BOUNDED DERIVED CATEGORIES AND REPETITIVE ALGEBRAS

DIETER HAPPEL, BERNHARD KELLER AND IDUN REITEN

Introduction

Let Λ be a finite dimensional algebra over a field k. It was proved in [H1] that there is a full and faithful embedding of the bounded derived category D^{b}(Λ) into the stable category \underline{mod}Λ̂ of finite dimensional modules over the repetitive algebra Λ̂. This embedding is an equivalence if and only if Λ has finite global dimension [H1]. The category D^{b}(Λ) is a triangulated category which does not have almost split triangles when Λ has infinite global dimension, whereas \underline{mod}Λ̂ is triangulated and always has almost split triangles.

The purpose of this paper is to investigate the relationship between D^{b}(Λ) and \underline{mod}Λ̂ from various points of view, which is of course meaningful only for algebras Λ of infinite global dimension. The most satisfactory results are obtained for Gorenstein algebras, especially for selfinjective algebras.

We investigate the embedding D^{b}(Λ) ⊂ \underline{mod}Λ̂ from the point of view of universal properties with respect to triangle functors to triangulated categories with almost split triangles, and also to which extent \underline{mod}Λ̂ is the smallest category containing D^{b}(Λ) with these properties. The first question has a positive answer for Gorenstein algebras, and is not true in general. The second question has a negative answer even for selfinjective algebras.

We also investigate the behavior of almost split triangles and irreducible maps under the embedding functor, and show that both are actually preserved. While it is known from [H2] what the end terms of almost split triangles in D^{b}(Λ) look like, and hence the left and right end terms of certain irreducible maps, we do not know in general so much about irreducible maps in D^{b}(Λ). However in the selfinjective case we show that there are no irreducible maps not associated with almost split triangles when (rad Λ)^{2} ≠ 0, and we describe them all when (rad Λ)^{2} = 0. We believe that also for arbitrary Λ there should be very few irreducible maps not associated with almost split triangles. As an application of our results we show that when Λ is selfinjective all the components of the AR-quiver of the category K^{b}(P) of bounded complexes of projective modules are of the form ℤA_{∞}.

The paper is organised as follows. In section 1 we give some background material from [H1] on the categories D^{b}(Λ) and mod Λ̂, including properties of almost split triangles. In section 2, we give an example showing that in general, the embedding D^{b}(Λ) ⊂ \underline{mod}Λ̂ is not universal among triangle functors from D^{b}(Λ) to triangulated categories with almost split triangles. We also show that the embedding mod Λ ⊂ \underline{mod}Λ̂ has a weak universal property with respect to triangle functors from D^{b}(Λ) to triangulated categories where the Nakayama functor becomes an equivalence. We deduce that if Λ is Gorenstein, there is a natural triangle functor from \underline{mod}Λ̂ to D^{b}(Λ). In section 3 we show that even when Λ is selfinjective, there is an infinite strictly descending chain of triangulated subcategories of \underline{mod}Λ̂ with almost split triangles and containing D^{b}(Λ). In section 4 we show that irreducible maps in D^{b}(Λ) stay irreducible in \underline{mod}Λ̂ , and give sufficient conditions for the existence of irreducible maps in D^{b}(Λ) of the form S[−1] → T where S and T are simple Λ-modules. In section 5 we show that almost split triangles in D^{b}(Λ) stay almost split in \underline{mod}Λ̂, and give the shape of the components of the AR-quiver of K^{b}(P) for selfinjective algebras. We also give necessary conditions for having irreducible maps not coming from almost split triangles for Gorenstein algebras, and deduce the result on irreducible maps in D^{b}(Λ) when Λ is selfinjective. In section 6 we deal with arbitrary finite dimensional algebras, and give some results supporting the suspicion that there are very few irreducible maps not associated with almost split triangles. We also show that the natural questions of a connection between irreducible maps between infinite complexes of projective modules and their finite parts have negative answers.

1. Preliminaries

In this section we will fix the notation and recall some of the results frequently used in the subsequent sections. For the proofs of the stated propositions we refer to [H1]. Let Λ be a finite dimensional algebra over a field k.

We denote by mod Λ the category of finitely generated left Λ-modules and by _{Λ}P (resp. _{Λ}I) the full subcategory of projective (resp. injective) Λ-modules. For a simple Λ-module S we denote by P (S) (resp. I(S) ) the projective cover (resp. injective envelope) of S. We denote by ν_{Λ} : _{Λ}P →_{Λ} I the Nakayama functor defined by ν_{Λ} = D Hom(−,_{Λ} Λ), where D is the duality with respect to k, and by ν_{Λ}^{−}: _{Λ}I →_{Λ} P the inverse Nakayama functor which is defined by ν_{Λ}^{−} = Hom(DΛ_{Λ}, −). We denote by D^{b}(Λ) the bounded derived category of mod Λ. The Nakayama functors ν_{Λ} and ν_{Λ}^{−} induce inverse equivalences of triangulated categories still denoted by ν_{Λ} : K^{b}(_{Λ}P) → K^{b}(_{Λ}I) and ν_{Λ}^{−}: K^{b}(_{Λ}I) → K^{b}(_{Λ}P). We denote by K^{−,b}(_{Λ}P) the homotopy category of complexes over _{Λ}P bounded above with bounded cohomology groups. Note that K^{−,b}(_{Λ}P) ≃ D^{b}(Λ). For a complex Z = (Z^{i}, d^{i}) in D^{b}(Λ) and n ∈ ℤ we always have a triangle Z_{≥n} → Z → Z_{<n} → Z_{≥n}[1] in D^{b}(Λ), where Z_{≥n}^{i} = Z^{i} for i ≥ n, Z_{<n} = Z^{i} for i < n, and zero otherwise, with the induced differentials.

To Λ we may associate the repetitive algebra Λ̂ and its category mod Λ̂ of finitely generated modules. The Λ̂-modules X are given by X = (X_{i}, f_{i}) where X_{i} ∈ mod Λ and X_{i} = 0 for almost all i, f_{i}: X_{i} → ν_{Λ}^{−}X_{i+1} such that f_{i}ν_{Λ}^{−}(f_{i+1}) = 0 for all i. A morphism of Λ̂-modules is defined in an obvious way. There is an automorphism νΛ̂ : mod Λ̂ → mod Λ̂ defined by (νΛ̂X)_{i} = X_{i+1}. The inverse is denoted by ν− ; Λ̂. The category mod Λ̂ is a Frobenius category in the sense of [H1]. The indecomposable projective-injective Λ̂-modules are given by I = P = (X_{i}, f_{i}) with X_{i} = P (S), X_{i+1} = I(S), f_{i} = id_{P (S)} and zero otherwise. Note that top P = νΛ̂ soc P. Clearly there are enough projectives. So for each X we obtain exact sequences 0 → X → I(X) → Ω− ; Λ̂X → 0 and 0 → ΩΛ̂X → P (X) → X → 0. We denote by \underline{mod}Λ̂ the stable category. This is a triangulated category where Ω− ; Λ̂ serves as a translation functor. If X ∈ \underline{mod}Λ̂, we may choose a representative again denoted by X ∈ mod Λ̂ without indecomposable projective direct summands. This fact will be used frequently later on.

There is a triangle functor μ: D^{b}(Λ) → \underline{mod}Λ̂ which is full and faithful such that μ extends the identity functor on mod Λ where mod Λ is embedded in D^{b}(Λ) (resp. \underline{mod}Λ̂) as complexes (resp. modules) concentrated in degree zero. It is known [H2] that μ is an equivalence if and only if gl. dim Λ < ∞.

In general we recall from [GK] the following description of Im μ. Z = (Z_{i}, g_{i}) ∈ Im μ if and only if there is some n ≥ 0 such that (Ω−n ; Λ̂Z)_{j} = 0 for j > 0 and (Ωn ; Λ̂Z)_{j} = 0 for j < 0. Also note that for a Λ̂-module Z = (Z_{i}, g_{i}) with Z_{i} = 0 for i < 0 also (Ωr ; Λ̂Z)_{j} = 0 for j < 0 and all r ≥ 0.

This has the following immediate consequence for Gorenstein algebras (see also [CZ]).

Corollary 1.1. Let Λ be a Gorenstein algebra. Then im μ = {Z = (Z_{i}, g_{i}) ∈ mod Λ̂ | pd_{Λ}Z_{i} < ∞ for i ≠ 0}

Proof. If Λ is a Gorenstein algebra then the modules of finite projective dimension coincide with the modules of finite injective dimension. Moreover this dimension is bounded by the projective dimension of DΛ_{Λ} which coincides with the injective dimension of _{Λ}Λ. Suppose that Z = (Z_{i}, g_{i}) ∈ mod Λ̂ satisfies pd_{Λ}Z_{i} < ∞ for i ≠ 0, then it follows immediately from the criterion mentioned above from [GK] that Z ∈ im μ. Conversely let Z = (Z_{i}, g_{i}) ∈ im μ and assume that Z = (Z_{i}, g_{i}) ∈ mod Λ̂ satisfies pd_{Λ}Z_{i} = ∞ for some i ≠ 0. We may assume that i > 0. Choose i maximal with this property. So pd_{Λ}Z_{j} < ∞ for j > i. By the first part of the proof and the fact that im μ is a triangulated category the factor module Z′ = (Z′_{j}, g_{j}) with Z′_{j} = Z_{j} for j ≤ i and Z′_{j} = 0 for j > i is contained in im μ. But then (Ω−n ; Λ̂Z′)_{i} ≠ 0 for all n ≥ 0, in contrast to [GK] □

Let C be a triangulated category which is Krull-Schmidt. Let Z be an object in C. We say that there is an almost split triangle ending at Z provided there is a triangle in C of the form

X ⟶^{u} Y ⟶^{v} Z ⟶^{w} X[1]

where (i) X is indecomposable, (ii) for all f: W → Z not split epi there is some g: W → Y with f = gv and (iii) w ≠ 0.

We refer to [H1] for equivalent formulations and the connection to irreducible maps.

In case there is an almost split triangle ending at Z, the starting term X is uniquely determined up to isomorphism. We then define τ_{C}Z = X.

It easily follows from the existence of almost split sequences in mod Λ̂ that \underline{mod}Λ̂ has almost split triangles. It is well known and can be shown using the definition of τ that for Z ∈ \underline{mod}Λ̂ indecomposable, then τΛ̂Z = νΛ̂Ω2 ; Λ̂X.

In the case of D^{b}(Λ) the following is known [H2]. Let Z ∈ D^{b}(Λ) be indecomposable. Then there is an almost split triangle X → Y → Z ⟶^{w} X[1] if and only if Z ∈ K^{b}(_{Λ}P). In this case τ_{D^{b}(Λ)}Z = ν_{Λ}Z[−1]. Thus D^{b}(Λ) has almost split triangles if and only if gl. dim Λ < ∞.

2. A counterexample and a weak universal property

Problem: Let Λ be an artin algebra and

μ : D^{b}(Λ) → \underline{mod} Λ̂

the embedding of [H1]. Let C be a triangulated category with Auslander-Reiten triangles and F : D^{b}(Λ) → C a triangle functor. Does there exist a triangle functor

G : \underline{mod} Λ̂ → C

such that F →^{∼} μG ?

The following example shows that the answer is no, in general.

Example: Let Λ be given as a factor algebra of a path algebra of a field k by an ideal:

[figure: T; β; S; α] , ⟨β^{2}, αβ⟩.

Let S, T be the two simple Λ-modules. Then

P (S) = ( \begin{matrix}S \\ T \end{matrix} ) and P (T) = ( \begin{matrix}T \\ T \end{matrix} )

are the indecomposable projective Λ-modules and I(S) = S and

I(T) = ( \begin{matrix}S T \\ T \end{matrix} )

the indecomposable injective Λ-modules. It is easy to see that Λ is not Gorenstein (I(T) is of infinite projective dimension). Let Γ = End_{Λ}(P (T)), so Γ = k[x]/(x^{2}) and let F = Hom(P (T), −) be a functor from mod Λ to mod Γ. Now F is exact, so F induces a functor D^{b}(Λ) → D^{b}(Γ). Since Γ is selfinjective, there is a functor [Ric]

π : D^{b}(Γ) → \underline{mod} Γ ,

so there is a triangle functor ϕ : D^{b}(Λ) → \underline{mod} Γ and \underline{mod} Γ has Auslander-Reiten triangles. We are now going to show that there is no triangle functor G : \underline{mod} Λ̂ → \underline{mod} Γ such that ϕ = μG.

Suppose there exists a triangle functor G : \underline{mod} Λ̂ → \underline{mod} Γ such that ϕ = μG. Let X = (X_{i}, f_{i}) be an object of \underline{mod} Λ̂ with X_{1} = S and X_{i} = 0 for i ≠ 1. Then Ω− ; Λ̂X = P (S), the stalk module concentrated in degree zero. So GΩ− ; Λ̂X = ϕ(P (S)) = T and

GΩ− ; Λ̂X ≅ Ω_{Γ}^{−}G(X) = GX ,

so GX ≅ T. Also G(S) = ϕ(S) = 0 and G(T) = ϕ(T) = T. But then also G(ΩΛ̂T) = T. Now ΩΛ̂T = (Y_{i}, f_{i}) where Y_{0} = T, Y_{1} = I(T) and f_{0} : T → P (T) the canonical map, and Y_{i} = 0 for i ≠ 0, 1. Consider the exact sequence in mod Λ:

0 → T → P (T) → T → 0.

It gives rise to a triangle

T [−1] ⟶^{f} T → P (T) → T (∗)

in D^{b}(Λ). Since ϕ(P (T)) = 0, the map ϕ(f) is invertible. Now we also have the exact sequence of mod Λ̂:

0 → Z → ΩΛ̂T → T → 0.

This gives rise to a triangle

Z → ΩΛ̂T ⟶^{μ(f)} T → Z[1] ,

which identifies with the image of the triangle (∗) under μ. Applying G then shows that GZ = 0 because μG(f) = ϕ(f) is invertible. Now Z = (Z_{i}, f_{i}), where Z_{1} = I(T) and Z_{i} = 0 for i ≠ 1. Let U = (U_{i}, f_{i}) with U_{1} = T and U_{i} = 0 for i ≠ 1. Then we obtain an exact sequence in mod Λ̂

0 → U → Z → U ⊕ X → 0 ,

which gives rise to a triangle

U → Z → U ⊕ X → U[1]

in \underline{mod} Λ̂ and so

GU → GZ → GU ⊕ GX → GU[1]

is a triangle in \underline{mod} Γ. Now GZ = 0 by the computation above and G(X) = T, so the triangle is of the form

GU → 0 → GU ⊕ T → GU[1] ,

a contradiction.

A weak universal property. As the above counterexample shows, the repetitive category is not the ‘universal triangulated category with Auslander-Reiten triangles containing the derived category’. However, we will see that if we take into account additional structure, we do get a weak universal property for the embedding

mod Λ → \underline{mod} Λ̂.

Roughly speaking this embedding is the ‘universal functor to a triangulated category where the Nakayama functor becomes an equivalence’. In the case where Λ is Gorenstein, we will use this property to construct a natural triangle functor from the stable category of the repetitive category to the bounded derived category.

Let us now construct the additional structure we need: For short, let us write M for mod Λ and R for mod Λ̂. We write Σ : M → M for the right exact extension of the Nakayama functor defined in section 1: Thus, we have ΣM = (DΛ) ⊗_{Λ} M for all M in M. We now define an exact functor R → R, which we will also denote by Σ. Namely, we put

ΣX = νΛ̂(ΩX) ,

where Ω is the syzygy functor R → R constructed as follows: If X is an object of R with structure maps f_{i}, i ∈ ℤ, we define the object P X to have the ith component

(Λ ⊗_{k} X_{i}) ⊕ (DΛ ⊗_{k} X_{i−1})

and the structure maps

[ \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ] : ν(P X)_{i} → (P X)_{i−1}.

Thus, the object P X is projective-injective. We define the canonical map P X → X to have the components

[can, g_{i−1}] : (P X)_{i} → X_{i}

where can is the canonical map from Λ⊗_{k}X_{i} to X_{i} and g_{i−1} is the map DΛ⊗_{k}X_{i−1} → X_{i} induced by ν(f_{i−1}). Thus, the map P X → X is a functorial projective right approximation of X. We define ΩX to be the kernel of P X → X.

The functor Σ : R → R is exact, preserves projective-injectives and induces an equivalence in the stable category (namely, the Serre functor). Moreover, if F_{0} : M → R denotes the canonical embedding, we have a morphism of functors

ϕ_{0} : F_{0}Σ → ΣF_{0}.

Namely, for an object M of M, the only non vanishing component of the morphism F_{0}Σ(M) → ΣF_{0}(M) is induced by the canonical map DΛ ⊗_{k} M → DΛ ⊗_{Λ} M. It is easy to check that if P is a projective Λ-module, then ϕ_{0}P becomes an isomorphism in the stable category of R. To summarize, we have

• a k-linear Frobenius category R endowed with an exact functor Σ : R → R preserving projective-injectives and inducing an equivalence in the stable category,

• an exact functor F_{0} : M → R endowed with a morphism

ϕ_{0} : F_{0}Σ → ΣF_{0}

such that ϕ_{0}P becomes an isomorphism in the stable category for each projective module P.

Theorem 2.1. Let E be a k-linear Frobenius category endowed with an exact functor Σ : E → E preserving projective-injectives and inducing an equivalence in the stable category. Let F : M → E be an exact functor endowed with a morphism ϕ : ΣF → F Σ such that ϕP becomes an isomorphism in the stable category for each projective module P. Then there is a triangle functor

G : \underline{R} → \underline{E} ,

such that G commutes with Σ up to isomorphism and the triangle

\xymatrix{M \ar[r]^{F_{0}} \ar[dr]_{F} & \underline{R} \ar[d]^{G} \\  & \underline{E}}

commutes up to isomorphism.

The theorem will be proved below. Note that it does not make any claim about uniqueness. In fact, one could obtain a more intrinsic formulation and a uniqueness statement by working in a more sophisticated framework based on towers of triangulated categories [K2], or derivators [G] or the homotopy category of dg categories [Ta] [To] [K1]. However, this would go beyond the scope of this article.

Corollary 2.2. Suppose that Λ is Gorenstein. Then there is a a triangle functor

G : \underline{mod} Λ̂ → D^{b}(Λ)

which commutes with the inclusion of mod Λ and such that we have a functorial isomorphism

Σ ∘ G →^{∼} G ∘ Σ ,

where Σ : \underline{mod} Λ̂ → \underline{mod} Λ̂ is the Serre functor and Σ : D^{b}(Λ) → D^{b}(Λ) the functor

M ↦ DΛ ⊗^{L}_{Λ} M.

Proof of the corollary. Let E be the category of right bounded complexes of projective Λ-modules with bounded homology. Then the stable category of E is triangle equivalent to the bounded derived category. For each Λ-bimodule B, write p(B) for a projective bimodule resolution of B. Let Σ : E → E be the (total) tensor product over Λ by the complex of bimodules p(DΛ). Let F_{0} be the functor taking a module M to p(Λ) ⊗_{Λ} M. To construct ϕ : F_{0}Σ → ΣF_{0}, it suffices to construct a quasi-isomorphism of bimodule complexes

ϕ̃: p(DΛ) ⊗_{Λ} p(Λ) → p(Λ) ⊗_{Λ} DΛ.

Indeed, since the morphism

p(DΛ) ⊗_{Λ} p(Λ) → DΛ

is a projective resolution, it lifts (in the homotopy category) along the quasi-isomorphism

p(Λ) ⊗_{Λ} DΛ → DΛ

and we define ϕ̃ to be a representative of a lift. If P is a projective module, then in the square (commutative in the homotopy category),

\xymatrix{p(DΛ) ⊗_{Λ} p(Λ) ⊗_{Λ} P \ar[r] \ar[d]_{ϕP} & DΛ ⊗_{Λ} P \ar[d]^{1} \\ p(Λ) ⊗_{Λ} DΛ ⊗_{Λ} P \ar[r] & DΛ ⊗_{Λ} P}

the two horizontal morphisms are quasi-isomorphisms and so the left vertical morphism is a homotopy equivalence. This means that ϕP becomes an isomorphism in the stable category. Thus, the hypotheses of the theorem are satisfied and we get, if Λ is Gorenstein, a natural triangle functor

G : \underline{mod} Λ̂ → D^{b}(Λ)

which extends the inclusion of mod Λ and commutes with Σ up to isomorphism of triangle functors.

Proof of the theorem. It is not hard to see that it suffices to define a functor with the required properties on the full subcategory of objects X of R with X_{i} = 0 for i > 0. Let X be such an object of R with structure maps f_{i} : ΣX_{i} → X_{i+1}, i ∈ ℤ. We define G_{1}X to be the complex over E with components Σ^{i}F (X_{−i}) and with the differential

Σ^{i}(F X_{−i}) → Σ^{i−1}(F X_{−i+1})

given by (Σ^{i−1}ϕX_{−i})(Σ^{i−1}F f_{i}). It is straightforward to check that the square of the differential vanishes and that with the natural definition of G_{1} on morphisms, we get a k-linear functor

G_{1} : R → C^{b}(E)

taking exact sequences of R to componentwise conflations of the category C^{b}(E) of bounded complexes over E. Moreover, the functor G_{1} takes an indecomposable projective injective object given by a projective P (put in degree −1, for simplicity of notation) and the identity ΣP → ΣP to a complex of the shape

\xymatrix{… \ar[r] & 0 \ar[r] & Σ(F P) \ar[r]^{ϕP} & F (ΣP) \ar[r] & 0 \ar[r] & …}

Now since E is a Frobenius category, we have a canonical triangle functor [KV] [Ric]

D^{b}(E) → \underline{E}

extending the natural projection functor E → \underline{E}. We define G_{2} to be the composition

C^{b}(E) → D^{b}(E) → \underline{E}

and we put G_{3} = G_{1} ∘ G_{2} : R → \underline{E}. Then G_{3} takes projective-injectives to zeroobjects: Indeed, a complex of the form

\xymatrix{… \ar[r] & 0 \ar[r] & Σ(F P) \ar[r]^{ϕP} & F (ΣP) \ar[r] & 0 \ar[r] & …}

is the cone over the morphism ϕP (between complexes concentrated in degree 0). Since ϕP becomes invertible in \underline{E} by assumption, the image of the cone under G_{2} is a zero object. Thus, G_{3} induces a k-linear functor G. It is clear from the construction that F_{0}G is isomorphic to F. Since G_{1} takes conflations to componentwise conflations and the projection C^{b}(E) → D^{b}(E) transforms each componentwise conflation into a canonical triangle, the functor G is in fact a triangle functor. Therefore, to construct a commutation isomorphism ΣG → GΣ it suffices to construct such a commutation isomorphism for Ω^{−1} ∘ Σ. Now in R, the composition Ω^{−1} ∘ Σ is isomorphic to the degree shifting functor νΛ̂. For an object X, the image G_{1}(νΛ̂X) is isomorphic to Σ(G_{1}X)[−1], where we denote by Σ the functor from C^{b}(E) to itself obtained by applying Σ : E → E to each component. Now the canonical triangle functor

D^{b}(E) → \underline{E}

is functorial with respect to exact functors preserving projective-injectives and thus canonically commutes with Σ. Moreover, since it is a triangle functor, it is compatible with shifts. So we get a canonical isomorphism

Ω^{−1}ΣG(X) →^{∼} G_{2}(Σ(G_{1}X)[−1]) →^{∼} G_{1}G_{2}ΣΩ^{−1}(X).

Note that for Gorenstein algebras we now have a positive answer to the question posed in the beginning of the section.

3. Infinite chain of subcategories

In this section we construct triangulated subcategories of \underline{mod}Λ̂ containing D^{b}(Λ) for Λ a selfinjective algebra. Recall from Section 1 that in this case D^{b}(Λ) can be identified with the full subcategory of \underline{mod}Λ̂ with objects X = (X_{i}, f_{i}) such that X_{i} is a projective Λ-module for i ≠ 0.

Lemma 3.1. Let Λ be a selfinjective algebra. Let I ⊆ ℤ and let C_{I} ⊂ {X = (X_{i}, f_{i}) ∈ \underline{mod}Λ̂ | X_{i} is a projective Λ-module if i ∉ I\}. Then the following hold

i) \underline{C}_{I} ⊆ \underline{mod}Λ̂ is a triangulated subcategory.

ii) 0 ∈ I if and only if D^{b}(Λ) ⊆ \underline{C}_{I}.

iii) If I, I′ ⊆ ℤ with I ⊆ I′, then \underline{C_{I}} ⊆ \underline{C}_{I′}.

Proof. If P is a projective-injective Λ̂ -module, then P ∈ C_{I}, since Λ is selfinjective. So \underline{C}_{I} ⊆ \underline{mod}Λ̂.

Let X ∈ \underline{C}_{I} and consider an exact sequence 0 → X → I(X) → Ω− ; Λ̂X → 0 with I(X) injective in mod Λ̂. Then for each i ∈ ℤ we have an exact sequence 0 → X_{i} → I(X_{i}) ⊕ ν^{−}I(X_{i+1}) → Z_{i} → 0 where I(X_{i}) is the Λ-injective envelope of X_{i}. If i ∉ I, the sequence splits, since X_{i} is projective, hence Ω− ; Λ̂X ∈ C_{I}. Thus \underline{C}_{I} is closed under the translation functor in \underline{mod}Λ̂. Finally, let X → Y → Z → C[1] be a triangle in \underline{mod}Λ̂ with X, Y ∈ \underline{C_{I}}. Then the triangle gives an exact sequence 0 → X → I(X) ⊕ Y → Z → 0 in \underline{mod}Λ̂. So for each j ∈ ℤ we obtain an exact sequence 0 → X_{i} → I(X)_{i} ⊕ Y_{i} → Z_{i} → 0 in mod Λ. If i ∉ I then X_{i}, Y_{i} are projective, so Z_{i} is projective, hence Z ∈ \underline{C}_{I}, so \underline{C}_{I} is a triangulated subcategory of \underline{mod}Λ̂.

(ii) and (iii) are obvious. □

Example 3.2. Let n ∈ ℕ and let I = (n + 1)ℤ. Let C_{n} = C_{I}. Then νn+1 ; Λ̂ is an automorphism on C_{n}. In fact, if X = (X_{i}, f_{i}) ∈ C_{n}, then (νn+1 ; Λ̂X)_{i} = X_{i+n+1}. Since j ∉ (n + 1)ℤ if and only if j + n + 1 ∉ (n + 1)ℤ we see that νn+1 ; Λ̂X ∈ C_{n}.

If we choose n + 1 = 2^{k} and let D_{k} = C_{n} we obtain a descending chain of subcategories ⋯ ⊆ \underline{D}_{2} ⊆ \underline{D}_{1} ⊆ \underline{mod}Λ̂ and clearly D^{b}(Λ) = ⋂_{i≥1} \underline{D}_{i}.

Let Λ be a symmetric algebra and let F = νΛ̂ΩΛ̂ be the Serre functor on \underline{mod}Λ̂. So for all X, Y ∈ mod Λ̂ we have η_{X,Y} : \underline{Hom}(X, Y) ⟶^{∼} D\underline{Hom}(Y, F X) natural in X and Y. We will show that F^{n+1} is a Serre functor on \underline{C}_{n}. For this we will construct η_{X}: F^{n+1}X → F X such that η_{X} is natural in X and for all X, Y ∈ \underline{C}_{n} we have that \underline{Hom}(Y, F^{n+1}X) ⟶^{∼} \underline{Hom}(Y, F X). This then implies that F^{n+1} is a Serre functor on \underline{C}_{n}, hence \underline{C}_{n} has Auslander-Reiten triangles.(Compare [RV])

Lemma 3.3. For all X ∈ mod Λ̂ there is an exact sequence 0 → K_{X} ⟶^{μ_{X}} F (X) ⟶^{π_{X}} X → 0 which is natural in X.

Proof. Since Λ is finite-dimensional, there is a functor P: mod Λ →_{Λ} P and an exact sequence 0 → Ω_{Λ}X ⟶^{α_{X}} P (X) ⟶^{β_{X}} X → 0 natural in X for X ∈ mod Λ (compare section 2). Now let X = (X_{i}, f_{i}) ∈ mod Λ̂. Applying ν− ; Λ̂ if necessary we may assume that X_{i} = 0 for i < 0. Now P extends to a functor P̃: mod Λ̂ →Λ̂ P and we have an exact sequence 0 → ΩΛ̂X → P̃(X) → X → 0. Explicitly we have for i ≥ 0 a commutative diagram of the form

\xymatrix{P (X_{i−1}) ⊕ Ω_{Λ}X_{i} \ar[r]^{^{( \begin{matrix}P (f_{i−1}) & 0 \\ α_{X_{i}} & Ω_{Λ}(f_{i}) \end{matrix} )}} \ar[d]_{_{( \begin{matrix}1 & 0 \\ 0 & α_{X_{i}} \end{matrix} )}} & P (X_{i}) ⊕ Ω_{Λ}X_{i+1} \ar[d]^{_{( \begin{matrix}1 & 0 \\ 0 & α_{X_{i+1}} \end{matrix} )}} \\ P (X_{i−1}) ⊕ P (X_{i}) \ar[r]^{_{( \begin{matrix}−P (f_{i−1}) & 0 \\ 1 & P (f_{i}) \end{matrix} )}} \ar[d]_{_{( \begin{matrix}o \\ β_{X_{i}} \end{matrix} )}} & P (X_{i}) ⊕ P (X_{i+1}) \ar[d]^{_{( \begin{matrix}0 \\ β_{X_{i+1}} \end{matrix} )}} \\ X_{i} \ar[r]^{f_{i}} & X_{i+1}}

The map π_{X}: F X → X is now defined by (π_{X})_{i}: P (X_{i}) ⊕ ΩX_{i+1} \underrightarrow{( \begin{matrix}β_{X_{i}} \\ 0 \end{matrix} )} X_{i}. Clearly π_{X} is surjective and K_{X} is described by the following commutative diagram

\xymatrix{Ω(X_{i}) ⊕ Ω_{Λ}(X_{i+1}) \ar[r]^{^{( \begin{matrix}Ω(f_{i}) & 0 \\ 1 & Ω_{Λ}(f_{i+1}) \end{matrix} )}} \ar[d]_{_{( \begin{matrix}α_{X_{i}} & 0 \\ 0 & 1 \end{matrix} )}} & Ω(X_{i+1}) ⊕ Ω(X_{i+2}) \ar[d]^{_{( \begin{matrix}α_{X_{i+1}} & 0 \\ 0 & 1 \end{matrix} )}} \\ P (X_{i}) ⊕ Ω(X_{i+1}) \ar[r]^{_{( \begin{matrix}P (f_{i}) & 0 \\ α_{X_{i}} & Ω_{Λ}(f_{i+1}) \end{matrix} )}} \ar[d]_{_{( \begin{matrix}β_{X_{i}} \\ 0 \end{matrix} )}} & P (X_{i+1}) ⊕ Ω(X_{i+2}) \ar[d]^{_{( \begin{matrix}β_{X_{i+1}} \\ 0 \end{matrix} )}} \\ X_{i} \ar[r]^{f_{i}} & X_{i+1}}

Since P, P̃ are functors, the exact sequence 0 → K_{X} ⟶^{μ_{X}} F (X) ⟶^{π_{X}} X → 0 is natural in X. □

So for each 1 ≤ i ≤ n we obtain an exact sequence

(∗)_{i} 0 → F^{i}K_{X} → F^{i+1}X → F^{i}X → 0

Now (∗)_{i} induces η_{X}: F^{n+1}X → F X natural in X. Thus for all X, Y ∈ mod Λ̂ we have η_{X,Y} : Hom(Y, F^{n+1}X) → Hom(Y, F X) natural in X and Y.

For the following lemma we need some notation. Let X ∈ mod Λ and denote by δ^{i}(X) = (Z_{j}, γ_{j}) the Λ̂-module with Z_{i−1} = Z_{i} = X, Z_{j} = 0 for j ≠ i−1, i, γ_{i−1} = 1_{X} and γ_{j} = 0 for j ≠ i − 1.

Lemma 3.4. If X ∈ C_{n}, then K_{X} ≃ ⨁_{i∈(n+1)ℤ} δ^{i}(ΩX_{i}) in \underline{mod}Λ̂

Proof. If X = (X_{i}, f_{i}) ∈ C_{n}, then by definition X_{i} is projective for i ∉ (n + 1)ℤ. Moreover it follows that δ^{i}(ΩX_{i}) is projective as a Λ̂-module if X_{i} is projective. It follows from the previous lemma that for each i we have that δ^{i}(ΩX_{i}) is a submodule of K_{X}. Explicitly consider the following commutative diagram

\xymatrix{ΩX_{i+1} \ar[r]^{1} \ar[d]^{( 0 1 )} & ΩX_{i+1} \ar[d]^{( 1 Ωf_{i+1} )} \\ ΩX_{i} ⊕ ΩX_{i+1} \ar[r]^{_{( \begin{matrix}Ωf_{i} & 0 \\ 1 & Ωf_{i+1} \end{matrix} )}} & ΩX_{i+1} ⊕ ΩX_{i+2}}

So if i ∉ (n+1)ℤ we see that δ^{i}(ΩX_{i}) is a direct summand of K_{X}. But then it follows from the description of K_{X} in the previous lemma that K_{X} ≃ ⨁_{i∈(n+1)ℤ} δ^{i}(ΩX_{i}) in

\underline{mod}Λ̂  □

Lemma 3.5. Let X ∈ mod Λ and i ∈ ℤ. Then i)F δ^{i}(X) ≃ δ^{i−1}(ΩX) in \underline{mod}Λ̂. ii)If Y ∈ C_{n}, then \underline{Hom}(Y, δ^{i}(X)) = 0, if i ∉ (n + 1)ℤ

Proof. i) Clearly, if X, Y ∈ mod Λ then ΩΛ̂δ^{i}(X) = δ^{i}(Ω_{Λ}X) in \underline{mod}Λ̂ and νΛ̂δ^{i}(Y) = δ^{i−1}(Y), so the assertion follows.

ii)Let Y = (Y_{i}, g_{i}) ∈ C_{n} and let φ ∈ Hom(Y, δ^{i}(X)). So we have the following commutative diagram with φ = (φ_{j})

\xymatrix{⋯ & Y_{i−2} \ar[r]^{g_{i−2}} \ar[d] & Y_{i−1} \ar[r]^{g_{i−1}} \ar[d]_{φ_{i−1}} & Y_{i} \ar[r]^{g_{i}} \ar[d]^{φ_{i}} & Y_{i+1} \\ ⋯ & 0 \ar[r] & X \ar[r]^{1_{X}} & X \ar[r] & 0}

Consider δ^{i}(P (X)) ⟶^{π} δ^{i}(X), then δ^{i}(P (X)) is a projective Λ̂-module. Since i ∉ (n + 1)ℤ, we have that Y_{i} is a projective Λ-module, so there is some α_{i}: Y_{i} → P (X) such that α_{i}π_{i} = φ_{i}, where π = (π_{j}) and π_{j} = 0 for j ≠ i − 1, i and π_{i−1} = π = β_{X}.

Set α_{i−1} = g_{i−1}α_{i} and α_{j} = 0 forj ≠ i−1, i. Then α = (α_{j}) is a map Y → δ^{i}(P (X)) such that απ = φ, hence \underline{Hom}(Y, δ^{i}(X)) = 0 □

Proposition 3.6. For all X, Y ∈ C_{n}, the natural transformation η_{X,Y} : \underline{Hom}(Y, F^{n+1}X) → \underline{Hom}(Y, F X) is an isomorphism

Proof. By the previous considerations we have for each X ∈ C_{n} and each 1 ≤ i ≤ n an exact sequence 0 → F^{i}K_{X} → F^{i+1}X → F^{i}X → 0 in mod Λ̂ which gives rise to a triangle F^{i}K_{X} → F^{i+1}X → F^{i}X → F^{i}K_{X}[1] in \underline{mod}Λ̂. Applying \underline{Hom}(Y, −) to this triangle for Y ∈ C_{n} gives an exact sequence

\underline{Hom}(Y, F^{i}K_{X}) → \underline{Hom}(Y, F^{i+1}X) → \underline{Hom}(Y, F^{i}X) → \underline{Hom}(Y, F^{i}K_{X}[1])

By the description of K_{X} and the previous lemma we see that \underline{Hom}(Y, F^{i}K_{X}) = 0 = \underline{Hom}(Y, F^{i}K_{X}[1]), hence \underline{Hom}(Y, F^{i+1}K_{X}) ⟶^{∼} \underline{Hom}(Y, F^{i}X).

For each 1 ≤ i ≤ n we also have an exact sequence 0 → K^{i} → F^{i+1}X → F X → 0 in mod Λ̂ which gives rise to a triangle K^{i} → F^{i+1}X → F X → K^{i}[1] in \underline{mod}Λ̂ where F^{i+1}X → F X is obtained from the composition F^{i+1}X → F^{i}X → F X. By the octahedral axiom we have a triangle F^{i}K_{X} → K^{i} → K^{i−1} → F^{i}K_{X}[1] . By induction and the previous considerations we have that \underline{Hom}(Y, K^{i}) = 0 = \underline{Hom}(Y, K^{i}[1]), hence \underline{Hom}(Y, F^{i+1}X) ≃ \underline{Hom}(Y, F X), thus η_{X,Y} is an isomorphism for all X, Y ∈ \underline{C}_{n}. □

As pointed out above this implies

Corollary 3.7. For each n ∈ ℕ the category \underline{C}_{n} has almost split triangles

4. Irreducible maps in D^{b}(Λ)

In this section we study the behavior of the embedding μ: D^{b}(Λ) → \underline{mod}Λ̂ under irreducible maps. If X = (X^{i}, f^{i}) ∈ D^{b}(Λ) we write μ(X) = (X_{i}, f_{i}). If X = (X^{i}, f^{i}) satisfies X^{i} = 0 for i < 0, then X_{i} = 0 for i < 0. Here of course, we assume that μ(X) has no projective-injective indecomposable summands. If X^{i} = 0 for i > 0 then X_{i} = 0 for i > 0.

We denote by mod^{≥0} Λ̂ = {(X_{i}, f_{i})|X_{i} = 0, i < 0} and mod^{>0} Λ̂ = {(X_{i}, f_{i})|X_{i} = 0, i ≤ 0}. The categories mod^{≤0} Λ̂ and mod^{<0} Λ̂ are defined analogously. Clearly mod^{≥0} Λ̂ and mod^{>0} Λ̂ are stable under ΩΛ̂. Moreover we clearly have Hom(X, Y) = 0 for X ∈ mod^{≥0} Λ̂ and Y ∈ mod^{<0} Λ̂. This yields the following easy lemma.

Lemma 4.1. Let X ∈ mod^{≥0} Λ̂ and Y ∈ mod^{<0} Λ̂. Then Ext1 ; Λ̂(X, Y) = 0.

Proof. We have Ext1 ; Λ̂(X, Y) ≃ \underline{Hom}(X, Ω− ; Λ̂Y) = 0, since Ω− ; Λ̂Y ∈ mod^{<0} Λ̂. □

Lemma 4.2. Let Z = (Z_{i}, f_{i}) ∈ mod^{≥0} Λ̂ with Z ∈ Im μ. Consider the exact sequence 0 → Z^{>0} → Z → Z_{0} → 0. Then νΛ̂Z^{>0} ∈ Im μ.

Proof. We verify the condition mentioned in section 1 from [GK]. Since νΛ̂Z^{>0} ∈ mod^{≥0} Λ̂ we have that Ωr ; Λ̂(νΛ̂Z^{>0}) ∈ mod^{≥0} Λ̂ for all r ≥ 0. The exact sequence 0 → Z^{>0} → Z → Z_{0} → 0 gives a triangle ΩΛ̂Z_{0} → Z^{>0} → Z → Z_{0} in \underline{mod}Λ̂. So for each n ≥ 0 we obtain a triangle

Ω−n ; Λ̂Z_{0} → Ω−n−1 ; Λ̂Z^{>0} → Ω−n−1 ; Λ̂Z → Ω−n−1 ; Λ̂Z_{0}

For each n ≥ 0 we clearly have Ω−n ; Λ̂Z_{0} ∈ mod^{≤0} Λ̂. Since Z ∈ Im μ there is n_{0} such that Ω−n ; Λ̂Z ∈ mod^{≤0} Λ̂ for all n ≥ n_{0} by [GK]. Let n ≥ n_{0} and assume that Ω−n−1 ; Λ̂Z^{>0} ∉ mod^{≤0} Λ̂. Then there is X ∈ mod^{>0} Λ̂ such that \underline{Hom}(X, Ω−n−1 ; Λ̂Z^{>0}) ≠ 0. Since \underline{Hom}(X, Ω−n ; Λ̂Z_{0}) = 0 = \underline{Hom}(X, Ω−n−1 ; Λ̂Z) we obtain a contradiction. □

Theorem 4.3. Let X, Y be indecomposable in D^{b}(Λ) and let f: X → Y be irreducible. Then μ(f): μX → μY is irreducible in \underline{mod}Λ̂.

Proof. We consider the almost split triangle (*) τΛ̂μY ⟶^{α} E ⟶^{β} μY ⟶^{γ} τΛ̂μY [1] in \underline{mod}Λ̂. Now μ(f) is not split epi, since f is irreducible, hence we get g: μX → E such that μ(f) = gβ. Let μX = (X_{i}, f_{i}) and μY = (Y_{i}, g_{i}). We may assume that μX, μY ∈ mod^{≥0} Λ̂. Now τΛ̂μY = νΛ̂Ω2 ; Λ̂μY = (Z_{i}, h_{i}) satisfies Z_{i} = 0 for i < −1, Z_{−1} = Ω_{Λ}^{2}Y_{0}. Thus E = (E_{i}, u_{i}) satisfies E_{i} = 0 for i < −1, E_{−1} = Ω_{Λ}^{2}Y_{0} and u_{−1} : Ω_{Λ}^{2}Y_{0} → ν^{−}E_{0}. Let P ⟶^{π} Ω_{Λ}^{2}Y_{0} be epi with P a projective Λ-module. Let Ẽ = (Ẽ_{i}, v_{i}) ∈ mod Λ̂ defined by Ẽ_{i} = 0 for i < −1, Ẽ_{−1} = P, Ẽ_{i} = E_{i} for i ≥ 0, v_{−1} = πu_{−1}, v_{i} = u_{i} for i ≥ 0.

Now Ω2 ; Λ̂μY ∈ Im μ, since Im μ is a triangulated subcategory. By Lemma 3.2 we have that νΛ̂E^{>0} ∈ Im μ. Since νΛ̂P ∈ Im μ we see that Ẽ ∈ Im μ. The construction of Ẽ clearly yields a triangle

νΛ̂μ(Ker π) ⟶^{δ} Ẽ ⟶^{ϵ} E ⟶^{η} νΛ̂μ(Ker π)[1]

The factorization μ(f) = gβ induces another factorization as follows:

[figure: μX; g̃; μX; g; νΛ̂μ(Ker π); δ; Ẽ; ϵ; β̃; E; η; β; νΛ̂μ(Ker π)[1]; μY; μY]

where β̃ = ϵβ. Since gη = 0 by Lemma 3.1 we obtain g̃ with g̃ϵ = g. Now g̃β̃ = g̃ϵβ = gβ = μ(f).

Since Ẽ ∈ Im μ and f is irreducible, we get that g̃ is a split mono or β̃ is a split epi. If β̃ is split epi, there is β̃_{1} : μY → Ẽ such that β̃_{1}β = 1_{μY}. Set β_{1} = β̃_{1}c. Then β_{1}β = β̃_{1}cβ = β̃_{1}β̃ = 1_{μY}, so β is a split epi, in contrast to (*) being an almost split triangle. So g̃ is split mono, hence there is some g̃_{1} : Ẽ → μX such that g̃g̃_{1} = 1_{μX}. Since Hom(ν_{Λ} Ker π, μX) = 0 , we have δg̃_{1} = 0, so there is some g_{1} : E → μX such that ϵg_{1} = g̃_{1}. Now gg_{1} = g̃ϵg_{1} = g̃g̃_{1} = 1_{μX} shows that g is split mono, hence X is an indecomposable direct summand of E and μ(f) is a component of β, hence μ(f) is irreducible. □

Next we show how certain irreducible maps in D^{b}(Λ) arise quite naturally from extensions of simple Λ-modules. This will be of interest in section 5.

Proposition 4.4. Let S and T be simple Λ-modules with Ext_{Λ}^{1}(S, T) ≠ 0. If rad P (S) and I(T)/T are both semisimple, then there is an irreducible map f: S[−1] → T in D^{b}(Λ).

Proof. We will show that there is an irreducible map φ: μ(S[−1]) → μ(T) in \underline{mod}Λ̂. For simplicity set μ(S) = S and μ(T) = T. Then S[−1] = ΩΛ̂S ≃ radΛ̂ P (S), where P (S) is the Λ̂-projective cover of S. We consider the almost split sequence in mod Λ̂ starting in S[−1] = radΛ̂ P (S). It is well-known [AR] that this is of the form

0 → radΛ̂ P (S) →Λ̂ P (S) ⊕ radΛ̂ P (S)/ socΛ̂ P (S) → P (S)/ socΛ̂ P (S) → 0

Clearly, socΛ̂ P (S) = ν− ; Λ̂S. Let 0 → S ⟶^{α} I(S) ⟶^{β} I(S)/S → 0 be exact in mod Λ, with I(S) the Λ-injective envelope of S. Applying the Nakayama functor ν_{Λ}^{−} = Hom(DΛ_{Λ}, −) yields an exact sequence

0 → Hom(DΛ_{Λ}, S) ⟶^{ν_{Λ}^{−}(α)} P (S) = Hom(DΛ_{Λ}, I(S)) ⟶^{ν_{Λ}^{−}(β)} Hom(DΛ_{Λ}, I(S)/S)

Let g be the composition of rad P (S) ⟶^{γ} P (S) ⟶^{ν_{Λ}^{−}(β)} ν_{Λ}^{−}I(S)/S, so g = γν_{Λ}^{−}β. Then radΛ̂ P (S)/ socΛ̂ P (S) = (Z_{i}, g_{i}) = Z with Z_{0} = rad_{Λ} P (S), Z_{1} = I(S)/S, g_{0} = g, and zero otherwise. By assumption, rad P (S) is semisimple and Ext_{Λ}^{1}(S, T) ≠ 0, so T is an indecomposable direct summand of rad P (S). Let 0 ≠ δ: I(T) → I(S) be a map which is not an isomorphism. Since I(T)/T is semisimple, δ factors over α, hence g(T) = 0, or equivalently T is an indecomposable direct summand of Z. Hence there is an irreducible map f: ΩΛ̂S → T in \underline{mod}Λ̂, so there is an irreducible map f: S[−1] → T in D^{b}(Λ) □

5. Components

We consider the embedding μ : D^{b}(Λ) → \underline{mod}Λ̂. The category \underline{mod}Λ̂ has almost split triangles, where for an indecomposable X ∈ \underline{mod}Λ̂, the translate τΛ̂X = νΛ̂[−2]X. In general, D^{b}(Λ) will not have almost split triangles. However it was shown in [H2] that for each P ∈ K^{b}(_{Λ}P) indecomposable there is an almost split triangle in D^{b}(Λ) of the form νP [−1] →^{u} E →^{v} P →^{w} νP where ν : K^{b}(_{Λ}P) → K^{b}(_{Λ}I) is the Nakayama functor. We will show first that this triangle is sent under μ to the almost split triangle in \underline{mod}Λ̂ ending at μ(P) and then apply this to determine the structure of the components of the Auslander-Reiten quiver of K^{b}(_{Λ}P) in case Λ is a Gorenstein algebra.

Lemma 5.1. If P ∈ K^{b}(_{Λ}P), then τΛ̂μ(P) ∈ Im μ.

Proof. Let P = (P^{i}, d^{i}) ∈ K^{b}(_{Λ}P). Since μ commutes with the translation functors we may assume that P^{i} = 0 for i > 0 and it is enough

to show that νΛ̂μ(P) ∈ Im μ. Since P ∈ K^{b}(_{Λ}P) there is m_{0} such that P^{m} = 0 for m < m_{0}. We proceed by induction on m_{0}. If m_{0} = 0, then P is a stalk complex concentrated in degree 0, so μ(P) is the stalk module P^{0} concentrated in degree zero. But νΛ̂P^{0} ≅ νP^{0}[1] shows that νΛ̂P^{0} ∈ Im μ. If m_{0} < 0, let P′ = (P^{′i}, d^{′i}) with P^{′i} = P^{i} for i < 0 and P^{′0} = 0, d^{′i} = d^{i} for i < −1 and d^{′i} = 0 for i ≥ −1 be the truncated complex.

We clearly have a map of complexes P′[−1] →^{u} P^{0} whose mapping cone is P. So we obtain a triangle P′[−1] → P^{0} → P → P′ in K^{b}(_{Λ}P). This yields a triangle νΛ̂μP′[−1] → νΛ̂P^{0} → νΛ̂μP → νΛ̂μP′ in \underline{mod}Λ̂.

By induction the first two terms belong to Im μ, hence so does the third, since μ is a triangle functor. □

Proposition 5.2. Let P ∈ K^{b}(_{Λ}P) and let νP [−1] →^{u} E →^{v} P →^{w} νP be the almost

split triangle in D^{b}(Λ) ending at P. Then μνP [−1] →^{μ(u)} μE →^{μ(v)} μP →^{μ(w)} μνP is the almost split triangle in \underline{mod}Λ̂ ending at μP.

Proof. Let

τΛ̂μP →^{u} F →^{v} μP →^{w} τΛ̂μP [1]  (∗)

be the almost split triangle in \underline{mod}Λ̂ ending at μP. By Lemma 5.1 there is X ∈ D^{b}(Λ) such that μX = τΛ̂μP [1]. So there is some w′ : P → X such that μw′ = w. Let

X[−1] →^{u′} E →^{v′} P →^{w′} X  (∗∗)

be a triangle in D^{b}(Λ). By construction μX[−1] → μE → μP →^{w} μX is isomorphic to (∗). Since μ is an embedding and (∗) is an almost split triangle, we infer that (∗∗) is the almost split triangle in D^{b}(Λ) ending at P. □

In the following let Λ be a Gorenstein algebra. Then the Nakayama functor ν : K^{b}(_{Λ}P) → K^{b}(_{Λ}I) is an endofunctor, hence K^{b}(_{Λ}P) has almost split triangles, which are almost split triangles in D^{b}(Λ), and therefore by Proposition 5.2 also almost split triangles in \underline{mod}Λ̂. Hence we get

Corollary 5.3. Let C be a connected component of the Auslander-Reiten quiver of K^{b}(_{Λ}P). Then C is a connected component of the Auslander-Reiten quiver of \underline{mod}Λ̂.

We will now investigate the shape of the components of the Auslander-Reiten quiver of K^{b}(_{Λ}P) for Λ a selfinjective algebra.

Theorem 5.4. Let Λ be a connected selfinjective algebra, which is not semisimple. Let C be a connected component of the Auslander-Reiten quiver of K^{b}(_{Λ}P). Then C is of the form ℤA_{∞}.

Proof. Let C be a connected component of the Auslander-Reiten quiver of K^{b}(_{Λ}P). By Corollary 5.3, C is a connected component of the Auslander-Reiten quiver of \underline{mod}Λ̂. In fact C is a connected component of the Auslander-Reiten quiver of modΛ̂, since otherwise there would exist an indecomposable projective Λ̂-module P such that rad P ∈ C, in particular rad P ∈ Im μ_{|K^{b}(_{Λ}P)}.

But it follows from the description of Im μ in Section 3 that Im μ_{|K^{b}(_{Λ}P)} = {(X_{i}, f_{i}) | X_{i} is a projective Λ-module}, so rad P ∉ Im μ_{|K^{b}(_{Λ}P)}, since Λ is not semisimple. Consider l : C → ℕ defined by l(X) = |X|, the length of X as a Λ̂-module. Then l is an additive function on C, since C is a component of the Auslander-Reiten quiver of modΛ̂: If P ∈ K^{b}(_{Λ}P) and μ(P) = (X_{i}, f_{i}) we have that X_{i} = 0 for |i| > m and some m and X_{i} is projective for |i| ≤ m. Since Λ is selfinjective there exists n ∈ ℕ such that ν_{Λ}^{n}P ≅ P for each projective Λ-module P. If μ(P) = (X_{i}, f_{i}), so that l(μP) = ∑_{i} |X_{i}|, then l(ΩΛ̂μP) = ∑_{i} |ν_{Λ}X_{i}|.

So l(μP) = l(Ωn ; Λ̂μP), hence l(τ n ; Λ̂μP) = l(μP), showing that l is a τΛ̂-periodic additive function. Let X ∈ C and let 0 ≠ f : P → X with P an indecomposable projective Λ̂-module. Since C does not contain any projective Λ̂-modules we obtain for each i a chain of irreducible maps X_{i} →^{f_{i}} X_{i−1} → ⋯ → X_{1} →^{f_{1}} X_{0} = X such that f_{i} … f_{1} ≠ 0 and X_{i} ∈ C. By Harada-Sai (see [ARS]) we know that the length of the indecomposable modules in C is unbounded; so l is unbounded on C. In particular C contains infinitely many τΛ̂-orbits. By [F] the tree class of C is A_{∞}. Trivially C does not contain any τΛ̂-periodic vertices. So C ≅ ℤA_{∞}. □

If Λ is a Gorenstein algebra, then in Section 2 we constructed a functor G : \underline{mod}Λ̂ → D^{b}(Λ) such that Gμ ≅ 1_{D}^{b}(Λ). Let ν̃ : D^{b}(Λ) → D^{b}(Λ) be the equivalence induced by ν_{Λ} = D Hom(−,_{Λ}Λ), then we obtain a commutative diagram

\underline{mod}Λ̂ ⟶^{G} D^{b}(Λ)
↓^{νΛ̂}  ↓^{ν̃[1]}
\underline{mod}Λ̂ ⟶ D^{b}(Λ)

Proposition 5.5. Let Λ be a Gorenstein algebra and X, Y ∈ D^{b}(Λ) indecomposable. If f : X → Y is irreducible and Y ∉ K^{b}(_{Λ}P), then X ≅ ν̃Y [−1].

Proof. We consider the almost split triangle in \underline{mod}Λ̂

τΛ̂μY →^{α} E →^{β} μY →^{γ} τΛ̂μY [1]  (∗)

Since f : X → Y is irreducible and by Theorem 4.3, also μ(f) is irreducible, we see that E ≅ μ(X) ⊕ C and β = (μ(f), g)^{t} for some g : C → μY. Since τΛ̂ = νΛ̂Ω2 ; Λ̂ and using the diagram above we see that G applied to (∗) yields a triangle in D^{b}(Λ)

ν̃Y [−1] →^{G(α)} X ⊕ GC →^{G(β)} Y →^{G(γ)} ν̃Y  (∗∗)

We claim that G(γ) = 0. Otherwise, let h : Z → Y be a map which is not a split epi, then μ(h) is not a split epi. But then μ(h)γ = 0, hence 0 = G(μ(h)γ) = hG(γ). Since Y and ν̃Y [−1] are indecomposable, (∗∗) would be an almost split triangle in D^{b}(Λ). Since Y ∉ K^{b}(_{Λ}P) this contradicts [H2]. So G(γ) = 0, hence G(β) is a split epi. Since X is not isomorphic to Y, we get that X ≅ ν̃Y [−1]. □

We will now show that for selfinjective algebras Λ irreducible maps in D^{b}(Λ) outside K^{b}(_{Λ}P) are rare. For this we will need the following easy fact, but first we will define the relevant class of algebras Λ_{n} for n ≥ 1. Let Λ_{1} = k[x]/(x^{2}) and Λ_{n} for n ≥ 2 defined by the following quiver

\xymatrix{ & 1 \ar[r]^{α_{1}} & 2 \ar[dr]^{α_{2}} \\ n \ar[ur]^{α_{n}} &  &  & 3}

with relations α_{i}α_{i+1} = 0 for 1 ≤ i ≤ n where α_{n+1} = α_{1}.

We collect the relevant information in the following well-known lemma (see [ARS])

Lemma 5.6. Let Λ be a basic selfinjective algebra which is not semisimple. (1) rad^{2} Λ = 0 if and only if Λ ≅ Λ_{n} for some n.

(2) If S(i) is a simple Λ_{n}-module, then νS(i) = S(i − 1) where S(0) = S(n).

(3) Λ_{n} is symmetric if and only if n = 1.

Theorem 5.7. Let Λ be a basic selfinjective algebra which is not semisimple. Let Y ∈ D^{b}(Λ) \ K^{b}(_{Λ}P) be indecomposable. There exists an irreducible map f : X → Y in D^{b}(Λ) if and only if Λ ≅ Λ_{n}, Y ≅ S(i − 1)[j], X ≅ S(i)[j − 1] for some 1 ≤ i ≤ n and j ∈ ℤ.

Proof. If Λ = Λ_{n} for some n, we have seen in Proposition 4.4 that for each arrow α_{i} we have an irreducible map νS(i)[−1] → S(i+1) in D^{b}(Λ). Since pd_{Λ_{n}} S(i+1) = ∞ we have S(i + 1) ∉ K^{b}(_{Λ}P).

Conversely, let f : X → Y be irreducible in D^{b}(Λ) and Y ∉ K^{b}(_{Λ}P). We choose Y ∈ K^{−,b}(_{Λ}P) and may assume that Y = (P^{i}, d^{i}) satisfies Y^{i} = 0 for i > 0 and H^{0}(Y) ≠ 0. By Proposition 5.5 we know that X ≅ ν̃Y [−1]. Since Λ is selfinjective, we have that ν_{Λ} is exact, hence ν̃Y = (ν_{Λ}P^{i}, ν_{Λ}d^{i}). Consider the triangle ν_{Λ}P^{0}[−1] →^{α} Y [−1] →^{β} Y [−1]_{≤0} →^{γ} ν_{Λ}P^{0}. Since Hom(ν_{Λ}P^{0}[−1], Y) = 0, there is some f̄ such that f = βf̄. H^{0}(Y) ≠ 0 implies that α ≠ 0. Thus β is not split mono, so f̄ is split epi, since f is irreducible. Hence H^{i}(f̄) : H^{i}(ν̃Y [−1]_{≤0} → H^{i}(Y) is split epi for all i. Since ν is exact, we have that H^{i}(ν̃Y) ≅ ν_{Λ}H^{i}(Y) for all i. Also we have that H^{i}(ν̃Y [−1]_{≤0}) = H^{i}(ν̃Y [−1]) for all i ≤ −1 and H^{i}(ν̃Y [−1]) = H^{i−1}(ν̃Y). Since Y ∈ K^{−,b}(_{Λ}P) there is n_{0} ≤ 0 such that H^{n}(Y) = 0 for all n ≤ n_{0}. Choose n_{0} maximal with this property, so H^{n_{0}}(Y) = 0 and H^{n_{0}+1}(Y) ≠ 0. We claim that Y ∈ modΛ or equivalently n_{0} = −1. Otherwise n_{0} ≤ −2. But then 0 = H^{n_{0}}(Y) = H^{n_{0}}(ν̃Y) = H^{n_{0}+1}(ν̃Y [−1]) = H^{n_{0}+1}(ν̃Y [−1]^{≤0}) ↠ H^{n_{0}+1}(Y), hence H^{n_{0}+1}(Y) = 0 in contrast to the choice of n_{0}. Hence n_{0} = −1, so Y is an indecomposable Λ-module. But then Y^{≤−1} ≅ Ω_{Λ}Y [1], so Y^{≤−1} is indecomposable, since Λ is selfinjective. But then ν̃Y [−1]^{≤0} is indecomposable, so f̄ is an isomorphism, so ν_{Λ}Ω_{Λ}Y ≅ Y. If Y is not simple, there is a proper epi Y →^{π} S for some simple S. So there is h : Y → I(S), with Ker h ≠ 0 and Coker h ≠ 0. Since Hom(I(S), Y [1]) = Ext_{Λ}^{1}(I(S), Y) = 0 we obtain a triangle

C_{h}[−1] →^{g} Y →^{h} I(S) → C_{h}

with C_{h}[−1] indecomposable and H^{0}(C_{h}[−1]) = Ker h, H^{1}(C_{h}[−1]) = Coker h. Since Hom(ν_{Λ}Y [−1], I(S)) = 0, there is f′ : ν_{Λ}Y [−1] → C_{h}[−1] such that f′g = f. Since h ≠ 0, g is not split epi, hence f′ is split mono, since f is irreducible. Since C_{h}[−1] is indecomposable, we have that f′ is an isomorphism, in contrast to H^{0}(C_{h}[−1]) ≠ 0 ≠ H^{1}(C_{h}[−1]), so X is a simple Λ-module. Since Y ≅ ν_{Λ}Ω_{Λ}Y, we see that Ω_{Λ}Y is a simple Λ-module. But then rad^{2} Λ = 0, since Λ is selfinjective and the assertion follows from Lemma 5.6. □

6. Behavior of irreducible maps

In this section we show that beyond the Gorenstein algebras the behavior of irreducible maps in D^{b}(Λ) is not so regular. In particular, we show that some natural conjectures have a negative answer.

For a non-zero map f : P → Q between indecomposable objects in D^{b}(Λ) but not in K^{b}(_{Λ}P) we investigate the connection between f : P → Q and f_{≥−n} : P_{≥−n} → Q_{≥−n} being irreducible for some n. We also give some sufficient condition for an irreducible map in modΛ not be irreducible in D^{b}(Λ).

We start with a general result on mapping cones of irreducible maps, where the analogous result in abelian categories is well known.

Proposition 6.1. Let X and Y be indecomposable in D^{b}(Λ), for a finite dimensional algebra Λ, and assume that we have an irreducible map f : X → Y. Then the mapping cone C_{f} is indecomposable.

Proof. This can be proved in a similar way as the abelian analog. Here we give a slightly shorter proof using Theorem 4.3. Let μ : D^{b}(Λ) → \underline{mod}Λ̂ be as usual the natural embedding. Then we know from Theorem 4.3 that μ(f) : μ(X) → μ(Y) is irreducible in \underline{mod}Λ̂. This is induced by an irreducible map f′ : μ(X) → μ(Y) in modΛ̂. If f′ is mono, we have an exact sequence 0 → μ(X) → μ(Y) → μ(Y)/μ(X) → 0, and if f′ is epi, we have an exact sequence 0 → Ker f′ → μ(X) → μ(Y) → 0. We know that in the first case μ(Y)/μ(X) is indecomposable and in the second case Ker f′ is indecomposable, see [ARS]. So in any case we have a triangle μ(X) →^{μ(f)} μ(Y) → Z in \underline{mod}Λ̂, where Z is indecomposable. Since μ(C_{f}) ≅ Z, it follows that C_{f} is indecomposable. □

Let 0 → A → B → C → 0 be an almost split sequence in modΛ. Then it is known that if id_{Λ} A ≤ 1 and pd_{Λ} C ≤ 1, then the sequence gives rise to an almost split triangle in D^{b}(Λ) [H1]. Consequently the corresponding irreducible maps f_{i} : A → B_{i} and g_{i} : B_{i} → C stay irreducible, where B = ⨁_{i=1}^{t} B_{i} with B_{i} indecomposable. But the normal behavior is that irreducible maps in modΛ do not stay irreducible in D^{b}(Λ). We illustrate this with the following result.

Proposition 6.2. Let Λ be a finite dimensional algebra, X and Y indecomposable Λ-modules with pd_{Λ} X < ∞ and pd_{Λ} Y ≥ pd_{Λ} X + 2. Then there is no irreducible map f : X → Y in D^{b}(Λ).

Proof. Assume that we have an irreducible map f : X → Y in modΛ, with X and Y indecomposable, pd X = i < ∞ and pd Y ≥ i + 2. Let P : ⋯ → P^{−(i+2)} → ⋯ → P^{−1} → P^{0} → 0 be a minimal projective resolution of Y and Q : 0 → Q^{−i} → ⋯ → Q^{−1} → Q^{0} → 0 a minimal projective resolution of X. Let C denote the complex

0 → P^{−(i+1)} → ⋯ → P^{−1} → P^{0} → 0. Then f : Q → P factors as Q →^{g} C →^{h} P,

since we have the commutative diagram

0 ⟶ Q^{−i} ⟶ … ⟶ Q^{−1} ⟶ Q^{0} ⟶ 0
↓  ^{f^{−i}}↓  ^{f^{−1}}↓  ^{f^{0}}↓
0 ⟶ P^{−(i+1)} ⟶ P^{−i} ⟶ … ⟶ P^{−1} ⟶ P^{0} ⟶ 0
∥  ∥  ∥  ∥
… ⟶ P^{−(i+2)} ⟶ P^{−(i+1)} ⟶ P^{−i} ⟶ … ⟶ P^{−1} ⟶ P^{0} ⟶ 0

We want to show that g is not a split monomorphism and h is not a split epimorphism. If g : Q → C was a split monomorphism, the induced map H^{0}(Q) = X → H^{0}(C) = Y would be a split monomorphism. Since X and Y are indecomposable nonisomorphic modules, this is impossible.

The diagram

Ω^{−(i+2)}Y
↓
P^{−(i+1)} ⟶ … ⟶ P^{−1} ⟶ P^{0}
↓
Y

gives rise to the triangle Ω^{−(i+2)}Y [i + 1] → C → Y [0] →^{α} Ω^{−(i+2)}Y [i + 2]. Here α is given by the sequence 0 → Ω^{−(i+2)}Y → P^{−(i+1)} → ⋯ → P^{−1} → P^{0} → Y → 0, which does not represent the zero element since pd_{Λ} Y ≥ i + 2. Hence, h : C → Y [0] or equivalently h : C → P is not a split epimorphism. It follows that f : Q → P is not irreducible. □

Note that if Λ is hereditary, then each irreducible map in modΛ stays irreducible in D^{b}(Λ). In this case pd X is 0 or 1, hence we can never have pd Y ≥ pd X + 2.

The next natural question is to which extent we have irreducible maps X → Y [1], where X and Y are indecomposable in modΛ, corresponding to elements of Ext_{Λ}^{1}(X, Y). Here we have seen some sufficient conditions in Section 3. Normally, we do not have such irreducible maps.

Proposition 6.3. Let f : X → Y [1] be an irreducible map, where X and Y are indecomposable Λ-modules. Then Y must be a summand of ΩX.

Proof. We have the factorization X →^{h} ΩX[1] →^{g[1]} Y [1] of f : X → Y [1], as is seen by considering the diagram

[figure: 0; ΩX; g; P_{X}; X; 0; 0; Y; E; X; 0]

Then h : X → ΩX[1] is not a split monomorphism since H^{0}(h) is not a split monomorphism. Since f : X → Y [1] is irreducible, it follows that g : ΩX → Y is a split epimorphism, so that Y is a summand of ΩX. □

We now give another situation where there are no irreducible maps, containing the case X → Y [2], corresponding to elements of Ext_{Λ}^{2}(X, Y), as a special case.

Proposition 6.4. Let P and Q be indecomposable objects in D^{b}(Λ) for a finite dimensional algebra Λ, represented by complexes of projective Λ-modules with no split exact summands, with P^{0} ≠ 0, P^{i} = 0 for i > 0 and Q^{i} = 0 for i ≥ −1. Then there is no irreducible map f : P → Q in D^{b}(Λ).

Proof. Let f : P → Q be a map in D^{b}(Λ). Consider the factorization of f given by

… ⟶ P^{−2} ⟶ P^{−1} ⟶^{a} P^{0} ⟶ 0  P
↓  ↓  ↓  ↓^{h}
… ⟶ P^{−2} ⟶ P^{−1} ⟶ 0 ⟶ 0  P_{≤−1}
↓  ↓  ↓^{g}
… ⟶ Q^{−2} ⟶ 0  Q

We have H^{0}(P) = P^{0}/ Im a, which is not zero since P has no split exact direct summands. Since H^{0}(P_{≤−1}) = 0, h : P → P_{≤−1} cannot be a split monomorphism.

Assume now that g is a split epimorphism, and consider the triangle P_{≤−1} →^{g} Q →^{u} C_{g} →. Then u : Q → C_{g} must be homotopic to 0, that is, we have the diagram where b_{−i}s_{−i} + s_{−(i+1)}c_{−(i+1)} = (0, 1) for all i ≥ 1. Using the same maps s_{i} we see

[figure: …; Q^{−3}; (0,1); b_{−2}; Q^{−2}; s_{−2}; (0,1); b_{−1}; 0; s_{−1}; …; P^{−2} ⊕ Q^{−3}; c_{−2}; P^{−1} ⊕ Q^{−2}; c_{−1}; 0]

that in the triangle P →^{f} Q →^{v} C_{f} →, the map v must be 0, so that f would also be a split epimorphism. Since P and Q are indecomposable, f would be an isomorphism, which is impossible because H^{0}(P) ≠ 0 and H^{0}(Q) = 0. We conclude that g is not a split epimorphism. Since we already have that h is not a split monomorphism, it follows that f : P → Q is not irreducible. □

The following sufficient condition for the mapping cone to be indecomposable will be useful. We formulate the result in the more general setting of Krull-Schmidt triangulated categories.

Lemma 6.5. Let f : P → Q be a map between indecomposable objects in a Krull-Schmidt triangulated category C with shift [1] and assume that f is not zero and not invertible. Complete to a triangle P →^{f} Q →^{g} C → P [1]. If Hom(Q, P [1]) = 0, then C is indecomposable.

Proof. Assume to the contrary that C is not indecomposable, and write C = ⨁_{i=1}^{r} Z_{i}, where r > 1 and each Z_{i} is indecomposable. Let g = (g_{1}, …, g_{r}) and h = (h_{1}, …, h_{r})^{t}. Then we know from [R] that g_{i} ≠ 0 and h_{i} ≠ 0 for each i = 1, …, r.

Consider the map

φ = ( \begin{matrix}1 & 0 & … & 0 \\ 0 & & & ⋮ \\ ⋮ & & & ⋮ \\ 0 & … & … & 0 \end{matrix} ) : C → C,

where 1 = 1_{Z_{1}}. We then have the diagram

P ⟶^{f} Q ⟶^{g} C ⟶^{h} P [1]
↓^{φ}
P ⟶^{f} Q ⟶^{g} C ⟶^{h} P [1]

Since by the assumption Hom(Q, P [1]) = 0, it follows that gφh = 0. Hence there is a map φ_{Q} : Q → Q such that φ_{Q}g = gφ. We have gφ = (g_{1}, 0, …, 0) and φ_{Q}g = (φ_{Q}g_{1}, …, φ_{Q}g_{r}), so that φ_{Q}g_{1} = g_{1} and φ_{Q}g_{i} = 0 for 2 ≤ i ≤ r. Since φ^{2} = φ, we have φ_{Q}^{n}g = gφ^{n} = gφ, so that φ_{Q}^{n}g_{1} = g_{1} and φ_{Q}^{n}g_{i} = 0 for 2 ≤ i ≤ r. Since Q is indecomposable, any map t : Q → Q is nilpotent or an isomorphism, so that we have a contradiction. It follows that C is indecomposable. □

We now consider the following question. If we have an irreducible map f : P → Q between unbounded complexes of projective modules, not objects in K^{b}(_{Λ}P), is then f_{≥−n} : P_{≥−n} → Q_{≥−n} irreducible for all n where f_{≥−n} is a nonzero map between indecomposable objects?

For selfinjective algebras Λ, the existence of an irreducible map f : P → Q not in K^{b}(_{Λ}P) implies that Λ is selfinjective with rad^{2} Λ = 0 and that we have f : S → T [1], where S and T are simple Λ-modules. In this case f_{≥−n} : P_{≥−n} → Q_{≥−n} is irreducible for n ≥ 2.

We now give an example which gives a negative answer to the above question. Let Λ be the path algebra of the quiver

[figure: 1; γ; 2; α]

with relations γα = 0, γ^{2} = 0. Denote by S the simple module at vertex 1 and by T the simple module at vertex 2. Then the indecomposable projective Λ-modules have Loewy series S S and T S, and we have T and {ST \atop S} for the indecomposable injectives. We know from Proposition 4.4 that the map f : T → S[1] is irreducible, and we can write this as f : P → Q given by

… ⟶ S S ⟶ S S ⟶ T S
∥  ∥  ↓
… ⟶ S S ⟶ S S ⟶ 0

We then have the following.

Proposition 6.6. Let Λ be as above. In the above notation we have that f : P → Q is irreducible, while f_{≥−1} : P_{≥−1} → Q_{≥−1} is a map between indecomposable objects which is not irreducible.

Proof. We have already seen that f : P → Q is irreducible. We now want to show that f_{≥−1} : P_{≥−1} → Q_{≥−1} is not irreducible. We have Q_{≥−1} = (S S)[1], and hence νQ_{≥−1} = {ST \atop S}, so that we have an almost split triangle {ST \atop S} → (S S →^{α} {ST \atop S}) → S S[1] →, where Im α = S. We claim that X = (S S →^{α} {ST \atop S}) is indecomposable. For this, it is sufficient to show that Hom({ST \atop S} , S S[1]) = 0 by Lemma 6.5, that is that Ext_{Λ}^{1}({ST \atop S} , S S) = 0. This follows by considering the injective resolution 0 → S S →^{h} {ST \atop S} →^{g} T → 0, which gives rise to the exact sequence Hom({ST \atop S} , {ST \atop S}) →^{φ} Hom({ST \atop S} , T) → Ext^{1}({ST \atop S} , S S) → 0, and using that φ is clearly an epimorphism. Hence we conclude that X is indecomposable. Alternatively we could prove that X is indecomposable by considering the homology of X and how it could decompose.

Since H^{0}(P_{≥−1}) = T while H^{0}(X) = S ⊕ T, P_{≥−1} cannot be isomorphic to X. Hence f_{≥−1} : P_{≥−1} → Q_{≥−1} is not irreducible. □

We now give an example of a nonzero map f : P → Q between indecomposable objects which is not irreducible, but such that f_{≥−n} : P_{≥−n} → Q_{≥−n} is an irreducible map between indecomposable objects for some n.

Let Λ = k[x]/(x^{3}), and consider the complexes of projective modules:

P  ⋯ → Λ →^{·x} Λ →^{·x^{2}} Λ →^{(x,0)} Λ ⊕ Λ →^{({x^{2} \atop x})} Λ
Q  ⋯ → Λ →^{·x} Λ →^{·x^{2}} Λ →^{·x} Λ → 0

where the right hand terms are in degree 0, as objects in D^{b}(Λ). Consider the map f : P → Q in D^{b}(Λ) induced by the commutative diagram

… ⟶ Λ ⟶^{·x} Λ ⟶^{·x^{2}} Λ ⟶^{(x,0)} Λ ⊕ Λ ⟶^{({x^{2} \atop x})} Λ
∥  ∥  ∥  ↓^{(^{1}_{0})}  ↓
… ⟶ Λ ⟶^{·x} Λ ⟶^{·x^{2}} Λ ⟶^{·x} Λ ⟶ 0

We have the following

Proposition 6.7. With the above notation and assumptions we have the following

(1) The induced map f_{≥−2} : P_{≥−2} → Q_{≥−2} is an irreducible map between indecomposable objects in D^{b}(Λ).

(2) The map f : P → Q is a map between indecomposable objects which is not irreducible.

Proof. (1) The map f_{≥−2} is given by the following diagram

0 ⟶ Λ ⟶^{(x,0)} Λ ⊕ Λ ⟶^{({x^{2} \atop x})} Λ ⟶ 0
∥  ↓^{(^{1}_{0})}  ↓
0 ⟶ Λ ⟶^{·x} Λ ⟶ 0 ⟶ 0

Since Λ = k[x]/(x^{3}) is symmetric, we have τ(Q_{≥−2}) = Q_{≥−2}[−1], and hence an almost split triangle Q_{≥−2}[−1] → E → Q_{≥−2} →^{α} Q_{≥−2}. The map α : Q_{≥−2} → Q_{≥−2} inducing the almost split triangle is easily seen to be given by the diagram

0 ⟶ Λ ⟶^{·x} Λ ⟶ 0
↓^{0}  _{·x^{2}}↓
0 ⟶ Λ ⟶^{·x} Λ ⟶ 0

For it is clear that the induced map is nonzero and is in the socle of End(Q_{≥−2}). Taking the mapping cone of α we obtain P_{≥−2}[1], so that E ≅ P_{≥−2}. This shows that f_{≥−2} : P_{≥−2} → Q_{≥−2} is irreducible.

We next show that P_{≥−2} is indecomposable. We give a proof which at the same time illustrates the previous theory, rather than giving a direct computational proof. We know from Theorem 5.4 that the components of the AR-quiver of K^{b}(_{Λ}P) are of the form ℤA_{∞}, and that the image of a component for K^{b}(_{Λ}P) is a component of the AR-quiver for \underline{mod}Λ̂. All Λ̂-modules in such a component C are given by projective modules, the same ones as for K^{b}(_{Λ}P). Then C is also a component for modΛ̂. This follows since any indecomposable projective object in modΛ̂ has an irreducible map to this object modulo its socle, and this object is not given by only projective modules.

If P_{≥−2} was not indecomposable, then Λ →^{·x} Λ would not be at the border of the ℤA_{∞}-component. Hence we would have an irreducible epimorphism starting at Λ →^{·x} Λ, which would then have to end at Λ, since the terms must be projective. But on the other hand we have an almost split triangle Λ[−1] → (Λ →^{·x^{2}} Λ) → Λ in K^{b}(_{Λ}P), which gives a contradiction.

(2) We first show that P and Q are indecomposable. This is obvious for Q. Assume P = P′ ⊕ P′′ is a nontrivial decomposition. Then we have P_{≥−2} = P′_{≥−2} ⊕ P′′_{≥−2}. Since H^{0}(P) = S, H^{−1}(P) = S S and H^{−i}(P) = 0 for i ≠ 0, 1, we must have, say P′ ≅ S and P′′ ≅ (S S)[1]. But then P′_{≥−2} and P′′_{≥−2} are both nonzero, contradicting that P_{≥−2} is indecomposable.

That f : P → Q is not irreducible follows since Λ is selfinjective and rad^{2} Λ ≠ 0 and P and Q are not in K^{b}(_{Λ}P). We could alternatively give a direct argument by considering the following factorization of the map f : P → Q:

… ⟶ Λ ⟶^{·x} Λ ⟶^{·x^{2}} Λ ⟶^{(x,0)} Λ ⊕ Λ ⟶^{({x^{2} \atop x})} Λ  P
∥  ∥  ↓^{(1,0)}  ∥  ∥  ↓^{g}
… ⟶ Λ ⟶^{·x} Λ ⟶^{(x^{2},0)} Λ ⊕ Λ ⟶^{(^{x}_{0}{0 \atop x^{2}})} Λ ⊕ Λ ⟶^{({x^{2} \atop x})} Λ  U
∥  ∥  ↓^{(^{1}_{0})}  ↓^{(^{1}_{0})}  ↓  ↓^{h}
… ⟶ Λ ⟶^{·x} Λ ⟶^{·x^{2}} Λ ⟶^{·x} Λ ⟶ 0  Q

and showing that g is not a split monomorphism and h is not a split epimorphism. The first claim follows directly by considering the homology of P and U, and the second claim is also not hard to show. □

References

[AR] Auslander, Maurice; Reiten, Idun, Representation theory of Artin algebras. IV. Invariants given by almost split sequences. Comm. Algebra 5 (1977), no. 5, 443–518.

[ARS] Auslander, Maurice; Reiten, Idun; Smalø, Sverre O., Representation theory of Artin algebras. Cambridge Studies in Advanced Mathematics, 36. Cambridge University Press, Cambridge, 1995.

[CZ] Chen Xiao-Wu; Zhang, Pu, Quotient triangulated categories, preprint.

[G] Alexandre Grothendieck, Les dérivateurs, Manuscript, 1990, edited electronically by M. Künzer, J. Malgoire and G. Maltsiniotis.

[GK] Geiss, Christof; Krause, Henning, On the notion of derived tameness. J. Algebra Appl. 1 (2002), no. 2, 133–157.

[F] Farnsteiner, Rolf, Stable representation quivers: Subadditive functions and Webb’s theorem. Preprint 2006.

[H1] Happel, Dieter, Triangulated categories in the representation theory of finite-dimensional algebras. London Mathematical Society Lecture Note Series, 119. Cambridge University Press, Cambridge, 1988.

[H2] Happel, Dieter, Auslander-Reiten triangles in derived categories of finite-dimensional algebras. Proc. Amer. Math. Soc. 112 (1991), no. 3, 641–648.

[K1] Keller, Bernhard, On differential graded categories, preprint, contribution to the proceedings of the ICM 2006.

[K2] , Derived categories and universal problems, Comm. in Algebra 19 (1991), 699–747.

[KV] Keller, Bernhard; Vossieck, Dieter, Sous les catégories dérivées, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 6, 225–228.

[RV] Reiten, Idun; Van den Bergh, Michel, Noetherian hereditary abelian categories satisfying Serre duality. J. Amer. Math. Soc. 15 (2002).

[Ric] Rickard, Jeremy, Derived categories and stable equivalence, J. Pure and Appl. Algebra 61 (1989), 303–317.

[Rin] Ringel, Claus M., Hereditary triangulated categories. Compositio Mathematica. To appear.

[Ta] Tabuada, Gonçalo, Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories, C. R. Math. Acad. Sci. Paris 340 (2005), no. 1, 15–19.

[To] Toën, Bertrand, The homotopy theory of dg-categories and derived Morita theory, to appear in Inv. Math., arXiv:math.AG/0408337.