CONSEQUENCES OF THE EXISTENCE OF

AUSLANDER-REITEN TRIANGLES WITH APPLICATIONS

TO PERFECT COMPLEXES FOR SELF-INJECTIVE ALGEBRAS

PETER WEBB

Abstract. In a k-linear triangulated category (where k is a field) we show that the existence of Auslander-Reiten triangles implies that objects are determined, up to shift, by knowing dimensions of homomorphisms between them. In most cases the objects themselves are distinguished by this information, a conclusion which was also reached under slightly different hypotheses in a theorem of Jensen, Su and Zimmermann. The approach is to consider bilinear forms on Grothendieck groups which are analogous to the Green ring of a finite group.

We specialize to the category of perfect complexes for a self-injective algebra, for which the Auslander-Reiten quiver has a known shape. We characterize the position in the quiver of many kinds of perfect complexes, including those of lengths 1, 2 and 3, rigid complexes and truncated projective resolutions. We describe completely the quiver components which contain projective modules. We obtain relationships between the homology of complexes at different places in the quiver, deducing that every self-injective algebra of radical length at least 3 has indecomposable perfect complexes with arbitrarily large homology in any given degree. We find also that homology stabilizes away from the rim of the quiver. We show that when the algebra is symmetric, one of the forms considered earlier is Hermitian, and this allows us to compute its values knowing them only on objects on the rim of the quiver.

1. Introduction

We will consider triangulated categories which are k-linear for some algebraically closed field k and for which the Krull-Schmidt theorem holds, meaning that every object is a finite direct sum of indecomposable objects, and endomorphism rings of indecomposable objects are local. We will also assume that Auslander-Reiten triangles exist. An important example of such a category for us is the category D^{b}(Λ-proj) of perfect complexes of Λ-modules with homotopy classes of maps, where Λ is a self-injective finite

2000 Mathematics Subject Classification. Primary 16G70; Secondary 18E30, 20C20. Key words and phrases. Auslander-Reiten triangle, self-injective algebra, perfect

complex.

dimensional k-algebra. Other categories of interest include cluster categories, the stable module category of Λ-modules when Λ is self-injective, and D^{b}(Λ-mod) when Λ is an algebra of finite global dimension.

We approach such triangulated categories from two angles. We first consider the role of the Auslander-Reiten triangles in the context of two related Grothendieck groups, analogous to the Green ring in group representation theory. These Grothendieck groups carry bilinear forms determined by the dimensions of homomorphisms between complexes. We show that the bilinear forms constructed in this way are non-degenerate in a certain sense (Corollary 3.2). We also show that for the second Grothendieck group we construct, in the case of D^{b}(Λ-proj) when Λ is symmetric, the form is Hermitian and the Auslander-Reiten triangles provide a set of elements dual to the basis determined by indecomposable complexes (Theorem 4.2). This development is very much in the spirit of [6].

We next pay attention to D^{b}(Λ-proj) when Λ is a self-injective algebra. To explain the notation, Λ-proj denotes the full subcategory of all Λ-modules whose objects are finitely generated projective, and D^{b}(Λ-proj) may be taken to be the category whose objects are finite complexes of finitely generated projective modules – the perfect complexes – with homotopy classes of chain maps as the morphisms. Such perfect complexes appear in many places in the representation theory of algebras and elsewhere, with many applications. They arise as tilting complexes in Rickard’s theory [11] of derived equivalences of rings. They are the compact objects in the bounded derived category of a ring [11, 6.3]. In the context of free group actions on topological spaces, a CW complex on which a group acts freely has a chain complex which is a perfect complex of modules for the group algebra (see [1] for a notable contribution in this area). In the theory of subgroup complexes of a finite group, the chain complex of the poset of non-identity p-subgroups, over a p-local ring, is homotopy equivalent to a perfect complex of modules for the group algebra [12]. These are just a few examples.

When Λ is self-injective it is known that D^{b}(Λ-proj) has Auslander-Reiten triangles [7], and furthermore that the Auslander-Reiten quiver has shape ℤA_{∞} [13]. We exploit this to study the position of perfect complexes of various kinds in their Auslander-Reiten quiver component. We completely describe all the complexes in quiver components which contain projective modules (Theorem 6.6). For complexes of length 2 or 3 we show that they all lie within distance 2 of the rim, and we describe which ones lie on the rim, which lie at distance 1 and which lie at distance 2 (Corollary 6.8 and Proposition 7.3). In general, complexes at distance n from the rim must have length at least n + 1 (Corollary 6.10). We prove this by studying the homology modules of complexes in the interior of the quiver component, showing that they have composition factors which are the union of the composition factors of the homology modules lying on the rim of the corresponding ‘wing’ in the quiver (Proposition 6.4). We show also that homology modules stabilize, in a certain sense, away from the rim of the quiver (Theorem 6.5), so that to each perfect complex there is associated a stabilization module.

Some of our conclusions can be expressed without the formalism of the triangulated categories. We prove that when Λ is a symmetric algebra of radical length at least 3 there are always indecomposable complexes with zero homology of arbitrarily large dimension (Theorem 6.13). We also show that for each Auslander-Reiten sequence of Λ modules there is a three-term perfect complex which has the three modules in the sequence as its homology groups (Corollary 7.2). Given three modules appearing in a short exact sequence it is not, in general, possible to realize them as homology of a perfect complex of length 3, so the fact that it can be done for Auslander-Reiten sequences is a positive result. The construction of the three-term complex also provides an algorithm (which works for an arbitrary finite dimensional algebra) for constructing Auslander-Reiten sequences on a computer.

Here is the organization of this paper in greater detail. In Section 2 we present the basic property of Auslander-Reiten triangles with respect to homomorphisms on which the subseqent arguments depend (Lemma 2.1). The setting is that of a Krull-Schmidt triangulated category with at least one Auslander-Reiten triangle. In Section 3 we define the first Grothendieck group in the same generality, and show that the defining property of the Auslander-Reiten triangles implies a non-singularity property of the bilinear form given by dimensions of homomorphisms (Corollary 3.2). We deduce that in many situations objects are determined by knowing the dimensions of homomorphisms to other objects (Corollary 3.3). This is something which is analogous to the approach of Benson and Parker [6] in their study of the Green ring of group representations and which has been proven for triangulated categories with slightly different conditions by Jensen, Su and Zimmermann [9].

In Section 4 we modify the Grothendieck group so that it becomes a ℚ(t)- vector space on which t acts as the shift operator. With respect to the field automorphism determined by t ↦ t^{−1} the earlier bilinear form on the first Grothendieck group induces a sesquilinear bilinear form on the modified group, which in case our triangulated category is the category of perfect complexes for a symmetric algebra turns out to be Hermitian (Theorem 4.2). In Section 5 we work with perfect complexes for a symmetric algebra, and use the Hermitian property of the form to calculate its values, showing that they only depend on values of complexes on the rim of the Auslander-Reiten quiver (Theorem 5.1).

The contents of Sections 6 and 7 have already been largely described, having to do with the position of complexes in the quiver and properties of their homology. Apart from the results so far mentioned we point out a characterization of complexes of length n + 1 which lie at the maximal distance n from the rim of the quiver (Corollary 6.10). These complexes only appear in quiver components containing projective modules, and they are completely described. We also show that truncated projective resolutions lie on the rim, except in one situation (Proposition 6.12). In Section 8 we show that rigid complexes lie on the rim of the quiver (Theorem 8.2).

Auslander-Reiten triangles were introduced by Happel in the 1980s, and his book [7] is a very good place to read about them and find background material on triangulated categories. The definition and first properties of these triangles can be found there. Happel also shows that in the bounded derived category an Auslander-Reiten triangle X → Y → Z → X[1] exists if and only if Z is isomorphic to a perfect complex, which happens if and only if X is isomorphic to a finite complex of finitely generated injective modules. From this it follows that Auslander-Reiten triangles exist in D^{b}(Λ-proj) when Λ is self-injective. He describes a construction of these triangles in the bounded derived category D^{b}(Λ-mod), which also works for the perfect complexes D^{b}(Λ-proj) and which we will use all the time. The construction uses the Nakayama functor ν : Λ-mod → Λ-mod which is described in [2]. This functor is extended to complexes by applying ν to each term and each morphism in the complex, and the resulting functor D^{b}(Λ-proj) → D^{b}(Λ-inj) is also denoted ν.

2. The basic lemma

We assume that C is a Krull-Schmidt k-linear triangulated category which is Hom-finite, that is, the homomorphism spaces between objects are finite dimensional over a field k. For convenience we assume that k is algebraically closed.

Many of our results depend upon the following observation, which follows directly from the definition of an Auslander-Reiten triangle.

Lemma 2.1. Let X → Y → Z → X[1] be an Auslander-Reiten triangle in a k-linear triangulated category C and let W be an indecomposable object of C. Consider the long exact sequence obtained by applying Hom_{C}(W, −) to the triangle.

(1) If W ≅ Z̸[r] for any r then the long exact sequence is a splice of short exact sequences

0 → Hom(W, X[n]) → Hom(W, Y [n]) → Hom(W, Z[n]) → 0.

(2) If W ≅ Z[r] for some r and Z ≅ Z̸[1] the long exact sequence is still the splice of short short exact sequences as above, except that the sequences for n = r and r + 1 combine to give a 6-term exact sequence whose middle connecting homomorphism δ has rank 1:

0 → Hom(W, X[r]) → Hom(W, Y [r]) → Hom(W, Z[r])
⟶^{δ} Hom(W, X[r + 1]) → Hom(W, Y [r + 1]) → Hom(W, Z[r + 1])
→0.

(3) If W ≅ Z ≅ Z[1] the long exact sequence becomes a repeating exact sequence with three terms:

Hom(W, X) → Hom(W, Y) → Hom(W, Z) ⟶^{δ} Hom(W, X) → ⋯

where the connecting homomorphism δ has rank 1. The dual result on applying Hom_{C}(−, W) to the triangle also holds.

Proof. In the long exact sequence

[figure: …; β_{r−1}; Hom(W, Z[r − 1]); Hom(W, X[r]); Hom(W, Y [r]); β_{r}; Hom(W, Z[r]); Hom(W, X[r + 1]); Hom(W, Y [r + 1]); β_{r+1}; Hom(W, Z[r + 1]); Hom(W, X[r + 2]); ⋯]

the map β_{r} is surjective unless W ≅ Z[r]. Thus if W ≅ Z̸[r] for all r, the connecting homomorphisms are all zero, which forces the long sequence to be a splice of short exact sequences. In the case where W = Z[r] for some r, the map β_{r} has image Rad End(W) and cokernel of dimension 1, from the lifting property of an Auslander-Reiten triangle. If Z ≅ Z̸[1] it follows that Z[s] ≅ Z̸[s + 1] for any s, so Z[r − 1] ≅ W̸ ≅ Z̸[r + 1] and both β_{r−1} and β_{r+1} are surjective, giving a six-term exact sequence. When Z ≅ Z[1] then all shifts of Z are isomorphic, as are the shifts of X and of Y. Evidently the long exact sequence becomes the three-term sequence shown. □

We deduce an immediate consequence of this lemma for the homology of complexes of Λ-modules. Observe that the functor which sends such a complex C to its nth homology group H_{n}(C) is representable, since H_{n}(C) = Hom_{D^{b}(Λ)}(Λ[n], C). Here Λ[n] denotes the complex which is zero in every position except for position n, where it is Λ.

Corollary 2.2. Let X → Y → Z → X[1] be an Auslander-Reiten triangle in D^{b}(Λ-mod) or D^{b}(Λ-proj) where Λ is a finite dimensional k-algebra.. The long exact sequence of homology groups of this triangle breaks up as the splice of short exact sequences of the form 0 → H_{n}(X) → H_{n}(Y) → H_{n}(Z) → 0 except when Z is a shift of a projective module.

Proof. This is immediate from Proposition 2.1 using the fact that homology is representable as Hom(Λ, −), so that the long exact sequence is a splice of short exact sequences if Z is not a projective module. □

3. The first Grothendieck group

Again let C be a Krull-Schmidt k-linear triangulated category which is Hom-finite, and where k is algebraically closed. We define A(C) to be the free abelian group with the isomorphism classes [C] of indecomposable objects C as basis. If C = C_{1} ⊕ ⋯ ⊕ C_{n} we put [C] = [C_{1}] + ⋯ + [C_{n}]. We define a bilinear form

⟨ , ⟩ : A(C) × A(C) → ℤ

by ⟨[C], [D]⟩ := dim Hom_{C}(C, D). If X → Y → Z → X[1] is a triangle in C we put Ẑ := [Z] + [X] − [Y] in A(C).

Proposition 3.1. Let X → Y → Z → X[1] be an Auslander-Reiten triangle in C and let W be an indecomposable object. If Z ≅ Z̸[1] then

⟨W, Ẑ⟩ = \begin{cases}1 & if W ≅ Z or Z[−1] \\ 0 & otherwise. \end{cases}

If Z ≅ Z[1] then

⟨W, Ẑ⟩ = \begin{cases}2 & if W ≅ Z \\ 0 & otherwise. \end{cases}

Proof. Observe that ⟨W, Ẑ⟩ is the alternating sum of the dimensions of the vector spaces in the (not necessarily exact) sequence

0 → Hom(W, X) ⟶^{α_{0}} Hom(W, Y) ⟶^{β_{0}} Hom(W, Z) → 0.

From Lemma 2.1 this is a short exact sequence unless W ≅ Z or Z[−1], and hence apart from these cases ⟨W, Ẑ⟩ = 0. Assuming that Z ≅ Z̸[1], if W ≅ Z then the sequence is exact except at the right, where Cokerβ_{0} has dimension 1, and if W ≅ Z[−1] then the sequence is exact except at the left, where Ker α_{0} has dimension 1. Thus in these cases ⟨W, Ẑ⟩ = 1. Finally when W ≅ Z ≅ Z[1] we see from Lemma 2.1 that both Ker α_{0} and Cokerβ_{0} have dimension 1, so that ⟨W, Ẑ⟩ = 2. □

Corollary 3.2. Suppose that C has Auslander-Reiten triangles. Let

ϕ : A(C) → A(C)^{∗} := Hom(A(C), ℤ)

be the map given by [W] ↦ ([X] ↦ ⟨[W], [X]⟩). Let I be the set of shift orbits of isomorphism classes of objects in C, and for each orbit O ∈ I let A_{O} be the span in A(C) of the [M] where M belongs to orbit O. Then ϕ(A(C)) is the direct sum ⨁_{O∈I} ϕ(A_{O}). Furthermore, if O is infinite or has odd length, then the restriction of ϕ to A_{O} is injective.

Proof. As W and Z range through the shifts of some single fixed object the values of ⟨W, Ẑ⟩ are the entries of the matrix

(\begin{matrix}. & . & . \\ & & & 1 \\ & & & 1 & 1 \\ & & & & 1 & 1 \\ & & & & & & . & . & . \end{matrix})

when the orbit is infinite. The matrix is zero except on the leading diagonal and the diagonal immediately below. In the case of a finite shift orbit of length > 1 the matrix is a circulant matrix with the same entries except for a 1 in the top right corner, and the matrix is (2) in the case of a shift orbit of length 1. If W and Z are not in the same shift orbit then ⟨W, Ẑ⟩ = 0. The columns of these matrices give the values of the function ϕ(W) on the Ẑ. This shows, first of all, that ϕ sends objects from different shift orbits to independent functions. Furthermore, the matrix indicated has independent columns if either it is infinite, or if it is finite of odd size, so that in these cases the corresponding shift orbit is mapped injectively to A(C)^{∗}. □

The last corollary is a statement about the non-degeneracy of the bilinear form we have constructed. We may reword it in the following more elementary fashion.

Corollary 3.3. Let C be a Krull-Schmidt k-linear triangulated category which is Hom-finite and which has Auslander-Reiten triangles. If W is an indecomposable object of C then the values of dim Hom(W, Z) as Z ranges over indecomposable objects determine the shift orbit to which W belongs. If the shift orbit containing W is either infinite or of odd length, then the isomorphism type of W is determined by the values of dim Hom(W, Z).

This corollary may be compared with a result of Jensen, Su and Zimmermann [9, Prop. 4], who proved something similar without the hypothesis that C should have Auslander-Reiten triangles, but with the additional requirement that objects W and W′ which we are trying to distinguish should have Hom(W, W′[n]) = 0 for some n. Their approach is based on an earlier result of Bongartz and is particularly appealing because it proceeds by elementary means. It is nevertheless interesting to see the connection between these results and the Auslander-Reiten triangles. For what it is worth, our result applies to categories such as cluster categories, or the stable module category of a group algebra, where the vanishing of a homomorphism space need not hold. In the case of the stable module category of a group algebra the phenomena which can occur were analyzed very fully by Benson and Parker in [6, Sec. 4]. For instance, their theory explains completely the following example. We state it without making reference to group algebras, but when k is a field of characteristic 5 the algebra in the example is isomorphic to the group algebra of a cyclic group of order 5.

Example 3.4. The stable module category for the ring k[X]/(X^{5}) has four indecomposable objects, namely the uniserial modules V_{i} = k[X]/(X^{i}) of dimension i where i = 1, 2, 3, 4. The dimensions of homomorphisms between these objects in the stable category are given in the following table.

V_{1} V_{2} V_{3} V_{4}
V_{1}  1 1 1 1
V_{2}  1 2 2 1
V_{3}  1 2 2 1
V_{4}  1 1 1 1

Modules V_{1} and V_{4} form an orbit of the shift operator, which is the inverse of the syzygy operator, and in this example they cannot be distinguished by dimensions of homomorphisms. The same is true of the modules V_{2} and V_{3}. We can, however, distinguish the shift orbits by means of dimensions of homomorphisms.

4. The second Grothendieck group and a Hermitian bilinear form

In the paper [6] of Benson and Parker it is shown that the almost split sequences give rise to elements of the Green ring which are dual to the standard basis of indecomposable modules with respect to the bilinear form given by dimensions of homomorphisms. We have seen in Section 3 that a similar statement is not immediately true for triangulated categories with Auslander-Reiten triangles, but that something close to this is true, in that the alternating sum of terms in an Auslander-Reiten triangle has non-zero product with two indecomposable objects, rather than just one.

We now show how to modify the bilinear form so that Auslander-Reiten triangles do indeed give dual elements to the standard basis. The approach requires us to modify the Grothendieck group as well. We will then show that for the category of perfect complexes over a symmetric algebra we obtain a Hermitian form, and when the algebra is a group algebra the form behaves well with respect to tensor product of complexes.

As before, let C be a Krull-Schmidt k-linear triangulated category which is Hom-finite, and where k is algebraically closed. We define

A(C)^{t} := (ℤ[t, t^{−1}] ⊗_{ℤ} A(C))/I

where t is an indeterminate and I is the ℤ[t, t^{−1}]-submodule generated by expressions

1 ⊗ [M[i]] − t^{i} ⊗ [M]

for all objects M in C and i ∈ ℤ. We put

A_{ℚ}(C)^{t} := ℚ(t) ⊗_{ℤ[t,t^{−1}]} A(C).

It simplifies the notation to write M instead of [M] in A(C)^{t} and in A_{ℚ}(C)^{t}. The tensor products are extension of scalars, and with this in mind we will write t^{i}M instead of t^{i} ⊗ [M]. Thus in A(C)^{t} and in A_{ℚ}(C)^{t} we have M[i] = t^{i}M, so that t acts as the shift operator.

Proposition 4.1. (1) A(C)^{t} ≅ ⨁_{shift orbits O} A_{O} where A_{O} is a free abelian group with basis the objects in orbit O, regarded as a ℤ[t, t^{−1}]- module via the action t^{i}M = M[i]. We understand that objects are taken up to isomrphism. If the shift orbit O is infinite then A_{O} ≅ ℤ[t, t^{−1}], while if the shift operator has order n on O then A_{O} ≅ ℤ[t, t^{−1}]/(t^{n} − 1) as ℤ[t, t^{−1}]-modules.

(2) For each shift orbit O (taken up to isomorphism) let M_{O} be an object in O. The M_{O} where O is infinite form a basis for A_{ℚ}(C)^{t} over ℚ(t).

Proof. (1) We have A(C) = ⨁_{O} A_{O} where A_{O} is the span of the objects in orbit O. Since each generator of I lies in some ℤ[t, t^{−1}]⊗_{ℤ}A_{O} it follows that A(C)^{t} is the direct sum of the images of the ℤ[t, t^{−1}] ⊗_{ℤ} A_{O} in it. Factoring out I has the effect of identifying each basis element M[i] of A_{O} with t^{i}M. The identification of these spaces when the shift operator has infinite or finite order is immediate.

(2) On tensoring further with ℚ(t) the summands ℤ[t, t^{−1}] become copies of ℚ(t), and the summands ℤ[t, t^{−1}]/(t^{n} − 1) become zero. □

On ℚ(t) we denote by the field automorphism specified by t ↦ t^{−1}. We will now assume that for every pair of objects M and N in C, Hom(M, N[i]) ≠ 0 for only finitely many i. We define a mapping

⟨ , ⟩^{t} : A_{ℚ}(C)^{t} × A_{ℚ}(C)^{t} → ℚ(t)

on basis elements M and N by

\begin{aligned}⟨M, N⟩^{t} : & = ∑_{i∈ℤ}t^{i} dim Hom_{C}(M, N[i]) \\ & = ∑_{i∈ℤ}t^{i}⟨M, N[i]⟩. \end{aligned}

We extend this definition to the whole of A_{ℚ}(C)^{t} × A_{ℚ}(C)^{t} so as to have a sesquilinear form with respect to the field automorphism , that is,

⟨a_{1}M_{1} + a_{2}M_{2}, N⟩^{t} = a_{1}⟨M_{1}, N⟩^{t} + a_{2}⟨M_{2}, N⟩^{t}

and

⟨M, b_{1}N_{1} + b_{2}N_{2}⟩^{t} = \overline{b_{1}}⟨M, N_{1}⟩^{t} + \overline{b_{2}}⟨M, N_{2}⟩^{t}

always hold.

We come to the main result of this section, which establishes the key properties of the form we have just defined, notably that Auslander-Reiten triangles give elements dual to the standard basis of indecomposables, and that for perfect complexes over a symmetric algebra the form is Hermitian.

Theorem 4.2. Let C be a Krull-Schmidt k-linear triangulated category which is Hom-finite, and where k is algebraically closed. Suppose that C satisfies the property that for every pair of objects M and N in C, Hom(M, N[i]) ≠ 0 for only finitely many i.

(1) The expression defining ⟨ , ⟩^{t} gives a well-defined sesquilinear form on A_{ℚ}(C)^{t} × A_{ℚ}(C)^{t}.

(2) Whenever X → Y → Z → X[1] is an Auslander-Reiten triangle in C and Ẑ = X + Z − Y in A_{ℚ}(C)^{t} we have

⟨M, Ẑ⟩^{t} = \begin{cases}0 & unless M ≅ Z[i] for some i, \\ (1 + t) & if M ≅ Z, \end{cases}

so that the element \frac{1}{1+t} Ẑ is dual on the right to Z. Similarly

⟨Ẑ, N⟩^{t} = \begin{cases}0 & unless N ≅ X[i] for some i, \\ (1 + t^{−1}) & if N ≅ X, \end{cases}

so that the element \frac{1}{1+t^{−1}} Ẑ is dual on the left to X.

(3) When C = D^{b}(Λ-proj) is the category of perfect complexes for a symmetric algebra Λ the form is Hermitian, in the sense that ⟨M, N⟩^{t} = \overline{⟨N, M⟩^{t}} always.

(4) When C = D^{b}(kG-proj) is the category of perfect complexes for a group algebra kG we have

⟨M ⊗_{k} U, N⟩^{t} = ⟨M, U^{∗} ⊗_{k} N⟩^{t}

and

⟨M, N⟩^{t} = ⟨M^{∗}, N^{∗}⟩^{t}.

Proof. (1) The expression which defines the form shows that

⟨M[j], N⟩^{t} = ∑_{i∈ℤ}t^{i} dim Hom_{C}(M[j], N[i])

= ∑_{i∈ℤ}t^{i+j} dim Hom_{C}(M[j], N[i + j])

= t^{j} ∑_{i∈ℤ}t^{i} dim Hom_{C}(M, N[i])

= t^{j}⟨M, N⟩^{t}

and by a similar calculation ⟨M, N[j]⟩^{t} = t^{−j}⟨M, N⟩^{t}. Thus the form vanishes when elements M[j] − t^{j}M are put in either side and consequently passes to a well defined sesquilinear form on A_{ℚ}(C)^{t}.

(2) Because of the hypothesis that Hom(M, N[i]) ≠ 0 for only finitely many i we can never have M ≅ M[1] for any M. We see by Proposition 3.1 that ⟨M, Ẑ⟩^{t} is only non-zero when M ≅ Z[j] for some j, and that in the expression for ⟨Z, Ẑ⟩^{t} the only non-zero terms are ⟨Z, Ẑ⟩+t⟨Z, Ẑ[1]⟩ = 1+t. The argument for ⟨Ẑ, N⟩^{t} is similar.

(3) When Λ is symmetric the Nakayama functor is the identity. The identity functor on D^{b}(Λ-proj) is a Serre duality, and we have Hom(M, N) ≅ Hom(N, M)^{∗}. Thus ⟨M, N⟩^{t} = \overline{⟨N, M⟩^{t}} when M and N are basis elements of A_{ℚ}(C)^{t}, and the same formula follows for arbitrary elements of A_{ℚ}(C)^{t} by the sesquilinear property of the form.

(4) The first two formulas follow from the identities Hom(M ⊗_{k} U, N) ≅ Hom(M, U^{∗} ⊗ N) and Hom(M, N) ≅ Hom(N^{∗}, M^{∗}). □

It is interesting at this point to compare the bilinear form ⟨M, N⟩^{t} we have constructed to another bilinear form which appears in [5, page 13]. A bilinear form is defined there as

⟨M, N⟩ := ∑_{i∈ℤ}(−1)^{i} dim Hom(M, N[i])

which is the specialization of ⟨M, N⟩^{t} on putting t = −1. We see from Theorem 4.2 part (2) that this is exactly the specialization which destroys the possibility of having dual elements in our sense.

5. Values of the Hermitian form on Auslander-Reiten quiver components

We assume in this section that Λ is a symmetric algebra over k and we consider the homotopy category of perfect complexes D^{b}(Λ-proj). In this situation we know from [7] that D^{b}(Λ-proj) has Auslander-Reiten triangles, and it is shown in [13] (see also [8]) that, provided Λ has no semisimple summand, all components of the Auslander-Reiten quiver of D^{b}(Λ-proj) have the form ℤA_{∞}. It follows from [7] (described also in [8]) that when Λ is a symmetric algebra the Auslander-Reiten triangles all have the form X → Y → X[1] → X[1], because the Nakayama functor is the identity. We say that a complex Z lies on the rim of the Auslander-Reiten quiver if, in the Auslander-Reiten triangle X → Y → Z → X[1], the complex Y is indecomposable.

We will show that the special shape of the Auslander-Reiten quiver and the fact that the bilinear form ⟨ , ⟩^{t} is Hermitian enable us to compute the values of the bilinear form given only its values on objects on the rim.

Assuming that Λ has no semisimple summand, we will label the objects in a component of the Auslander-Reiten quiver of D^{b}(Λ-proj) as shown in Figure 1. Objects on the rim (the top row) are the shifts of a single object C_{0}, and at distance n from the rim the objects are shifts of an indecomposable C_{n}, which is chosen so that there is a chain of irreducible morphisms C_{0} → C_{1} → ⋯ → C_{n}.

In the next result we write σ_{r} := 1 + t + t^{2} + ⋯ + t^{r} ∈ ℤ[t, t^{−1}] when r ≥ 1, and σ_{0} := 1.

Theorem 5.1. Assume Λ is a symmetric k-algebra with no semisimple summand, and let C_{0} and D_{0} be indecomposable objects in D^{b}(Λ-proj) which lie on the rim of their quiver components. Then the values of the form ⟨C_{m}, D_{n}⟩^{t} on objects in the same components as C_{0} and D_{0} are entirely determined by knowing ⟨C_{0}, D_{0}⟩^{t}. Specifically, if C_{0} and D_{0} lie in different

C_{0}[−1]  C_{0}  C_{0}[1]  C_{0}[2]
↘  ↗  ↘  ↗  ↘  ↗
⋯  C_{1}[−1]  C_{1}  C_{1}[1]  ⋯
↗  ↘  ↗  ↘  ↗  ↘
C_{2}[−2]  C_{2}[−1]  C_{2}  C_{2}[1]
↘  ↗  ↘  ↗  ↘  ↗
⋮  ⋮  ⋮

Figure 1. Auslander-Reiten quiver component of perfect complexes for a symmetric algebra

quiver components then

⟨C_{m}, D_{n}⟩^{t} = σ_{m}\overline{σ}_{n}⟨C_{0}, D_{0}⟩^{t}

while

⟨C_{m}, C_{n}⟩^{t} = σ_{m}\overline{σ}_{n} (⟨C_{0}, C_{0}⟩^{t} − \frac{(1 + t)(1 − t^{μ})}{1 − t^{μ+1}})

where μ is the maximum of m and n.

Proof. Step 1: we show that if C_{m} is not a shift of any D_{i} where 0 ≤ i ≤ n−1 then ⟨C_{m}, D_{n}⟩^{t} = \overline{σ}_{n}⟨C_{m}, D_{0}⟩^{t}. We proceed by induction on n. The result is true when n = 0. When n = 1 the calculation is special because D_{1} is adjacent to the rim. Since C_{m} is not a shift of D_{0} we have

0 = ⟨C_{m}, D̂_{0}⟩^{t}
= ⟨C_{m}, D_{0}⟩^{t} + ⟨C_{m}, D_{0}[−1]⟩^{t} − ⟨C_{m}, D_{1}[−1]⟩^{t}
= (1 + t)⟨C_{m}, D_{0}⟩^{t} − t⟨C_{m}, D_{1}⟩^{t}.

From this we deduce that

⟨C_{m}, D_{1}⟩^{t} = t^{−1}(1 + t)⟨C_{m}, D_{0}⟩^{t} = \overline{σ}_{1}⟨C_{m}, D_{0}⟩^{t}.

Now suppose that n ≥ 2 and the result holds for smaller values of n. We have

0 = ⟨C_{m}, D̂_{n−1}⟩^{t}
= ⟨C_{m}, D_{n−1}⟩^{t} + ⟨C_{m}, D_{n−1}[−1]⟩^{t} − ⟨C_{m}, D_{n}[−1]⟩^{t} − ⟨C_{m}, D_{n−2}⟩^{t}
= (1 + t)⟨C_{m}, D_{n−1}⟩^{t} − t⟨C_{m}, D_{n}⟩^{t} − ⟨C_{m}, D_{n−2}⟩^{t}.

This is a recurrence relation for ⟨C_{m}, D_{n}⟩^{t} starting with the values already obtained when n = 0 and 1, and it is solved by

⟨C_{m}, D_{n}⟩^{t} = \overline{σ}_{n}⟨C_{m}, D_{0}⟩^{t}.

Step 2: We deduce that if D_{n} is not a shift of any C_{i} where 0 ≤ i ≤ m − 1 then ⟨C_{m}, D_{n}⟩^{t} = σ_{m}⟨C_{0}, D_{n}⟩^{t}. This follows from Step 1 on exploiting the fact that the form is Hermitian, for

⟨C_{m}, D_{n}⟩^{t} = \overline{⟨D_{n}, C_{m}⟩^{t}} = \overline{\overline{σ_{m}}⟨D_{n}, C_{0}⟩^{t}} = σ_{m}⟨C_{0}, D_{n}⟩^{t}.

Step 3: We put together Steps 1 and 2 to obtain the first statement of the Proposition, which applies when C_{0} and D_{0} lie in different quiver components.

Step 4: We treat the case of two objects in the same quiver component similarly, taking account of the fact that the values ⟨C_{m}, Ĉ_{n}⟩^{t} are not always zero. First

1 + t = ⟨C_{0}, Ĉ_{0}⟩^{t}
= ⟨C_{0}, C_{0}⟩^{t} + ⟨C_{0}, C_{0}[−1]⟩^{t} − ⟨C_{0}, C_{1}[−1]⟩^{t}
= (1 + t)⟨C_{0}, C_{0}⟩^{t} − t⟨C_{0}, C_{1}⟩^{t}

so that

⟨C_{0}, C_{1}⟩^{t} = t^{−1}(1 + t)(⟨C_{0}, C_{0}⟩^{t} − 1) = \overline{σ}_{1}(⟨C_{0}, C_{0}⟩^{t} − 1)

which agrees with the formula we have to prove. By exactly the same calculation as was used in Step 1, taking n ≥ 2 and Ĉ_{n−1} = D̂_{n−1} and using the fact that 0 = ⟨C_{0}, Ĉ_{n−1}⟩^{t} we obtain the same recurrence

0 = (1 + t)⟨C_{0}, C_{n−1}⟩^{t} − t⟨C_{0}, C_{n}⟩^{t} − ⟨C_{0}, C_{n−2}⟩^{t}

valid when n ≥ 2, and this has solution

⟨C_{0}, C_{n}⟩^{t} = \overline{σ}_{n} (⟨C_{0}, C_{0}⟩^{t} − \frac{(1 + t)(1 − t^{n})}{1 − t^{n+1}}) .

Now if m ≤ n, since C_{m} is not a shift of any C_{i} where 0 ≤ i ≤ n − 1, by Step 2 we deduce

⟨C_{m}, C_{n}⟩^{t} = σ_{m}\overline{σ}_{n} (⟨C_{0}, C_{0}⟩^{t} − \frac{(1 + t)(1 − t^{n})}{1 − t^{n+1}}) .

Finally if m > n we use the Hermitian property to deduce

⟨C_{m}, C_{n}⟩^{t} = \overline{⟨C_{n}, C_{m}⟩^{t}}

= \overline{σ_{n}\overline{σ}_{m} (⟨C_{0}, C_{0}⟩^{t} − \frac{(1 + t)(1 − t^{m})}{1 − t^{m+1}})}

= σ_{m}\overline{σ}_{n} (⟨C_{0}, C_{0}⟩^{t} − \frac{(1 + t)(1 − t^{m})}{1 − t^{m+1}})

which is what we have to prove. □

6. The position in the quiver of perfect complexes with prescribed properties

In this section we assume that Λ is a self-injective algebra over k (not necessarily a symmetric algebra). We will also suppose that Λ has no direct summand which is a semisimple algebra, so as before it follows from [13] that all components of the Auslander-Reiten quiver of D^{b}(Λ-proj) have the form ℤA_{∞}. We will describe the position in the quiver of several kinds of indecomposable perfect complexes: those with one or two or (in the next section) three non-zero terms and those whose sequence of homology modules has a non-zero term sandwiched between two zero terms. This allows us to identify truncated projective resolutions as lying on the rim of the quiver, except in one specific case where they are adjacent to the rim. We do this by identifying aspects of the structure of complexes at a given distance from the rim: their length, and the composition factors of their homology modules. We will see that at a suitable distance from the rim the homology modules of the indecomposable complexes stabilize in a single module which we may call the stabilization module, and that provided Λ has radical length at least 3, this stabilization module can be made arbitrarily large. This latter result seems surprising, even for rings Λ such as k[X]/(X^{3}). By comparison, indecomposable perfect complexes for k[X]/(X^{2}) cannot have homology modules of arbitrarily large dimension.

Before getting started on this we recall some basic facts about perfect complexes which are well known. They are proven (in the dual context of complexes of injectives) in [10, B2] except for the final statement, which is an easy exercise.

Proposition 6.1. Every perfect complex over a finite dimensional algebra is homotopy equivalent to a complex whose terms all have minimal dimension among complexes in the same homotopy equivalence class. Such a minimal complex is unique up to isomorphism, and is a summand of every perfect complex homotopy equivalent to it. When the algebra is self-injective the minimal complex is characterized by the fact that every boundary map with non-zero target has kernel without non-zero projective summands, and it has the property that its top and bottom homology are non-zero.

By the length of a perfect complex of Λ-modules we will mean the number of non-zero terms in the minimal complex chain homotopy equivalent to the given complex. Since Λ is supposed to be self-injective, this equals the difference between the degrees of the highest and lowest non-zero homology of the complex.

We will continue to use the fact, established in [7], that Auslander-Reiten triangles X → Y → Z → X[1] have X[1] = νZ where ν is the Nakayama functor. Over a self-injective algebra Λ the Nakayama functor is exact, since it is the composite of the Auslander-Bridger duality Hom_{Λ}(−, Λ) which is exact because Λ is injective, and the ordinary duality Hom_{k}(−, k) which is always exact. Thus ν preserves the length of complexes, and H_{n}(X) = 0 if and only if H_{n}(νX) = 0.

Let P_{S} be the projective cover of a simple Λ-module S. We start by describing part of the quiver component containing P_{S}, regarded as a complex whose only non-zero term is P_{S} in degree 0. The approach in the next proposition is by direct calculation, and it will be used later in Theorem 6.6 where we give complete information about this quiver component. For each Λ-module M we will denote by P_{M} the complex P_{1} → P_{0} with terms in degrees 1 and 0 such that P_{1} → P_{0} → M is the start of a minimal projective resolution of M. Thus H_{0}(P_{M}) ≅ M and H_{1}(P_{M}) ≅ Ω^{2}M. We will be particularly interested in the complex P_{Ω^{−1}S}, which is the complex P_{S} → νP_{S} where the map sends the simple top of P_{S} isomorphically to the socle of νP_{S} (= the injective hull of S). We will also be interested in the complex H_{S} which is defined to be ν^{−1}P_{S} → P_{S} → νP_{S} in degrees 1, 0 and -1, where both maps send the top of one projective isomorphically to the socle of the next. It has homology Ω Soc P_{S}, Rad P_{S}/ Soc P_{S}, Ω^{−1}S in these degrees. We recall that the module H_{S} := Rad P_{S}/ Soc P_{S} is known as the heart of P_{S}.

Proposition 6.2. Let S be a simple module for a self-injective algebra Λ and let P_{S} be the projective cover of S. The Auslander-Reiten quiver of perfect complexes has the following shape close to P_{S}:

νP_{S}[−1]  P_{S}  ν^{−1}P_{S}[1]
↘  ↗  ↘  ↗
⋯ P_{Ω^{−1}S}[−1] ν^{−1}P_{Ω^{−1}S} ⋯
↗ ↘ ↗ ↘
H_{S}
⋮  ⋮

Thus perfect complexes of length 1 lie on the rim of the Auslander-Reiten quiver, and the complexes of length 2 of the form P_{Ω^{−1}S} = (P_{S} → νP_{S}) lie at distance 1 from the rim.

Proof. The construction of the Auslander-Reiten triangles comes from the description in [7]. Thus we have an Auslander-Reiten triangle νP_{S}[−1] → P_{Ω^{−1}S}[−1] → P_{S} → νP_{S} where the final map sends the top of P_{S} to the socle of νP_{S} since this map is not homotopic to zero and lies in the socle of Hom(P_{S}, νP_{S}) and P_{Ω^{−1}S} is the mapping cone of this map. The complex P_{Ω^{−1}S} is indecomposable (or zero in case P_{S} = S), because if it were not it would have to be the direct sum of two one-term complexes P_{S} and νP_{S} and would have projective homology. However, the homology of P_{Ω^{−1}S} is Ω(S) and Ω^{−1}(S) in degrees 1 and 0, and is not projective.

We construct the middle term of the Auslander-Reiten triangle starting at P_{Ω^{−1}S}[−1] as the mapping cone of a map ν^{−1}P_{Ω^{−1}S}[−1] → P_{Ω^{−1}S}[−1] which is not homotopic to zero and which lies in the socle of such maps under the action of the endomorphism ring of either complex. Such a map is

(\begin{matrix}P_{S} \\ α ↑ \\ ν^{−1}P_{S} \end{matrix}) ⟶^{ν(α)} ; ⟶^{0} (\begin{matrix}νP_{S} \\ ν(α) ↑ \\ P_{S} \end{matrix} )

and its mapping cone is a complex

ν^{−1}P_{S} \underrightarrow{(\begin{matrix}α \\ 0 \end{matrix})} P_{S} ⊕ P_{S} ⟶^{(ν(α),ν(α))} νP_{S}

The middle term has a direct summand {(x, −x) | x ∈ P_{S}} ≅ P_{S} which is a direct complement to the subcomplex whose middle term is the first P_{S} direct summand. From this we see that the mapping cone is the direct sum as complexes P_{S} ⊕ H_{S}. This completes the description of this part of the Auslander-Reiten quiver. □

Corollary 6.3. Let Λ be a self-injective algebra. Each quiver component of D^{b}(Λ-proj) contains at most one indecomposable projective module regarded as a complex in degree 0.

Proof. Any indecomposable projective module lies on the rim of its quiver component, which consists of that module and its shifts with the Nakayama functor applied, so no other projectives are possible. □

Assuming that the algebra Λ is self-injective and not necessarily symmetric, the complexes in a quiver component have the form shown in Figure 2, which is similar to Figure 1 but takes into account the non-identity Nakayama functor. We assume as always that Λ has no semisimple summand.

νC_{0}[−1]  C_{0}  ν^{−1}C_{0}[1]  ν^{−2}C_{0}[2]
↘  ↗  ↘  ↗  ↘  ↗
⋯  νC_{1}[−1]  C_{1}  ν^{−1}C_{1}[1]  ⋯
↗  ↘  ↗  ↘  ↗  ↘
ν^{2}C_{2}[−2]  νC_{2}[−1]  C_{2}  ν^{−1}C_{2}[1]
↘  ↗  ↘  ↗  ↘  ↗
⋮  ⋮  ⋮

Figure 2. Auslander-Reiten quiver component of perfect complexes for a self-injective algebra

We will obtain a lot of information from the homology of complexes in a quiver component. Taking zero homology of every complex in the component we obtain modules

H_{0}(ν^{j}C_{i}[−j]) ≅ ν^{j}H_{j}(C_{i}),

using the fact that ν is exact. Thus every homology module of each complex C_{0}, C_{1}, … appears in the picture, up to a twist by the Nakayama functor, on taking zero homology in this fashion. This is shown in Figure 3, which we call the homology diagram of the quiver component.

νH_{1}(C_{0})  H_{0}(C_{0}) ν^{−1}H_{−1}(C_{0}) ν^{−2}H_{−2}(C_{0})
↘  ↗  ↘  ↗  ↘  ↗
⋯  νH_{1}(C_{1})  H_{0}(C_{1}) ν^{−1}H_{−1}(C_{1}) ⋯
↗  ↘  ↗  ↘  ↗  ↘
ν^{2}H_{2}(C_{2})  νH_{1}(C_{2})  H_{0}(C_{2}) ν^{−1}H_{−1}(C_{2})
↘  ↗  ↘  ↗  ↘  ↗
⋮  ⋮  ⋮

Figure 3. Homology diagram of an Auslander-Reiten quiver component of perfect complexes for a self-injective algebra

In the next results we will examine the pattern of modules in the homology diagram. We will do this both for quiver components which do and which do not contain projective modules. The basic idea is the same in both cases, but it is a little more complicated in the case of components which contain a projective. For reasons of clarity we consider quiver components which do not contain a projective module first.

For any object in a ℤA_{∞} quiver component, by the wing of that object we mean the part of the quiver forming a triangle with the given object as as vertex and a segment of the rim of the quiver as its opposite edge. We call that segment of the rim the wing rim, being the part of the rim which lies in the wing.

Proposition 6.4. Let Λ be a self-injective algebra with no semisimple summand and consider the homology diagram of a quiver component of D^{b}(Λ-proj) which does not contain a projective module. Every mesh in the diagram labels the modules in a short exact sequence of H_{0} modules. The composition factors (taken with multiplicity) of H_{0} of any complex are the union of the composition factors of the H_{0} of objects lying on the wing rim of the given object.

Proof. Since the shift and the Nakayama functor are exact, it suffices to consider a mesh containing an object C_{n} at distance n from the rim in Figure 2. By Lemma 2.1, since homology is representable as Hom(Λ, −) and there is no projective module in the quiver component, its zero homology appears in a short exact sequence 0 → H_{0}(C_{n}) → H_{0}(ν^{−1}C_{n−1}[1]) ⊕ H_{0}(C_{n+1}) → H_{0}(ν^{−1}C_{n}[1]), where if n = 0 we take C_{−1} = 0. This establishes that each mesh corresponds to a short exact sequence.

To prove the statement about composition factors we proceed by induction on n. On the rim when n = 0 the result holds. In case n = 1 the exact sequence of H_{0} modules implies that the composition factors of the middle term are the union (with multiplicities) of the composition factors of the two outer terms, which is what we have to prove. When n ≥ 2 the composition factors of H_{0}(C_{n}) are those of the outer terms in the mesh above it, with the composition factors of the top term in the mesh removed. By induction these are the composition factors from two parts of the rim (factors common to both parts taken twice) with a copy of the factors from the intersection of those parts of the rim removed. This gives exactly the desired result. □

Theorem 6.5. Let Λ be a self-injective algebra and consider a quiver component of D^{b}(Λ-proj) which does not contain a projective module. Adjacent to zeros on the rim of the homology diagram, this diagram has the shape shown below, with the maps between identically labelled terms being isomorphisms. Sufficiently far from the rim the homology modules stabilize to a module Σ as shown. Thus given two positions in the quiver for which the wing rims of the homology diagram have the same non-zero terms, the zero homology modules at those positions are isomorphic.

0  0  A_{0}  •  ⋯  •  B_{0}  0
↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗
⋯ 0 A_{0} A_{1} ⋯ ⋯ B_{1} B_{0}
↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘
0 A_{0} A_{1} ⋯ • ⋯ B_{1} B_{0}
↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗
⋯ A_{0} A_{1} ⋯ A_{n−1} B_{n−1} ⋯ B_{1}
↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘
A_{0} A_{1} ⋯ A_{n−1} Σ B_{n−1} ⋯ B_{1}
↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗
⋯ A_{1} ⋯ A_{n−1} Σ Σ B_{n−1} ⋯
↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘
A_{1}  ⋯  A_{n−1}  Σ  Σ  Σ  B_{n−1}  ⋯
⋮  ⋮  ⋮  ⋮  ⋮  ⋮  ⋮

Proof. We show that a zero on the rim of the homology diagram gives rise to homology groups in the pattern

(6.1)  \begin{aligned}rim: & 0 & A_{0} & ⋯ \\ ↘ & ↗ & ↘ \\ A_{0} & A_{1} \\ ↘ & ↗ & ↘ \\ A_{1} & A_{2} \\ ↘ & ↗ \\ & & A_{2} \end{aligned}

where each of the mappings A_{i} → A_{i} is an isomorphism. Starting with a zero-homology module A_{0} on the rim which is adjacent to a zero homology module as shown, there is a short exact sequence of H_{0} modules of the Auslander-Reiten triangle at that point by Corollary 2.2: 0 → 0 → U → A_{0} → 0 where initially U is unknown. From this sequence we deduce that the mapping U → A_{0} is an isomorphism, which determines U. This fills in the term labelled A_{0} at distance 1 from the rim. Looking at the square of which this is the left-hand end and labelling the right-hand term A_{1}, we obtain a short exact sequence 0 → A_{0} → A_{0} ⊕ V → A_{1} → 0 for some unknown V, again from Corollary 2.2 applied to H_{0} of an Auslander-Reiten triangle. The component map A_{0} → A_{0} is an isomorphism, so the sequence splits and we deduce that the component map V → A_{1} is an isomorphism. This fills in V as A_{1}. Proceeding in this way we establish the pattern shown on the left side of diagram 6.1, and the pattern on the right adjacent to zeros on the rim of the homology diagram is established similarly.

The entries on the rim of the homology diagram are the various homology modules of a single perfect complex and its shifts, so they are eventually zero to the left and to the right. Sufficiently far from the rim the left and right patterns we have described overlap, and this happens in the region where we show modules Σ. At this point all morphisms in the diagram are isomorphisms, showing that the homology stabilizes. The positions labelled Σ are precisely those for which the wing rims in the homology diagram have the same non-zero terms, and this establishes the last statement of the theorem. □

We now describe the homology diagram of a quiver component containing a projective module and then use this to describe completely the entire quiver component.

Theorem 6.6. Let S be a simple module for a self-injective algebra Λ and let P_{S} be the projective cover of S. We assume S ≠ P_{S}.

(1) The homology diagram of the quiver component containing P_{S} is as follows:

0  0  × P_{S} ×  0  0
↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗
⋯ 0 Rad P_{S} P_{S}/ Soc P_{S} 0 ⋯
↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘
0 Rad P_{S} H_{S} P_{S}/ Soc P_{S} 0
↘ ↗ ↘ ↗ ↘ ↗ ↘ ↗
⋯ Rad P_{S} H_{S} H_{S} P_{S}/ Soc P_{S} ⋯
↗ ↘ ↗ ↘ ↗ ↘ ↗ ↘
Rad P_{S}  H_{S}  H_{S}  H_{S}  P_{S}/ Soc P_{S}
⋮  ⋮  ⋮  ⋮

Here the module H_{S} = Rad P_{S}/ Soc P_{S} is the heart of P_{S}. Every mesh in the diagram labels the terms in a short exact sequence of zero homology modules, except for the two meshes labelled ×. In every case the short exact sequence is split except for the one underneath P_{S}, which is an Auslander-Reiten sequence. Sufficiently far from the rim the zero homology stabilizes to be H_{S}.

(2) With the notation of Figure 2 the complex C_{n} at distance n from the rim of the quiver component containing P_{S} is

ν^{−n}P_{S} → ⋯ → ν^{−2}P_{S} → ν^{−1}P_{S} → P_{S}

where each map sends the simple top of a projective module isomorphically to the socle of the next module. The irreducible morphisms between these complexes and their Auslander-Reiten translates are represented by the obvious inclusions and surjections between such complexes of adjacent lengths.

Proof. (1) The fact that all meshes correspond to short exact sequences except the ones marked × follows from Lemma 2.1 and the description of the quiver component given in Proposition 6.2, as in the proof of Proposition 6.4. The zero homology of the complexes shown in Proposition 6.2 has already been computed before that proposition, and accounts for terms on the rim and the mesh below P_{S}. The facts that the remaining meshes correspond to split short exact sequence with maps between identically labelled terms being isomorphisms, and that the homology stabilizes at H_{S}, follow by the same argument as in Theorem 6.5. We may deduce that the mesh below P_{S} corresponds to an Auslander-Reiten sequence by observing from our earlier calculation that the component maps to P_{S} and H in the mesh are mono and epi, while the component maps in the mesh from P_{S} and H are epi and mono. This identifies the sequence as a well-known Auslander-Reiten sequence [3, Sect. 4.11]. We will also see in Proposition 7.2 by a different argument that the short exact sequence under P_{S} is an Auslander-Reiten sequence.

(2) We show that C_{n} can be identified as stated by induction on n. We see from Proposition 6.2 that the expressions for C_{0}, C_{1} and C_{2} are correct. Now suppose that n ≥ 2, that the result is true for n and smaller values, and consider C_{n+1}. From the general structure of the quiver shown in Figure 2 the mapping cone of ν^{−1}C_{n} → C_{n} is C_{n+1} ⊕ ν^{−1}C_{n−1}[1]. The mapping cone has the form

ν^{−n−1}P_{S} → (ν^{−n}P_{S})^{2} → ⋯ → (ν^{−2}P_{S})^{2} → (ν^{−1}P_{S})^{2} → P_{S}

and ν^{−1}C_{n−1}[1] has the form

ν^{−n}P_{S} → ⋯ → ν^{−2}P_{S} → ν^{−1}P_{S}

so that

C_{n+1} ≅ ν^{−n−1}P_{S} → ν^{−n}P_{S} → ⋯ → ν^{−2}P_{S} → ν^{−1}P_{S} → P_{S}.

From the homology diagram in part (1) we see that the homology modules of C_{n+1} are ν^{−n−1} Rad P_{S}, ν^{−n}H_{S}, …, ν^{−1}H_{S}, P_{S}/ Soc P_{S} and by a dimension count it follows that each map in C_{n+1} sends the simple top of each projective to the socle of the next. □

We deduce more corollaries from these patterns. Throughout we assume that Λ is a self-injective algebra with no semisimple summand and we consider the Auslander-Reiten quiver of perfect complexes.

Corollary 6.7. If complexes on the rim of a quiver component have length t then complexes at distance r from the rim have length t + r.

Proof. This follows from Theorems 6.5 and 6.6, since the length of a complex is determined from its extreme non-vanishing homology modules, and so equals the length of the part of the row in the homology diagram where the terms are non-zero. These theorems show that this length grows by 1 with each step away from the rim. □

Corollary 6.8. Any (indecomposable) perfect complex with two non-zero terms, not of the form P_{S} → νP_{S} for some simple module S (where the map sends the top of P_{S} isomorphically to the socle of νP_{S}), lies on the rim of its quiver component.

Proof. If it didn’t lie on the rim, the complexes on the rim of its quiver component would have to have length 1 by Corollary 6.7, so would be shifts of a projective module P_{S} for some simple module S. In that case the 2-term complexes next to the rim have the form which has been excluded. □

Corollary 6.9. In an indecomposable complex at distance n from the rim, any non-zero homology module must occur as part of a string of at least n+1 non-zero consecutive homology modules.

Proof. The result is true for quiver components containing a projective module by Theorem 6.6. For other components, non-zero homology comes about from non-zero homology on the rim, by Proposition 6.4. Each non-zero term on the rim gives rise to n+1 non-zero consecutive homology modules at distance n. This is the only way we get non-zero homology at distance n from the rim and so every non-zero homology module must be part of a string of n + 1 non-zero homology modules. □

Corollary 6.10. The only perfect complexes of length n + 1 which are at distance n from the rim of their quiver component are the complexes

ν^{−n}P_{S} → ⋯ → ν^{−2}P_{S} → ν^{−1}P_{S} → P_{S}

for some simple module S, where every mapping sends a simple top isomorphically to the next simple socle. All other complexes of length n + 1 are closer to the rim than this.

Proof. By Corollary 6.7 all complexes of length n + 1 must lie at distance n or less from the rim. If such a complex lies at distance n then by the same corollary the complexes on the rim of the quiver component have length 1, and so are projective modules P_{S} in a single degree. We saw in Theorem 6.6 that the complexes at distance n from the rim in such a quiver component have the form stated. □

Corollary 6.11. Any indecomposable complex X with a sequence of homology modules of the form H_{d+1}(X) = 0, H_{d}(X) ≠ 0 and H_{d−1}(X) = 0 must lie on the rim of its quiver component.

Proof. If X were not on the rim, non-zero homology strings would all have to have length at least 2 by Corollary 6.9. Since the complex has a non-zero homology string of length 1, it must lie on the rim. □

Proposition 6.12. Let M be an indecomposable module, and let P_{n} → ⋯ → P_{1} → P_{0} → M → 0 be the start of a minimal resolution of M. Then the complex P_{n} → ⋯ → P_{1} → P_{0} is indecomposable. It lies on the rim of its quiver component unless M = Ω^{−1}S for some simple module S and n = 1, in which case the complex lies at distance 1 from the rim of its quiver component.

Proof. If the complex were to decompose as a direct sum of two complexes, one of the summands would have M as its zero homology. That summand would have projective terms, and would be the start of a smaller projective resolution of M. Since we chose a minimal projective resolution, this cannot happen.

If n > 1 then the homology of the complex has its zero homology group isolated in the sense of Corollary 6.11 and so the complex lies on the rim of its quiver component. The other possibility is that n ≤ 1 so that the complex has one or two terms. If it has one term (n = 0) then M is projective and lies on the rim of the quiver by Proposition 6.2. If n = 1 then by hypothesis the complex is not isomorphic to P_{S} → νP_{S} and so lies on the rim by Corollary 6.8. □

The next result has some independent significance, in that it makes no reference to the Auslander-Reiten quiver.

Theorem 6.13. Let Λ be a self-injective algebra of radical length at least 3. There exist indecomposable perfect complexes with degree zero homology of arbitrarily large dimension.

Proof. We may construct a perfect complex with arbitrarily many non-zero homology modules, and for some H ≠ 0 having terms 0, H, 0 somewhere in the list of homology groups. For example, we may take an indecomposable projective module P_{S} of radical length at least 3, form a complex

P_{Rad P_{S}} → P_{S} → νP_{S} → ν^{2}P_{S} → ⋯ →

where the first map is the projective cover of the radical of P_{S}, and after that each map identifies the top of a module with the socle of the next. Such a complex can be made to have arbitrarily many non-zero homology modules, and the second homology module from the left is zero. Such a complex must lie on the rim of its quiver component by Corollary 6.11. Considering homology at some distance from the rim the stabilizing module Σ which appears in Corollary 6.5 has composition length which is the sum of the lengths of all homology modules of the starting complex. We conclude that this may be made arbitrarily large. □

7. Perfect complexes with prescribed homology and Auslander-Reiten sequences

We start with a result which makes a connection between Auslander-Reiten sequences and Auslander-Reiten triangles. It turns out to be a well known result in the context of modules, expressed in the language of complexes. For this result Λ can be any finite dimensional algebra, not necessarily one that is self-injective.

Proposition 7.1. Let M be a non-projective indecomposable Λ-module and let P_{M} = (P_{1} → P_{0}) be the first two terms of a minimal projective resolution of M, so that H_{0}(P_{M}) ≅ M and H_{1}(P_{M}) ≅ Ω^{2}M. Consider the Auslander-Reiten triangle νP_{M}[−1] → E_{M} → P_{M} → νP_{M} where ν is the Nakayama functor. Then the part of the long exact sequence in homology of this triangle, whose terms are the zero homology groups, is the Auslander-Reiten sequence of Λ-modules 0 → τM → E_{M} → M → 0. The non-zero homology groups of the mapping cone complex E_{M} are H_{1}(E_{M}) = Ω^{2}M, H_{0}(E_{M}) = E_{M} and H_{−1}(E_{M}) = νM.

Proof. We know from [2, IV.2.4] that νP_{M}[−1] has homology modules

H_{−1}(P_{M}[−1]) = νM  and  H_{0}(P_{M}[−1]) = τM.

By Corollary 2.2, since M is not projective, the long exact sequence in homology of the triangle breaks up as a splice of short exact sequences, of which the 0-homology sequence has the form

0 → τM → H_{0}(E_{M}) → M → 0.

This sequence has the correct end terms to be an Auslander-Reiten sequence. We show first that the sequence has the lifting property of Auslander-Reiten sequences, and then later that it is not split. This will establish that it is indeed an Auslander-Reiten sequence.

Suppose that we have an indecomposable module N and a morphism N → M which is not an isomorphism. Take the start of a minimal projective resolution Q = (Q_{1} → Q_{0}) of N and lift the morphism N → M to a morphism of complexes Q → P_{M}. This mapping is not split epi, because on taking zero homology it is not split epi. It therefore lifts to a morphism Q → E_{M} and on taking zero homology we deduce that the original morphism lifts to a morphism N → H_{0}(E_{M}).

To show that the sequence of zero homology groups is not split, suppose to the contrary that we have a splitting map M → H_{0}(E_{M}) = Z_{0}(E_{M})/B_{0}(E_{M}) with image U/B_{0}(E_{M}) for some submodule U ⊆ Z_{0}(E_{M}). By the projective property of P_{0} this lifts to a map P_{0} → U and hence also to a map P_{1} → (E_{M})_{1}, so that we have a chain map P_{M} → E_{M}. Composing with the map E_{M} → P_{M} we obtain an endomorphism of P_{M} which lifts the identity on M. By Fitting’s Lemma some power of this endomorphism has image a summand of P_{M}, which again must be the start of a resolution of M. By minimality of P_{M} this summand must be the whole of P_{M}, and so the endomorphism of P_{M} is an isomorphism and we have split the Auslander-Reiten triangle – a contradiction.

We have now shown that taking zero homology gives an Auslander-Reiten sequence. To identify the top and bottom homology groups of E_{M} we use the fact again from Corollary 2.2 that the degree 1 and degree −1 terms in the long exact homology sequence give short exact sequences, and they are

0 → 0 → H_{1}(E_{M}) → H_{1}(P_{M}) = Ω^{2}M → 0

and

0 → H_{0}(νP_{M}) = νM → H_{−1}(E_{M}) → 0 → 0.

This completes the proof. □

We comment that the construction we have just described gives a way to implement the construction of Auslander-Reiten sequences on a computer, without explicitly computing Ext groups and extensions corresponding to Ext classes (although the ingredients in the construction are closely related to this). Note also that we do not assume that Λ is self-injective here, so the algorithm works for any finite dimensional algebra. The algorithm is as follows, and relies on the construction of Auslander-Reiten triangles given in [7]:

(1) Given a module M, compute the first two terms P_{M} = (P_{1} → P_{0}) in a minimal projective resolution of M.

(2) Compute the effect of the Nakayama functor νP_{M}. (For some algebras this is particularly easy. For example, with symmetric algebras the Nakayama functor is the identity.)

(3) Compute a chain map P_{M} → νP_{M} which is not homotopic to zero, and which is annihilated by Rad(End(P_{M})). This can be realized always as a chain map which sends P_{0} to the socle of νP_{0} and which sends P_{1} to 0. The map P_{0} → Soc νP_{0} must vanish on the image of Rad End(M) in M/ Rad M ≅ P_{0}/ Rad P_{0}.

(4) Compute the mapping cone E_{M}[1] of the map in (3).

(5) Compute the zero homology of the sequence νP_{M}[−1] → E_{M} → P_{M}.

The above algorithm has similar ingredients in it to the more obvious approach of computing τM as H_{1}(νP_{M}), computing Ext^{1}(M, τM), finding an element in the socle under the action of End M and then constructing the corresponding extension as a pushout, but some parts of this are streamlined, in that we do not have to compute the whole Ext group, and the construction of the resulting extension is done automatically within the mapping cone construction.

It is intriguing that when Λ is a symmetric algebra, the homology modules of the 3-term complex E_{M} we have just constructed are the modules which appear in an Auslander-Reiten sequence, as we observe in the next corollary.

Corollary 7.2. If Λ is a symmetric algebra and M is a non-projective indecomposable Λ-module, there is a 3-term perfect complex E_{M} with H_{1}(E_{M}) = τM, H_{0}(E_{M}) = E and H_{−1}(E_{M}) = M, where 0 → τM → E_{M} → M → 0 is the Auslander-Reiten sequence of Λ-modules terminating at M Proof. When Λ is symmetric the Nakayama functor is the identity and τM ≅ Ω^{2}M. Thus the complex E_{M} of Proposition 7.2 is perfect and has the desired homology. □

It is not always possible to construct perfect complexes with prescribed homology groups. One constraint on the homology groups is that their alternating sum must lie in the span of projective modules in the appropriate Grothendieck group. Even when this condition is satisfied, it is still not always possible to realize a list of groups as homology groups, and we mention in this context the result of Carlsson and Allday-Puppe, for which a different approach has been given recently by Avramov, Buchweitz, Iyengar and Miller [4]: if P is a perfect complex of kG-modules for an elementary abelian p-group G = C_{p}^{r}, where k is a field of characteristic p, then

∑_{i} Loewy length H_{i}(P) ≥ r + 1.

This non-existence result complements the content of Corollary 7.2 and allows us to see that even with three modules which appear in a short exact sequence, it is not always possible to realize those modules as the homology of a perfect complex of length 3.

We can now characterize the positions of complexes of length 3 in the Auslander-Reiten quiver.

Proposition 7.3. Let Λ be a self-injective algebra, M an indecomposable Λ-module and E_{M} the 3-term complex which is the mapping cone constructed above.

(1) The complex E_{M} is indecomposable unless M = Ω^{−1}S for some simple module S, in which case E_{M} is the direct sum of an indecomposable 3-term complex and P_{S} as a complex in degree 0.

(2) An indecomposable 3-term perfect complex lies on the rim of its quiver component unless, up to shift, it is either of the form E_{M} for some M ≅ Ω̸^{−1}S (in which case it lies at distance 1 from the rim) or it is the indecomposable 3-term summand of E_{Ω^{−1}S} for some simple Λ-module S (in which case it lies at distance 2 from the rim).

Proof. (1) The complexes P_{M} lie on the rim of the quiver by Corollary 6.8 unless M = Ω^{−1}S for some simple module S. Apart from this exception, E_{M} is the middle term of the Auslander-Reiten triangle with third term P_{M}, and it is indecomposable since P_{M} lies on the rim. When M = Ω^{−1}S we have seen the decomposition of E_{M} in Theorem 6.6.

(2) We have seen from Corollary 6.10 that indecomposable 3-term complexes lie at distance at most 2 from the rim, and the ones at distance 2 have the form ν^{−1}P_{S} → P_{S} → νP_{S}. If a 3-term complex lies at distance 1 from the rim, the complexes on the rim must have length 2, and are (up to shift) of the form P_{M} with M ≅ Ω̸^{−1}S for any simple S, by Corollary 6.8. Apart from these cases, 3-term complexes must lie on the rim. □

Thus, if a 3-term perfect complex has homology modules which are not the terms in an Auslander-Reiten sequence of Λ-modules (to within a projective summand of the middle term) then the complex lies on the rim of the Auslander-Reiten quiver of perfect complexes.

8. The position of rigid complexes in the Auslander-Reiten quiver

We will show that over a symmetric algebra an indecomposable complex C which is rigid, i.e. Hom(C, C[1]) = 0, must lie on the rim of the quiver. It follows that summands of tilting complexes lie on the rim, but this could be deduced from Rickard’s theory of tilting complexes, since they a tilting complex is the image of the regular representation under a derived equivalence, and we have already seen that projective modules lie in the rim.

We will deduce our result from the following lemma on Auslander-Reiten triangles, whose analogue for Auslander-Reiten sequences is also true by the same argument.

Lemma 8.1. Let A → B → C → A[1] be an Auslander-Reiten triangle in a Krull-Schmidt triangulated category C and suppose that Hom(A, C) = 0 Then A and C lie on the rim of the Auslander-Reiten quiver of C.

Proof. Suppose to the contrary that B = B_{1} ⊕ B_{2}. We show that it is not possible for both of the composite maps A → B_{1} → C[1] and A → B_{2} → C[1] to be zero as follows. Let W be any object and consider the long exact sequence

[figure: …; Hom(W, C[−1]); Hom(W, A); α; Hom(W, B_{1}) ⊕ Hom(W, B_{2}); β; Hom(W, C); Hom(W, A[1]); ⋯]

If both composites A → B_{i} → C were zero for i = 1, 2, then both composites Hom(W, C) → Hom(W, B_{i}) → Hom(W, C) would be zero. It would follow that both projections of Ker β = Im α onto the two summands Hom(W, B_{i}) lie in Ker β and hence

α Hom(W, A) = (Ker β ∩ Hom(W, B_{1})) ⊕ (Ker β ∩ Hom(W, B_{2}))

is a direct sum of End(W)-modules. Take W = A. Because End(A) is local we deduce that α Hom(A, A) has a unique simple quotient as an End Amodule. From this it follows that one of the summands Ker β ∩ Hom(A, B_{i}) is zero and that α Hom(A, A) ⊆ Hom(A, B_{j}) for the other suffix j. Now α(1_{A}) is represented by the two component maps A → B_{1} and A → B_{2}, so one of these must be zero. However, these component maps are irreducible morphisms and are never zero. This contradiction shows that one of the composite maps A → B_{1} → C and A → B_{2} → C is non-zero (and hence both are). We complete the proof by invoking the hypothesis that there is no non-zero map A → C. This further contradiction establishes that B is indecomposable. □

We can now deduce our application to rigid complexes.

Theorem 8.2. Let C be an indecomposable perfect complex for a symmetric algebra Λ with the property that Hom(C, C[1]) = 0 Then C lies on the rim of its component in the Auslander-Reiten quiver.

Proof. The Auslander-Reiten triangle starting at C has the form C → X → C[1] → C[1] since the Nakayama functor is the identity. Since Hom(C, C[1]) = 0, the result follows from Lemma 8.1. □

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E-mail address: webb@math.umn.edu

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA