THE TRIANGULATED STRUCTURE OF THE STABLE DERIVED CATEGORY

Wayne W. Wheeler Mathematics, I.A.S. Australian National University GPO Box 4 Canberra ACT 2601 wheeler@pell.anu.edu.au

24 July 1992

Introduction

The study of the stable derived category of a finite-dimensional self-injective algebra A over a field k was begun in [7] and [8]. As the name suggests, this category can be regarded as a sort of hybrid version of the stable category and the derived category of A. Although the stable derived category \overline{D}^{b}(A) is not abelian, we show that it does at least carry a triangulated structure, and we consider various questions related to this structure.

In addition to describing the triangulation on \overline{D}^{b}(A), Section 1 develops a strong analogy with the derived category D^{b}(A). The derived category is usually defined by starting with a category C^{b}(A) of chain complexes. In a process analogous to localization in a commutative ring, one localizes the corresponding homotopy category K^{b}(A) by inverting those maps that induce isomorphisms on cohomology, and the result is the derived category D^{b}(A). By working instead with an appropriate quotient \overline{C}^{b}(A) of the category C^{b}(A), one can localize with respect to the maps that induce stable isomorphisms on cohomology, and the result is the stable derived category \overline{D}^{b}(A). Although this is not the definition used in [7] and [8], we show in Section 1 that it defines an equivalent category.

The remaining two sections are devoted to considering Auslander–Reiten triangles in \overline{D}^{b}(A). These are certain distinguished triangles first considered in [2] that are analogous to almost split sequences. The results of Section 2 show that an object of \overline{D}^{b}(A) determines an Auslander–Reiten triangle if and only if it is isomorphic to a bounded complex of projective modules. If \overline{C}^{b}(P_{A}) denotes the full subcategory consisting of all such objects, then there is a very close connection between the Auslander–Reiten triangles of \overline{C}^{b}(P_{A}) and those of the homotopy category K^{b}(P_{A}) of bounded complexes of projective modules.

In Section 3 we describe the Auslander–Reiten quiver of the category \overline{C}^{b}(P_{A}). Because of the connection with K^{b}(P_{A}), we obtain as a by-product a description of the Auslander–Reiten quiver of this category. Specifically, we prove the following result.

Typeset by AMS-TEX

Theorem. Every component of the Auslander–Reiten quiver of K^{b}(P_{A}) containing more than one vertex is isomorphic to ℤA_{∞}. In \overline{C}^{b}(P_{A}) every component of the Auslander–Reiten quiver is isomorphic to ℤA_{∞}.

We remark in addition that an indecomposable complex P^{•} in K^{b}(P_{A}) lies in a component containing only one vertex if and only if P^{•} consists of a simple projective module concentrated in a single degree.

In the following work the letter k always denotes a field, and A is a finitedimensional self-injective algebra over k. All categories and functors considered are assumed to be k-linear, and all A-modules are finite-dimensional. Finally, the reader should be warned that we have deviated here from the convention in [7] and [8] by using cochain complexes: if P^{•} is any complex, then the differential ∂ is assumed to raise the degree by one.

§1. Localization and the Stable Derived Category

Let A be a finite-dimensional self-injective algebra over a field k. Let C(A) denote the category of chain complexes of finitely generated right A-modules; C^{+}(A), C^{−}(A), and C^{b}(A) denote the full subcategories of complexes that are bounded below, bounded above, or bounded above and below respectively. The full subcategory of C^{+}(A) consisting of objects with bounded cohomology is denoted C^{+,b}(A), and C^{−,b}(A) is defined similarly. Let P_{A} denote the category of all finitely generated projective right A-modules. Then we write C(P_{A}) for the full subcategory of C(A) consisting of complexes of objects in P_{A}. The categories C^{+}(P_{A}), C^{−}(P_{A}), C^{b}(P_{A}), C^{+,b}(P_{A}), and C^{−,b}(P_{A}) are defined similarly. In addition, the category C(A) determines a corresponding homotopy category K(A) and a derived category D(A). Each of the above subcategories of C(A) likewise determines a homotopy category and a derived category. If f is a chain map, we will write [f] for the homotopy class of f.

Finally, we let mod-A denote the category of finitely generated right A-modules, and we define the stable category \overline{mod}-A to be the quotient of mod-A by the subcategory of objects in P_{A}. In other words, the objects of \overline{mod}-A are the same as the objects of mod-A; if U and V are objects in \overline{mod}-A and P (U, V) ⊆ Hom_{A}(U, V) denotes the vector subspace consisting of those homomorphisms that factor through a projective A-module, then the morphisms from U to V in \overline{mod}-A are defined by \overline{Hom}_{A}(U, V) = Hom_{A}(U, V)/P (U, V). If f ∈ Hom_{A}(U, V), then we will write \overline{f} for the corresponding morphism in \overline{Hom}_{A}(U, V).

Following [5], we will say that a short exact sequence 0 → K^{•} → L^{•} → M^{•} → 0 in C(A) is proper if the corresponding sequence of n-cocycles 0 → Z^{n}(K^{•}) → Z^{n}(L^{•}) → Z^{n}(M^{•}) → 0 is exact for all n. When this condition is satisfied, it is easy to check that the coboundaries and the cohomology also give short exact sequences for all n (Proposition XII.11.2 of [5]). A proper monomorphism is a monomorphism occurring in a proper short exact sequence, and a proper epimorphism is defined similarly. A complex in C(A) is a proper projective complex if it satisfies the usual projective lifting property for all proper epimorphisms. Proper injectives can be described similarly. By Lemmas XII.11.4 and XII.11.5 of [5], the category C(A) has enough proper projectives, and any indecomposable proper projective is isomorphic to either

⋯ ⟶ 0 ⟶ P ⟶^{1_{P}} P ⟶ 0 ⟶ ⋯

or

⋯ ⟶ 0 ⟶ P ⟶ 0 ⟶ ⋯

for some indecomposable projective A-module P. Since A is a self-injective algebra, the proper projectives coincide with the proper injectives in C(A). We note that proper projective complexes were called homology-projective in [7]; we will use both terms interchangeably. If a complex X^{•} has no homology-projective summands, then it is said to be h.p.-free.

For each object X^{•} of C(A), fix a proper injective hull ι_{X^{•}} : X^{•} → I(X^{•}), and let Ω^{−1}X^{•} = Coker ι_{X^{•}}. Then there is a proper short exact sequence

0 ⟶ X^{•} ⟶^{ι_{X^{•}}} I(X^{•}) ⟶^{π_{X^{•}}} Ω^{−1}X^{•} ⟶ 0.

Note that when X^{•} consists of a module in a single degree, this definition agrees with the usual definition of Ω^{−1}.

Now let P^{•} denote the collection of all proper projective objects in C(A), and let E denote the proper short exact sequences. Then (C(A), E) is a Frobenius category, so the corresponding stable category \overline{C}(A) = C(A)/P^{•} is triangulated by Theorem I.2.6 of [2]. The functor Ω^{−1} is an autoequivalence on \overline{C}(A) that provides the translation functor.

As in the case of the stable category \overline{mod}-A, we will write \overline{f} for the image in \overline{C}(A) of a morphism f in C(A). Note that if \overline{f} : X^{•} → Y^{•} is a morphism in \overline{C}(A), then on coboundaries, cocycles, and cohomology \overline{f} induces well-defined maps B^{n}(\overline{f}) = \overline{B^{n}(f)} : B^{n}(X^{•}) → B^{n}(Y^{•}), Z^{n}(\overline{f}) = \overline{Z^{n}(f)} : Z^{n}(X^{•}) → Z^{n}(Y^{•}), and H^{n}(\overline{f}) = \overline{H^{n}(f)} : H^{n}(X^{•}) → H^{n}(Y^{•}) in \overline{mod}-A for all n.

Lemma 1.1. For any integer n the functors B^{n}, Z^{n}, and H^{n} : \overline{C}(A) → \overline{mod}-A are exact.

Proof. We consider the proof for H^{n} since it is similar for the other functors.

To show that H^{n} maps triangles in \overline{C}(A) into triangles in \overline{mod}-A, it suffices to consider the triangle arising from a diagram of proper short exact sequences:

0 ⟶ W^{•} ⟶ I(W^{•}) ⟶ Ω^{−1}W^{•} ⟶ 0
f↓  ↓  ∥
0 ⟶ X^{•} ⟶_{g} Y^{•} ⟶_{h} Ω^{−1}W^{•} ⟶ 0.

Because the sequences are proper, we get the following commutative diagram with exact rows in mod-A:

0 ⟶ H^{n}(W^{•}) ⟶ H^{n}(I(W^{•})) ⟶ H^{n}(Ω^{−1}W^{•}) ⟶ 0
H^{n}(f)↓  ↓  ∥
0 ⟶ H^{n}(X^{•}) ⟶_{H^{n}(g)} H^{n}(Y^{•}) ⟶_{H^{n}(h)} H^{n}(Ω^{−1}W^{•}) ⟶ 0.

Since H^{n}(I(W^{•})) is injective, one can easily check that there is a natural isomorphism η : H^{n}Ω^{−1} → Ω^{−1}H^{n} so that we get a triangle

H^{n}(W^{•}) ⟶^{H^{n}(\overline{f})} H^{n}(X^{•}) ⟶^{H^{n}(\overline{g})} H^{n}(Y^{•}) ⟶^{η_{W^{•}} H^{n}(\overline{h})} Ω^{−1}H^{n}(W^{•})

in \overline{mod}-A. Hence H^{n} is exact, as desired. □

Definition. A morphism \overline{f} : X^{•} → Y^{•} in \overline{C}(A) is said to be a stable quasiisomorphism if the induced maps H^{n}(\overline{f}) are isomorphisms in \overline{mod}-A for all n.

We write \overline{Qis} for the collection of all stable quasi-isomorphisms.

Let C be a triangulated category with translation functor T. Recall that a family S of morphisms in C is said to be a multiplicative system compatible with the triangulation if the following conditions are satisfied: (FR1) If s, t ∈ S and st is defined, then st ∈ S. Moreover, the identity map 1_{X} is

in S for every object X in C. (FR2) Suppose f : X → Y and s : Z → Y are morphisms with s ∈ S. Then there

is a commutative diagram

W ⟶ Z
t↓  ↓^{s}
X ⟶_{f} Y

with t ∈ S. The dual statement also holds. (FR3) If f ∈ Hom_{C}(X, Y), then the following conditions are equivalent.

(1) There is a morphism s : Y → Y′ such that s ∈ S and sf = 0.

(2) There is a morphism t : X′ → X such that t ∈ S and ft = 0. (FR4) A morphism s lies in S if and only if T (s) lies in S. (FR5) Let (X, Y, Z, α, β, γ) and (X′, Y′, Z′, α′, β′, γ′) be triangles in C. Suppose

f : X → X′ and g : Y → Y′ are morphisms such that α′f = gα. If f, g ∈ S, then there is an h ∈ S such that the following diagram commutes:

X ⟶^{α} Y ⟶^{β} Z ⟶^{γ} T X
f↓  ^{g}↓  ↓^{h}  ↓^{T f}
X′ ⟶_{α′} Y′ ⟶_{β′} Z′ ⟶_{γ′} T X′.

Proposition 1.2. \overline{Qis} is a multiplicative system in \overline{C}(A) compatible with the triangulation.

Proof. The condition (FR1) is obvious. To prove (FR2), let \overline{f} : X^{•} → Y^{•} and \overline{s} : Z^{•} → Y^{•} be morphisms with \overline{s} ∈ \overline{Qis}. By Lemma I.2.3 of [2], we may assume without loss of generality that s is a proper monomorphism so that there is a proper short exact sequence

0 ⟶ Z^{•} ⟶^{s} Y^{•} ⟶^{g} C^{•} ⟶ 0.

Then for each n there is a short exact sequence

0 ⟶ H^{n}(Z^{•}) ⟶^{H^{n}(s)} H^{n}(Y^{•}) ⟶^{H^{n}(g)} H^{n}(C^{•}) ⟶ 0.

Since H^{n}(\overline{s}) is an isomorphism in \overline{mod}-A, we see that H^{n}(C^{•}) is projective for all n. If we now embed \overline{g}\overline{f} : X^{•} → C^{•} into a distinguished triangle, then we get a morphism of triangles

W^{•} ⟶^{\overline{t}} X^{•} ⟶^{\overline{g}\overline{f}} C^{•} ⟶ Ω^{−1}W^{•}
↓  \overline{f}↓  ↓^{1_{C^{•}}}  ↓
Z^{•} ⟶_{\overline{s}} Y^{•} ⟶_{\overline{g}} C^{•} ⟶ Ω^{−1}Z^{•},

and we must show that \overline{t} ∈ \overline{Qis}. Because H^{n} : \overline{C}(A) → \overline{mod}-A is exact and H^{n}(C^{•}) is projective, we conclude that H^{n}(\overline{t}) is an isomorphism for all n. Thus \overline{t} ∈ \overline{Qis}, as desired. The dual condition in (FR2) follows, not surprisingly, by a dual argument.

To prove (FR3), suppose that \overline{f} : X^{•} → Y^{•} and \overline{s} : Y^{•} → Y′^{•} are morphisms with \overline{s} ∈ \overline{Qis} and \overline{s}\overline{f} = 0. Consider a triangle of the form

Z^{•} ⟶^{\overline{v}} Y^{•} ⟶^{\overline{s}} Y′^{•} ⟶ Ω^{−1}Z^{•}.

Since \overline{s}\overline{f} = 0, there is a map \overline{g} : X^{•} → Z^{•} such that \overline{f} = \overline{v}\overline{g}, and there is a triangle

X′^{•} ⟶^{\overline{t}} X^{•} ⟶^{\overline{g}} Z^{•} ⟶ Ω^{−1}X′^{•}.

Then \overline{f}\overline{t} = \overline{v}\overline{g}\overline{t} = 0, so it remains to show that \overline{t} ∈ \overline{Qis}. But since \overline{s} ∈ \overline{Qis}, we know that H^{n}(Z^{•}) is projective for all n and hence \overline{t} ∈ \overline{Qis}, as desired. The proof that condition (ii) of (FR3) implies (i) is similar.

Condition (FR4) is an easy consequence of the existence of a natural isomorphism H^{n}Ω^{−1} ≅ Ω^{−1}H^{n}, and (FR5) follows from the fact that H^{n} : \overline{C}(A) → \overline{mod}-A is exact for all n. □

Definition. We define the stable derived category \overline{D}(A) to be the localization of \overline{C}(A) with respect to \overline{Qis}.

It should be noted that the stable quasi-isomorphisms in \overline{C}^{+}(A) also form a multiplicative system compatible with the triangulation, so we can consider the localization \overline{D}^{+}(A). Similarly, we can form the categories \overline{D}^{−}(A), \overline{D}^{b}(A), \overline{D}^{+}(P_{A}), and so on. The following lemma, which is stated for the natural functor \overline{D}^{−}(P_{A}) → \overline{D}^{−}(A), also has analogues for these other categories so that they can all be considered as subcategories of \overline{D}(A).

Lemma 1.3. The natural functor \overline{D}^{−}(P_{A}) → \overline{D}^{−}(A) is fully faithful.

Proof. Let P^{•} be an object of \overline{C}^{−}(P_{A}), and suppose that X^{•} is an object of \overline{C}^{−}(A) such that there is a stable quasi-isomorphism \overline{s} : X^{•} → P^{•}. We claim that there is an object Q^{•} of \overline{C}^{−}(P_{A}) and a morphism \overline{f} : Q^{•} → X^{•} such that \overline{s}\overline{f} is a stable quasi-isomorphism. It suffices to consider the case in which X^{•} and P^{•} are h.p.-free. Then s is actually a quasi-isomorphism, and the dual of Lemma 4.5 of [4] implies that s has a right homotopy inverse f : P^{•} → X^{•}. Since sf : P^{•} → P^{•} is homotopic to 1_{P^{•}}, it follows that 1_{P^{•}} − sf factors through a homology-projective complex in C^{−}(P_{A}) so that \overline{sf} = 1_{P^{•}} in \overline{C}^{−}(P_{A}). Proposition 3.3 of [4] now implies that the functor \overline{D}^{−}(P_{A}) → \overline{D}^{−}(A) is fully faithful. □

Lemma 1.4. A morphism \overline{f} : P^{•} → Q^{•} in \overline{C}^{−}(P_{A}) is an isomorphism if and only if \overline{f} ∈ \overline{Qis}.

Proof. The result is proven for the category \overline{C}^{−,b}(P_{A}) in Proposition 3.2 of [8], but the proof given there works equally well for the category \overline{C}^{−}(P_{A}). □

Proposition 1.5. The natural functor \overline{C}^{−}(P_{A}) → \overline{D}^{−}(A) is an equivalence of triangulated categories.

Proof. We know by Lemma 1.3 that the natural functor \overline{D}^{−}(P_{A}) → \overline{D}^{−}(A) is fully faithful. But all stable quasi-isomorphisms in \overline{C}^{−}(P_{A}) are actually isomorphisms by Lemma 1.4, so \overline{C}^{−}(P_{A}) = \overline{D}^{−}(P_{A}). Thus it is only necessary to show that every object of \overline{D}^{−}(A) is isomorphic in \overline{D}^{−}(A) to an object in \overline{C}^{−}(P_{A}).

Let X^{•} be an object of \overline{D}^{−}(A). We may assume that X^{•} is h.p.-free. Then there is a complex P^{•} in C^{−}(P_{A}) and a quasi-isomorphism f : X^{•} → P^{•}, so \overline{f} is a stable quasi-isomorphism. This completes the proof. □

Note that a similar argument applies for the other subcategories of \overline{D}^{+}(A) and \overline{D}^{−}(A). In particular, the functor \overline{C}^{−,b}(P_{A}) → \overline{D}^{b}(A) is a triangle-equivalence. In fact, in [7] and [8] we used the notation \overline{D}^{b}(A) for the category called \overline{C}^{−,b}(P_{A}) here.

§2. Auslander–Reiten Triangles in \overline{D}^{b}(A)

The goal of this section is to determine those objects of \overline{D}^{b}(A) for which there is a corresponding Auslander–Reiten triangle. Because \overline{D}^{b}(A) is triangle-equivalent to \overline{C}^{−,b}(P_{A}) by the results of the previous section, we will work instead with \overline{C}^{−,b}(P_{A}). We begin by recalling the definition of an Auslander–Reiten triangle in a triangulated category.

Definition. A triangle

X ⟶^{u} Y ⟶^{v} Z ⟶^{w} T X

is said to be an Auslander–Reiten triangle if the following conditions are satisfied: (1) X and Z are indecomposable.

(2) w ≠ 0.

(3) If f : W → Z is not a split epimorphism, then there is a map f′ : W → Y such that vf′ = f.

Our objective is to show that there is an Auslander–Reiten triangle of the form

Q_{1}^{•} ⟶ Q_{2}^{•} ⟶ P^{•} ⟶ Ω^{−1}Q_{1}^{•}

in \overline{C}^{−,b}(P_{A}) if and only if P^{•} is isomorphic to a bounded, indecomposable complex in \overline{C}^{−,b}(P_{A}). In fact, these are exactly the objects for which the category K^{−,b}(P_{A}) has Auslander–Reiten triangles (Theorem 1.4 of [3]), and the Auslander–Reiten triangles in the two categories are closely related, as we shall see.

An argument similar to the one used to prove the following proposition appears in [3], where it gives the analogous result for the homotopy category K^{−,b}(P_{A}).

Proposition 2.1. Let P^{•} be an indecomposable object of \overline{C}^{−,b}(P_{A}), and suppose that

Δ :  Q_{1}^{•} ⟶^{\overline{u}} Q_{2}^{•} ⟶^{\overline{v}} P^{•} ⟶^{\overline{w}} Ω^{−1}Q_{1}^{•}

is an Auslander–Reiten triangle in \overline{C}^{−,b}(P_{A}). Then P^{•} is bounded.

Proof. Suppose that P^{•} is not bounded. Choose an integer N with H^{i}(Ω^{−1}Q_{1}^{•}) = 0 for i ≤ N. Let P_{[N,∞)}^{•} be the subcomplex of P^{•} defined by setting P_{[N,∞)}^{i} = P^{i} for i ≥ N and P_{[N,∞)}^{i} = 0 for i < N. Then there is a chain map f : P_{[N,∞)}^{•} → P^{•} given by f^{i} = 1_{P^{i}} for i ≥ N and f^{i} = 0 for i < N, and \overline{f} cannot be a split epimorphism because P_{[N,∞)}^{•} is bounded. Since Δ is an Auslander–Reiten triangle, it follows that \overline{w}\overline{f} = 0. By Proposition 2.4 of [7] we see that wf is homotopic to a chain map that factors through a complex Q^{•} in \overline{C}^{−,b}(P_{A}) such that all boundary maps in Q^{•} are zero. Let ξ : P_{[N,∞)}^{•} → Q^{•} and λ : Q^{•} → Ω^{−1}Q_{1}^{•} be chain maps such that there is a homotopy s with ∂s + s∂ = wf − λξ. Now consider the map w′ : P^{•} → Ω^{−1}Q_{1}^{•} defined by setting

w′^{i} = \{ \begin{matrix}w^{i} & for i < N \\ w^{N} − s^{N+1}∂ & for i = N \\ 0 & for i > N. \end{matrix}

It is easy to check that w′ is a chain map with \overline{w}′ = \overline{w}. Since w′^{i} = 0 for i > N and H^{i}(Ω^{−1}Q_{1}^{•}) = 0 for i ≤ N, it follows that w′ is homotopic to 0. Hence \overline{w} = \overline{w}′ = 0, and we have a contradiction. □

Let DV denote the k-dual of an A-module V, and let ν = D Hom_{A}(−, A) be the Nakayama functor. Because A is self-injective, ν is exact. Moreover, ν is an autoequivalence on the category P_{A} of finitely generated projective A-modules, and a quasi-inverse is given by Hom_{A}(DA, −). Theorem 1.4 of [3] shows that if P^{•} is a bounded, indecomposable complex in K^{−,b}(P_{A}), then P^{•} determines an Auslander–Reiten triangle of the form

(∗)  νP^{•}[−1] ⟶ Q^{•} ⟶ P^{•} ⟶ νP^{•}.

Proposition 2.2. Let P^{•} be a bounded, indecomposable object of C^{−,b}(P_{A}) that is not contractible. Then there is an almost split sequence in C^{−,b}(P_{A}) of the form

(1)  0 ⟶ νP^{•}[−1] ⟶^{u} Q^{•} ⟶^{v} P^{•} ⟶ 0.

Moreover, if P^{•} is not homology-projective, then this sequence is proper.

Proof. Let w : P^{•} → νP^{•} be a chain map such that the homotopy class [w] is the map occurring in the Auslander–Reiten triangle (∗) determined by P^{•}. Let Q_{1}^{•} be the projective cover of νP^{•} in the category C(A) so that Q_{1}^{•} is contractible and Q_{1}^{n} = νP^{n−1} ⊕ νP^{n}. Then there is a commutative diagram

(2)  \begin{aligned}0 & ⟶ & νP^{•}[−1] & ⟶^{u} & Q^{•} & ⟶^{v} & P^{•} & ⟶ & 0 \\ & & ∥ & & ↓ & & ↓^{−w} \\ 0 & ⟶ & νP^{•}[−1] & ⟶ & Q_{1}^{•} & ⟶ & νP^{•} & ⟶ & 0, \end{aligned}

and we claim that the top row is almost split in C^{−,b}(P_{A}). Indeed, there is a triangle

(3)  νP^{•}[−1] ⟶^{[u]} Q^{•} ⟶^{[v]} P^{•} ⟶^{[w]} νP^{•}

in K^{−,b}(P_{A}), and this is an Auslander–Reiten triangle by the choice of w. Thus the exact sequence in the top row of (2) is not split.

Let P_{1}^{•} be an indecomposable object of C^{−,b}(P_{A}), and suppose that f : P_{1}^{•} → P^{•} is not a split epimorphism. If P_{1}^{•} is contractible, then it is a projective object in the category of chain complexes. Hence there is a chain map f′ : P_{1}^{•} → Q^{•} with vf′ = f. On the other hand, if P_{1}^{•} is not contractible, then it is a nonzero object in K^{−,b}(P_{A}), and [f] is not a split epimorphism. Since (3) is an Auslander–Reiten triangle, there is a chain map f_{1} : P_{1}^{•} → Q^{•} such that [vf_{1}] = [f]. Hence f − vf_{1} factors through a contractible complex P_{2}^{•} of projective modules. Let ξ : P_{1}^{•} → P_{2}^{•} and λ : P_{2}^{•} → P^{•} be chain maps with f − vf_{1} = λξ. Since P_{2}^{•} is projective in the category of chain complexes, there is a chain map λ′ : P_{2}^{•} → Q^{•} with vλ′ = λ. Set f′ = f_{1} + λ′ξ. Then vf′ = vf_{1} + vλ′ξ = vf_{1} + λξ = f, as desired.

Finally, we must show that (1) is a proper short exact sequence if P^{•} is not homology-projective. Since the sequence of n-cocycles corresponding to a short exact sequence of chain complexes is always left exact, it suffices to show that the induced map Z^{n}(v) : Z^{n}(Q^{•}) → Z^{n}(P^{•}) is an epimorphism for all n. Let Q_{n}^{•} denote the complex such that Q_{n}^{i} = 0 for i ≠ n and Q_{n}^{n} is the projective cover of Z^{n}(P^{•}). Then the epimorphism Q_{n}^{n} → Z^{n}(P^{•}) gives a chain map g : Q_{n}^{•} → P^{•}, and g is not a split epimorphism if P^{•} is not homology-projective. Hence we can write g = vg′ for some chain map g′ : Q_{n}^{•} → Q^{•}. Since the image of g′ must lie in Z^{n}(Q^{•}), it follows that Z^{n}(v) is an epimorphism. This completes the proof. □

The following result is now immediate from Proposition 2.2.

Corollary 2.3. Let P^{•} be a bounded, indecomposable object of C^{−,b}(P_{A}) that is not contractible, and let

0 ⟶ νP^{•}[−1] ⟶^{u} Q^{•} ⟶^{v} P^{•} ⟶ 0

denote the corresponding almost split sequence in C^{−,b}(P_{A}).

(1) There is an Auslander–Reiten triangle in K^{−,b}(P_{A}) of the form

νP^{•}[−1] ⟶^{[u]} Q^{•} ⟶^{[v]} P^{•} ⟶ νP^{•}.

(2) If P^{•} is not homology-projective, then there is an Auslander–Reiten triangle in \overline{C}^{−,b}(P_{A}) of the form

νP^{•}[−1] ⟶^{\overline{u}} Q^{•} ⟶^{\overline{v}} P^{•} ⟶ Ω^{−1}νP^{•}[−1].

The remainder of this section is devoted to determining the Auslander–Reiten triangles corresponding to certain specific complexes in K^{b}(P_{A}) that will be important in the next section. We begin with a general lemma.

Lemma 2.4. Let P^{•} be a nonzero indecomposable object of K^{b}(P_{A}), and let n be the smallest index such that P^{n} ≠ 0. If a triangle

(1)  νP^{•}[−1] ⟶ Q^{•} ⟶ P^{•} ⟶^{[w]} νP^{•}

is an Auslander–Reiten triangle, then w is homotopic to a map f such that f^{i} = 0 for i ≠ n and f^{n} induces a retraction of P^{n}/ Rad P^{n} onto a simple summand of soc(νP^{n}).

Proof. Write P^{n}/ Rad P^{n} ≅ soc(νP^{n}) = S ⊕ T, where S is simple. Then P^{n} = P_{S} ⊕ P_{T}, where P_{S} and P_{T} are projective covers of S and T respectively. Suppose that (1) is an Auslander–Reiten triangle. Let P′ be the projective cover of Rad P_{S}⊕ P_{T} ⊆ P^{n}, and let π : P′ → P^{n} be the corresponding map. Consider the complex Q_{1}^{•} with Q_{1}^{n} = P′ and Q_{1}^{i} = P^{i} for i ≠ n; the boundary maps in Q_{1}^{•} are given by ∂π : Q_{1}^{n} → Q_{1}^{n+1} in degree n and by the boundary map for P^{•} in all other degrees. Then there is a chain map g : Q_{1}^{•} → P^{•} given by g^{n} = π and g^{i} = 1_{P^{i}} for i ≠ n, and g is clearly not a split epimorphism. Hence [wg] = 0. It follows that w is homotopic to a chain map f : P^{•} → νP^{•} such that f^{i} = 0 for i ≠ n and f^{n} vanishes on the image of π. Since f ≠ 0, we see that f^{n} induces a retraction of P^{n}/ Rad P^{n} onto S. This completes the proof. □

Now fix a projective indecomposable A-module P that is not simple, and let d : P → νP be any A-homomorphism that induces an isomorphism P/ Rad P ≅ soc(νP). For n ≥ 0 let Q_{n}^{•} be the complex

⋯ ⟶ 0 ⟶ P ⟶^{d} νP ⟶^{νd} ⋯ ⟶^{ν^{n−1}d} ν^{n}P ⟶ 0 ⟶ ⋯

with Q_{n}^{0} = P. It is easy to see that Q_{n}^{•} is indecomposable.

Lemma 2.5. If β ∈ End_{A}(P), then for any n there is a chain map α : Q_{n}^{•} → Q_{n}^{•} with α^{0} = β. Moreover, α is an automorphism if and only if β is an automorphism.

Proof. We begin by proving the existence of a chain map α with α^{0} = β. If β is not an automorphism, then dβ = 0, and the desired chain map is obtained by setting α^{0} = β and α^{i} = 0 for i ≠ 0. Thus we may assume that β is an automorphism.

Set α^{i} = 0 for i < 0, and set α^{0} = β. Assume by induction that we have defined automorphisms α^{i} : Q_{n}^{i} → Q_{n}^{i} for i ≤ m in such a way that the α^{i} commute with the boundary maps in Q_{n}^{•}. If Q_{n}^{m+1} = 0, then we can simply set α^{m+1} = 0. If Q_{n}^{m+1} ≠ 0, then (ν^{m}d)α^{m} and ν^{m}d both induce isomorphisms ν^{m}P/ Rad ν^{m}P ≅ soc(ν^{m+1}P). Since ν^{m+1}P is injective, there is an automorphism α^{m+1} : ν^{m+1}P → ν^{m+1}P such that (ν^{m}d)α^{m} = α^{m+1}(ν^{m}d). Hence we get the desired chain map.

Finally, since P and Q_{n}^{•} are both indecomposable, their endomorphisms are either nilpotent or invertible. Hence α is an automorphism if and only if β is an automorphism. □

Now let w : Q_{n}^{•} → νQ_{n}^{•} be the chain map given by w^{0} = d and w^{i} = 0 for i ≠ 0. We claim that there is an Auslander–Reiten triangle in K^{b}(P_{A}) of the form

νQ_{n}^{•}[−1] ⟶ Q^{•} ⟶ Q_{n}^{•} ⟶^{[w]} νQ_{n}^{•}.

Indeed, suppose that γ : Q_{n}^{•} → νQ_{n}^{•} is a chain map such that [γ] is the third map in an Auslander–Reiten triangle. By Lemma 2.4 we may assume that γ^{i} = 0 for i ≠ 0, and γ^{0} induces an isomorphism P/ Rad P ≅ soc(νP). Let S = P/ Rad P so that S is a simple A-module, and let π : P → S denote the canonical epimorphism. Then we can factor w^{0} as w^{0} = w′π for some map w′ : S → νP. It follows that γ^{0} factors as γ^{0} = w′β′π for some automorphism β′ of S. Because P is projective, there is an automorphism β of P with β′π = πβ, and Lemma 2.5 implies that there is an automorphism α of Q_{n}^{•} with α^{0} = β. Hence γ = wα, and we get an isomorphism of triangles

νQ_{n}^{•}[−1] ⟶ Q′^{•} ⟶ Q_{n}^{•} ⟶^{[γ]} νQ_{n}^{•}
∥  ↓  ↓^{[α]}  ∥
νQ_{n}^{•}[−1] ⟶ Q^{•} ⟶ Q_{n}^{•} ⟶_{[w]} νQ_{n}^{•}.

In particular, the bottom row of this diagram is an Auslander–Reiten triangle, as claimed.

Finally, it is easy to check that for n > 0 the mapping cone of w is isomorphic to νQ_{n−1}^{•} ⊕ Q_{n+1}^{•}[1]. Thus we have proven the following result.

Proposition 2.6. There are Auslander–Reiten triangles in K^{b}(P_{A}) of the form

νQ_{n}^{•}[−1] ⟶ νQ_{n−1}^{•}[−1] ⊕ Q_{n+1}^{•} ⟶ Q_{n}^{•} ⟶^{[w]} νQ_{n}^{•}

for n > 0 and

νQ_{0}^{•}[−1] ⟶ Q_{1}^{•} ⟶ Q_{0}^{•} ⟶^{[w]} νQ_{0}^{•}

for n = 0.

§3. The Auslander–Reiten Quiver

Our main goal in this section is to give a complete description of the connected components of the Auslander–Reiten quiver of \overline{C}^{b}(P_{A}). Because the Auslander– Reiten triangles in \overline{C}^{b}(P_{A}) are so closely related to those of K^{b}(P_{A}) by Corollary 2.3, we will obtain at the same time a description of the quiver for K^{b}(P_{A}). We begin by recalling some of the relevant definitions.

Let C be an additive category, and suppose that U and V are indecomposable objects of C. A morphism f : U → V is said to be irreducible if f is not an isomorphism and for any factorization f = f_{1}f_{2}, either f_{2} is a split monomorphism or f_{1} is a split epimorphism. If C has an Auslander–Reiten triangle or an almost split sequence terminating in V, then the irreducible maps into V can be described in terms of this triangle or exact sequence. For example, suppose that

0 ⟶ W ⟶ X ⟶^{g} V ⟶ 0

is almost split. Then f : U → V is irreducible if and only if there is a split monomorphism ι : U → X such that f = gι. The analogous result also holds for Auslander–Reiten triangles (Proposition I.4.3 of [2]).

In particular, suppose that P_{0}^{•} and P^{•} are indecomposable objects of C^{b}(P_{A}), and let f : P_{0}^{•} → P^{•} be irreducible. Then Corollary 2.3 gives the following conclusions:

(1) If P_{0}^{•} and P^{•} are not contractible, then [f] : P_{0}^{•} → P^{•} is irreducible in K^{b}(P_{A}).

(2) If P_{0}^{•} and P^{•} are not homology-projective, then \overline{f} : P_{0}^{•} → P^{•} is irreducible in \overline{C}^{b}(P_{A}).

The following proposition describes the irreducible maps in C^{b}(P_{A}) involving a contractible complex. It says in effect that if we consider any almost split sequence in C^{b}(P_{A}), then the middle term contains no contractible summands except in the case where the last term consists of a simple projective module in a single degree.

Proposition 3.1. Let P^{•} and Q^{•} be indecomposable complexes in C^{b}(P_{A}), and suppose that P^{•} is contractible. If there is an irreducible map from Q^{•} to P^{•} or from P^{•} to Q^{•}, then there is an integer n such that P^{n} is simple.

Proof. It suffices by duality to consider the case in which there is an irreducible map f : P^{•} → Q^{•}. By shifting P^{•} and Q^{•} if necessary, we may assume that Q^{i} ≠ 0 if and only if −m ≤ i ≤ 0. Let M = H^{0}(Q^{•}).

First suppose that M = 0. Then Q^{•} is a contractible complex concentrated in degrees −1 and 0. Since f ≠ 0, it follows that P^{•} is concentrated either in degrees −1 and 0 or in degrees 0 and 1. In the first case it is easy to see that f factors through the complex

⋯ ⟶ 0 ⟶ P^{0} ⟶^{f^{0}} Q^{0} ⟶ 0 ⟶ ⋯

concentrated in degrees −1 and 0. In the second case f factors through the complex with Q^{0} in degree zero and 0 in all other degrees. Thus we get a contradiction.

Now suppose that M ≠ 0, and let ι_{M} : M → I(M) be the injective hull of M. If π : Q^{0} → M denotes the canonical projection, then we can define a new complex Q_{1}^{•} in C^{b}(P_{A}) by setting Q_{1}^{i} = Q^{i} for i ≠ 1, Q_{1}^{1} = I(M), ∂_{1}^{i} = ∂^{i} for i ≠ 0, and ∂_{1}^{0} = ι_{M}π : Q_{1}^{0} → Q_{1}^{1}. The chain map ϵ : Q_{1}^{•} → Q^{•} with ϵ^{i} = 1_{Q^{i}} for −m ≤ i ≤ 0 is an epimorphism in C^{b}(P_{A}). Since P^{•} is a projective object it follows that there is a chain map g : P^{•} → Q_{1}^{•} with ϵg = f. But f is irreducible and ϵ is certainly not a split epimorphism, so it follows that g must be a split monomorphism. Since Q_{1}^{•} is indecomposable, g is actually an isomorphism. In particular, it follows that P^{i} = 0 for i ≠ 0, 1, and Q^{i} = 0 for i ≠ 0.

Let d : Q^{0} → νQ^{0} be a map inducing an isomorphism Q^{0}/ Rad Q^{0} ≅ soc(νQ^{0}). Then f : P^{•} → Q^{•} factors as

⋯ ⟶ 0 ⟶ P^{0} ⟶^{1_{P^{0}}} P^{0} ⟶ 0 ⟶ ⋯
↓  ^{f^{0}}↓  ↓^{df^{0}}  ↓
⋯ ⟶ 0 ⟶ Q^{0} ⟶^{d} νQ^{0} ⟶ 0 ⟶ ⋯
↓  ^{1_{Q^{0}}} ↓  ↓  ↓
⋯ ⟶ 0 ⟶ Q^{0} ⟶ 0 ⟶ 0 ⟶ ⋯,

so the irreducibility of f implies that the middle complex is contractible and hence Q^{0} is simple. Since f^{0} : P^{0} → Q^{0} is nonzero, it must be an isomorphism. This completes the proof. □

Now suppose that C is a triangulated category satisfying the Krull–Schmidt Theorem and having Auslander–Reiten triangles. Then the Auslander–Reiten quiver of C is a quiver whose vertices are isomorphism classes of indecomposable objects of C. There is an arrow U → V in the quiver whenever there is an irreducible map from U to V. Associated with each such arrow is an ordered pair of numbers (a_{UV} , a′_{UV}), called the valuation. If Irr (U, V) = Rad (U, V)/Rad^{2}(U, V) is the space of irreducible maps, then a_{UV} is the length of Irr (U, V) as a left End(V)-module, and a′_{UV} is the length as a right End(U)-module. We can also describe these numbers in terms of Auslander–Reiten triangles: if

W ⟶ X ⟶ U ⟶ T W

is an Auslander–Reiten triangle in C, then a_{W V} = a′_{V U} is the multiplicity of V as a summand of X (See Section I.4 of [2], where the Auslander–Reiten quiver is referred to simply as the quiver of a Krull–Schmidt category).

By Corollary 2.3 we know that the Auslander–Reiten quiver of \overline{C}^{b}(P_{A}) is obtained from that of K^{b}(P_{A}) by deleting the vertices corresponding to homologyprojective objects. These Auslander–Reiten quivers are valued stable representation quivers, so the structure theorem of Riedtmann [6] implies that each connected component is of the form ℤT/Γ, where T is a labelled tree, ℤT is the covering quiver of T, and Γ is an admissible group of automorphisms of ℤT. The labelled tree is uniquely determined by the quiver and is called its tree class. We shall be primarily interested in quivers with tree class A_{1} or A_{∞}. A connected quiver is of type A_{1} if and only if it contains a single vertex and no arrows. The quiver ℤA_{∞} has vertices {(m, n) | m ∈ ℤ, n ∈ ℕ}; there are arrows (m, n) → (m + 1, n + 1) and (m, n + 1) → (m, n) for all m ∈ ℤ and n ∈ ℕ, and each arrow carries the valuation (1, 1).

Let Q be a valued stable representation quiver with translation τ, and let Q_{0} denote the set of vertices of Q. For any vertices u, v ∈ Q_{0} with an arrow u → v, let (a_{uv}, a′_{uv}) be the corresponding valuation. If v ∈ Q_{0}, let v^{−} be the set of all vertices w such that there is an arrow w → v. A subadditive function on Q is defined to be a function f : Q_{0} → ℕ such that

f(v) + f(τv) ≥ ∑_{w∈v^{−}} a′_{wv}f(w)

for all v ∈ Q_{0}. If equality always holds, then f is said to be additive.

If P^{•} is a complex in C^{b}(P_{A}), then there are complexes P_{1}^{•} and P_{2}^{•} such that P_{1}^{•} has no contractible summands, P_{2}^{•} is contractible, and P^{•} = P_{1}^{•} ⊕ P_{2}^{•}. Let ℓ(P^{•}) denote the composition length of the module ⨁_{n∈ℤ} P_{1}^{n}. Then ℓ is constant on isomorphism classes in K^{b}(P_{A}).

Proposition 3.2. Let P^{•} be an indecomposable homology-projective complex in K^{b}(P_{A}) that is not contractible, and let C be the component of the Auslander– Reiten quiver containing P^{•}.

(1) If ℓ(P^{•}) = 1, then P^{•} is the only object in the component C, and C is of type A_{1}.

(2) If ℓ(P^{•}) > 1, then C ≅ ℤA_{∞}, and ℓ is unbounded on C.

Proof. First suppose that ℓ(P^{•}) = 1. Then P^{•} consists of a simple projective module in a single degree, and there is an Auslander–Reiten triangle

P^{•}[−1] ⟶ 0 ⟶ P^{•} ⟶^{1_{P^{•}}} P^{•}.

Thus there are no arrows in C terminating at P^{•}. Similarly, there are no arrows starting at P^{•}, and (1) follows.

Now suppose that ℓ(P^{•}) > 1. We may assume without loss of generality that P^{0} ≠ 0 and P^{i} = 0 for i ≠ 0. Then P^{0} is a projective indecomposable Amodule that is not simple. Let d : P^{0} → νP^{0} be a map inducing an isomorphism P^{0}/ Rad P^{0} ≅ soc(νP^{0}). For n ≥ 0 let Q_{n}^{•} be the complex with Q_{n}^{i} = ν^{i}P^{0} for 0 ≤ i ≤ n and Q_{n}^{i} = 0 otherwise; the boundary maps are given by ν^{i}d : Q_{n}^{i} → Q_{n}^{i+1} for 0 ≤ i ≤ n − 1. In particular, Q_{0}^{•} = P^{•}. Now Proposition 2.6 implies that Q_{1}^{•} is the only predecessor of Q_{0}^{•} in C; if n > 0, then the predecessors of Q_{n}^{•} are νQ_{n−1}^{•}[−1] and Q_{n+1}^{•}. More generally, for any i ∈ ℤ the complex ν^{i}Q_{1}^{•}[−i] is the only predecessor of ν^{i}Q_{0}^{•}[−i]; if n > 0, then the predecessors of ν^{i}Q_{n}^{•}[−i] are ν^{i+1}Q_{n−1}^{•}[−i − 1] and ν^{i}Q_{n+1}^{•}[−i]. Thus it follows that C ≅ ℤA_{∞}. Moreover, ℓ is unbounded on C since it is unbounded on {Q_{n}^{•} | n = 0, 1, 2, … }. This completes the proof. □

The next two lemmas are well-known in the case of modules (For example, see Proposition 4.13.5 and Lemma 4.14.1 of [1]). We omit the proofs since they are the same for the category C^{b}(P_{A}).

Lemma 3.3. Suppose that P^{•} and Q^{•} are indecomposable complexes in C^{b}(P_{A}) that are not contractible. Let f : P^{•} → Q^{•} be a nonzero chain map that is not an isomorphism. Suppose that there is no chain of irreducible maps from P^{•} to Q^{•} of length less than n. Then there exists a chain of irreducible maps

P^{•} = P_{0}^{•} ⟶^{g_{1}} P_{1}^{•} ⟶^{g_{2}} ⋯ ⟶ P_{n−1}^{•} ⟶^{g_{n}} P_{n}^{•}

and a map h : P_{n}^{•} → Q^{•} with hg_{n} ⋯ g_{1} ≠ 0.

Lemma 3.4. Let P_{0}^{•}, …, P_{2^{n}−1}^{•} be indecomposable objects of C^{b}(P_{A}) that are not contractible. If ℓ(P_{i}^{•}) ≤ n for all i and f_{i} : P_{i−1}^{•} → P_{i}^{•} is not an isomorphism for 1 ≤ i ≤ 2^{n} − 1, then f_{2^{n}−1} ⋯ f_{2}f_{1} = 0.

Auslander has proven that any infinite component of the Auslander–Reiten quiver of a finite-dimensional algebra contains modules with an arbitrarily large number of composition factors (Theorem 4.14.2 of [1]). The proof of the next proposition is based on Auslander’s argument.

Proposition 3.5. Let C be a connected component of the Auslander–Reiten quiver of K^{b}(P_{A}). If C is not of type A_{1}, then ℓ is unbounded on C.

Proof. Suppose that there is an integer n such that ℓ(P^{•}) ≤ n for all P^{•} ∈ C. Fix some indecomposable complex P^{•} ∈ C, and let m be the smallest index such that P^{m} ≠ 0. Choose some indecomposable summand Q of P^{m}, and let π : P^{m} → Q denote the projection onto Q. Let Q^{•} be the complex with Q^{m} = Q and Q^{i} = 0 for i ≠ m. Then π : P^{•} → Q^{•} is a chain map, and π is not homotopic to zero since P^{•} has no contractible summands.

Suppose that Q^{•} ∉ C. Then π cannot be an isomorphism, and by Lemma 3.3 there is a chain of irreducible maps

P^{•} = P_{0}^{•} ⟶^{g_{1}} P_{1}^{•} ⟶^{g_{2}} ⋯ ⟶^{g_{2^{n}−1}} P_{2^{n}−1}^{•}

and a map h : P_{2^{n}−1}^{•} → Q^{•} with hg_{2^{n}−1} ⋯ g_{1} ≠ 0. But this contradicts Lemma 3.4, so it follows that Q^{•} ∈ C. By Proposition 3.2 either C is of type A_{1} or ℓ is unbounded on C. This completes the proof. □

Proposition 3.6. The tree class is of type A_{1} or A_{∞} for every connected component of the Auslander–Reiten quiver of K^{b}(P_{A}); for \overline{C}^{b}(P_{A}) it is always of type A_{∞}.

Proof. If P_{0}^{•} is any complex in C^{b}(P_{A}), write P_{0}^{•} = P_{1}^{•} ⊕ P_{2}^{•}, where P_{1}^{•} has no contractible summands and P_{2}^{•} is contractible. Define ℓ′(P_{0}^{•}) to be the number of projective indecomposable summands of the module ⨁_{n∈ℤ} P_{1}^{n} (counting multiplicities).

Let P^{•} be an indecomposable complex in K^{b}(P_{A}) that is not contractible. If P^{•} consists of a simple projective module in a single degree, then ℓ′(P^{•}) = ℓ(P^{•}) = 1, and the tree class of the component containing P^{•} is of type A_{1} by Proposition 3.2. In \overline{C}^{b}(P_{A}), of course, such an object is isomorphic to zero so that this case does not arise.

Now suppose that P^{•} does not consist of a simple projective module in a single degree. Let C be the component of the Auslander–Reiten quiver of K^{b}(P_{A}) containing P^{•}. Then C contains at least one other complex, so it is not of type A_{1}. Proposition 3.5 implies that ℓ is unbounded on C and hence so is ℓ′. Let

0 ⟶ νP^{•}[−1] ⟶ Q_{1}^{•} ⟶ P^{•} ⟶ 0

denote the almost split sequence in C^{b}(P_{A}) terminating at P^{•}. By Proposition 3.1 we know that Q_{1}^{•} has no contractible summands. Since ℓ′(P^{•}) = ℓ′(νP^{•}[−1]), it follows that 2ℓ′(P^{•}) = ℓ′(Q_{1}^{•}) and hence

2ℓ′(P^{•}) = ∑_{Q^{•}∈(P^{•})^{−}} a′_{Q^{•}P^{•}} ℓ′(Q^{•}).

Thus ℓ′ defines an unbounded additive function on the tree class of C, so the tree class is of type A_{∞} by Theorem 4.5.8 of [1]. Since the Auslander–Reiten quiver of \overline{C}^{b}(P_{A}) is obtained from that of K^{b}(P_{A}) by deleting the homology-projectives, the same holds for \overline{C}^{b}(P_{A}) in this case. Thus we are done. □

Finally, we end by proving the theorem stated in the introduction.

Theorem 3.7. Every component of the Auslander–Reiten quiver of K^{b}(P_{A}) containing more than one vertex is isomorphic to ℤA_{∞}. In \overline{C}^{b}(P_{A}) every component of the Auslander–Reiten quiver is isomorphic to ℤA_{∞}.

Proof. Let C be either a component of the Auslander–Reiten quiver of K^{b}(P_{A}) containing more than one vertex or a component of the Auslander–Reiten quiver of \overline{C}^{b}(P_{A}). Then we know by Proposition 3.6 that C has tree class A_{∞} so that C is isomorphic to ℤA_{∞}/Γ for some admissible group of automorphisms Γ of ℤA_{∞}. Now C clearly contains no periodic vertices since the functor P^{•} ↦ νP^{•}[−1] shifts the degrees in which the complex is concentrated. If γ ∈ Γ, then γ preserves the set of vertices at the end of ℤA_{∞}, so it must act as the identity on each vertex in this set. It is now easy to see that γ = 1 so that C ≅ ℤA_{∞}, as desired. □

Acknowledgements

Portions of the work presented here were done during visits to the University of Warwick, the University of Virginia, and the University of Sydney, and I would like to express my gratitude for the hospitality shown to me at all three institutions. In addition, particular thanks are due to Professor Leonard Scott for a number of helpful conversations regarding categories, functors, and almost split sequences.

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