On the Four Terms in the Middle Theorem for almost split sequences

Henning Krause

Recently Liu [L] has proved the following result, which generalizes remarkably the wellknown Four Terms in the Middle Theorem of Bautista and Brenner [BB].

Liu’s Theorem Let Λ be an artin algebra and let 0 → X → ∐_{i=1}^{r} Y_{i} → Z → 0 be an almost split sequence in the category of finitely generated Λ-modules such that all Y_{i}’s are indecomposable. Suppose that X has a projective predecessor and Z has an injective successor in the Auslander–Reiten quiver Γ_{Λ} of Λ. Then r ≤ 4 and r = 4 implies that one of the Y_{i}’s is projective-injective and the others are neither projective nor injective.

We will show that Liu’s proof can be translated into a purely combinatorial one. Before we state our combinatorial result which implies the above-mentioned theorem, let us fix some terminology.

Let Γ = (Γ_{0}, Γ_{1}) be a quiver, that is, a locally finite oriented graph with set of vertices Γ_{0} and set of arrows Γ_{1}. Suppose that Γ contains neither loops nor multiple arrows. Given a vertex x, denote by x^{+} the set of vertices y such that there is an arrow x → y; the set x^{−} consists of all vertices y such that there is an arrow y → x. A pair (Γ, τ) is called a translation quiver, if τ: Γ′_{0} → Γ_{0} is an injective map for some subset Γ′_{0} ⊆ Γ_{0}, satisfying (τx)^{+} = x^{−} for all x ∈ Γ′_{0}. It is convenient to put τ^{0}x = x for all x ∈ Γ_{0}. The vertices in Γ_{0} \ Γ′_{0} are called projective; those in Γ_{0} \ τΓ′_{0} are called injective. Let δ: Γ_{1} → ℕ × ℕ be a map and denote the values by δ(α) = (δ_{x,y}, δ′_{x,y}) for each arrow α: x → y. The triple (Γ, τ, δ) is called a valued translation quiver, if the following conditions are satisfied for all non-projective vertices x.

(δ1) δ′_{τx,y} = δ_{y,x} for all y ∈ x^{−}.

(δ2) δ_{τx,y} = δ′_{y,x} for all y ∈ x^{−}. Finally, a map ℓ: Γ_{0} → ℕ is called an additive length function for (Γ, τ, δ), if the following conditions are satified for all vertices x.

(ℓ1) ℓ(x) + ℓ(τx) = ∑_{y∈x^{−}} δ_{y,x}ℓ(y), if x is non-projective.

(ℓ2) ℓ(x) > ∑_{y∈x^{−}} δ_{y,x}ℓ(y), if x is projective.

(ℓ3) ℓ(x) > ∑_{y∈x^{+}} δ′_{y,x}ℓ(y), if x is injective.

The following is the main result of this note.

Theorem Let (Γ, τ, δ) be a valued translation quiver and ℓ be an additive length function. Let x be a vertex having a projective predecessor and an injective successor. Then ∑_{y∈x^{+}} δ′_{x,y} ≤ 4 and equality implies that x is non-injective and that x^{+} contains a projective vertex.

The following consequence is the combinatorial version of Liu’s theorem.

Corollary Let (Γ, τ, δ) be a valued translation quiver and ℓ be an additive length function. Let x be a non-injective vertex such that x has a projective predecessor and τ^{−1}x has an injective successor. Then ∑_{y∈x^{+}} δ′_{x,y} ≤ 4 and equality implies that x^{+} contains a projectiveinjective vertex y_{0} with δ′_{x,y_{0}} = 1 and x^{+} \ {y_{0}} contains neither a projective nor an injective vertex.

It is well-known that the Auslander–Reiten quiver of an artin algebra together with the usual length function satisfies all the assumptions of the preceeding result. Applying the corollary in this situation one obtains Liu’s theorem.

The rest of this paper is devoted to proving our combinatorial result, using a series of six lemmas. We stress that all the assertions of the lemmas can be found in a representation theoretic formulation in Liu’s paper [L].

Recall that given a path y = x_{n} → ⋯ → x_{1} → x_{0} = x of length n ≥ 1, the vertex y is called a predecessor of x and the vertex x is called a successor of y. If the path is sectional, i.e. τx_{i} ≠ x_{i+2} for all x_{i}, 0 ≤ i ≤ n − 2, lying in Γ′_{0}, then the vertex y is called a sectional predecessor of x and the vertex x is called a sectional successor of y.

Lemma 1 Let x be a vertex such that τ^{r}x is non-projective for all r ≥ 0. Then all predecessors of x are non-projective if and only if all sectional predecessors of τ^{r}x are non-projective for all r ≥ 0.

Proof : Suppose there is a path x_{n} → ⋯ → x_{0} = x with projective x_{n}. Choose i minimal such that y = τ^{s}x_{i} is projective for some s ≥ 0. We obtain a sectional path y → ⋯ → τ^{r}x, for some r ≥ s, by induction on i.

Lemma 2 Let x_{n} → ⋯ → x_{1} → x_{0} = x be a sectional path of length n ≥ 1. If ∑_{y∈x^{−}} δ_{y,x}ℓ(y) − ℓ(x_{1}) ≥ ℓ(x), then x_{n} is non-projective.

Proof : Clearly ∑_{y∈x^{−}} δ_{y,x}ℓ(y) − ℓ(x_{1}) ≥ ℓ(x_{0}) implies that x_{0} is non-projective and ℓ(τx_{0}) = ∑_{y∈x^{−}} δ_{y,x}ℓ(y) − ℓ(x_{0}) ≥ ℓ(x_{1}). Hence, x_{1} is non-projective. Since τx_{0} ≠ x_{2}, it follows that ∑_{y∈x_{1}^{−}} δ_{y,x_{1}}ℓ(y) − ℓ(x_{2}) ≥ ℓ(τx_{0}) ≥ ℓ(x_{1}). Proceeding by induction, one shows that x_{n} is non-projective.

Lemma 3 Let x be a non-projective vertex satisfying ℓ(τx) ≥ ℓ(y) for all y ∈ x^{−}. Then all sectional predecessors of x are non-projective.

Proof : Let x_{n} → ⋯ → x_{1} → x_{0} = x be a sectional path of length n ≥ 1. By assumption ℓ(τx) ≥ ℓ(x_{1}). Therefore ∑_{y∈x^{−}} δ_{y,x}ℓ(y) − ℓ(x_{1}) = ℓ(x) + ℓ(τx) − ℓ(x_{1}) ≥ ℓ(x). Using Lemma 2 we conclude that x_{n} is non-projective.

Lemma 4 Let y and y′ be not necessarily distinct elements in x^{+} for some vertex x. Suppose that δ′_{x,y} ≥ 2, if y = y′. Then ℓ(x) ≥ ℓ(y) + ℓ(y′) implies that y, y′ and x are non-projective and ℓ(τx) ≥ ∑_{z∈x^{−}} δ_{z,x}ℓ(z) − ℓ(τy) − ℓ(τy′).

Proof : The assumption ℓ(x) ≥ ℓ(y) + ℓ(y′) implies that y and y′ are non-projective and ℓ(τy) + ℓ(τy′) ≥ 2ℓ(x) − ℓ(y) − ℓ(y′) ≥ ℓ(x). Since δ_{τy,x} = δ′_{x,y} ≥ 2, if y = y′, it follows that x is non-projective and ℓ(τx) = ∑_{z∈x^{−}} δ_{z,x}ℓ(z) − ℓ(x) ≥ ∑_{z∈x^{−}} δ_{z,x}ℓ(z) − ℓ(τy) − ℓ(τy′).

Lemma 5 Let y_{1}, …, y_{n} be non-projective vertices in x^{+} for some vertex x. Let ε_{i}, 1 ≤ i ≤ n be integers satisfying 1 ≤ ε_{i} ≤ δ′_{x,y_{i}} for all i and ∑_{i} ε_{i} = 4. Suppose that either x is noninjective with ℓ(x) ≥ ℓ(τ^{−1}x), or ℓ(x) ≥ ∑_{i} ε_{i}ℓ(y_{i}). Then x and all its predecessors are non-projective.

Proof : By our assumptions, we have 2ℓ(x) ≥ ∑_{i} ε_{i}ℓ(y_{i}). Choosing integers ε′_{i}, ε′′_{i} ≥ 0 satisfying ε_{i} = ε′_{1} + ε′′_{i} for all i and ∑_{i} ε′_{i} = 2 = ∑_{i} ε′′_{i}, we have ℓ(x) ≥ ∑_{i} ε′_{i}ℓ(y_{i}) or ℓ(x) ≥ ∑_{i} ε′′_{i}ℓ(y_{i}). Therefore x is non-projective by Lemma 4. Let z ∈ x^{−} and define γ_{i} = ε_{i} − 1, if z = τy_{i} and γ_{i} = ε_{i} otherwise. Then

ℓ(τx) ≥ ℓ(z) + ∑_{i} γ_{i}ℓ(τy_{i}) − ℓ(x) ≥ ℓ(z) + ∑_{i} γ_{i}(ℓ(x) − ℓ(y_{i})) − ℓ(x) ≥ ℓ(z).

Therefore any sectional predecessor of x is non-projective by Lemma 3. In particular, τy_{i} is non-projective for all i and

ℓ(τx) ≥ ∑_{i} ε_{i}ℓ(τy_{i}) − ℓ(x) ≥ ∑_{i} ε_{i}(ℓ(x) − ℓ(y_{i})) − ℓ(x) ≥ ℓ(x).

By induction, τ^{r}x and all sectional predecessors of τ^{r}x are non-projective for all r ≥ 0. Now the assertion follows from Lemma 1.

Lemma 6 Let x be a vertex having an injective sectional successor. Suppose that either r = ∑_{y∈x^{+}} δ′_{x,y} > 4, or r = 4 and x^{+} contains no projective vertex. Then all predecessors of x are non-projective.

Proof : Let x = x_{0} → ⋯ → x_{n} be a sectional path of length n ≥ 1 with x_{n} injective. Then ℓ(x) ≥ ∑_{y∈x^{+}} δ′_{x,y}ℓ(y) − ℓ(x_{1}) by the dual of Lemma 2. If r > 4, then the assertion follows from Lemma 5. Therefore assume r = 4 and x^{+} ⊆ Γ′_{0}. For any choice of integers ε_{y}, y ∈ x^{+} satisfying 0 ≤ ε_{y} ≤ δ′_{x,y} for all y, ε_{y} ≤ δ′_{x,y} − 1 for y = x_{1} and ∑_{y∈x^{+}} ε_{y} = 2, we have ℓ(x) ≥ ∑_{y∈x^{+}} ε_{y}ℓ(y). Therefore x is non-projective and ℓ(τx) ≥ ∑_{z∈x^{−}} δ_{z,x}ℓ(z) − ∑_{y∈x^{+}} ε_{y}ℓ(τy) by Lemma 4. We conclude that ℓ(τx) ≥ ℓ(z) for all z ∈ x^{−}. Thus all sectional predecessors of x are non-projective by Lemma 3. In particular, x^{−} ⊆ Γ′_{0}. We have also ∑_{y∈x^{+}} δ′_{τx,τy} = ∑_{y∈x^{+}} δ_{τy,x} = ∑_{y∈x^{+}} δ′_{x,y} = 4, and so, since x has an injective successor, ℓ(τx) ≥ ℓ(x) by the dual of Lemma 5. Therefore τx and all predecessors of τx are non-projective by Lemma 5. This finishes the proof.

We are now ready to prove the main result.

Proof of the Theorem: From the fact that x has an injective successor, we obtain r ≥ 0 such that either z = τ^{−r}x has an injective sectional successor or z is injective. This follows from the dual statement of Lemma 1. Choose r minimal and assume first that z has an injective sectional successor. Then ∑_{y∈x^{+}} δ′_{x,y} = ∑_{y∈x^{+}} δ′_{z,τ^{−r}y} ≤ ∑_{y∈z^{+}} δ′_{z,y} ≤ 4 by Lemma 6. Moreover, equality implies that z^{+} = τ^{−r}(x^{+}) and z^{+} contains a projective vertex. But this is only possible for r = 0. Therefore x^{+} contains a projective vertex and hence x is non-injective. Now assume that z is injective. We have ℓ(z) ≥ ∑_{y∈x^{+}} δ′_{z,τ^{−r}y}ℓ(τ^{−r}y) and ∑_{y∈x^{+}} δ′_{x,y} = ∑_{y∈x^{+}} δ′_{z,τ^{−r}y} < 4 follows from Lemma 5, since τ^{−r}(x^{+}) ⊆ Γ′_{0}.

Proof of the Corollary: Combine the theorem and its dual with the following well-known observation.

Lemma 7 Let x be a vertex such that x^{+} contains a projective and an injective vertex. Then x^{+} contains a projective-injective vertex y_{0} with δ′_{x,y_{0}} = 1 and x^{+} \ {y_{0}} contains neither a projective nor an injective vertex.

Proof : Suppose y_{0} ∈ x^{+} is projective. Then x is non-injective and we obtain ∑_{x^{+}\{y_{0}}} δ′_{x,y}ℓ(y)+ (δ′_{x,y_{0}} − 1)ℓ(y_{0}) = ℓ(τ^{−1}x) + ℓ(x) − ℓ(y_{0}) ≤ ℓ(τ^{−1}x). Therefore x^{+} \ {y_{0}} contains no injective vertex and y_{0} is projective-injective by assumption. Moreover, δ′_{x,y_{0}} − 1 = 0, since y_{0} is injective. The dual argument shows that x^{+} \ {y_{0}} contains no projective vertex.

The following example which was suggested by R. Betzler and R. Schmidmeier illustrates the difference between Liu’s theorem and the theorem presented in this note.

Example Let k be a field and denote by Λ the k-algebra given by the following quiver with relations:

[figure: 5; 4; 1; 2; 3; 0; α_{1}; α_{2}; α_{3}; β_{1}; β_{2}; β_{3}; α_{1}β_{1} = α_{2}β_{2} = α_{3}β_{3}.; ,]

Denote by P_{i} the indecomposable projective Λ-module corresponding to the vertex i and let S_{i} = P_{i}/ rad P_{i}, 1 ≤ i ≤ 4. There is an almost split sequence

0 → rad P_{4} → S_{1} ∐ S_{2} ∐ S_{3} ∐ P_{4} → P_{4}/ soc P_{4} → 0

and an irreducible map P_{4} → P_{5}. The projective P_{0} is a predecessor and the injective P_{5} is a successor of rad P_{4} in the Auslander-Reiten quiver Γ_{Λ} of Λ, but P_{4}/ soc P_{4} has no injective successor.

This example shows that in Liu’s theorem the assumption “Z has an injective successor” cannot be replaced by “X has an injective successor” since the middle term S_{1} ∐ S_{2} ∐ S_{3} ∐ P_{4} has no projective-injective summand. However, the example satisfies the assumptions of the theorem presented in this note.

References

[BB] R. Bautista, S. Brenner, Replication numbers for non-Dynkin sectional subgraphs in finite Auslander–Reiten quivers and some properties of Weyl roots, Proc. London Math. Soc. 43 (1983), 429–462.

[L] S. Liu, Almost split sequences for non-regular modules, Preprint (1993).

Fakultät für Mathematik, Universität Bielefeld, Germany.

October 1993