A HOMOLOGICAL CHARACTERIZATION OF PIECEWISE HEREDITARY ALGEBRAS

DIETER HAPPEL AND DAN ZACHARIA

Abstract. Let Λ be a finite dimensional algebra over a field k. We will show here that Λ is piecewise hereditary if and only if s.gl.dim Λ is finite.

Let Λ be a finite dimensional algebra over a field k. We denote by mod Λ the category of finitely generated left Λ−modules. Recall that Λ is said to be piecewise hereditary, if there exists a hereditary, abelian category H such that the bounded derived categories D^{b}(Λ) and D^{b}(H) are equivalent as triangulated categories [H1], [HRS2]. The categories H occurring in this situation have been described in [H2] and [HR], but we do not have to make use of these investigations. Over twenty years ago Ringel proposed to consider for a finite dimensional algebra the strong global dimension of Λ. The definition will be recalled below. This concept was taken up in [S] and further pursued in [KSYZ], where also the question was raised whether the class of piecewise hereditary algebras coincides with the class of algebras being of finite strong global dimension. In [KSYZ] this was proved provided that rad^{2}Λ = 0, where rad Λ denotes the radical of Λ. The main result of this article shows that the question mentioned above has an affirmative answer for all finite dimensional algebras.

To recall the definition of the strong global dimension of a finite dimensional algebra Λ we first want to define the length of a complex. For this let 𝔞 be an additive k−category which is Krull-Schmidt (compare [R1]). Let C^{b}(𝔞) be the category of bounded complexes over 𝔞. This is a Frobenius category in the sense of [H1]. We denote by K^{b}(𝔞) the corresponding stable (or homotopy) category. If X^{•} = (X^{i}, d^{i}) ∈ K^{b}(𝔞) is a complex we may consider a preimage X̄^{•} = (X̄^{i}d̄^{i}) of X^{•} in C^{b}(𝔞) without indecomposable projective direct summands. Clearly X̄^{•} is uniquely determined by X^{•} up to isomorphism an C^{b}(𝔞). Thus the following is well-defined. If 0 ≠ X^{•} ∈ K^{b}(𝔞) there exists r ≤ s such that X̄^{r} ≠ 0 ≠ X̄^{s} and X̄^{i} = 0 for i < r and i > s. Then by definition the length of X^{•} is defined as ℓ(X^{•}) = s−r. In the following we will always identify X^{•} with X̄^{•}.

1991 Mathematics Subject Classification. 16E05, 16E10, 16G10.

These results were obtained during a visit of the first author at Syracuse University. He would like to thank the second author for the hospitality during his stay.

If Λ is a finite dimensional algebra we denote by _{Λ}P the subcategory of mod Λ containing the projective Λ−modules. Then we define the strong global dimension of Λ by

s.gl.dim Λ = sup {ℓ(P^{•}) | P^{•} ∈ K^{b}(_{Λ}P) indecomposable}

For example, if Λ is a finite dimensional hereditary algebra then it follows from [H1] that s.gl.dim Λ = 1. Also note that for the minimal projective resolution P^{•}(X) of an indecomposable Λ−module X of finite projective dimension we will obtain that ℓ(P^{•}(X)) = proj.dim X. Also we point out that gl.dim Λ < ∞ if s.gl.dim Λ < ∞.

Note that we slightly deviate from the definition given in [S], where the width of a complex is used instead of the length. To be precise if d = s.gl.dim Λ < ∞ then the strong global dimension of Λ in the sense of [S] would be d + 1. The example of hereditary algebras suggests that our definition may be more natural.

We will give now an outline of the paper and at the same time give the ideas behind the proof of the main result. In section one we briefly recall some notation and elementary facts of derived categories which will be used in the later sections. It was already noticed in [KSYZ] that a piecewise hereditary algebra has finite strong global dimension. We will give a direct proof in section 3 avoiding the use of repetitive algebras. The converse strongly relies on a characterization of piecewise hereditary algebras given by Ringel in [R2]. He shows that an algebra Λ is not piecewise hereditary if and only if for each indecomposable object X ∈ D^{b}(Λ) there exists a path in D^{b}(Λ) from X[1] → ⋯ → X, where X[1] is the shift of X in the triangulated category D^{b}(Λ). For the definition of a path in a triangulated category we refer to section two. We will show in section three that paths from X[1] to X can be used to construct indecomposable complexes in K^{b}(_{Λ}P) of arbitrary length. The main technical tool for doing this is provided in section two, where we show that a path of length s between indecomposables X, Y in a triangulated category C with finite dimensional Hom-spaces can be shortened to a path of length two between X[t] and Y for some 0 ≤ t ≤ s − 2.

We denote the composition of morphisms f : X → Y and g : Y → Z in a given category K by fg. The notation and terminology introduced here will be fixed throughout this article. For unexplained representation-theoretic terminology we refer [ARS] and [R1].

1. Preliminaries

In this section we briefly recall some of the notation we will use for derived categories and state some useful facts involving triangles in triangulated categories involving nonzero and noninvertible maps between indecomposable objects.

Let A be an abelian k−category and let D^{b}(A) be its bounded derived category. For X^{•} ∈ D^{b}(A) we denote by X^{•}[1] the shift in the triangulated category D^{b}(A). We have an embedding of A into D^{b}(A) by sending X ∈ A to the stalk complex concentrated in degree 0 with stalk X. We denote by A[0] the image of this embedding. Then for each i ∈ ℤ we also have A ≃ A[i] ∈ D^{b}(A). For a complex X^{•} = (X^{i}, d^{i}) ∈ D^{b}(A) we denote by H^{i}(X^{•}) = ker d^{i}/im d^{i−1} the i−th cohomology space.

For the convenience of the reader we present the following simple lemma which is used in the section three. We keep the notation and conventions for categories of complexes over an additive category 𝔞.

Lemma 1.1. Let 𝔞 be an additive category which is Krull-Schmidt. Let 0 ≠ X ∈ 𝔞 and 0 ≠ Y^{•} = (Y^{i}, d^{i}) ∈ K^{b}(𝔞) with Y^{r} ≠ 0 ≠ Y^{0} for some r ≤ 0 and Y^{i} = 0 for i < r and i > 0.

(i) If Hom_{K^{b}(𝔞)}(X, Y^{•}) ≠ 0, then Y^{0} ≠ 0.

(ii) If Hom_{K^{b}(𝔞)}(Y^{•}, X) ≠ 0, then Y^{0} ≠ 0.

(iii) Hom_{K^{b}(𝔞)}(Y^{•}, Y^{r}[−r]) ≠ 0.

(iv) Hom_{K^{b}(𝔞)}(Y^{0}, Y^{•}) ≠ 0.

Proof: Trivially (i) and (ii) hold, since K^{b}(𝔞) is the stable category of C^{b}(𝔞). We will show (iii). Clearly (iv) then follows by dual considerations. Let π^{•} : Y^{•} → Y^{r}[−r] be defined by π^{r} = id_{Y^{r}} and π^{j} = 0 for j ≠ r. Clearly π^{•} is map of complexes, since Y^{i} = 0 for i < r. If Hom_{K^{b}(𝔞)}(Y^{•}, Y^{r}[−r]) = 0, then π^{•} is homotopic to zero. In particular there is h : Y^{r+1} → Y^{r} with d^{r}h = id_{Y^{r}}. So Y^{•} has indecomposable projective direct summands in C^{b}(𝔞), in contrast to the convention agreed upon in the introduction.

For the rest of this section we denote by C a triangulated k−category which is Krull-Schmidt and has the property that for all X, Y ∈ C the dimension of Hom_{C}(X, Y) is finite. For a map f : X → Y in C we have a triangle X → Y → C_{f} → X[1] in C. The object C_{f} is uniquely determined up to isomorphism and is called the cone of f. In the proof of the main lemma in the next section we will need some results about maps occuring in triangles. The following lemma collects these results.

Some are taken from [H1] and [R2] and are stated here again without proof. We will use the following notation for the components of a map u : X → E and a map v : E → Y. Let E = ⊕_{i=1}^{r}E_{i} with E_{i} indecomposable. Then we denote by μ_{i} : E_{i} → E the canonical split mono and by π_{i} : E → E_{i} the canonical split epi. Then we define for 1 ≤ i ≤ r the map u_{i} : X → E_{i} by uπ_{i}. The components v_{i} of the map v : E → Y are defined by μ_{i}v.

Lemma 1.2. (i) The following are equivalent for a triangle

X ⟶^{f} Y ⟶^{u} Z ⟶^{v} X[1]

(a) f is split mono.

(b) u is split epi.

(c) v = 0. (ii) If f : X → Y is nonzero and not invertible for X, Y indecom-

posable, and let

X ⟶^{f} Y ⟶^{u} ⊕_{i=1}^{r}Z_{i} ⟶^{v} X[1]

be a triangle with Z_{i} indecomposable, then the components u_{i} of u are nonzero and noninvertible and the components v_{i} of v are nonzero and noninvertible.

(iii) If r > 1 in the situation of (ii), then u_{i}v_{i} is nonzero for each i.

Proof: For (i) we refer to [H1] and for (ii) we refer to [R2]. In order to show (iii), let us assume that there is 1 ≤ i ≤ r with u_{i}v_{i} = 0. So 0 = u_{i}v_{i} = uπ_{i}μ_{i}v. Thus there is φ_{i} : Y → Y with φ_{i}u = uπ_{i}μ_{i}. So φ_{i}u_{j} = φ_{i}uπ_{j} = 0 for j ≠ i and φ_{i}u_{i} = u_{i}. By (ii) we have that u_{j} ≠ 0, hence φ_{i} is neither an isomorphism nor zero. Since r > 1 there is j ≠ i. Since Y is indecomposable we infer that φ_{i} is nilpotent, hence there is n such that φ_{i}^{n} = 0, in contrast to φ_{i}^{n}u_{i} = u_{i} ≠ 0, so u_{i}v_{i} ≠ 0 for all i.

We will now present a generalization of a result in [HKR] which follows as Corollary 1.4

Proposition 1.3. If f : X → Y is nonzero and not invertible for X, Y indecomposable, and let

X ⟶^{f} Y ⟶^{u} C_{f} ⟶^{v} X[1]

be a triangle. If the induced map f^{∗} : Hom_{C}(Y, X[1]) → Hom_{C}(X, X[1]) is injective, then C_{f} is indecomposable.

Proof: Assume to the contrary that C_{f} = C_{1} ⊕ C for C_{1} indecomposable. By 1.2 (iii) we have that u_{1}v_{1} ≠ 0. Since f^{∗} is injective we infer that 0 ≠ f^{∗}(u_{1}v_{1}) = fu_{1}v_{1}. Now fu_{1} = fuπ_{1} = 0, since fu = 0, for the composition of any two consecutive maps in a triangle is zero (compare [H1]). So we obtain a contradiction. Thus C_{f} is indecomposable.

Corollary 1.4. Let X, Y ∈ C and let f ∈ Hom_{C}(X, Y) be nonzero and noninvertible. If Hom_{C}(Y, X[1]) = 0, then C_{f} is indecomposable.

2. Paths in triangulated categories

In this section we present the proof of our main ingredient of the proof of the main Theorem of this article. This is concerned with shortening paths in a triangulated category C which we assume to be Krull-Schmidt and such the dimension of Hom_{C}(X, Y) is finite for all X, Y ∈ C. Note that this assumption is satisfied for C = D^{b}(Λ) for a finite dimensional algebra Λ, the case of interest to us in the next section.

Following Ringel [R2] a path in C is a sequence X_{0}, …, X_{s} of indecomposable objects in C such that either Hom_{C}(X_{i}, X_{i+1}) ≠ 0 for 0 ≤ i ≤ s − 1 or X_{i+1} = X_{i}[1]. We will call a sequence X_{0}, …, X_{s} of indecomposable objects in C a strong path in C if Hom_{C}(X_{i}, X_{i+1}) ≠ 0 for 0 ≤ i ≤ s − 1 As already pointed out in [R2] these two notions are in many cases equivalent For this we recall from [R2] the following lemma and its proof.

Lemma 2.1. Let C be a triangulated category where for each indecomposable X ∈ C there exists an indecomposable object Y ∈ C such that there is a nonzero, noninvertible map f : X → Y. Then each path in C can be refined to a strong path in C.

Proof: In fact, let X_{0}, …, X_{s} be a path in C. There is nothing to show if Hom_{C}(X_{i}, X_{i+1}) ≠ 0,. So assume there is i with X_{i+1} = X_{i}[1]. By assumption there is Y indecomposable in C and f : X_{i} → Y nonzero and noninvertible. Consider a triangle

X_{i} ⟶^{f} Y ⟶^{u} C_{f} ⟶^{v} X_{i}[1]

Then by 1.2 (ii) we obtain for each indecomposable summand C_{j} of C_{f} a strong path X_{i} → Y → C_{j} → X_{i}[1]. So the given path can be refined to a strong path.

We include a proof of the following easy lemma for the convenience of the reader.

Lemma 2.2. Let Λ be a connected finite dimensional algebra which is not semi-simple. Then for each indecomposable object X ∈ D^{b}(Λ) there exist an indecomposable object Y ∈ D^{b}(Λ) and a nonzero, noninvertible map f : X → Y

Proof: Let X^{•} ∈ D^{b}(Λ) be indecomposable. By applying the shift functor if necessary we may assume that H^{i}(X^{•}) = 0 for i > 0 and that H^{0}(X^{•}) ≠ 0. Let τ_{≤0}X^{•} be the complex defined by (τ_{≤0}X^{•})^{i} = 0 for i > 0 and (τ_{≤0}X^{•})^{i} = X^{i} for i < 0 and (τ_{≤0}X^{•})^{0} = ker d^{0} with the induced differentials. Let (τ_{>0}X^{•}) be the complex defined by (τ_{>0}X^{•})^{i} = 0 for i < 0 and (τ_{≤0}X^{•})^{i} = X^{i} for i > 0 and (τ_{≤0}X^{•})^{0} = im d^{0} with the induced differentials. Then we obtain a triangle where u is the obvious map

τ_{≤0}X^{•} ⟶^{u} X^{•} → τ_{>0}X^{•} → τ_{≤0}X^{•}[1].

The long exact sequence in cohomology shows that u is an isomorphism in D^{b}(Λ), since (τ_{>0}X^{•}) is acyclic.

Hence we may assume that X^{i} = 0 for i > 0. Now H^{0}(X^{•}) = coker d^{−1}. So we obtain a map f : X^{•} → coker d^{−1}. Considering cohomology shows that H^{0}(f) ≠ 0, hence f is nonzero. So there exists an indecomposable direct summand Z of coker d^{−1} and a nonzero map f′ : X^{•} → Z. If f′ is noninvertible, we are done. Otherwise we are reduced to the case that X = X^{•} is an indecomposable Λ−module. If X is not simple, there exists a simple S and a nonzero, noninvertible map X → S. If X is simple and not injective there exists an indecomposable injective I and a nonzero, noninvertible map S → I. So assume that S is simple and injective. Since Λ is connected and not semi-simple, there is a simple Λ−module S′ with Ext_{Λ}^{1}(S, S′) ≠ 0. But then Hom_{D^{b}(Λ)}(S, S′[1]) ≠ 0. Any nonzero map in Hom_{D^{b}(Λ)}(S, S′[1]) is clearly noninvertible.

The following corollary will be used in section three.

Corollary 2.3. Let Λ be a connected finite dimensional algebra which is not semi-simple. Then any path in D^{b}(Λ) can be refined to a strong path in D^{b}(Λ).

Proof: This follows immediately from 2.1 and 2.2.

The following lemma is the key ingredient in the proof of the main result of this article.

Lemma 2.4. Let X_{0}, X_{1}, X_{2}, X_{3} be a strong path in C. Then there exist 0 ≤ t ≤ 1, an indecomposable object Y ∈ C and a strong path X_{0}[t] → Y → X_{3}.

Proof: Given X_{0} and X_{3} we may choose a strong path from X_{0} to X_{3} of minimal length. Assume that the minimal length of these paths is three. Otherwise there is an indecomposable object Y ∈ C and a strong path X_{0} → Y → X_{3}. We will show that then there exists an indecomposable object Y ∈ C and a strong path X_{0}[1] → Y → X_{3}. For this we choose amongst the strong paths of minimal length

X_{0} ⟶^{f} X_{1} ⟶^{g} X_{2} ⟶^{h} X_{3}

a strong path ω such that dim_{k}ker g^{∗} is minimal, where

g^{∗} : Hom_{C}(X_{2}, X_{1}[1]) → Hom_{C}(X_{1}, X_{1}[1])

is the map induced by g.

Since ω is of minimal length, we see that Hom_{C}(X_{0}, X_{2}) = 0 and also that Hom_{C}(X_{1}, X_{3}) = 0. Consider the following triangle in C.

(∗) C_{g}[−1] ⟶ X_{1} ⟶^{g} X_{2} ⟶ C_{g}

Applying Hom_{C}(X_{0}, −) and Hom_{C}(−, X_{3}) to the triangle (∗) shows that Hom_{C}(X_{0}, C_{g}[−1]) ≠ 0 and that Hom_{C}(C_{g}, X_{3}) ≠ 0. Thus we see that also Hom_{C}(X_{0}[1], C_{g}) ≠ 0. Note that we also may assume that Hom_{C}(X_{1}[1], X_{3}) = 0, since otherwise we would obtain a strong path X_{0}[1] → X_{1}[1] → X_{3}.

We claim that C_{g} is indecomposable. Assume to the contrary that C_{g} = ⊕_{i=1}^{r}C_{i} with C_{i} indecomposable and that r > 1. From above we have that Hom_{C}(C_{g}, X_{3}) ≠ 0. Thus there is an indecomposable direct summand C_{i} of C_{g} such that Hom_{C}(C_{i}, X_{3}) ≠ 0. Note that we also may assume that Hom_{C}(X_{0}, C_{i}[−1]) = 0. Since otherwise we obtain a strong path X_{0}[1] → C_{i} → X_{3}. The canonical split mono μ_{i} : C_{i} → C_{g} induces the following map of triangles

X_{1} ⟶^{α} Y ⟶^{β} C_{i} ⟶^{v_{i}} X_{1}[1]
∥  ^{φ}↓  ^{μ_{i}}↓  ∥
X_{1} ⟶^{g} X_{2} ⟶^{u} C_{g} ⟶^{v} X_{1}[1].

Note that Y [1] is the cone of v_{i} and φ exists since C is a triangulated category. Applying Hom_{C}(−, X_{3}) to the upper triangle and using Hom_{C}(X_{1}[1], X_{3}) = 0 shows that Hom_{C}(Y, X_{3}) ≠ 0. Applying Hom_{C}(X_{0}, −) to the upper triangle and using Hom_{C}(X_{0}, C_{i}[−1])) = 0 shows that Hom_{C}(X_{0}, Y) ≠ 0. If Y is indecomposable we obtain a strong path X → Y → X_{3}, in contrast to the minimality of ω. Let Y = ⊕_{j=1}^{t}Y_{j} with Y_{j} indecomposable and t > 1. Since Hom_{C}(Y, X_{3}) ≠ 0 there is j such that Hom_{C}(Y_{j}, X_{3}) ≠ 0. The minimality of ω implies that g is not invertible. Thus by 1.2 (ii) v_{i} is nonzero and noninvertible for 1 ≤ i ≤ r. Thus again by 1.2 (ii) we infer that β_{j} is nonzero and noninvertible. So we then obtain a strong path X_{0} → X_{1} → Y_{j} → X_{3}, which clearly again is minimal.

Applying Hom_{C}(−, X_{1}[1]) to the commutative diagram of triangles yields the following commutative diagram of exact sequences

Hom_{C}(C_{g}, X_{1}[1]) ⟶ Hom_{C}(X_{2}, X_{1}[1]) ⟶^{g^{∗}} Hom_{C}(X_{1}, X_{1}[1])
μ_{i}^{∗}↓  ^{φ^{∗}}↓  ∥
Hom_{C}(C_{i}, X_{1}[1]) ⟶^{β^{∗}} Hom_{C}(Y, X_{1}[1]) ⟶^{α^{∗}} Hom_{C}(X_{1}, X_{1}[1]).

Since μ_{i} is split mono, we infer that μ_{i}^{∗} is split epi by 1.2 (i). So the commutative diagram of exact sequences implies that we obtain a surjective map ker g^{∗} → ker α^{∗}, hence dim ker g^{∗} ≥ dim ker α^{∗}.

The canonical split epi π_{j} : Y → Y_{j} induces the following map of triangles

X_{1} ⟶^{α} Y ⟶^{β} C_{i} ⟶^{v_{i}} X_{1}[1]
∥  ^{π_{j}} ↓  ↓  ∥
X_{1} ⟶^{α_{j}} Y_{j} ⟶ C_{α_{j}} ⟶ X_{1}[1].

Applying Hom_{C}(−, X_{1}[1]) to this commutative diagram of triangles yields the following commutative diagram of exact sequences

0 ⟶ ker α_{j}^{∗} ⟶ Hom_{C}(Y_{1}, X_{1}[1]) ⟶^{α_{j}^{∗}} Hom_{C}(X_{1}, X_{1}[1])
ψ↓  ^{π_{j}^{∗}}↓  ∥
0 ⟶ ker α^{∗} ⟶ Hom_{C}(Y, X_{1}[1]) ⟶^{α^{∗}} Hom_{C}(X_{1}, X_{1}[1]),

where ψ is the induced map. Since π_{j} is split epi, we infer that π_{j}^{∗} is mono by 1.2 (i). Thus ψ is mono. Thus we obtain that dim ker g^{∗} ≥ dim ker α^{∗} ≥ dim ker α_{j}^{∗} By minimality of dim ker g^{∗} we obtain that dim ker g^{∗} = dim ker α_{j}^{∗}, hence also dim ker g^{∗} = dim ker α^{∗}. This implies that the surjection ker g^{∗} → ker α^{∗} is an isomorphism. Thus φ^{∗} is injective. Let s ≠ i. By 1.2 (iii) we know that u_{s}v_{s} ≠ 0. Thus φ^{∗}(u_{s}v_{s}) ≠ 0. But φ^{∗}(u_{s}v_{s}) = φu_{s}v_{s} = φuπ_{s}v_{s} = βμ_{i}π_{s}v_{s} = 0, since π_{s}μ_{i} = 0 for s ≠ i.

So we obtain a contradiction. Thus C_{g} is indecomposable. As shown above we have that Hom_{C}(X_{0}, C_{g}[−1]) ≠ 0 and that Hom_{C}(C_{g}, X_{3}) ≠ 0. Thus we have a strong path X_{0}[1] → C_{g} → X_{3}. This finishes the proof.

We point out that easy examples show that a strong path X_{0} → X_{1} → X_{2} → X_{3} may not be shortened to a strong path X_{0} → Y → X_{3}. So the case t = 1 in 2.4 will occur.

Theorem 2.5. Let X_{0}, …, X_{s} be a strong path in C. Then there exist 0 ≤ t ≤ s − 2, an indecomposable object Y ∈ C and a strong path X_{0}[t] → Y → X_{s}.

Proof: This follows by successive application of 2.4.

We want to mention an application to hereditary algebras.

Corollary 2.6. Let Λ be a finite dimensional connected hereditary algebra which is not semi-simple. Let X, Y be indecomposable Λ−modules. Then there exist 0 ≤ m ≤ 2, an indecomposable object Z ∈ D^{b}(Λ) and a strong path X → Z → Y [2].

Proof: Since Λ is connected there is by [R2] an n ≥ 0 and a path from X to Y [n]. Since Λ is not semi-simple we obtain by 2.3 a strong path from X to Y [n]. By 2.5 we there is t ≥ 0, an indecomposable Z ∈ D^{b}(Λ) and a strong path X → Z → Y [n−t]. Since Λ is hereditary we clearly have that 0 ≤ n − t ≤ 2.

3. Piecewise hereditary algebras

The aim of this section is to prove the main result in this article mentioned in the introduction. First we gill an alternative proof to the one given in [KSYZ] of the fact that piecewise hereditary algebras have finite strong global dimension. For this we have to recall a little bit of notation. If Λ is piecewise hereditary there exists a hereditary, abelian category H and a triangle equivalence F : D^{b}(Λ) → D^{b}(H) (compare [H1] and [HRS2]). We may assume that F is normalized in the sense that there exists r ≥ 0 such that for an indecomposable Λ−module X we have F (X) ∈ ⋃_{i=0}^{r} H[i] and that there exist indecomposable Λmodules X, Y such that F (X) ∈ H[0] and F (Y) ∈ H[r]. Note that such normalized equivalences are not unique and that the value of r may depend on the choice of H. By [HRS2] there exists for each 0 ≤ i ≤ r a simple Λ−module S_{i} such that F (S_{i}) ∈ H[i]. Thus r + 1 ≤ rk K_{0}(Λ), where K_{0}(Λ) denotes the Grothendieck group of Λ.

Proposition 3.1. Let Λ be a finite dimensional piecewise hereditary algebra. Then s.gl.dim Λ ≤ rk K_{0}(Λ) + 1. In particular s.gl.dim Λ < ∞.

Proof: Let F : D^{b}(Λ) → D^{b}(H) be a normalized triangle equivalence. Let P^{•} = (P^{i}, d^{i}) ∈ K^{b}(_{Λ}P) be indecomposable with length ℓ(P^{•}) = t. Applying the shift functor in K^{b}(_{Λ}P), if necessary, we may assume that P^{0} ≠ 0 and that P^{i} = 0 for i > 0. Thus P^{i} = 0 for i < −t and P^{−t} ≠ 0. By 1.1 we have that Hom_{K^{b}(_{Λ}P)}(P^{0}, P^{•}) ≠ 0 and that Hom_{K^{b}(_{Λ}P)}(P^{•}, P^{−t}[t]) ≠ 0. Now F (P^{−t}[t]) ∈ ⋃_{i=t}^{r+t} H[i] and F (P^{0}) ∈ ⋃_{i=0}^{r} H[i], since F is normalized. Since P^{•} is indecomposable there is s ∈ ℤ such that F (P^{•}) ∈ H[s]. Since H is hereditary we have that for indecomposable objects X, Y ∈ D^{b}(H) with Hom_{D^{b}(H)}(F (P^{•}), X) ≠ 0 and Hom_{D^{b}(H)}(Y, F (P^{•})) ≠ 0 that X ∈ H[s] ∪ H[s + 1] and that Y ∈ H[s − 1] ∪ H[s]. So Hom_{K^{b}(_{Λ}P)}(P^{0}, P^{•}) ≠ 0 yields 0 ≤ s ≤ r + 1 and Hom_{K^{b}(_{Λ}P)}(P^{•}, P^{−t}[t]) ≠ 0 yields t−1 ≤ s ≤ r +1. Thus t ≤ r +2, hence the assertion.

We point out that the bound given in 3.1 is not optimal. A more detailed analysis provides better bounds, but we refrain here from going into the details.

Theorem 3.2. Let Λ be a finite dimensional algebra. Then Λ is piecewise hereditary if and only if s.gl.dim Λ < ∞.

Proof: If Λ is piecewise hereditary, then s.gl.dim Λ < ∞ by 3.1. Conversely, assume that s.gl.dim Λ < ∞. In particular we have that gl.dim Λ < ∞. So we may identify D^{b}(Λ) with K^{b}(_{Λ}P). Assume to the contrary that Λ is not piecewise hereditary. Clearly we may assume that Λ is basic and connected and that Λ ≄ k. Then by [R2] and 2.3 there exists for each indecomposable X ∈ K^{b}(_{Λ}P) a strong path from X[1] to X in K^{b}(_{Λ}P). Thus for each n ≥ 1 there exists a strong path from X[n] to X in K^{b}(_{Λ}P). By 2.5 there exist t ≥ 0, an indecomposable object Q_{n,t}^{•} ∈ K^{b}(_{Λ}P) and a strong path X[n + t] → Q_{n,t}^{•} → X. Let P be an indecomposable projective Λ−module. Using the considerations above, we obtain for each n ≥ 1 an integer t ≥ 0 and an indecomposable object Q_{n,t}^{•} ∈ K^{b}(_{Λ}P) with Hom_{K^{b}(_{Λ}P)}(P [n + t], Q_{n,t}^{•}) ≠ 0 and Hom_{K^{b}(_{Λ}P)}(Q_{n,t}^{•}, P) ≠ 0. So Q_{n,t}^{−(n+t)} ≠ 0 ≠ Q_{n,t}^{0} by 1.1. Hence ℓ(Q_{n,t}^{•}) ≥ n + t. Thus we have constructed indecomposable complexes in K^{b}(_{Λ}P) of arbitrary length, a contradiction.

It may be an interesting problem to study the class of piecewise hereditary algebras Λ where gl.dim Λ = s.gl.dim Λ = d. If d = 1 this is clearly the class of finite dimensional hereditary algebras which are not semi-simple. The following deals with the case d = 2. For details on quasitilted algebras we refer to [HRS1].

Proposition 3.3. Let Λ be a finite dimensional algebra. Then the following are equivalent.

(i) gl.dim Λ = s.gl.dim Λ = 2.

(ii) Λ is a quasitilted algebra which is not hereditary.

Proof: If Λ is a quasitilted algebra which is not hereditary we clearly have that gl.dim Λ = 2. Moreover we have a triangle equivalence F : D^{b}(Λ) → D^{b}(H) for some hereditary, abelian category H such that F (P) ∈ H[0] for each indecomposable projective Λ−module P. The proof of 3.1 then shows that s.gl.dim Λ ≤ 2, hence gl.dim Λ = s.gl.dim Λ = 2.

Conversely, assume that gl.dim Λ = s.gl.dim Λ = 2. Then clearly Λ is not hereditary. Since gl.dim Λ = 2 it is enough to show that each indecomposable Λ−module X has either proj.dim_{Λ}X ≤ 1 or inj.dim_{Λ}X ≤ 1. Assume that there is an indecomposable Λ−module X with both proj.dim_{Λ}X = 2 and inj.dim_{Λ}X = 2. We claim that then there also exists an indecomposable Λ−module Y with proj.dim_{Λ}Y = 2 and Hom_{Λ}(Y,_{Λ}Λ) ≠ 0. By [R1] we know that inj.dim_{Λ}X = 2 implies that Hom_{Λ}(τ_{Λ}^{−}X,_{Λ}Λ) ≠ 0. Let

(∗) 0 → X → E → τ_{Λ}^{−}X → 0

be the Auslander-Reiten sequence starting at X. From above we have that Hom_{Λ}(τ_{Λ}^{−}X,_{Λ}Λ) ≠ 0. So there is an indecomposable direct summand E′ of E with Hom_{Λ}(E′,_{Λ}Λ) ≠ 0. Let α : X → E′ be the irreducible map given by (∗). If α is mono, then proj.dim_{Λ}E′ = 2, for gl.dim Λ = 2. If α is epi, then also Hom_{Λ}(X,_{Λ}Λ) ≠ 0. So there is an indecomposable Λ−module Y with proj.dim_{Λ}Y = 2 and Hom_{Λ}(Y,_{Λ}Λ) ≠ 0. Let P be an indecomposable projective Λ−module with Hom_{Λ}(Y, P) ≠ 0 and let 0 ≠ β ∈ Hom_{Λ}(Y, P). Let P^{•} be the minimal projective resolution of Y. Then β induces a nonzero element in Hom_{K^{b}(_{Λ}P)}(P^{•}, P). By 1.3 we have that the mapping cone C_{β} is indecomposable and clearly ℓ(C_{β}) = 3, a contradiction to s.gl.dim Λ = 2.

References

[ARS] M. Auslander, I. Reiten, S. Smalø, Representation theory of artin algebras, Cambridge University Press 1995.

[H1] D. Happel, Triangulated categories in the representation theory of finite dimensional algebras, London Math. Soc. Lecture Notes Series 119 (1988).

[H2] D. Happel, A characterization of hereditary categories with tilting object, Invent. Math. 144 (2001), no. 2, 381–398.

[HKR] D. Happel, B. Keller, I. Reiten, Bounded derived categories and repetitive algebras, preprint.

[HR] D. Happel, I. Reiten, Hereditary abelian categories with tilting object over arbitrary base fields, J. Algebra 256 (2002), no. 2, 414–432.

[HRS1] D. Happel, I. Reiten, S. Smalø, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 575 (1996).

[HRS2] D. Happel, I. Reiten, S. Smalø, Piecewise hereditary algebras, Arch. Math. 66 (1996), 182 - 186.

[KSYZ] O. Kerner, A. Skowronski, K. Yamagata, D. Zacharia, Finiteness of the strong global dimension of radical square zero algebras, CEJM 2(1) 2004, 103 - 111.

[R1] C. M. Ringel, Tame algebras and integral quadratic forms, Springer Lecture Notes in Mathematics 1099, Heidelberg 1984.

[R2] C. M. Ringel, Hereditary triangulated categories, to appear Comp. Math.

[S] A. Skowronski, On algebras with finite strong global dimension, Bull. Polish. Acad. Sci. 35 (1987), 539 - 547.

Fakultät für Mathematik, Technische Universität Chemnitz, 09107 Chemnitz, Germany. email: happel@mathematik.tu-chemnitz.de

Department of Mathematics, Syracuse University, Syracuse, NY 13244-0001, USA. email: zacharia@mailbox.syr.edu