Contemporary Mathematics Volume 00, 1997

The phantom menace in representation theory

Birge Huisgen-Zimmermann

1. Introduction and prerequisites

Our principal goal in this overview is to explain and motivate the concept of a phantom in the representation theory of finite dimensional algebras. In particular, we will exhibit the key role of phantoms towards understanding how a full subcategory A of Λ -mod is embedded into Λ -mod, in terms of maps leaving or entering A. (Here Λ is a finite dimensional algebra over a field K, and Λ -mod denotes the category of all finitely generated left Λ-modules.) Very roughly speaking, phantoms serve the following dual purpose in this connection: On one hand, they represent an effective tool for systematically tackling the question of whether or not A is functorially finite in Λ -mod, or – in less technical terms – whether all objects in Λ -mod have best (right/left) approximations in A, in a sense to be made precise momentarily. On the other hand, the effective A-phantoms of a given object X ∈ Λ -mod capture – within a minimal frame – a condensed picture of the relations of those objects in A which have nontrivial homomorphisms to X.

For a preview of the central definitions, let A be closed under finite direct sums and C a subcategory of A. Recall that f ∈ Hom_{Λ}(A, X) is called a C-approximation of X inside A in case A belongs to A and all maps in Hom_{Λ}(C, X) with C ∈ C factor through f [29]. In case C = A, we re-encounter the classical right A-approx-imations of X as introduced by Auslander-Smalø [4] and, independently, Enochs [19]. Whenever such right approximations exist, there is a unique candidate of minimal K-dimension, which thus serves as ‘best’ (right) approximation of X in A. (The concept of left A-approxiamation is dual; we will suppress the qualifier ‘right’ in the sequel, since we will leave dualization of our notions and results to the reader.) Note, however, that C-approximations of X inside A need not exist in general. One can, a priori, rely on their availability only when C is finite. So if C is countable, for instance, say C = {C_{n} | n ∈ ℕ}, it is natural to consider {C_{1}, …, C_{n}}-approximations of X inside A, and to explore whether they ‘converge’ to an object in A. In general they will not, but rather keep growing in dimension. So the question arises whether one should drop the requirement that the expected limits be finitely generated. Of course, if we do not insist on finitely generated

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

1991 Mathematics Subject Classification. 16D70, 16D90, 16E10, 16G10, 16G20.

While writing this paper, the author was partially supported by a grant from the National Science Foundation

approximating objects, the factorization problem per se is trivialized; namely – to again refer to our countable test situation – forming the direct sum of a full collection of {C_{1}, …, C_{n}}-approximations of X inside A, where n traces ℕ, will always yield the domain of a homomorphism through which all maps C_{n} → X can be factored. Yet, such sums will be highly redundant with respect to this stipulation as a rule. The factorization problem thus re-gains interest if we require that the redundancy be trimmed off. The minimality condition which Happel and the author imposed [29] to produce condensed images of the ‘mapping behavior’ of A relative to X, is as follows: An A-phantom of X is a module H ∈ Λ -Mod such that each finitely generated submodule H′ of H has the following property: there exists a finite subcategory A′ ⊆ A – which will usually depend on H′ – such that H′ occurs as a subfactor of every A′-approximation of X inside A. On one hand, this definition is quite loose, in that the class of A-phantoms of X is obviously closed under subfactors and direct limits (of directed systems). On the other hand, for finite classes A′ ⊆ A, one expects a large number of A′-approximations of X inside A, whence the requirement that H′ should make an appearance in all of them places major pressure on H′. The slack in the definition is useful towards testing whether X has a classical A-approximation, for the following reason: The existence of classical approximations obviously forces all A-phantoms of X to be subfactors of the minimal A-approximation of X. In fact, the class of phantoms contains infinite dimensional objects – or, equivalently, objects of unbounded finite dimensions – if and only if X fails to have a classical A-approximation. So the easier it is to construct (infinite dimensional) phantoms, the better, as long as they are only to serve as indicators for the classical existence problem. On the other hand, in case this problem has already been resolved in the negative, small phantoms are no longer of interest. One is led to investigate a class of upgraded phantoms that are fairly close in spirit to the classical approximations, except for usually being infinite dimensional: namely, the ‘effective’ phantoms. Given a subcategory C ⊆ A, a C-effective phantom H of X is a direct limit of a directed system of objects in A which is a phantom and comes equipped with a homomorphism H → X, through which all homomorphisms C → X for C ∈ C can be factored. For existence of effective phantoms, see Section 4.

On the side, we mention that, in algebraic topology, there has long been interest in phantom maps, namely in maps f : X → Y of CW-complexes with the property that, for each n ∈ ℕ, the restriction of f to the n-skeleton X^{n} of X is null-homotopic (see [40] for a survey and [14] for a modified notion of a phantom map). Around the same time as Happel and the author introduced the concepts outlined above, Benson and Gnacadja [8] defined phantom maps in the context of group representations as follows: Given a finite group G and a field K of positive characteristic, a homomorphism X → Y of KG-modules is said to be a phantom map if its restriction to every finitely generated submodule of M factors through a projective module. These phantom maps share a decisive feature with the universal maps accompanying the effective phantoms we mentioned in the preceding paragraph: Namely, they are comparatively well understood on restrictions to finitely generated subobjects of the domain, while globally, they often display qualitatively new properties.

Let us add a bit more history. The approximation theory that gave rise to the work recorded in this overview has its roots in the discovery that injective envelopes and projective covers are highly useful auxiliary objects in the structural analysis of more general modules. In their seminal paper of 1953, Eckmann and Schopf [18] showed that every module can be embedded as an essential submodule into an injective module, and in 1962, Gabriel [23] extended their findings to the context of Grothendieck categories. The more delicate existence question for projective covers, on the other hand, was resolved by Bass [6] in 1960, in his famous Theorem P. Put in slightly different terms, these results address best approximations of a module X by an object from the category of all injective, resp. projective, modules. The passage to more general subcategories of modules was performed by Auslander-Smalø [4] and Enochs [19], almost simultaneously, in the early eighties. The former team was primarily interested in categories of finitely generated modules over Artin algebras and spoke about right and left approximations, while Enochs focused on more general settings, such as coherent or noetherian rings, calling the decisive maps covers and envelopes, depending on whether they leave or enter the considered module category. The artinian line was first used towards sufficient conditions for the existence of almost split sequences in subcategories by Auslander and Smalø [5] in 1981; a decade later, this line obtained an important boost through its applications to homology and tilting theory, first observed by Auslander-Reiten [3]. Meanwhile, the latter approach was pursued separately by Asensio Mayor, Belshoff, Enochs, Guil-Asensio, Martínez Hernández, Rada, Saorín, del Valle, Xu, and others (see [2, 7, 20, 39, 41, 42, 52, 53]). Finally, we mention that in [38], Levy independently used minimal approximations from a category of particulary accessible objects, in order to study modules over pullback rings.

In the sequel, our favorite category will be P^{∞}(Λ -mod), the category of all finitely generated left Λ-modules of finite projective dimension; it is contained in the category P^{∞}(Λ -Mod) having as objects all Λ-modules of finite projective dimension. To place the new concepts (of Section 4) into context, we will discuss and exemplify (in Sections 2,3) the massive impact of contravariant finiteness of P^{∞}(Λ -mod) on the representation theory of Λ. The most striking results in this direction are Theorems 2 and 3 of Section 3 which, in case of existence, identify the minimal P^{∞}(Λ -mod)-approximations of the simple modules as the basic building blocks of arbitrary objects in P^{∞}(Λ -Mod) – and as the indicators of little and big finitistic dimensions; these theorems combine results of Auslander and Reiten [3] with more recent work of Smalø and the author [35]. The resulting program of accessing the structure of objects in P^{∞}(Λ -Mod) can be carried out in an ideally explicit format when Λ is left serial [12]. This first model situation will be presented at the end of Section 3. The second class of algebras for which the theory developed yields particularly complete and smooth results is the class of string algebras. In that case, answers to essentially all homological questions one might pose can be provided in the form of a finite number of ‘characteristic’ P^{∞}(Λ -mod)-phantoms, one for each of the simple left modules. This is the content of ongoing joint work of the author with Smalø [36]; it is sketched in Section 5. We include a brief history of the theory of string algebras, since these algebras have established their role as excellent display cases of the representation-theoretic methods developed during the past decades.

Since our primary goal is to familiarize the reader with the type of information stored in phantoms, and since we consider examples as one of the keys to an intuitive grasp, there will be major emphasis on explicit illustrations. In fact, we start by setting up a sequence of ‘test examples’ to which we keep returning as the discussion proceeds.

This survey is a revised and updated version of a series of lectures given at the 30th Symposium on Ring Theory and Representation Theory, which took place in Nagano, Japan, in October 1997; written records of the talks were subsequently distributed among the conference participants. The author would like to thank M. Saorín for references to the history of the non-artinian thread of the subject.

There are people indeed [ …] for whom all the things that have a fixed value, assessable by others, fortune, success, high positions, do not count; what they must have is phantoms.

Marcel Proust, Remembrance of Things Past

Content overview.

2. Contravariant finiteness and first examples

3. Homological importance of contravariant finiteness and a model application of the theory

4. Phantoms. Definitions, existence, and basic properties

5. Phantoms over string algebras – another model setting

Prerequisites.

Throughout, Λ = KΓ/I will be a path algebra modulo relations with Jacobson radical J, and the vertex set of Γ will be identified with a full set of primitive idempotents of Λ. By Λ -Mod we will denote the category of all left Λ-modules and by Λ -mod the full subcategory of finitely generated modules. Moreover, P^{∞}(Λ -Mod) and P^{∞}(Λ -mod) will be the subcategories of Λ -Mod and Λ -mod, respectively, having as objects the modules of finite projective dimension. The suprema of the projective dimensions attained on P^{∞}(Λ -Mod) and P^{∞}(Λ -mod) will be labeled l Fin dim Λ and l fin dim Λ, respectively.

Given a path p in KΓ, we will denote by start(p) and end(p) the starting and end points of p, respectively, and our convention for multiplying paths p, q ∈ KΓ is as follows: “qp” stands for “q after p”.

Our most important auxiliary devices will be labeled and layered graphs of finite dimensional modules. Since our graphing conventions differ to some extent from those of other authors (in particular, they are akin to, but not the same as, the module diagrams studied by Alperin [1] and Fuller [22]), we include an informal description of our graphs for the convenience of the reader.

The graphs we use are based on sequences of ‘top elements’ of a module M which are K-linearly independent modulo JM. Here x ∈ M is a top element of M if x ∉ JM and x = ex for one of the primitive idempotents e corresponding to the vertices of Γ; in this situation we will also say that x is a top element of type e.

Let Λ = KΓ/I be the algebra presented in Example C.1 of Section 2 below. That the indecomposable projective module Λe_{6} have graph

[figure: 6; ρ; σ; 5; δ; ϵ; 5; 5; 5]

with respect to the top element e_{6} means that J^{3}e_{6} = 0 and Je_{6}/J^{2}e_{6} ≅ J^{2}e_{6} ≅ S_{5} ⊕ S_{5}, that ρe_{6} and σe_{6} are K-linearly independent modulo J^{2}e_{6}, and that δρe_{6} and ϵρe_{6} are K-linearly independent (modulo J^{3}e_{6}), while δσe_{6} = ϵδe_{6} = 0 in Λe_{6}.

Generally, the entries in each row of a graph of a module M record the composition factors in the radical layers J^{r}M/J^{r+1}M in the correct multiplicities. Whenever we present the graph of an indecomposable projective module Λe, we tacitly assume that the corresponding top element of Λe is chosen to be e. In our first example the choice of top element of Λe_{6} does not influence the graph, but it will in other situations. For instance, the module M = (Λe_{6} ⊕ Λe_{6})/U, where U = Λ(ρe_{6}, ρe_{6}), has graph

[figure: 6; x_{1}; σ; ρ; 6; x_{2}; ρ; σ; 5; 5; δ; ϵ; 5; 5; 5]

relative to the top elements x_{1} = (e_{6}, 0) and x_{2} = (0, e_{6}), while its graph relative to the top elements y_{1} = x_{1} + x_{2} and y_{2} = x_{2} is

[figure: 6; y_{1}; σ; 6; y_{2}; ρ; σ; 5; ⨁; 5; δ; ϵ; 5; 5; 5]

Note that the graph of a module need not determine the module in question, up to isomorphism. For example, each of the modules M_{k} = Λe_{6}/U_{k}, where U_{k} = Λ(σ − kρ)e_{6} with k ∈ K^{∗}, has graph

6 σ ρ 5

with respect to the top element e_{6} + U_{k} of M_{k}, while M_{k_{1}} ≅ M̸_{k_{2}} if k_{1} ≠ k_{2}.

For a slightly more involved example, consider the factor module N = (Λe_{6} ⊕ Λe_{11})/V, where V is the submodule generated by (ρ, ρψ + δρψ) and (ϵρ − σ, 0). Relative to the top elements x_{1} = (e_{6}, 0) and x_{2} = (0, e_{11}), the module N has graph

[figure: 6; ρ; σ; 11; ψ; 6; ρ; 5; ϵ; δ; 5; 5]

indicating – next to the previously discussed information – that σx_{1} is equal to a nonzero scalar multiple of ϵρx_{1}, modulo J^{4}M = 0, and ρx_{1} equals a nonzero scalar multiple of ρψx_{2}, modulo J^{3}M.

To enlarge the family of objects in Λ -mod which possess labeled and layered graphs relative to suitable sequences of top elements, we also allow for graphs with ‘pools’ along the following model. That a module A ∈ Λ -mod has graph

[figure: 1; x_{1}; π; 9; x_{9}; μ; 10; x_{10}; ν; 11; x_{11}; ψ; 12; x_{12}; χ; 7; ξ; 8; η; τ; 6; ρ; 6; ρ; 2; 2; 2; 5; 5; 5]

is to encode the following information: The module A is generated by top elements x_{i} of type e_{i} (i = 1, 9, 10, 11, 12) which are K-linearly independent modulo JA (here automatically satisfied, since e_{i} ≠ e_{j} for i ≠ j) such that J^{3}A = 0, and

(a) JA/J^{2}A ≅ S_{7} ⊕ S_{8} ⊕ S_{6}^{2}, with μx_{9}, νx_{10}, ψx_{11}, χx_{12} being K-linearly independent modulo J^{2}A;

(b) J^{2}A ≅ S_{2}^{2} ⊕S_{5}^{2}, and the “pooled elements” πx_{1}, ξμx_{9}, ηνx_{10} are K-linearly dependent, while any two of these elements are K-linearly independent; analogously, τνx_{10}, ρψx_{11}, ρχx_{12} are K-linearly dependent, with any subset of two K-linearly independent.

Note that in this particular example, the module A is determined up to isomorphism by its graph, since the coefficients arising in the mentioned linear dependence relations can be adjusted by suitably modifying the top elements by scalar factors.

It is clear that we will not lose information if we omit the labels on edges

\begin{matrix}i \\ j \end{matrix}

with the property that there is a unique arrow i → j in Γ. Moreover, it should be self-explanatory that certain countably generated left Λ-modules can be communicated by graphs as well. The infinite graph

[figure: 6; σ; 6; ρ; σ; 6; ρ; σ; · · · · · ·; 5; 5; 5; _{· · · · · ·}]

for instance, goes with a module B, uniquely determined up to isomorphism, which is generated by top elements x_{1}, x_{2}, x_{3}, … of type e_{6} which are K-linearly independent modulo JB such that ρx_{1} = 0 and σx_{i} = ρx_{i+1} for all i ≥ 1, and such that the elements σx_{i}, i ∈ ℕ, are K-linearly independent (modulo J^{2}B = 0).

2. Contravariant finiteness and first examples

In 1980, Auslander and Smalø [4] introduced the following definitions – along with the duals – in connection with their search for conditions ensuring existence of almost split sequences. It turned out that existence is guaranteed in full, extensionclosed subcategories A of Λ -mod which are both co- and contravariantly finite in the sense recalled below. Even though their results have been applied on numerous occasions during the 1980’s, it was not until the 1990’s that the concept of contravariant finiteness reached a high level of popularity, due to its links with homology and tilting. The spark in the tinder barrel was a paper by Auslander and Reiten [3]. We will describe their homological results in Section 3.

Definitions. Let A ⊆ Λ -mod be a full subcategory and X ∈ Λ -mod.

(1) A right A-approximation of X is a homomorphism f : A → X with A ∈ A such that each g ∈ Hom(B, X) with B ∈ A factors through f, or equivalently, such that the following sequence of functors induced by f is exact:

Hom_{Λ}(−, A)|_{A} ⟶ Hom_{Λ}(−, X)|_{A} ⟶ 0.

(Since we will hardly mention the dual concept of ‘left approximation’, we will systematically suppress the qualifier ‘right’ when discussing approximations.)

(2) The subcategory A is said to be contravariantly finite (in Λ -mod) if each X ∈ Λ -mod has an A-approximation, i.e., if each of the functors Hom_{Λ}(−, X)|_{A} is finitely generated in the category of all contravariant functors from A to Ab.

Suppose that X has an A-approximation. By a slight abuse of language, we will then also refer to the source of this map as an approximation. Not surprisingly, the A-approximations of X of minimal length are all isomorphic. Indeed, as was shown by Auslander and Smalø [4], given a minimal A-approximation f : A → X and any A-approximation f′ : A′ → X, there exists a split embedding g : A → A′ which makes the following diagram commute:

[figure: A; g; f; A′; f′; X]

It is thus justified to refer to \underline{the} minimal A-approximation of X in case of existence.

If A is a resolving subcategory of Λ -mod, i.e., if A contains all projectives in Λ -mod and is closed under extensions and kernels of epimorphisms, then the simples play a prominent role in checking contravariant finiteness. Indeed:

Theorem 1. [3] Suppose A is a resolving subcategory of Λ -mod. Then A is contravariantly finite in Λ -mod if and only if each of the simple left Λ-modules has an A-approximation. □

Since, clearly, our favorite category P^{∞}(Λ -mod) is resolving, this will provide a convenient test for contravariant finiteness. As we will see in the next section, the minimal P^{∞}(Λ -mod)-approximations of the simples are the basic structural building blocks for the objects in P^{∞}(Λ -Mod) in case of existence, which makes it a priority to understand the structure of these particular approximations.

To provide examples, we begin with some

Well Known Facts.

• A subcategory A ⊆ Λ -mod is contravariantly finite in case it is ‘very big’ or ‘very small’. Indeed, if A = Λ -mod, then clearly contravariant finiteness is guaranteed, the minimal approximations being the identity maps. So, in particular, P^{∞}(Λ -mod) is contravariantly finite provided that Λ has finite global dimension. On the other hand, if A has finite representation type, i.e., if there exist objects A_{1}, …, A_{n} ∈ A such that each object in A is a direct sum of copies of the A_{i}, we have contravariant finiteness as well [3]. To construct an A-approximation of a module X, simply add up as many copies of each A_{i} as the K-dimension of Hom_{Λ}(A_{i}, X) indicates. In particular, if A is the category of all finitely generated projectives in Λ -mod, the minimal A-approximations are precisely the projective covers. This latter category coincides with P^{∞}(Λ -mod) precisely when l fin dim Λ = 0.

• [3] If Λ is stably equivalent to a hereditary algebra Λ′ (meaning that the stable category Λ-\underline{mod}, obtained as a factor category from Λ -mod by killing the maps that factor through projectives, is equivalent to the corresponding stable category Λ′-\underline{mod}), then P^{∞}(Λ -mod) is contravariantly finite. This hypothesis is, in particular, satisfied if J^{2} = 0, and in that case, the minimal approximation A(X) of any Λ-module X can be readily pinned down as follows [32]: A(X) = P/(JP)_{fin}, where P is the projective cover of X and (JP)_{fin} is the direct sum of those homogeneous components of JP which have finite projective dimension.

• [12] If Λ is left serial, then P^{∞}(Λ -mod) is always contravariantly finite. The minimal approximations of the simple left modules arising in this situation will be described in the next section.

• The first example for which P^{∞}(Λ -mod) was shown not to be contravariantly finite is due to Igusa-Smalø-Todorov [37]. It is a monomial relation algebra with J^{3} = 0 and l fin dim Λ = 1 which is closely related to the Kronecker algebra, its K-dimension exceeding that of the latter algebra only by 2. In this example, the right finitistic dimension is 0, which, in view of our first remark, demonstrates that the right-hand category P^{∞}(mod-Λ) \underline{is} contravariantly finite in mod-Λ. Thus contravariant finiteness of P^{∞}(−) is not left-right symmetric.

• In [29], very general criteria for the failure of contravariant finiteness of P^{∞}(Λ -mod) were developed. We refer the reader to the examples given there and in [34].

The emerging picture indicates that, while hardly any of the traditionally considered classes of finite dimensional algebras enjoy – en bloc – the property that P^{∞}(−) is contravariantly finite, the positive case is ubiquitous. In fact, the condition of having contravariantly finite P^{∞}(−) appears to slice diagonally through the prominent classes of finite dimensional algebras.

First Installment of Nonstandard Examples.

We will next present a first set of examples which are to informally communicate prototypical phenomena ensuring that a given simple module has a P^{∞}(Λ -mod)- approximation or fails to have such an approximation. These specific algebras will then continue to serve us as illustrations along the way. We will follow with proofs of some of the positive instances, but defer the discussion of the negative instances to Section 4.

Examples A. Our first example shows that, for each natural number n, there exists a finite dimensional monomial relation algebra Λ and a simple S ∈ Λ -mod such that P^{∞}(Λ -mod) is contravariantly finite and the minimal P^{∞}(Λ -mod)-ap-proximation of S is a direct sum of n distinct nonzero indecomposable components.

A.1. Fix n ∈ ℕ, and let Λ = KΓ/I be the monomial relation algebra with quiver Γ

[figure: 1; α; a_{1}; α_{1}; ⋮; 2; γ; δ; 3; ϵ; a_{n−1}; α_{n−1}; a_{n}; α_{n}]

having indecomposable projective left modules with graphs

[figure: 1; α; 2; γ; δ; 3; ϵ; a_{1}; α_{1}; a_{n}; α_{n}; 2; γ; 3; 3; 3; 2; δ; _{· · · · · ·}; 2; δ; 3; 3; 3]

Then P^{∞}(Λ -mod) is contravariantly finite, and the minimal P^{∞}(Λ -mod)-approx-imations A_{1}, A_{2}, A_{3}, A_{a_{1}}, …, A_{a_{n}} of the simple modules S_{1}, S_{2}, S_{3}, S_{a_{1}}, …, S_{a_{n}} have the following graphs (which determine the corresponding modules up to isomorphism).

[figure: 1; a_{1}; 1; a_{n}; 2; A_{1} :; ⨁; ⋯; ⨁; A_{2} :; 2; 2; 3; 3; 3; a_{i}; 1; A_{3} :; A_{a_{i}} :; (1 ≤ i ≤ n); 3; 2]

A.2. Now let Λ = KΓ′/I′, where Γ′ is obtained from the quiver Γ of A.1 by removing the arrow γ, and the ideal I′ ⊆ KΓ′ is such that the graphs of Λe_{3}, Λe_{a_{1}}, …, Λe_{a_{n}} are as under A.1, whereas Λe_{1} and Λe_{2} have graphs

1 α  2δ and 2  3

respectively.

Then S_{1} = Λe_{1}/Je_{1} has minimal P^{∞}(Λ -mod)-approximation

[figure: 1; a_{1}; a_{2}; · · · · · ·; a_{n}; 2]

S_{2} has minimal P^{∞}(Λ -mod)-approximation Λe_{2}, and S_{a_{1}}, …, S_{a_{n}} belong to P^{∞}(Λ -mod). In particular, P^{∞}(Λ -mod) is again contravariantly finite.

Examples B.

B.1. Let Λ = KΓ/I, where Γ is the quiver

[figure: 2; β; 4; γ; 1; α; ρ; 7; 8; 6; 3; σ; 5; τ]

and the ideal I ⊆ KΓ contains the relation γβα − τσρ, together with suitable monomial relations, such that the indecomposable projective left Λ-modules have graphs

[figure: 1; 2; 3; 4; 5; 6; 7; 8; 2; 3; 4; 5; 6; 6; 6; 4; 5; 4; 5; 4; 5; 6; 6; 6; 6; 6]

Then S_{i} ∈ P^{∞}(Λ -mod) for i = 2, 3, the minimal P^{∞}(Λ -mod)-approximations of S_{4}, S_{5}, S_{6} are Λe_{4}, Λe_{5}, and Λe_{6}, respectively, while none of S_{1}, S_{7}, S_{8} has a P^{∞}(Λ -mod)-approximation. In fact, in Section 4, we will see that there is no object A ∈ P^{∞}(Λ -mod) such that all of the homomorphisms from the modules in the following subclass of P^{∞}(Λ -mod) – call it C – to S_{1} factor through A.

[figure: 7; y_{n}; ⋯; 8; z_{1}; 7; y_{1}; 1; x; 1; x′; 8; z′_{1}; 7; y′_{1}; 8; z′_{2}; ⋯; 7; y′_{n}; 5; 4; ⋯; 4; 5; 2; ⨁; 3; 4; 5; 4; ⋯; 4; 5; 4; 5]

B.2. Now let Λ be the factor algebra of the algebra described in B.1, modulo the ideal generated by e_{7} and e_{8}. Then the graphs of the indecomposable projective modules Λe_{1}, …, Λe_{6} remain unchanged, but this time P^{∞}(Λ -mod) is contravariantly finite. Indeed, the minimal P^{∞}(Λ -mod)-approximations of S_{2}, …, S_{6} are as above, while S_{1} has the following minimal P^{∞}(Λ -mod)-approximation:

1  1 2 ⨁ 3 4  5

The next example shows that the structure of the minimal P^{∞}(Λ -mod)-approx-imations of the simple modules need by no means be as simplistic as in the previous instances, not even in situations where the indecomposable projective modules are of a simplistic makeup.

Examples C.

C.1. This time, let Λ = KΓ/I be the monomial relation algebra with quiver Γ

[figure: 1; π; 4; ω; 2; β; α; γ; 3; ι; 9; μ; 7; ξ; 10; ν; 8; η; τ; 11; ψ; 6; ρ; σ; 5; δ; ϵ; 12; χ]

and choose the ideal I ⊆ KΓ of relations so that the indecomposable projective left Λ-modules have the following graphs:

[figure: 1; 2; α; β; γ; 3; 4; 5; δ; ϵ; 6; ρ; σ; 7; 8; 9; 10; 11; 12; 2; α; 3; 3; 3; 3; 2; β; γ; 5; 5; 5; δ; ϵ; 5; 2; β; 2; γ; 5; 7; 8; 6; ρ; 6; ρ; 3; 3; 3; 5; 5; 3; 3; 2; β; 2; γ; 5; 5; δ; 5; ϵ; 3; 3; 5; 5]

In this example, P^{∞}(Λ -mod) is again contravariantly finite, and the minimal approximations A_{1}, …, A_{12} of the simples S_{1}, …, S_{12} are determined by their graphs as follows:

[figure: 1; 9; 10; 11; 12; 1; 4; A_{1} :; 7; 8; 6; 6; ⨁; 2; 2; 2; 2; 5; 5; 5]

A_{2} ≅ Λe_{2}  A_{3} ≅ Λe_{3}  A_{4} : [figure: 1; 4; 2]

[figure: 12; 11; 5; δ; ϵ; 11; 12; 6; σ; 11; 12; A_{5} :; 6; 6; 6; 6; A_{6} :; 6; 6; 5; 5; 5; 5; 5; 5; 5; 5; 5]

[figure: 7; 1; 10; 11; 12; 12; 11; 8; 1; 9; A_{7} :; 2; _{.................}_{.................}\begin{matrix}. \\ . \\ . \\ . \\ . \\ . \\ . \end{matrix}_{. .. . ...... .}_{. . . . . . . . . ........}; 2; 8; 6; 6; A_{8} :; 6; 6; 2; _{.................}_{.................}\begin{matrix}. \\ . \\ . \\ . \\ . \\ . \\ . \end{matrix}_{. .. . .......}_{. . . . . . . . . ........}; 2; 7; 2; 5; 5; 5; 5; 5; 5; 2]

A_{9} ≅ S_{9}  A_{10} ≅ S_{10}  A_{11} ≅ A_{12} : [figure: 11; 12; 6; 6; 5]

C.2. Finally, let Λ be the factor algebra of the algebra under C.1, modulo the ideal generated by e_{11} and e_{12}. Note that the graphs of the indecomposable projectives Λe_{1}, …, Λe_{10} remain the same as in C.1, since e_{11} and e_{12} are sources of Γ. This time, P^{∞}(Λ -mod) fails to be contravariantly finite, S_{1}, S_{5}, S_{6}, S_{8} being precisely those simples which have lost their P^{∞}(Λ -mod)-approximations in the passage to the smaller algebra. As we will see in Section 4, the homomorphisms onto S_{5} from the modules of the following P^{∞}(Λ -mod)-family can, for instance, not all be factored through a fixed A ∈ P^{∞}(Λ -mod):

[figure: 6; σ; ⋯; 6; ρ; σ; 6; ρ; σ; 6; ρ; σ; 6; ρ; σ; ⋯; 6; ρ; σ; 5; ⋯; 5; 5; 5; 5; 5; ⋯; 5; 5]

(On the other hand, observe that they can be factored through the module A_{6} over the algebra in C.1.)

To sketch justifications for some of the claims in the preceding examples, we first spell out an obvious sufficient condition for a simple module S = Λe/Je to have a P^{∞}(Λ -mod)-approximation. Indeed, this is the case provided that the following is true: There exist indecomposable modules T_{1}, …, T_{m} in P^{∞}(Λ -mod) with top elements x_{i} ∈ T_{i} of type e such that for each indecomposable object X ∈ P^{∞}(Λ -mod) having a top element x of type e, there exists a factor module X/Y with \overline{x} = x+Y ≠ 0 which can be embedded into some T_{j} in such a fashion that \overline{x} is mapped to x_{j}. If there exist T_{1}, …, T_{m} as stipulated, we know moreover that the indecomposable direct summands of the minimal P^{∞}(Λ -mod)-approximations of S are all recruited from the T_{i}.

Ad Example A.1. In arguing that A_{1} is the minimal P^{∞}(Λ -mod)-approx-imation of S_{1}, we bypass the facts that l fin dim Λ = 1 (use the method of [30], for instance), that the module A_{1} has finite projective dimension and satisfies the required minimality condition. To verify that A_{1} is a P^{∞}(Λ -mod)-approximation of S_{1}, we let X ∈ P^{∞}(Λ -mod) be indecomposable and endowed with a top element

x of type e_{1}. Then αx ≠ 0, since otherwise the module with graph {2 \atop 3} would be a direct summand of Ω^{1}(X), which is impossible. If γαx ≠ 0, then X ≅ Λx ≅ Λe_{1} (use indecomposability and finite projective dimension), and X/ soc X embeds into

each of the modules [figure: 1; a_{i}; 2] for i = 1, …, n. So suppose that γαx = 0. In this

case, J^{2}X = 0, the socle of X being homogeneous of type e_{2}, and Ω^{1}(X) ≅ (Λe_{2})^{r} for some r. One infers the existence of a factor module X/Y with graph

[figure: 1; \overline{x}; a_{j}; 2]

for some j. □

Ad A.2. To see that, also in this example, the exhibited module A_{1} is the

minimal P^{∞}(Λ -mod)-approximation of S_{1}, observe that Ω^{1}(A_{1}) = ( 2δ 3 )^{n} =

(Λe_{2})^{n}, whence A_{1} ∈ P^{∞}(Λ -mod). The rest of the argument is similar to the one given above. □

Ad B.2. Once more, we will show that A_{1} is a P^{∞}(Λ -mod)-approximation of S_{1} (minimality being clear then). Any indecomposable module in P^{∞}(Λ -mod) containing Λe_{1} is clearly isomorphic to Λe_{1}, and the only proper nonzero factor modules of Λe_{1} which embed into indecomposable modules X ∈ P^{∞}(Λ -mod) are the direct summands of A_{1}, as well as

[figure: 1; 1; 2; 3; and; 2; 3; 4; 5]

Observe that none of these factors of Λe_{1} has a proper extension to an indecomposable module in P^{∞}(Λ -mod). Since clearly the former factors are in turn factor modules of the latter, our claim follows. □

All direct summands of the minimal approximations of the various simple modules S = Λe/Je exhibited above contain factor modules of Λe which are minimal with respect to the property of being top-embeddable into modules of finite projective dimension. (We say that a monomorphism A → B is a top-embedding if it induces a monomorphism A/JA → B/JB.) This is generally true for the minimal P^{∞}(Λ -mod)-approximations of simple modules, whenever Λ is either left serial or a string algebra (see Sections 3 and 5), a fact which greatly facilitates resolving the existence question (always positive in case of a left serial algebra and algorithmically decidable for string algebras) and the construction of such approximations. We conclude this section with an easy example showing that this cannot be expected to hold in general. In Section 5, we will follow up with an example demonstrating that the mentioned asset of string algebras is not shared by arbitrary special biserial algebras either.

Example D. Let Λ = KΓ/I be the monomial relation algebra with the following indecomposable projective left modules:

[figure: 1; 2; 3; 4; 5; 6; 2; 3; 4; 3; 4; 2; 2; 3; 4; 3; 4]

Then P^{∞}(Λ -mod) is contravariantly finite, the minimal P^{∞}(Λ -mod)-approxima-tion A_{1} of S_{1} having the following graph:

[figure: 1; 5; 1; 6; 2; ⨁; 2; 3]

Note that the second summand of A_{1} does not contain a proper factor module of Λe_{1}, whereas Λe_{1}/ soc(Λe_{1}) is the (unique in this case) minimal factor module of Λe_{1} which can be top-embedded into an object of P^{∞}(Λ -mod).

3. Homological importance of contravariant finiteness and a model application

As surfaced in [31], [33] and [48], non-finitely generated modules of finite projective dimension may display structural phenomena which are completely different from those encountered in finitely generated modules of finite projective dimension. (We labeled them ‘domino effects’ in [31].) In particular, there may be objects in P^{∞}(Λ -Mod) whose projective dimension exceeds l fin dim Λ by any predetermined positive number, even when l fin dim Λ = 1 [48]. Moreover, the left cyclic finitistic dimension l cyc fin dim Λ, i.e., the supremum of those projective dimensions which are attained on the cyclic modules in P^{∞}(Λ -mod), may be strictly smaller than l fin dim Λ. In fact, for each natural number n, there exists a finite dimensional algebra Λ_{n} such that l fin dim Λ_{n} is not attained on any n-generated module (see [33]). However, in case P^{∞}(Λ -mod) is contravariantly finite, all of the left finitistic dimensions of Λ coincide, and the objects of the big category P^{∞}(Λ -Mod) are as well understood as those of the small P^{∞}(Λ -mod).

The following notation will be convenient: Given objects A_{1}, …, A_{n} in Λ -mod, let filt(A_{1}, …, A_{n}) be the full subcategory of Λ -mod the objects of which are those modules which have filtrations with consecutive factors among A_{1}, …, A_{n}. More precisely, X belongs to filt(A_{1}, …, A_{n}) if and only if there exists a chain X = X_{0} ⊇ X_{1} ⊇ ⋯ ⊇ X_{m} = 0 such that each of the factors X_{i}/X_{i+1} is isomorphic to some A_{j(i)}. Moreover, \overrightarrow{filt}(A_{1}, …, A_{n}) will stand for the closure of filt(A_{1}, …, A_{n}) under direct limits in Λ -Mod.

Concerning the structure of the finitely generated modules of finite projective dimension in case A = P^{∞}(Λ -mod) is contravariantly finite, Auslander and Reiten proved the following result in the more general context of an arbitrary resolving subcategory A.

Theorem 2. [3] Suppose that A is a resolving contravariantly finite subcategory of Λ -mod, and that A_{1}, …, A_{n} are the minimal A-approximations of the simple left Λ-modules. Then a module X belongs to A if and only if X is a direct summand of an object in filt(A_{1}, …, A_{n}). □

Of course, this theorem, applied to A = P^{∞}(Λ -mod), yields the following consequence:

Corollary 3. [3] If P^{∞}(Λ -mod) is contravariantly finite and A_{1}, …, A_{n} are the minimal P^{∞}(Λ -mod)-approximations of the simple left Λ-modules, then

l fin dim Λ = l cyc fin dim Λ = max{p dim A_{1}, …, p dim A_{n}}.□

Due to the fact that contravariant finiteness of P^{∞}(Λ -mod) places demands only on the finitely generated modules, the strong impact which this condition has on non-finitely generated modules may come as a surprise. In fact, the structure theory for objects in P^{∞}(Λ -mod) extends smoothly to P^{∞}(Λ -Mod).

Theorem 4. [35] Again suppose that P^{∞}(Λ -mod) is contravariantly finite, and let A_{1}, …, A_{n} be as in the corollary. Then

P^{∞}(Λ -Mod) = \overrightarrow{filt}(A_{1}, …, A_{n}),

and, in particular,

l Fin dim Λ = l fin dim Λ = max{p dim A_{1}, …, p dim A_{n}}.□

Thus, contravariant finiteness of P^{∞}(Λ -mod) resolves the notorious quandary of locating objects in Λ -Mod on which l Fin dim Λ is attained; this search may be a very difficult task, even when finiteness of l Fin dim Λ is guaranteed in advance. The helpfulness of the above theory will be displayed to full advantage in our examples.

Examples of Section 2 revisited.

For the moment, we will only determine the finitistic dimensions of those algebras displayed which give rise to contravariantly finite categories P^{∞}(−). By the preceding discussion, the objects of Λ -Mod having finite projective dimension are precisely the direct limits of the objects in filt(A_{1}, …, A_{n}) with the A_{i} as shown in Section 2.

Ad A.1. It is clear that p dim A_{1} = p dim A_{a_{i}} = 1 for i = 1, …, n, whereas p dim A_{2} = p dim A_{3} = 0. Hence, l Fin dim Λ = l fin dim Λ = 1.

Ad A.2. In this example, the minimal P^{∞}(Λ -mod)-approximation of each simple S_{a_{i}} centered in the vertex a_{i} coincides with S_{a_{i}} and has projective dimension 1, as does the minimal approximation of S_{1}. The minimal approximations of S_{2} and S_{3} are again identical with their projective covers. So, once more, l Fin dim Λ = l fin dim Λ = 1.

Ad B.2. Here A_{1}, A_{4}, A_{5}, A_{6} are projective, while p dim A_{2} = p dim A_{3} = 1, and we obtain the same conclusion as before. □

We will briefly digress from our main line of thought for another corollary of the preceding theorem. The existence theorem for internal almost split sequences [4] which we quoted at the outset can be strengthened for the category A = P^{∞}(Λ -mod) as follows.

Corollary 5. If P^{∞}(Λ -mod) is contravariantly finite, then P^{∞}(Λ -mod) is also covariantly finite and thus has almost split sequences.

Proof. By a result of Crawley-Boevey [15], it suffices to show that arbitrary direct products of objects in the category P^{∞}(Λ -mod) belong to its closure \overrightarrow{P^{∞}}(Λ -mod) under direct limits in Λ -Mod. But in view of the theorem, contravariant finiteness of P^{∞}(Λ -mod) entails the equality P^{∞}(Λ -Mod) = \overrightarrow{P^{∞}}(Λ -mod) and, in view of the finiteness of l Fin dim Λ, this guarantees closedness of this latter subcategory under direct products. □

As a class of examples of algebras Λ with very rich and complex module categories, for which the above program of zeroing in on the structure of the objects in P^{∞}(Λ -Mod) works to perfection, we will present the left serial algebras. Recall that a split algebra Λ = KΓ/I is called left serial in case no more than one arrow leaves any given vertex of Γ; equivalently, this means that the indecomposable projective left Λ-modules are all uniserial. To describe the minimal P^{∞}(Λ -mod)- approximations of the simple modules in this situation, we require the following definition.

Definition. Suppose that T_{1}, …, T_{m} is a sequence of nonzero uniserial left Λ-modules, and let p_{i} be a mast of T_{i}, namely a path in KΓ of maximal length with p_{i}T_{i} ≠ 0. A left Λ-module T is called a saguaro on (p_{1}, …, p_{m}) if

(i) T ≅ (⨁_{1≤i≤m} T_{i})/U, where U ⊆ ⨁_{1≤i≤m} JT_{i} is generated by a sequence of elements of the form q_{i}t_{i} − q′_{i+1}t_{i+1}, 1 ≤ i ≤ m − 1, where t_{i} ∈ T_{i} are suitable top elements and q_{i}, q′_{i} are right subpaths of the masts p_{i} such that q_{i}t_{i} ≠ 0, and q′_{i+1}t_{i+1} ≠ 0; moreover, we require that

(ii) each T_{j} embeds canonically into T via

T_{j} ⟶^{can} ( ⨁_{1≤i≤m} T_{i})/U ≅ T.

The uniserial modules T_{i} are called the trunks of T.

We will identify T with (⨁_{1≤i≤m} T_{i})/U. To avoid ambiguities, we will denote the canonical images of the trunks T_{i} inside T by T̂_{i} and the canonical images of the top elements t_{i} by t̂_{i}. Any such sequence (t̂_{1}, …, t̂_{m}) will be called a canonical sequence of top elements for T.

Note that saguaros are particularly amenable to graphing, the shape of their graphs explaining their name (they share shape and name with a cactus found in the Sonoran desert, Cereus giganteus). In fact, the definition forces them to be glued together in a very straightforward fashion from their uniserial trunks: Layered and labeled graphs relative to a canonical sequence of top elements always exist (not only over left serial algebras), and are built on the pattern illustrated below.

[figure: 1; t̂_{1}; 2; t̂_{2}; 2; t̂_{3}; 4; t̂_{4}; 1; t̂_{5}; 2; 2; 2; 3; 3; 3; 4]

Here T = (⨁_{i=1}^{5} T_{i})/U, where the trunks T_{i} = Λt_{i} of T have graphs

1  2  2  4  1 2  2  2  2  2 3  3  3  3  3 3  3  3  3  3 4  4  4  4  4

relative to the top elements t_{i}.

Of course, over left serial algebras, the graphs of the uniserial left modules are necessarily edge paths (in particular, uniserials have unique masts), so that the graphs of saguaros simplify to trees. It turns out that, in this situation, all minimal P^{∞}(Λ -mod)-approximations of simple modules are recruited from the class of saguaros with simple socles. In fact, these approximating saguaros can be pinned down explicitly and even constructed algorithmically on the basis of quiver and relations.

Theorem 6. [12] Suppose that Λ = KΓ/I is a left serial algebra. Then P^{∞}(Λ -mod) is contravariantly finite, and the minimal P^{∞}(Λ -mod)-approxima-tions of the simple left Λ-modules are saguaros with simple socles.

More precisely, the minimal P^{∞}(Λ -mod)-approximation of a simple left Λmodule S = Λe/Je can be described as follows: If Λe/C is the (unique) minimal nonzero factor module of Λe which has finite projective dimension, there is a unique saguaro A(S) of maximal length in P^{∞}(Λ -mod) such that Λe/C is a trunk of A(S) and soc A(S) is simple. Moreover, the canonical epimorphisms A(S) → S, which map Λe/C onto S and send the other trunks of A(S) to zero, are minimal P^{∞}(Λ -mod)-approximations. □

To refer back to the concluding remark of Section 2: In the setting of the theorem, Λe/C actually coincides with the nonzero factor module of Λe which is minimal with respect to top-embeddability into a module of finite projective dimension.

Actually, not only is P^{∞}(Λ -mod) always contravariantly finite in the left serial case, but so are the categories P^{(d)} = P^{(d)}(Λ -mod) consisting of the finitely generated left Λ-modules of projective dimensions at most d. Moreover, the minimal P^{(d)}-approximations of the simples are again saguaros, and the sequences of these saguaros for 1 ≤ d ≤ l fin dim Λ record the homological properties of Λ with high precision, the case d = l fin dim Λ leading back to P^{∞}(Λ -mod).

Example E. Let Λ be a left serial algebra whose indecomposable projective modules are represented by the following graphs.

1 2  3  4  5  6  7  8  9 10 11 12 13 14 2 3  4 12 2  3  4  3  8  8  6 13 14 14 3 4 12 13 3  4 12 4  3  3  3 14 14 4 12 13 14 4 12 13 12 4  4  4 13  12 13 14 13 12 12 12 13

The evolution of the P^{(d)}-approximations of the simple left Λ-module S_{1} is graphically represented below. We exhibit the minimal P^{(1)}-, P^{(2)}-, P^{(3)}-approximations of S_{1} from left to right; the last coincides with the minimal P^{∞}(Λ -mod)-approxi-mation, since the left finitistic dimension of Λ is 3.

[figure: 1; 5; 6; 8; 7; 1; 5; 6; 8; 1; 5; 11; 9; 10; 2; 2; 2; 6; 8; 3; 3; 3; 4]

4. Phantoms. Definitions, existence, and basic properties

The objects discussed in this section were introduced by Happel and the author in [29]. As mentioned in the introduction, their purpose is twofold: In the first place, they serve as indicators as to whether or not a given subcategory A ⊆ Λ -mod is contravariantly finite. Their second role is that of retaining the kind of information which is stored in minimal A-approximations whenever they exist, within potentially infinite dimensional frames; this role is played most satisfactorily by the ‘effective’ phantoms.

Since the concept of a phantom is possibly not easily translated into an intuitive picture, we break the definitions into several parts, and add a fairly extensive discussion at each step. The subsequent detailed analysis of the examples of Section 2 should also help the reader to see that we are dealing with objects that arise very naturally when the relations of a module X are being compared with those of the objects in a subcategory of Λ -mod.

Definition, Part I. (Relative approximations) Let A be a full subcategory of Λ -mod, and C a subcategory of A. Moreover, let X ∈ Λ -mod.

A C-approximation of X inside A is a homomorphism f : A → X with A ∈ A such that each map in Hom_{Λ}(C, X) with C ∈ C factors through f. Again, we will loosely refer to the object A as a C-approximation of X inside A.

Clearly, whenever X has a minimal A-approximation, A_{0} say, the classical A-approximations of X are precisely the {A_{0}}-approximations of X inside A. In particular, we then obtain A_{0} as a C-approximation of X inside A, where C is a \underline{finite} subcategory of A. Moreover, A_{0} is a direct summand of any \{A_{0}\}-approxi-mation of X inside A, and so, a fortiori, is a subfactor of any such approximation. On the other hand, given a finite subcategory C ⊆ A, the module X will have C-ap-proximations inside A provided that we require A to be closed under finite direct sums: Just sum up a sufficient number of copies of the objects in C. In other words, approximations relative to \underline{finite} classes are always available, also in case there are no classical A-approximations of X.

Definition, Part II. (Phantoms) Retain the notation of Part I, and suppose, in addition, that A is closed under finite direct sums.

A finitely generated module H ∈ Λ -mod is an A-phantom of X in case (*) there is a finite subcategory A′ ⊆ A such that H occurs as a subfactor of

every A′-approximation of X inside A.

More generally, an arbitrary module H ∈ Λ -Mod will be called an A-phantom of X if each of its finitely generated submodules satisfies (*). Of course, the choices of the finite subclasses A′ ⊆ A will vary with the finitely generated submodules H′ of H.

The class of all A-phantoms of X is clearly closed under subfactors, and consequently is closed under direct limits of direct systems as well. In fact, a module H ∈ Λ -Mod is an A-phantom of X if and only if H is the direct limit of a direct system of finitely generated A-phantoms of X. This may make the class of A-phantoms enormous: Indeed, the class of all A-phantoms of a simple module S may encompass the entire class of indecomposables H in Λ -Mod with S ↪ H/JH. This slack in the definition of phantoms has the advantage of facilitating their construction. Often, the particular structure of phantoms is irrelevant – their sheer size is enough to signal non-existence of traditional A-approximations of X. In fact, if X has an A-approximation then the class of A-phantoms of X coincides with the set of subfactors of the minimal such approximation. Consequently, the existence of phantoms of X of unbounded lengths is enough to guarantee non-existence of traditional A-approximations. Of course, in terms of encapsulating further information, the usefulness of phantoms so generously defined is moderate. Hence, we single out a subclass of phantoms which are more strongly tied to the category A and carry a full complement of information on how a given subcategory C ⊆ A relates to X.

Definition, Part III. (Effective phantoms) Keep the notation of Part II, and denote by \overrightarrow{A} the closure of A under direct limits (of direct systems) in Λ -Mod. Moreover, fix a subcategory C ⊆ A.

An A-phantom H ∈ \overrightarrow{A} is called effective relative to C if there exists a homomorphism h : H → X with the property that each map in Hom_{Λ}(C, X) with C ∈ C factors through h.

In a self-explanatory extension of of the term ‘approximation’ to subcategories of Λ -Mod, the effectiveness condition thus calls for h : H → X to be a C-approxi-mation of X inside \overrightarrow{A}. In case X has a traditional A-approximation, the minimal such approximation is clearly the only effective A-phantom of X relative to A. Otherwise, existence of interesting phantoms, effective or not, is not immediately clear, but is guaranteed by the following result.

Theorem 7. [29] Let A ⊆ Λ -mod be a full subcategory which is closed under finite direct sums. For X ∈ Λ -mod, the following conditions are equivalent:

(1) X fails to have an A-approximation.

(2) X has A-phantoms of infinite K-dimension.

(3) There exist countable subclasses C ⊆ A such that X has infinite dimensional A-phantoms which are effective relative to C. □

Next we discuss our ‘test examples’ in light of the new concepts.

Ad B.1. Here are two infinite dimensional P^{∞}(Λ -mod)-phantoms of S_{1}:

[figure: 1; α; 7; 8; 7; ⋯; 1; ρ; 8; 7; 8; ⋯; U :; 2; β; 5; 4; 5; ⋯; V :; 3; σ; 4; 5; 4; ⋯; 4; 5]

Note that U ⊕ V is in turn a P^{∞}(Λ -mod)-phantom of S_{1}, since there is no object in P^{∞}(Λ -Mod) having a graph with subgraph

[figure: 8; 1; 7; 4; 3; 2; 5; 5; 4]

In fact, the phantom U ⊕ V is effective relative to the class C ⊆ P^{∞}(Λ -mod) exhibited in B.1.

We will justify only that U is a P^{∞}(Λ -mod)-phantom of S_{1}. Consider the class of objects

[figure: 1; x; 7; y_{1}; 8; z_{1}; 7; y_{2}; ⋯; 7; y_{n}; 8; z_{n}; U_{n} :; 2; 5; 5; 5; ⋯; 5; 5; 5; 4]

in P^{∞}(Λ -mod), and observe that U = \underrightarrow{lim} U_{n}. Hence, it suffices to show that, for each n ∈ ℕ, the module U_{n} is a submodule of each {U_{n}}-approximation of S_{1} inside P^{∞}(Λ -mod). To see this, fix n, and let A be any {U_{n}}-approximation of S_{1} inside P^{∞}(Λ -mod); say f : A → S_{1} has the factorization property of the definition, and g ∈ Hom(U_{n}, A) factors the canonical epimorphism U_{n} → S_{1}. Let a = g(x), where x is the unique top element of type e_{1} of U_{n}. Then a is a top element of A. Due to the fact that finiteness of the projective dimension of A entails either βαa ≠ 0 or σρa ≠ 0, our factorization requirement forces βαa to be nonzero. Consequently, g(βαx) = βαa. If the arrows 7 → 4 and 7 → 5 are named χ and ψ, respectively, we deduce βαa = f(χy_{1}) ≠ 0, and hence f(y_{1}) ≠ 0. The element b_{1} = f(y_{1}) is a top element of A (necessarily of type e_{7}), since 7 is a source of Γ. To prevent the syzygy Ω^{1}(A) from having a summand S_{5} (the latter being incompatible with finite projective dimension), we require 0 ≠ ψ(b_{1}) = μf(z_{1}), where μ is the arrow 8 → 5 and z_{1} is as indicated in the above figure. Setting c_{1} = f(z_{1}) and iterating this type of argument, we see that 0 ≠ νc_{1} = f(χy_{2}), where ν is the arrow 8 → 4. Set b_{2} = f(y_{2}). It is readily checked that the top elements b_{1}, b_{2} are K-linearly independent modulo JA, and an obvious induction on m ≤ n gives us sequences of top elements, b_{1}, …, b_{n} of type e_{7}, and c_{1}, …, c_{n} of type e_{8} in A, both series being K-linearly independent modulo JA. It is now straightforward to deduce that the submodule of A generated by a and the b_{i}, c_{i} has the same graph as U_{n}. But this graph clearly determines the corresponding module up to isomorphism, which completes the argument.

Each module from the subclass D = {D_{n} | n ∈ ℕ} of P^{∞}(Λ -mod), with D_{n} determined by the graph

[figure: 7; y_{1}; 8; z_{1}; 7; y_{2}; ⋯; 8; z_{n}; 5; 4; 5; ⋯; 5; 4]

is a P^{∞}(Λ -mod)-phantom of S_{7}, and consequently \underrightarrow{lim} D_{n} provides us with an infinite dimensional P^{∞}(Λ -mod)-phantom of S_{7} which, moreover, is effective relative to the class D. The simple module S_{8} behaves similarly.

Ad C.2. An infinite dimensional P^{∞}(Λ -mod)-phantom of S_{1} is determined (uniquely, up to isomorphism) by the graph

[figure: 1; 9; 10; 6; ρ; σ; 6; ρ; σ; 6; ρ; σ; · · · · · ·; 7; 8; 5; 5; 5; _{· · · · · ·}; 2; 2; 2; 5]

This phantom is effective relative to the class of modules in P^{∞}(Λ -mod) obtained by chopping suitable ‘infinite tails’ off the given graph.

An example of an infinite dimensional P^{∞}(Λ -mod)-phantom of S_{8}, finally, is

[figure: · · · · · ·; 6; σ; ρ; 6; σ; ρ; 8; 1; 9; _{· · · · · ·}; 5; 5; 5; 2; _{.................}_{.................}\begin{matrix}. \\ . \\ . \\ . \\ . \\ . \\ . \end{matrix}_{. .. . ...... .}_{. . . . . . . . . ........}; 2; 7; 2]

This concludes our discussion of Examples A,B,C. The following criterion from [29] is hovering in the background of most of the infinite dimensional phantoms displayed so far.

Criterion for failure of contravariant finiteness of A.

Let Λ = KΓ/I be a split finite dimensional algebra, and, once more, suppose that A ⊆ Λ -mod is closed under finite direct sums. The simple module S = Λe_{1}/Je_{1} centered in the vertex e_{1} fails to have an A-approximation in case the following holds:

The vertex e_{1} can be supplemented to a sequence e_{1}, …, e_{m} of distinct vertices of Γ, together with sequences p_{1}, …, p_{m}, q_{1}, …, q_{m} in J, where p_{i} = p_{i}e_{i} and q_{i} = q_{i}e_{i} are such that conditions (1) and (2) below are satisfied:

(1) For each n ∈ ℕ, there exists a module M_{n} ∈ A having a graph that contains a subgraph of the form

[figure: e_{1}; p_{1}; e_{2}; q_{2}; p_{2}; e_{3}; q_{3}; p_{3}; e_{m}; q_{m}; p_{m}; e_{1}; n repetitions; ; q_{1}; p_{1}; e_{1}; q_{1}; p_{1}; e_{2}; q_{2}; p_{2}; e_{m}; q_{m}; p_{m}; •; •; •; •; •; •; •; •; •; •; •]

(2) Given any object A in A, the top elements of A of type e_{1} are not annihilated by p_{1}, and

(p_{i}a = q_{i+1}b ≠ 0 ⟹ p_{i+1}b ≠ 0)

for a, b ∈ A and 1 ≤ i ≤ m; here p_{m+1} = p_{1} and q_{m+1} = q_{1}. □

In fact, the hypotheses of the criterion yield an infinite dimensional factor module of \underrightarrow{lim} M_{n} which is an A-phantom of S; it has a graph containing the ‘infinite extension to the right’ of the graph in condition (1) as a subgraph.

5. Phantoms over string algebras

In this section, we present a preview of ongoing joint work with S. O. Smalø. A special class of string algebras Λ_{m,n}, for m, n ≥ 2 – those on the quiver

\xymatrix{α \ar[r] & • & β \ar[l]}

and subject to the relations αβ = βα = 0 and α^{m} = β^{n} = 0 – was first singled out by Gelfand and Ponomarev in the late 1960’s, as being intimately related to the representation theory of the Lorentz group [28]; in fact, classifying the finitely generated indecomposable modules over the algebras Λ_{m,n} amounts to a classification of the Harish-Chandra modules over the Lorentz group. Taking this route, Gelfand and Ponomarev gave a hands-on structural description of the finitely generated indecomposable objects in Λ_{m,n}-mod. In particular, their findings show that over an algebraically closed base field the algebras Λ_{m,n} are tame. Actually, this work was preceded by an investigation of Szekeres [49] into the structure of the finitely generated modules over certain factor rings of ℤ[x], which anticipates many of the ideas re-encountered in later work on biserial algebras. In the mid-seventies, Gabriel presented a categorical reinterpretation of the Gelfand-Ponomarev approach (see [24]), which in turn caused Ringel to recognize that these methods are applicable in a far wider context: In a first round of generalizations, he used them to describe the finite dimensional indecomposable representations of the dihedral 2-groups in characteristic 2 [43]; this work appeared in tandem with a paper of Bondarenko containing roughly the same information [11]. Next, Donovan and Freislich applied these methods to the ‘biserial’ algebras introduced below [17]. Clearly, the algebras Λ_{m,n} considered by Gelfand and Ponomarev belong to the subclass of ‘special beserial’ algebras; in fact, they are even ‘string algebras’.

Definitions. (see [51] and [13]) (1) Λ is called biserial if each indecomposable projective left or right Λ-module P has the following property: rad P = U + V, where U, V are uniserial (possibly trivial) with U ∩ V either zero or simple.

(2) Λ is special biserial provided Λ is of the form KΓ/I such that

• Given any vertex e of Γ, there are at most two arrows entering e and at most two arrows leaving e, and

• Given any arrow α of Γ, there exists at most one arrow β such that βα does not belong to I, and at most one arrow γ such that αγ does not belong to I.

Moreover, Λ is a string algebra if, in addition, Λ is a monomial relation algebra, meaning that I can be generated by certain paths in Γ.

Clearly, special biserial algebras are biserial. All finite dimensional biserial algebras over algebraically closed fields are known to be tame: the special biserial case was completed by Wald and Waschbüsch in [51], while the general biserial situation was settled much later by Crawley-Boevey [16] on the basis of an alternate description of biserial algebras due to him and Vila-Freyer [50] and a remarkable result of Geiss [25] (saying that algebras with tame degenerations are always tame).

All the while, special biserial algebras have continued to provide challenges which, in spite of the availability of a highly explicit classification of the indecomposables, were far from resolvable at a glance. We mention only a few such lines, together with a selection of references, not aiming at completeness:

Let Λ be biserial.

• When does Λ have finite representation type? (See [47].)

• What does the Auslander-Reiten quiver of Λ look like? (See, e.g., [13], [21], and [26].)

• Characterize the auto-equivalences of the category Λ -mod. (See [9] and [10].)

• Describle the generic modules over string algebras and the maps among them. (See [45]).

New sources of symmetric biserial algebras can be found in [46].

Here we will provide an overview of a complete, constructive solution to the problem as to which string algebras Λ have the property that P^{∞}(Λ -mod) is contravariantly finite. Our proof of the answer – not given here – makes full use of the description of the indecomposable objects in Λ -mod, as reviewed graphically below.

Theorem 8. (Its evolution can be traced in [28], [43], [17], [51]) Let Λ = KΓ/I be a special biserial algebra over an algebraically closed field K. Then each indecomposable object in Λ -mod is either a band module or a string module. Here the string modules are those with graphs of the form

[figure: •; q_{1}; p_{1}; · · · · · ·; •; q_{m}; p_{m}; •; •; _{· · · · · ·}; •; •]

where the p_{i}, q_{i} are paths in KΓ \ I, with q_{1} and p_{m} possibly trivial, such that firstarrow(q_{i}) ≠ firstarrow(p_{i}) for 1 ≤ i ≤ m, and lastarrow(p_{i}) ≠ lastarrow(q_{i+1}) for 1 ≤ i ≤ m − 1. (Note that some of these conditions may be void, namely in case the relevant paths are trivial.)

The band modules are characterized by their graphs, paired with irreducible vector space automorphisms, as follows: The pertinent graphs are of the form

[figure: •; x_{1}; q_{1}; p_{1}; •; y_{1}; q_{r}; p_{r}; •; x_{2}; q_{1}; p_{1}; •; y_{2}; q_{r}; •; x_{s}; q_{1}; p_{1}; •; y_{s}; q_{r}; p_{r}; •; _{..........}_{.........}_{.........}_{.........}_{.........}_{............}_{.................}_{...........}_{.........}_{.........}_{.........}_{.........}_{.......}\begin{matrix}. \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \end{matrix} \begin{aligned}& & & & & & & . \\ & & & & & & & . \\ & & & & & & & . \\ & & & & & & . \\ & & & & & & . \\ & & & & & & . \\ & & & & & . \\ & & & & . \\ & & & & . \\ & & & & . \\ & & & & . \\ & & & & . \\ & & & & . \\ & & & & . \\ & & & . \\ & & . \\ & . \\ . \\ . \\ . \\ . \\ . \end{aligned} \begin{aligned}& & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & & . \\ & & & & & & & & & & & & & . \\ & & & & & & & & & & & & . \\ & & & & & & & & & & & . \\ & & & & & & & & & & . \\ & & & & & & & & & . \\ & & & & & & & & . \\ & & & & & & & . \\ & & & & & & . \\ & & & & & . \\ & & & & . \\ & & & . \\ & & . \\ & . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \end{aligned}_{....... . . . . . . . .... . .......}; •; •; •; •; •; •; •; •; •]

where p_{i}, q_{i} are are nontrivial paths in KΓ \ I with firstarrow(q_{i}) ≠ firstarrow(p_{i}) for all i and lastarrow(p_{i}) ≠ lastarrow(q_{i+1}) for i < r, and also lastarrow(p_{r}) ≠ lastarrow(q_{1}). The nature of the dotted pool is specified by the dependence relation

p_{r}y = ∑_{i=1}^{s} k_{i}q_{1}x_{i},

where [ \begin{matrix}0 & ··· & 0 & k_{1} \\ 1 & ··· & 0 & k_{2} \\ & ... & & ⋮ \\ 0 & ··· & 1 & k_{s} \end{matrix} ] is the Frobenius companion matrix of an irreducible automor-

phism of K^{s}.

Moreover, all modules having one of the above descriptions are indecomposable. □

This classification can be completed with a suitable uniqueness statement which does not impinge on our present results. For our main theorem, string modules will be of particular relevance. A generalized string module will be a module X ∈ Λ -Mod which arises as a direct limit of a countable directed system of string modules, each embedded into its successor, i.e., any module X having a graph of one of the following forms

[figure: · · · · · ·; •; •; · · · · · ·; •; •; · · · · · ·; _{· · · · · ·}; •; •; •; _{· · · · · ·}; •; •; •; _{· · · · · ·}]

such that each finite segment is the graph of a string module. (Here the dotted edges may but need not appear, depending on whether the graph is one-/two-sided infinite or finite.) These modules were also considered by Ringel in [44] as “modules associated with ℕ-words of ℤ-words” over the alphabet Γ_{0} ∪ Γ_{0}^{−1}, where Γ_{0} is the vertex set of the quiver Γ of Λ. It should be self-explanatory what we mean by a left and right periodic generalized string module; we merely emphasize that we allow the “left” and “right” periods to differ and that we regard termination as a period.

We are now in a position to state the main new result of this section.

Theorem 9. [36] Let Λ = KΓ/I be a finite dimensional string algebra. For any simple S ∈ Λ -mod, there exists a generalized string module H = H(S) which is uniquely determined by Γ and I, together with a canonical homomorphism f : H → S, having the following properties: (I) The following statements are equivalent:

(i) dim_{K} H < ∞;

(ii) S has a classical P^{∞}(Λ -mod)-approximation;

(iii) f : H → S is the minimal P^{∞}(Λ -mod)-approximation of S. (II) H always belongs to P^{∞}(Λ -Mod) and is a P^{∞}(Λ -mod)-phantom of S. In fact, the map f : H → S makes H an effective P^{∞}(Λ -mod)-phantom with respect to the class S^{∞}(Λ -mod) of all string modules of finite projective dimension. (III) H is left and right periodic and can be constructed from Γ and I in fewer than 3|Γ_{0}| steps. □

Given a simple module S = Λe/Je over a string algebra Λ, we will call the module H = H(S) of the theorem the characteristic P^{∞}(Λ -mod)-phantom of S. After illustrating the theory with a first example, we will give an inductive description of the finite segments of a graph

[figure: ⋯; •; p̃_{3}; q̃_{2}; •; p̃_{2}; q̃_{1}; e; x_{0}; p̃_{1}; p_{1}; •; q_{1}; p_{2}; •; q_{2}; p_{3}; ⋯; ⋯; •; •; •; •; •; •; ⋯]

of the characteristic P^{∞}(Λ -mod)-phantom of S. Sending the top element marked x_{0} to e + Je in S and sending the other top elements to zero will then yield a map f : H → S as stipulated in the theorem.

Example F. Let Λ = KΓ/I be the string algebra with the following indecomposable projective left Λ-modules

[figure: 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 2; 3; 11; 4; 12; 5; 4; 5; 2; 3; 9; 6; 6; 9; 10; 10; 11; 12; 11; 12; 4; 5; 10; 3; 2; 4]

Then the characteristic P^{∞}(Λ -mod)-phantom H_{1} of S_{1} has graph

[figure: 7; 8; 1; x_{0}; 6; 1; x_{1}; 6; ⋯; 6; 9; 6; 3; 2; 3; 2; ⋯; 3; 2]

and the homomorphism f : H_{1} → S_{1} which sends x_{0} to a nonzero element of S_{1} and the x_{i} for i ≥ 1 to zero has the properties described in Theorem 9. In particular, S_{1} does not have a P^{∞}(Λ -mod)-approximation.

The characteristic P^{∞}(Λ -mod)-phantom of S_{7}, on the other hand, has graph

7 6 3

and thus coincides with the minimal P^{∞}(Λ -mod)-approximation of S_{7}. These graphs are obtained via the algorithm which we describe next. □

Recall that we refer to a module X as a top-embeddable submodule of Y if there exists a monomorphism f : X → Y which induces a monomorphism X/JX → Y/JY. Dually, we call X a socle-faithful factor module of Y if there exists an epimorphism f : Y → X which induces an epimorphism soc Y → soc X.

Description of the characteristic phantom of a simple module S ∈ Λ -mod, where Λ is a finite dimensional string algebra.

Let S = Λe/Je. The following are the steps of an algorithmic procedure for constructing H = H(S), but here we will not discuss the algorithmic nature, nor prove that the quantities stipulated in the process exist.

\underline{Step 1}. Let p_{1} and p̃_{1} be paths starting in e which have minimal lengths ≥ 0 such that

[figure: e; p̃_{1}; p_{1}; •; •]

is the graph of a string module which can be top-embedded into an object in S^{∞}(Λ -mod). In particular, we have p_{1} ≠ p̃_{1} unless both of these paths are trivial, and startarrow(p_{1}) ≠ startarrow(p̃_{1}) if both are nontrivial.

If both p_{1} and p̃_{1} are trivial, we set H = S. Otherwise, we proceed to \underline{Step 2}. Let q_{1} and q̃_{1} be paths ending in end(p_{1}) and end(p̃_{1}), respectively, and

having maximal lengths ≥ 0 with the property that

[figure: •; q̃_{1}; e; p̃_{1}; p_{1}; •; q_{1}; •; •]

is the graph of a string module which arises as a socle-faithful factor module of an object in S^{∞}(Λ -mod). In case p_{1} is trivial, set q_{1} = e, and deal symmetrically with q̃_{1}.

If both q_{1} and q̃_{1} are trivial, i.e., if the graph of Step 2 coincides with that of Step 1, we let H be the string module having this graph. Otherwise, we proceed to

\underline{Step 3}. Let p_{2} and p̃_{2} be paths starting in start(q_{1}) and start(q̃_{1}), respectively, which have minimal lengths ≥ 0 with the property that

[figure: •; p̃_{2}; q̃_{1}; e; p̃_{1}; p_{1}; •; q_{1}; p_{2}; •; •; •; •]

is the graph of a string module which can be top-embedded into some object in S^{∞}(Λ -mod), … etc.

After fewer than 3|Γ_{0}| steps, this procedure either terminates or has become periodic on both sides. It is easy to recognize when one has hit a left or right period: Indeed, if startarrow(p_{i}) = startarrow(p_{j}) for some i < j, then p_{i+r} = p_{j+r} and q_{i+r} = q_{j+r} for all r ≥ 0, the symmetric criterion holding for the other side. □

As one gleans from this inductive description of the phantoms H = H(S) of the simple modules S, string algebras merit their name also from a homological viewpoint. Indeed, the phantoms H(S) depend only on the string modules of finite projective dimension, and the K-dimensions of the H(S) in turn determine whether or not P^{∞}(Λ -mod) is contravariantly finite. One may wonder whether our emphasis on string modules is just dictated by convenience and whether band modules play a similar homological role. They do not. In fact, there exist string algebras of positive little finitistic dimension which have no nontrivial (finitely generated) band modules of finite projective dimension. Another asset of string algebras that arises as a byproduct of our main theorem we record somewhat more formally.

Corollary 10. Suppose that S is a simple module over a finite dimensional string algebra Λ. If S has a P^{∞}(Λ -mod)-approximation, then the minimal one is a string module and, in particular, is indecomposable. □

The corollary does not carry over to arbitrary special biserial algebras, as the next example demonstrates.

Example G. Let Λ = KΓ/I be the special biserial algebra with indecomposable projectives

[figure: 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 2; 3; 4; 5; 6; 6; 7; 8; 8; 9; 3; 2; 4; 5; 6; 6; 7; 8; 6]

Then the minimal P^{∞}(Λ -mod)-approximation of S_{1} is as follows:

[figure: 1; 1; 1; 1; 9; 10; 1; 2; ⨁; 3; ⨁; 2; 3; ⨁; 2; 3; ⨁; 2; 3; 4; 5; 4; 5]

In particular, we observe that the factor modules of Λe_{1} contained in the two rightmost summands are not minimal with respect to top-embeddability into objects of P^{∞}(Λ -mod). Thus the alternation “choose a minimal factor module embeddable into a module in P^{∞}(Λ -mod), then choose a maximal essential extension arising as a factor module of a module in P^{∞}(Λ -mod)” which leads to the minimal approximations of the simples over string algebras in case of existence, cannot be expected to achieve this goal in the more general situation. □

To give another illustration of our algorithm, we conclude with an example of a string algebra Λ and a simple left Λ-module S, the characteristic phantom of which is twosided infinite with left/right periods reached at different steps of the algorithm.

Example H. Let Λ = KΓ/I be the string algebra with the following indecomposable projective left modules:

[figure: 0; 1; 2; 3; 4; 5; 6; 7; 8; 2; 1; 1; 2; 4; 1; 5; 4; 5; 7; 2; 8; 7; 8; 4; 1; 7; 2; 9; 10; 11; 12; 13; 14; 15; 16; 5; 11; 6; 12; 11; 12; 9; 15; 6; 12; 15; 15; 9; 5; 2; 5; 8; 12; 15; 11; 2; 5]

Then the characteristic P^{∞}(Λ -mod)-phantom of S = Λe_{0}/Je_{0} has a graph as follows:

[figure: ⋯; 14; 10; 14; 10; 0; 3; 13; 16; 13; 16; ⋯; ⋯; 6; 12; 6; 12; 6; 1; 9; 15; 9; 15; 9; ⋯; 2; 5]

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Department of Mathematics, University of California, Santa Barbara, CA 93106, USA

    E-mail address: birge@math.ucsb.edu