + y^{l−1}m_{l}),

consequently ker g ⊂ Im f. And our sequence is a projective resolution of M. There is a commutative diagram with exact rows:

R ⊗_{Λ} Hom_{Λ}(M, Λ) ⟶^{f′} R ⊗_{Λ} Hom_{Λ}(M, Λ) ⟶^{g′} Hom_{Λ}(M, Λ) ⟶ 0

↓h  ↓h  ↓

Hom_{R}(R ⊗_{Λ} M, R) ⟶^{f^{∗}} Hom_{R}(R ⊗_{Λ} M, R) ⟶^{g^{∗}} Tr M ⟶ 0

where the top line is the projective resolution of the R-module Hom_{Λ}(M, Λ), and h is the map given by h(r ⊗ θ)(s ⊗ m) = srθ(m), for r, s ∈ R, m ∈ M.

Now h is an isomorphism for any finitely generated free Λ-module, as one sees by reducing to the case M = Λ. Thus Tr M ≃ Hom_{Λ}(M, Λ). On the other hand since Λ is a commutative and also symmetric algebra, and M is free over Λ, it follows that Hom_{Λ}(M, Λ) ≃ DM as R-modules.

2.4. Proposition. For each indecomposable M in C there is an almost split sequence

0 ⟶ M ⟶^{l} E ⟶^{π} M ⟶ 0

in ModR, then also in C.

Proof. M is finitely presented in ModR by the exact sequence given at the beginning of the proof of 2.3. Now we claim:

(∗) Any non-zero submodule of R has infinite k-dimension. Since y^{m}, m = 1, 2, ⋯, are not zero divisors. Therefore Hom_{R}(Λ, R) = 0, since dim_{k}Λ = 2.

¿From (∗) we know that M is not a projective R-module. Otherwise the sequence R ⊗_{Λ} M ⟶^{g} M ⟶ 0 induced in 2.3 splits. Assume that M ≃ Λ^{s}, then M is a

submodule of R^{s}, a contradiction to (∗).

Finally End_{R}(M) is local, since it is a finite-dimensional k-algebra, the idempotents have lifting property,and the idempotents split in R(A), therefore in C. Then rad(End_{R}(M)) is the unique maximal right ideal of End_{R}(M). Denote by P (M, M) the ideal of End_{R}(M) consisting of the morphisms decomposed through projectives, then P(M, M) = 0 from (∗).

By [A,II. 5.1] there exists a unique almost split sequence ending at M in ModR:

0 ⟶ DTr M ⟶^{f} E ⟶^{g} M ⟶ 0

. But here DTr M ≃ M.

2.5.Corollary. (1) dimF (M) = 2dimM for any M in R(A). (2) φ : M → N is a proper monomorphism (epimorphism) in R(A), iff F (φ) : F (M) → F (N) is an monomorphism (epimorphism) in modR. (3) A sequence of maps 0 → M ⟶^{φ} E ⟶^{ψ} N → 0 in R(A) is proper exact iff

0 ⟶ F (M) ⟶^{F (φ)} F (E) ⟶^{F (ψ)} F (N) ⟶ 0

is an exact sequence in modR. (4) φ : M → N is an irreducible map with M, N indecomposable in R(A). Then φ is either proper monomorphic or proper epimorphic, and dimM ≠dimN.

Proof. (2) φ is a proper monomorphism(epimorphic) iff rankφ_{X} =dimM(N) ≤ dimN(M), iff rankF (φ) =dimF (M)(F (N)) ≤ dimF (N)(F (M)), iff F (φ) is monomorphic(epimorphic). (3) In the two sequences φ_{X} is monomorphic iff F (φ) is so, and ψ_{X} is epimorphic iff F (ψ) is so. imφ_{X} ⊆kerψ_{X}, iff imF (φ) ⊆ kerF (ψ). Finally under the above conditions, dim(imφ_{X})=dim(kerψ_{X}) iff dimM+dimN=dimE iff dimF (M)+ dimF (N)=dimF (E) iff dim(imF (φ))=dim(kerF (ψ)). (4) By 2.2, F (φ) : F (M) → F (N) is an irreducible map in C. By 2.4 the irreducible maps in C are also those in ModR. Since ModR is an abelian category having images, F (φ) is either monomorphic or epimorphic. And dimF (M) ≠ F (N), otherwise F (φ) would be an isomorphism. The conclusion follows from (2) and (1).

2.6.The proof of the theorem.

Let M be any indecomposable object in R(A), then there exists an almost split sequence

0 ⟶ F (M) ⟶^{ι′} E′ ⟶^{π′} F (M) ⟶ 0 in C

by 2.4. Therefore by 2.5(3) there exists a proper almost split sequence

0 ⟶ M ⟶^{ι} E ⟶^{π} M ⟶ 0 in R(A).

And M is neither proper projective nor proper injective in R(A).

3. Proof of the shape of components

3.1. Lemma Let C be the category defined in 2.1, M be any fixed indecomposable object in C. Then for any positive integer n there exists a chain of indecomposables and irreducible maps:

M_{n} ⟶^{φ_{n}} M_{n−1} ⟶ ⋯ ⟶ M_{2} ⟶^{φ_{2}} M_{1} ⟶^{φ_{1}} M in C,

such that φ_{1}φ_{2} ⋯ φ_{n} is non zero.

Proof. First consider the almost sequence 0 → M ⟶ E_{1} ⟶^{π_{1}} M → 0. Assume E_{1} ≃ M_{11}⊕⋯⊕M_{1t_{1}} with M_{11}, ⋯, M_{1t_{1}} indecomposable, and π_{1} = (φ_{1,1}, ⋯, φ_{1,t_{1}}) : M_{11} ⊕ ⋯ ⊕ M_{1,t_{1}} → M with φ_{1,j} : M_{1j} → M being an irreducible map. If we have an integer i ≥ 1, and E_{i} ≃ M_{i1} ⊕ ⋯ ⊕ M_{it_{i}} with M_{ij}, j = 1, ⋯ t_{i} indecomposable (some of them could be isomorphic). Let 0 → M_{ij} ⟶ E_{i+1,j} ⟶^{π_{i+1,j}} M_{ij} → 0 be the almost split sequences for i = 1, ⋯, t_{i}. If E_{i+1,j} ≃ M_{i+1,j,1} ⊕ ⋯ ⊕ M_{i+1,j,t_{i+1,j}}, then π_{i+1,j} = (φ_{i+1,j,1}, ⋯, φ_{i+1,j,t_{i+1,j}}) : M_{i+1,j,1} ⊕ ⋯ ⊕ M_{i+1,j,ti+1,j} → M_{ij} with φ_{i+1,j,h} : M_{i+1,j,h} → M_{ij}, h = 1, ⋯, t_{i+1,j} being irreducible. Let E_{i+1} = E_{i+1,1} ⊕ ⋯ ⊕ E_{i+1,t_{i}}, π_{i+1} =diag(π_{i+1,1}, ⋯, π_{i+1,t_{i}}) : E_{i+1} = E_{i+1,1} ⊕⋯⊕E_{i+1,t_{i}} → M_{i,1} ⊕⋯⊕M_{i,t_{i}} = E_{i}. Give new indices 1, 2, ⋯, t_{i+1} for the direct summands of E_{i+1} ≃ ⊕_{j=1}^{t_{i}}⊕_{h=1}^{t_{i+1,j}} M_{i+1,j,h} occording to the original ordering. Simply denote by φ_{i+1,l_{i+1},l_{i}} : M_{i+1,l_{i+1}} → M_{i,l_{i}} the irreducible map φ_{i+1,j,h} when M_{i+1,j,h} = M_{i+1,l_{i+1}} in the new indices (in this case M_{i,l_{i}} = M_{i,j}). Then

π_{1}π_{2} ⋯ π_{n} = (φ_{1,l_{1}}φ_{2,l_{2},l_{1}} ⋯ φ_{n−1,l_{n−1},l_{n−2}}φ_{n,l_{n},l_{n−1}})_{(l_{1},l_{2},···,l_{n−1},l_{n})} : ⊕_{l_{n}=1}^{t_{n}}M_{n,l_{n}} → M

is an epimorphism. There exists at least one sequence of indices, say

φ_{1,l_{1}^{0}}φ_{2,l_{2}^{0},l_{1}^{0}} ⋯ φ_{n−1,l_{n−1}^{0},l_{n−2}^{0}}φ_{n,l_{n}^{0},l_{n−1}^{0}} ≠ 0.

3.2.Proof of the second part of the theorem.

The AR-quiver of R(A) coincides with the AR-quiver of C, which is a stable valued translation quiver. Let Γ be any connected component of the AR-quiver of C. There is an additive function dim_{k} : Γ → ℕ. And Γ is periodic. Then Cartan class of Γ is either a Dynkin diagram, an Euclidean diagram, or one of A_{∞}, A_{∞}^{∞}, B_{∞}, C_{∞}, D_{∞} by [PHR, p.289, Th.]. But the additive function dim_{k} on Γ is unbounded, otherwise Lemma 3.1 would contradict to Harada–Sai lemma. Therefore the Cartan class of Γ is A_{∞} and Γ ≃ ℤA_{∞}/ < τ >, a homogeneous tube.

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Acknowledgement

This is a team work. The original idea and proof was given by 3 authors by a concrete description of the category R(A). The present proof is given by Crawley-Boevey. The first and fourth authors thank to the Beijing Normal University and the Universidad Nacional Autónoma de Mexico for supporting visits to Beijing and Mexico.

Raymundo Bautista, Institute of Mathematics UNAM. Mexico D. F. 04510, Mexico. e-mail: raymundo@servidor.unam.mx

William Crawley-Boevey, Department of Pure Mathematics University of Leeds, Leeds, LS2 9JT England, e-mial: w.crawley-boevey@leeds.ac.uk

Tiangang Lei and Yingbo Zhang, Department of Mathematics, Beijing Normal University, Beijing 100875, P.R. China. e-mail: wangky@bnu.edu.cn