Canadian Mathematical Society Conference Proceeding Volume 00, 0000

Analyzing the Structure of Representations via Approximations

BIRGE HUISGEN ZIMMERMANN

Dedicated to the memory of Maurice Auslander.

Abstract. Primarily this paper presents an expository report on alternatives to the traditional methods of classifying representations of finite dimensional algebras. Some new results illustrating such alternatives for algebras with only finitely many isomorphism types of uniserial modules are included.

1. Introduction and notation

We discuss several avenues of approach to the structure of representations of a finite dimensional algebra Λ by comparing the target objects with the objects from a more thoroughly understood reference class. Our starting point is the concept of a ‘right 𝔄-approximation’ of a left Λ-module relative to a subcategory 𝔄 of Λ -mod, as introduced by Auslander and Smalø in [3] (although originally not in this terminology), as well as the subsequent work on the subject by Auslander and Reiten [2]. See Section 2 for the pertinent definitions.

Due to [2], the following holds for any resolving contravariantly finite subcategory 𝔄 of Λ -mod: If there are n simple left Λ-modules, up to isomorphism, and if A_{1}, …, A_{n} are their minimal right 𝔄-approximations, then a left Λ-module belongs to 𝔄 if and only if it is isomorphic to a direct summand of a module M having a filtration M = M_{0} ⊇ M_{1} ⊇ ⋯ ⊇ M_{l} = 0 with M_{i}/M_{i+1} ∈ {A_{1}, …, A_{n}}. In other words, approximating the simple modules by objects in 𝔄 yields basic structural information about arbitrary objects in 𝔄. To give a flavor of the usefulness of this type of information, e.g., towards a homological understanding of 𝔄, we point out that, in the described situation,

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society 0000-0000/00 $1.00 + $.25 per page

1991 Mathematics Subject Classification. 16D90, 16E10, 16G10, 16G20, 16G60, 16P10.

This work was partially supported by an N.S.F. grant.

This paper is in final form and no version of it will be submitted for publication elsewhere.

we have sup{p dim A | A ∈ 𝔄} = sup{p dim A_{i} | 1 ≤ i ≤ n}. However, the applicability of this structure theorem is bounded by the following facts: Numerous module categories of interest fail to be contravariantly finite; on top of it, deciding whether a given subcategory 𝔄 of Λ -mod has this property is a difficult task in general. Even when the question of contravariant finiteness has been resolved in the positive, it may still be extremely challenging to pin down the minimal right 𝔄-approximations of the simple objects.

In response to these obstacles, we suggest three lines of investigation, all based on the concept of approximation or derivatives thereof. The present report is partly an overview of existing results in these directions, illustrated by examples, partly an extension of ideas in the literature (e.g., most of Section 8 is new), and partly a program to be pursued more fully in the future. Roughly, the three lines of approach are as follows.

(1) Find manageable positive and negative criteria for contravariant finiteness of subcategories of Λ -mod, and systematically broaden the classes of algebras and module categories for which the ‘classical’ approximation theory can be brought to bear. In particular, this calls for an enlarged arsenal of techniques to find and describe minimal right 𝔄-approximations in the case of contravariant finiteness and for a study of ‘typical shapes’ occurring among the minimal approximations of the simple modules, in dependence of Λ and 𝔄. This line is illustrated by Sections 3, 4, and 7 below.

(2) Given a class ℭ of representations which cannot be mastered in the classical mode of direct sum decompositions and a complete listing of the indecomposable summands occurring, scan the objects of ℭ by comparing them with the objects of a reference class which is tailored to measure for this purpose. More precisely, construct a contravariantly finite subcategory 𝔄 of Λ -mod so that the following classification of ℭ relative to 𝔄 yields useful information on ℭ; namely, modules C and D are considered ‘similar relative to 𝔄’ provided they have isomorphic minimal right 𝔄-approximations. This, usually coarser, alternative to the traditional classification in terms of precise structural data naturally yields information of varying degrees of precision, ranging from the very rough – if 𝔄 is small – to a complete picture – if ℭ ⊆ 𝔄.

Here we illustrate this idea for ℭ = P^{∞}(Λ -mod), the category of all modules of finite projective dimension in Λ -mod. The objects of the reference category we use are glued together from uniserial building blocks (Sections 5,6,8). Sections 5 and 6 provide the tools: The first addresses the classification of the uniserial left Λ-modules in an informal sketch of some of the author’s results in [14,15,16]. The second proposes a pattern for pasting uniserial modules together, while keeping a tight grip on the structure of the new objects created; our choice of pattern is justified in Section 7. In Section 8, we deal with algebras of finite uniserial type for which the reference categories constructed in Section 7 have finite representation type. We conclude with concrete instances of the outlined approximation strategy.

(3) In a variation of our theme, we describe infinite dimensional substitutes for minimal 𝔄-approximations as developed by the author and Happel in [10]. These encode the same type of information as finite dimensional approximations do when they exist. To outline the underlying idea we require a few concepts, the first of which is only a slight extension of the original definition of right 𝔄-approximations. If 𝔅 ⊆ 𝔄 and M is a left Λ-module, a map f : A → M is called a (right) 𝔅-approximation of M inside 𝔄 if A ∈ 𝔄 and each map g ∈ Hom_{Λ}(B, M) for B ∈ 𝔅 factors through f. Furthermore, a Λ-module H is called an 𝔄-phantom of M if there exists a nonempty finite subclass 𝔅 of 𝔄 such that each 𝔅-approximation of M inside 𝔄 has H as a subfactor; direct limits of such modules H – not required to be finitely generated – are again named phantoms. We are particularly interested in phantoms H_{j} with the property that all homomorphisms B → M with B running through a specified subclass 𝔅_{j} of 𝔅 factor through H_{j}. Given a chain 𝔅_{1} ⊆ 𝔅_{2} ⊆ 𝔅_{3} ⊆ … of subclasses of 𝔅, the corresponding sequence of phantoms H_{1}, H_{2}, H_{3}, … will not stabilize in general, and so a direct limit is the natural way to integrate the information represented by such a sequence. To visualize this type of information, suppose that M = S is simple; then a phantom H_{j} as above communicates – in the most compressed form possible within the category 𝔄 – the relations of those objects B ∈ 𝔅_{j} which contain S in their tops.

The key fact in this connection is that nontrivial phantoms always exist. In fact, a module M ∈ Λ -mod fails to have an 𝔄-approximation in the traditional sense if and only if M has 𝔄-phantoms of infinite vector space dimension.

In Section 2 we review the basics of contravariantly finite subcategories, concentrating on the point of view to be pursued later. In particular, we will omit all results linking functorially finite categories to tilting theory or relative derived functors. Since the concept of covariant finiteness of a subcategory 𝔄 of Λ -mod is dual to that of contravariant finiteness, we restrict our attention to the latter; the lines of investigation we discuss can be dualized accordingly. The following convention will therefore not lead to ambiguities: namely, we will refer to right 𝔄-approximations simply as 𝔄-approximations.

Throughout, Λ will be a finite dimensional algebra over a field K with Jacobson radical J. We assume Λ to be split, i.e., Λ is of the form KΓ/I, where Γ is a quiver and I is an admissible ideal of the path algebra KΓ. Our convention for composing paths in KΓ is as follows: if p and q are paths, then pq stands for ‘p after q’ if the endpoint of q coincides with the starting point of p, and pq = 0 otherwise. A path u is called a right subpath (respectively, a left subpath) of p in case there exists a path v such that p = vu (respectively, p = uv); both left and right subpaths will also be referred to as subpaths of p.

Whenever we mention primitive idempotents of Λ, we mean those associated with the vertices of Γ; in fact, we identify these vertices with the corresponding idempotents. Given a left Λ-module M, a top element of M is an element x ∈ M \ JM such that x = ex for some primitive idempotent e; in that case we also say that x is a top element of type e of M.

Like numerous other authors (see, e.g., [1] and [7]), we find it convenient to represent certain Λ-modules by graphs. A special breed of such graphs, based on sequences of top elements of the pertinent modules, will make a great deal of information available at a glance. We introduce these labeled and layered graphs informally by means of some illustrative examples.

Example 1.1. Let Λ = KΓ/I, where Γ is the quiver

[figure: 1; α; τ; 2; γ; β; 3; ϵ; δ; 4; σ]

and I an ideal in KΓ. That a left Λ-module M have (layered and labeled) graph

[figure: 1; α; τ; 2; β; γ; 2; γ; 3; δ; ϵ; 3; ϵ; 4]

relative to top elements x_{1} of type e_{1} and x_{2} of type e_{2} means that:

M/JM ≅ S_{1} ⊕ S_{2}; JM/J^{2}M ≅ S_{2};
J^{2}M/J^{3}M ≅ J^{3}M/J^{4}M ≅ S_{3}; J^{4}M ≅ S_{4};
αx_{1} and βx_{2} each generate JM modulo J^{2}M;

τx_{1}, γx_{2}, γαx_{1} (and hence also γβx_{2}) each generate J^{2}M modulo J^{3}M; δγαx_{1} generates J^{3}M modulo J^{4}M;

ϵγαx_{1} and ϵδγαx_{1} each generate J^{4}M.

Finally, whenever p is a path in Γ starting in e_{i} which is not recorded under x_{i} in the graph of M, the product px_{i} is zero. So, in particular, the numbers in the m-th ‘layer’ (= row) of a graph of M stand for the simple composition factors of J^{m}M/J^{m+1}M if we start our count with m = 0.

On the other hand,

[figure: 2; β; 3; δ; 2; γ; 3; δ; 3; δ; ϵ; 3; δ; 3; ϵ; 4]

is not the graph of a left Λ-module in case ϵδ^{3} = 0 in Λ.

It is often convenient to communicate the ideal I factored out of the path algebra KΓ by giving the graphs of the indecomposable projective modules Λe_{i}. The algebra Λ with quiver Γ as above and with the following indecomposable projectives will recur in Section 6.

[figure: 1; α; τ; 2; γ; β; 3; ϵ; δ; 4; σ; 2; γ; β; 3; ϵ; δ; 3; ϵ; δ; 2; γ; 4; σ; 3; δ; ϵ; 2; β; γ; 3; ϵ; δ; 2; γ; 4; 3; ϵ; 4; 3; ϵ; 3; δ; ϵ; 2; β; 3; ϵ; δ; 4; 2; 3; δ; 4; 3; ϵ; 3; ϵ; 4; 4; 3; ϵ; 4; 2; 4; 3; 3; ϵ; 4; 4; 4; 4]

We extend these conventions as follows: That N ∈ Λ -mod has the following graph

[figure: 1; α; α; 2; β; 4; σ; 2; β; 2; β; 2; 2; 2; 2]

is to signify that J^{2}N = S_{2}^{3}, and that there are top elements x_{1}, x_{2}, x_{3} of the obvious types such that αx_{1}, βαx_{1}, βx_{2}, βσx_{4} generate J^{2}N, with every choice of three of these four elements K-linearly independent, while the set of four is K-linearly dependent. □

Finally, P^{∞}(Λ -mod) will stand for the full subcategory of Λ -mod having as objects the finitely generated modules of finite projective dimension, and P^{∞}(Λ -Mod) will be the analogous subcategory of Λ -Mod. As usual, l fin dim Λ denotes the supremum of the projective dimensions attained on P^{∞}(Λ -mod).

2. Contravariantly finite subcategories of Λ -mod

The notions of co- and contravariantly finite subcategories of Λ -mod were first introduced by Auslander and Smalø in [3], in connection with the existence of almost split sequences inside a subcategory 𝔄 and with preinjective/preprojective partitions of 𝔄. The following definitions evolved in [3], [4] and [2].

Throughout, we let 𝔄 ⊆ Λ -mod be a full subcategory which is closed under isomorphisms, direct summands, and finite direct sums.

Definitions 2.1. (1) Given M ∈ Λ -mod, a homomorphism f : A → M with A ∈ 𝔄 is called a right 𝔄-approximation of M if each g ∈ Hom_{Λ}(B, M) with B ∈ 𝔄 factors through f. Left 𝔄-approximations are defined dually.

(2) 𝔄 is called contravariantly finite (resp., covariantly finite) in Λ -mod if each module M ∈ Λ -mod has a right (resp., left) 𝔄-approximation.

In the following we will focus on contravariant finiteness and will briefly write ‘𝔄-approximation’ for ‘right 𝔄-approximation’. In [2] it is shown that, whenever M ∈ Λ -mod has an 𝔄-approximation, any two 𝔄-approximations of minimal Kdimension are isomorphic. We therefore refer to the minimal 𝔄-approximation of M in that case.

The nomenclature in Definition 2.1(2) stems from the fact that 𝔄 is contravariantly finite in Λ -mod if and only if the restricted contravariant Hom-functor Hom(−, M)|_{𝔄} : 𝔄 → Ab is finitely generated for each object M ∈ Λ -mod. The flavor of this finiteness condition of 𝔄 relative to Λ -mod is caught in the following easy argument showing that each subcategory 𝔄 ⊆ Λ -mod of finite representation type is contravariantly (as well as covariantly) finite in Λ -mod [3, Proposition 4.2]: Indeed, if A_{1}, …, A_{m} ∈ 𝔄 are such that each object in 𝔄 is a finite direct sum of copies of the A_{i}, then given M ∈ Λ -mod, the homomorphism ⊕_{i=1}^{m} ⊕_{j=1}^{d_{i}} f_{ij} : ⊕_{i=1}^{m}A_{i}^{d_{i}} → M is an 𝔄-approximation of M provided that, for each i ≤ m, the family (f_{ij})_{j≤d_{i}} is a K-basis of Hom_{Λ}(A_{i}, M); here A^{d} denotes a direct sum of d copies of a module A.

One of the first results to draw attention to these finiteness conditions for subcategories of Λ -mod was the following theorem of Auslander and Smalø.

Theorem 2.2. [4, Theorem 2.4] If 𝔄 is covariantly and contravariantly finite in Λ -mod, then 𝔄 has almost split sequences. □

The following two results, due to Auslander and Reiten, are pivotal in our present investigation.

Theorem 2.3. [2, Proposition 3.7] If 𝔄 ⊆ Λ -mod is a resolving subcategory (meaning that 𝔄 contains the projectives in Λ -mod and is closed under extensions and kernels of epimorphisms), then 𝔄 is contravariantly finite in Λ -mod if and only if each of the simple left Λ-modules has an 𝔄-approximation. □

Theorem 2.4. [2, Proposition 3.8] Suppose that 𝔄 is a resolving, contravariantly finite subcategory of Λ -mod. Moreover, suppose that Λ -mod contains n isomorphism classes of simple modules and that A_{1}, …, A_{n} are the minimal 𝔄-approximations of representative simples. Then a left Λ-module X belongs to A precisely when X is a direct summand of a module having a filtration with consecutive factors in {A_{1}, …, A_{n}}. □

In the next section, we give a first instalment of ‘positive’ examples, i.e., of instances of contravariantly finite subcategories. A powerful tool for extending the list of such subcategories is the following theorem of Sikko and Smalø, which subsumes earlier results by Grecht, Vossieck, de la Peña-Simson, Ringel, and Smalø.

Theorem 2.5. [22, Theorem 2.6] If 𝔄 and 𝔅 are contravariantly finite subcategories of Λ -mod, then the category of all direct summands of extensions of objects in 𝔄 by objects in 𝔅 has the same property. □

3. First instalment of positive examples

Roughly speaking, a subcategory 𝔄 ⊆ Λ -mod is likely to be contravariantly finite if it is either very small or very large. In the former situation one usually obtains very rough approximations, in the latter one can approach the target objects very closely with objects from 𝔄. The extreme cases are that of a category of finite representation type on one end – we already know such a category to be contravariantly finite in Λ -mod – and the case 𝔄 = Λ -mod on the other end. While the latter does not hold much interest, the former does. For example, in case the objects of 𝔄 are precisely the finitely generated projective left Λmodules, the minimal 𝔄-approximations coincide with projective covers. So in particular, if l fin dim Λ = 0, the category P^{∞}(Λ -mod) is contravariantly finite in Λ -mod. The most interesting cases of contravariant finiteness of P^{∞}(Λ -mod), however, are those where l fin dim Λ > 0, while gl dim Λ = ∞.

As was already observed by Auslander and Reiten in [2], P^{∞}(Λ -mod) is contravariantly finite in Λ -mod whenever Λ is stably equivalent to a hereditary algebra. This includes the case where J^{2} = 0, a situation in which it is easy to describe the minimal P^{∞}(Λ -mod)-approximations. According to [12], they look as follows: Given any X ∈ Λ -mod with JX = 0, write X = X_{fin} ⊕ X_{inf}, where X_{fin} is the sum of those simple submodules of X which have finite projective dimension, and X_{inf} is the sum of those simples which have infinite projective dimension. Then, given any M ∈ Λ -mod with projective cover P → M say, the induced epimorphism P/Ω^{1}(M)_{fin} → M is the minimal P^{∞}(Λ -mod)-ap-proximation of M. In particular, the minimal P^{∞}(Λ -mod)-approximation of a simple module S = Λe/Je has the form Λe/(Je)_{fin} → S. To be more specific, if Λ = KΓ/⟨all paths of length 2⟩, where Γ is the quiver

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

the graph of the minimal P^{∞}(Λ -mod)-approximation of S_{1} is the following ‘brush’:

[figure: 1; α_{1}; α_{2}; α_{4}; α_{5}; •; •; •; •]

Moreover, Auslander and Reiten proved that for each Gorenstein algebra Λ, i.e., for each algebra Λ which has finite injective dimension on both sides, the category P^{∞}(Λ -mod) is contravariantly finite in Λ -mod (see [2, p. 150]). This result applies to a particularly interesting situation to which the author was alerted by Kirkman and Kuzmanovich: Namely, let O ⊆ M_{n}(F) be a tiled classical order over a DVR D with quotient field F; that O be tiled means that O contains a full set of n primitive orthogonal idempotents. Moreover, let π be a uniformizing parameter of D and observe that then Λ = O/πO is a finite dimensional algebra over the residue class field K = D/(π) of D. If O has finite left and right injective dimension – a fortiori, if gl dim O < ∞ – one of the classical change of rings theorems (see [20, Theorem 205]) yields that Λ is a Gorenstein algebra. So in particular P^{∞}(Λ -mod) is contravariantly finite in Λ -mod whenever gl dim O < ∞; hence, l fin dim Λ = sup{p dim A_{i} | 1 ≤ i ≤ n}, where the A_{i} are the minimal P^{∞}(Λ -mod)-approximations of the simple left Λ-modules. On the other hand, l fin dim Λ = l fin dim O − 1 = gl dim O − 1 if gl dim O < ∞, due to [9]. Via approximation theory, this equality may provide access to the possible values of gl dim O, which have been the object of investigation for a long time (see [23], [18], [19], [21], [6] for more detail). A summary of the results to date pertaining to the global dimensions of tiled classical orders over DVRs can be found in [13].

Problem 3.1. Let Λ = O/πO for a tiled classical order O over a DVR. Give an explicit description of the minimal P^{∞}(Λ -mod)-approximations of the simple left Λ-modules in terms of the valuated quiver of O (see [24]) or, equivalently, in terms of quiver and relations of Λ.

4. Negative examples and a criterion for failure of contravariant finiteness

The first example of a finite dimensional algebra Λ for which P^{∞}(Λ -mod) fails to be contravariantly finite is due to Igusa, Smalø, Todorov [17]. It is a monomial relation algebra with vanishing radical cube which, in addition, is biserial. (In Section 7, we will see that, by contrast, P^{∞}(Λ -mod) is always contravariantly finite in Λ -mod when Λ is left serial.) However, the conclusion that contravariant finiteness of P^{∞}(Λ -mod) in Λ -mod be a rare occurrence would be precipitous. In fact, the condition cuts diagonally across the standard groupings of finite dimensional algebras, being extremely sensitive to changes in quiver and relations.

Criterion 4.1. [10] Let 𝔄 be a full subcategory of Λ -mod. Moreover, suppose that e_{1}, …, e_{m} are pairwise orthogonal primitive idempotents of Λ, and p_{1}, …, p_{m}, q_{1}, …, q_{m} elements of J, with p_{i} = p_{i}e_{i} and q_{i} = q_{i}e_{i}, such that the following conditions are satisfied:

(1) For each n ∈ ℕ, there is a module M_{n} ∈ 𝔄, together with a sequence x_{n1}, …, x_{nn} of top elements of M_{n} which are K-linearly independent modulo JM_{n} and have the property that p_{r(i)}x_{ni} = q_{r(i+1)}x_{n,i+1} is nonzero for 1 ≤ i ≤ n, where r(i) ∈ {1, …, m} is congruent to i modulo m.

(2) Whenever A ∈ 𝔄, the following hold:

(i) If x ∈ A is a top element of type e_{1}, then p_{1}x ≠ 0, and (ii) If y, z ∈ A are such that p_{r(i)}y = q_{r(i+1)}z ≠ 0, then p_{r(i+1)}z ≠ 0.

It follows that S_{1} = Λe_{1}/Je_{1} does not have an 𝔄-approximation. □

Example 4.2. [17] Let 𝔄 = P^{∞}(Λ -mod), where Λ = KΓ/I is the monomial relation algebra with quiver

[figure: 1; β; α; 2; γ]

such that the indecomposable projective left Λ-modules have graphs

[figure: 1; α; β; 2; γ; 2; γ; 2; 1; 1]

Set m = 1, p_{1} = β, q_{1} = α, and S_{i} = Λe_{i}/Je_{i}. Then the modules M_{n} ∈ P^{∞}(Λ -mod) with graphs

[figure: 1; β; 1; α; β; ⋯; 1; α; β; 2; 2; ⋯; 2; 2]

having n linearly independent top elements of type e_{1} modulo the radical clearly satisfy Condition (1) of Criterion 4.1.

Moreover, Condition (2i) of the criterion is trivially satisfied, in view of the fact that p dim S_{2} = ∞; indeed, given any A ∈ Λ -mod with top element x = e_{1}x and p_{1}x = βx = 0, the simple module S_{2} is a direct summand of Ω^{1}(A). As for (2ii), whenever p_{1}y = βy = q_{1}z = αz ≠ 0, the element e_{1}z is a top element of A of type e_{1}. Thus, S_{1} does not have a P^{∞}(Λ -mod)-approximation. □

To the following example we will refer back in Sections 8 and 9. It is an instance where Criterion 4.1 applies only for m > 1.

Example 4.3. This time, let Λ = KΓ/I, where Γ is the quiver

[figure: 1; α; β; 3; 2; 4; 5; 6; γ; δ]

and I ⊂ KΓ is such that the indecomposable projective left Λ-modules have the following graphs:

[figure: 1; α; β; 2; 3; 4; 5; 6; γ; δ; 2; 4; 3; 3; 5; 5; 2; 4; 3; 5]

Set m = 2, p_{1} = β, p_{2} = γ, q_{1} = α, q_{2} = δ, and for n ∈ ℕ, let M_{n} ∈ P^{∞}(Λ -mod) be the module with graph

[figure: 1; β; 6; δ; γ; 1; α; β; ⋯; •; 4; 2; 4; ⋯; •; •]

having n linearly independent top elements modulo the radical, of types alternating between 1 and 6. It is straightforward to see that Condition (2) of Criterion 4.1 is satisfied as well, so that P^{∞}(Λ -mod) again fails to be contravariantly finite in Λ -mod. □

Our two final examples illustrate the instability – under minor modifications of quiver and/or relations – of the condition that P^{∞}(Λ -mod) be contravariantly finite in Λ -mod. The first is a variant of the Igusa-Smalø-Todorov example (4.2); the second results from a further slight alteration of the relations.

Example 4.4. Let Λ = KΓ/I where Γ is

[figure: 3; δ; 1; β; α; 2; γ]

and I is such that the indecomposable projective left Λ-modules have graphs

[figure: 1; α; β; 2; γ; 3; δ; 2; γ; 2; 1; 1; α; 1; 2; γ; 1]

This algebra Λ contains the algebra of Example 4.2, and the modules M_{n} defined in 4.2 remain modules of finite projective dimension over the new algebra. However, this time P^{∞}(Λ -mod) is contravariantly finite in Λ -mod; see [10]. □

Example 4.5. The quiver of the algebra Λ is the same as that of Example 4.4, but we delete one of the relations in the previous example, to the effect that the indecomposable projective left Λ-modules take on the forms

[figure: 1; α; β; 2; γ; 3; δ; 2; γ; 2; 1; 1; α; β; 1; 2; γ; 2; 1]

Again, Criterion 4.1 (with m = 1, p_{1} = β and q_{1} = α) readily yields that S_{1} fails to have a P^{∞}(Λ -mod)-approximation. □

5. First intermezzo: Uniserial representations

We give a rough sketch of results from [14,15,16] which will provide the foundation for the construction of several useful contravariantly finite subcategories of Λ -mod, the objects of which are well understood. The pivotal problems addressed are the following:

(I) Classify the uniserial left Λ-modules in terms of manageable isomorphism invariants.

(II) Characterize the split algebras of finite uniserial type, i.e., characterize those algebras Λ = KΓ/I for which there are only finitely many uniserial Λmodules up to isomorphism.

As the author learned in the meantime, Auslander has proposed these problems since the mid-70’s. We include an excerpt of an email message of July 1993 from him to the author. “I have been raising the question of the classification or description of uniserial modules for artin algebras for many years now. I believe the first time I raised the question in public was at the special session on representation theory at the winter meeting of the AMS in Atlanta around 1975. The big shot group representation people, like [ …] assured me that they would have an answer by the afternoon. I am still waiting. I have raised the question repeatedly since then [ …] As to motivation. One reason I am interested in these modules, aside from the fact that they should in some sense be the simplest nonprojective modules, is the fact that they have bounded length for a given algebra. Therefore the second Brauer-Thrall is true for algebras with an infinite number of uniserial modules. Hence it would be interesting to know for which algebras there are only a finite number [ …] Secondly, I am interested where the uniserial modules occur in AR-quivers and preprojective partitions since as you mentioned other modules can be in some sense approximated by uniserial modules. [ …]”

The placement of uniserial modules in the Auslander-Reiten quiver will not be discussed in this intermezzo. We just mention that Axel Boldt is working on this subject in his dissertation, and has already settled the question in the hereditary case.

A preliminary subdivision of the class of uniserial left Λ-modules is in terms of their ‘masts’.

Definition 5.1. Given a uniserial left Λ-module U of length l + 1, any path p of length l in KΓ with pU ≠ 0 is called a mast of U.

Observe that, if U is uniserial with consecutive simple composition factors J^{i}U/J^{i+1}U ≅ Λe(i)/Je(i) for 0 ≤ i ≤ l, and with layered and labeled graph G, then the masts of U correspond precisely to those edge paths in G which pass exactly once through each of these vertices from top to bottom. E.g., a uniserial module with graph

e(0)_{α_{1}}  _{α_{2}}  _{γ} e(1)_{β} e(2)

has masts βα_{1} and βα_{2}.

The starting point of our approach to uniserial representations is the following theorem which we state somewhat informally.

Theorem 5.2. [14,15] Let p ∈ KΓ be a path.

(1) There is an affine algebraic variety V_{p}, not necessarily irreducible, which parametrizes the isomorphism types of the uniserial left Λ-modules with mast p in a natural fashion. Somewhat more precisely, there exists a canonical surjection

Φ_{p} : V_{p} → {isomorphism types of uniserials in Λ -mod with mast p}.

In particular, V_{p} is nonempty if and only if there exists a uniserial left Λ-module with mast p. (Polynomials for this variety can be readily determined on the basis of a ‘coordinatization’ Λ = KΓ/I.)

(2) If p : e(0) → e(1) → ⋯ → e(l) does not have a right subpath of positive length from e(0) to e(0), the map Φ_{p} is bijective, and the points of V_{p} serve as isomorphism invariants of the uniserial modules with mast p. More sharply, if e(0) recurs t times among the vertices e(1), …, e(l), then each fibre of Φ_{p} is contained in a closed subvariety of V_{p} of dimension at most t.

(3) If V_{p} ⊆ 𝔸^{d} and t is as under (2), there exists a system of equations

S_{p}(X, Y, Z) = S_{p}(X_{1}, …, X_{d}, Y_{1}, …, Y_{d}, Z_{1}, …, Z_{t}),

linear in Z_{1}, …, Z_{t}, with the following property: Two points k, k′ ∈ V_{p} belong to the same fibre of Φ_{p} if and only if the linear system S_{p}(k, k′, Z) is consistent. (Just as V_{p}, the system S_{p}(X, Y, Z) can be effectively computed from Γ and I.) □

The traditional variety describing the Λ-modules of a fixed K-dimension d clearly contains the points corresponding to the d-dimensional uniserial modules as an open subvariety. However, these varieties are far too large and unwieldy to serve the purpose of an effective classification. By contrast, the varieties which we will consider here fit the collection of uniserials with a fixed composition series rather tightly. In fact, the bit of slack which may occur in the presence of certain oriented cycles is quite harmless; each such cycle just adds a copy of 𝔸^{1} to the pertinent variety.

We will not give the equations defining the varieties V_{p} here, but will instead give a very rough answer to the question of what data of the uniserial module Φ_{p}(Q) are recorded by the coordinates of a point Q ∈ V_{p}. Suppose that p = α_{l} ⋯ α_{1}, where the α_{i} are arrows. Given any uniserial U ∈ Λ -mod with mast p and a top element x, the products α_{m} ⋯ α_{1}x, 0 ≤ m ≤ l, clearly form a K-basis of U. Roughly speaking, the points of V_{p} corresponding to U under Φ_{p} are strings of coordinate vectors of elements qx, relative to this basis, where q runs through certain paths in KΓ (sufficiently many to pin down U up to isomorphism). So, in contrast to the classical varieties of representations, the bases used to record coordinates depend on the multiplication of the pertinent modules. This makes it less than obvious that the sets of points in affine space which we consider are actually defined by polynomials; moreover, unicity questions become more difficult than in the classical situations. What we gain is a transparent connection between points in affine space and representations. In particular, the graphs of the uniserial representations are available at a glance from the corresponding points on the varieties V_{p}. For details, we refer to [14,15]. We include an example to illustrate the correspondence of Theorem 5.2, again suppressing all technical detail.

Example 5.3. Let Γ be the quiver

[figure: 1; α_{1}; γ_{1}; 2; β_{1}; 3; α_{2}; γ_{2}; 4; β_{2}; 5; α_{3}; γ_{3}; 6; β_{3}; 7; α_{4}; γ_{4}; 8; β_{4}; 9; α_{5}; γ_{5}; 10; β_{5}; 11]

and p the path q_{5}q_{4}q_{3}q_{2}q_{1}, where q_{i} = β_{i}α_{i} for 1 ≤ i ≤ 5. Define Λ = KΓ/I, where I is the ideal generated by the following relations:

γ_{5}γ_{4}q_{3}q_{2}q_{1} − q_{5}q_{4}γ_{3}γ_{2}γ_{1} + q_{5}q_{4}q_{3}q_{2}γ_{1}, q_{5}q_{4}q_{3}q_{2}γ_{1} − q_{5}q_{4}q_{3}γ_{2}q_{1},
q_{5}q_{4}q_{3}q_{2}γ_{1} − q_{5}q_{4}γ_{3}q_{2}q_{1}, q_{5}γ_{4}q_{3}q_{2}q_{1} − γ_{5}q_{4}q_{3}q_{2}q_{1}.

Then V_{p} = V (X_{21}X_{22} − X_{11}X_{12}X_{13} + X_{11}, X_{11} − X_{12}, X_{11} − X_{13}, X_{21} − X_{22}). Observe that V ≅ V (X_{2}^{2} − X_{1}(X_{1}^{2} − 1)) is the elliptic curve with ℝ-graph

A host of questions supplementing Problems I and II pose themselves at this point. We will briefly address a few of the most immediate among these.

Additional Questions 5.4. (1) Suppose that p, q ∈ KΓ are two paths of length l, both passing through the sequence of vertices (e(0), …, e(l)). Then the uniserial modules with mast p and those with mast q have the same sequence of composition factors, namely (Λe(0)/Je(0), …, Λe(l)/Je(l)). What can be said about the intersection Φ_{p}(V_{p}) ∩ Φ_{q}(V_{q})?

(2) The varieties V_{p} are defined by polynomials over K which depend on Γ and I; the labeling as such is, in fact, tied to the coordinatization of Λ. Is there geometric information on the uniserial left Λ-modules with fixed sequence of composition factors which does not depend on the coordinatization of Λ?

(3) Which affine varieties arise as varieties of uniserial modules with a fixed sequence of composition factors?

Answers (sketch). (1) [14, Theorem D] Given any pair of irreducible components U_{p} ⊆ V_{p} and U_{q} ⊆ V_{q}, the intersection D = Φ_{p}(U_{p}) ∩ Φ_{q}(U_{q}) is either empty or else Φ_{p}^{−1}(D) and Φ_{q}^{−1}(D) are dense open subsets of U_{p} and U_{q} respectively, and there exists an isomorphism of varieties Ψ : Φ_{p}^{−1}(D) → Φ_{q}^{−1}(D) which makes the following diagram commutative:

[figure: Φ_{p}^{−1}(D); Ψ; Φ_{p}; Φ_{q}^{−1}(D); Φ_{q}; D]

In particular, U_{p} and U_{q} are birationally equivalent in the latter case. (2) [16] Let 𝕊 = (S(0), …, S(l)) be a sequence of simple left Λ-modules. If Γ has no double arrows, there is at most one path p of length l passing

through a sequence of primitive idempotents of Λ corresponding to the simples S(i) and, in the positive case, the variety V_{p} = V_{𝕊} is determined up to isomorphism by the K-algebra isomorphism type of Λ.

In the general case, start with a coordinatization Λ = KΓ/I and let (e(0), …, e(l)) be the sequence of vertices of Γ corresponding to the simple modules S(i). Denote by V_{𝕊} the set of the birational equivalence classes of the irreducible components of the V_{p}’s, where p runs through the paths of length l passing through (e(0), …, e(l)). Then V_{𝕊} is uniquely determined by the isomorphism type of Λ.

(3) Each affine variety occurs as a variety V_{p} = V_{𝕊}, even under the additional requirements that the map Φ_{p} be bijective and that V_{p} be determined up to isomorphism by the corresponding algebra Λ. More precisely: Given any affine algebraic variety V over K, there exists an acyclic quiver Γ without double arrows, together with a path p in KΓ, such that V ≅ V_{p}. □

To close in on the structure of the algebras of finite uniserial type, we require several additions to our conceptual framework (see [15]). Since, in the sequel, we will focus on algebras with the stronger property that all the varieties V_{p} be finite, we will content ourselves with the far more straightforward characterization of these latter algebras. This characterization will be preceded by a strong necessary condition for finite uniserial type.

Theorem 5.5. [15] If Λ has finite uniserial type, then the following condition (N) is satisfied: Whenever α : e → e′ is an arrow in Γ and p : e → e′ a mast of positive length, the path p belongs to KΓα ∪ αKΓ, that is, p is of the form

[figure: e; α; e′; c′; or; e; α; c; e′]

where c′, c are oriented cycles which may be trivial.

A coordinate-free rendering of condition (N) is as follows: If there exists a uniserial left Λ-module W of length 2 with top S and socle S′, and if U is any uniserial left Λ-module of length l ≥ 2 with top S and socle S′, then either U/J^{2}U ≅ W or else J^{l−2}U ≅ W.

Condition (N), in turn, implies that:

(a) Γ has no double arrows, meaning that each uniserial left Λ-module has a unique mast.

(b) Given any uniserial module U with mast p, each graph of U results from the superposition of graphs of the form

[figure: •; p; e; c; α; e; α; •; •]

under identification of the edge path p; here c is an oriented cycle of positive length and α an arrow such that αc is a subpath of p. □

The algebras we wish to describe turn out to be precisely those for which the second option in Condition (N) of Theorem 5.5 is excluded.

Theorem 5.6. [15] Given any algebra Λ = KΓ/I, the following statements are equivalent:

(1) For each path p ∈ KΓ, the variety V_{p} is finite.

(2) For each path p ∈ KΓ, the variety V_{p} is either empty or a singleton.

(3) Whenever α : e → e′ is an arrow in Γ and p : e → e′ a mast of positive length, the path p is equal to c′α, where c′ is an oriented cycle which may be trivial, i.e., p has the form

[figure: e; α; e′; c′]

In coordinate-free terms: If there exists a uniserial left Λ-module W of length 2 with top S and socle S′, and if U is any uniserial left Λ-module of length ≥ 2 with top S and socle S′, then U/J^{2}U ≅ W.

(4) There is a 1–1 correspondence between the isomorphism types and the graphs of the uniserial left Λ-modules.

(5) The only graphs of uniserial left Λ-modules are edge paths. □

Clearly every left serial algebra satisfies condition (5) of Theorem 5.6. Natural instances of algebras satisfying the equivalent conditions of Theorem 5.6 are, moreover, the algebras Λ = O/πO where O is a tiled classical order over a DVR with uniformizing parameter π (see Section 3).

6. Second intermezzo: Saguaros

Let Λ = KΓ/I be a path algebra having a quiver Γ without double arrows, which means that each uniserial module has a unique mast in KΓ.

Definition 6.1. Suppose that T_{1}, …, T_{m} is a sequence of non-zero uniserial left Λ-modules, and let p_{i} be the mast of T_{i}, respectively. A left Λ-module T is called a saguaro^{1} on (T_{1}, …, T_{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 in T via

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

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

In the sequel, 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} 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. By a slight abuse of language, we will say that the graph of a saguaro T displays a canonical sequence of top elements if the simple summands of T/JT shown in the uppermost layer of the graph are generated by the terms of such a canonical sequence. It is clear that layered and labeled graphs displaying a canonical sequence of top elements always exist. To give an example, the following is a graph of a saguaro T over the algebra Λ of Example 1.1. Since Γ has no double arrows in this case, we can omit the labels on the edges without losing information.

1The name is that of a cactus found in the Sonoran Desert.

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

Here T = (⨁_{i=1}^{5} T_{i})/U, where U is generated by the elements αt_{1} − βt_{2}, β^{2}t_{2} − δγt_{3}, γt_{3} − γt_{4}, and δ^{2}γt_{4} − δ^{2}γt_{5}. The trunks of T are the uniserials T_{1}, …, T_{5} in Λ -mod which, relative to suitable top elements t_{i} ∈ T_{i}, 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

respectively.

In Section 8, we will need slight upgrades of some of the observations proved in [5]; wherever additional care is required, we include the short proofs for the convenience of the reader.

Observation 6.2. (On the role of scalars.) If T ≅ (⨁_{1≤i≤m} T_{i})/U is a saguaro, where U is generated by the relations q_{i}t_{i} − q′_{i+1}t_{i+1}, 1 ≤ i ≤ m − 1, as in Definition 6.1, and if k_{1}, … k_{m−1} are non-zero scalars, then

T ≅ (⨁_{i=1}^{m} T_{i})/( ∑_{i≤m−1} Λ(q_{i}t_{i} − k_{i}q′_{i+1}t_{i+1})). □

Observation 6.3. (On intersections of trunks.) Let T = ∑_{1≤i≤m} T̂_{i} be a saguaro on (T_{1}, …, T_{m}) with q_{i}t̂_{i} = q′_{i+1}t̂_{i+1}, 1 ≤ i ≤ m − 1, as in Definition 6.1. Then

T̂_{i} ∩ T̂_{i+1} = Λq_{i}t̂_{i} = Λq′_{i+1}t̂_{i+1} ≠ 0.

In particular, Soc(T̂_{1}) = Soc(T̂_{2}) = ⋯ = Soc(T̂_{m}) ⊆ Soc(T). Moreover, whenever i < j,

T̂_{i} ∩ T̂_{j} = T̂_{i} ∩ (∑_{l≥j} T̂_{l}) = (∑_{l≤i} T̂_{l}) ∩ T̂_{j}.

Proof. The first line of equalities is immediate from the definition. That Λq_{i}t̂_{i} ≠ 0 is a consequence of the facts that q_{i}t_{i} is nonzero in T_{i} by condition (i) of Definition 6.1, and that T_{i} is isomorphic to its canonical image T̂_{i} in T by condition (ii) of that definition. This clearly implies that Soc(T̂_{i}) = Soc(T̂_{i+1}) for all i.

To check that T̂_{i} ∩ T̂_{j} ⊇ T̂_{i} ∩ (∑_{l≥j} T̂_{l}) for i < j, let λt̂_{i} = ∑_{l≥j} λ_{l}t̂_{l} for suitable elements λ and λ_{l} in Λ. Note that, whenever μ ∈ Λ and k < m are such that μq_{k}t_{k} is equal to zero in T_{k}, then μq′_{k+1}t_{k+1} is zero in T_{k+1} by condition (ii) of the definition. Using this fact and condition (i), we obtain λt_{i} − ∑_{l≥j} λ_{l}t_{l} = ∑_{k≥i} μ_{k}(q_{k}t_{k} − q′_{k+1}t_{k+1}) in ⊕_{l≤m}T_{l} for certain μ_{k} ∈ Λ. It follows that λt_{i} = μ_{i}q_{i}t_{i}, and hence that λt̂_{i} = μ_{i}q′_{i+1}t̂_{i+1} ∈ T̂_{i+1} ∩ (∑_{l≥j} T̂_{l}), and an obvious induction on i completes the proof. □

Note that the inclusion Soc(T̂_{i}) ⊆ Soc(T) is proper, in general.

Observation 6.4. (Additional information on intersections of trunks.) Let T in Λ -mod be a saguaro on (T_{1}, …, T_{m}) with canonical sequence of top elements t̂_{1}, …, t̂_{m}, where T_{i} has mast p_{i}. For any two indices i < j in {1, …, m} there then exist right subpaths a of p_{i} and b of p_{j} such that at̂_{i} = bt̂_{j} and T̂_{i} ∩ T̂_{j} = Λat̂_{i} = Λbt̂_{j}.

Proof. We proceed by induction on j −i. The case where j −i = 1 is covered by Observation 6.3, so we may assume j − i ≥ 2. By the induction hypothesis, we can then find a right subpath u of p_{i+1} and a right subpath v of p_{j} with ut̂_{i+1} = vt̂_{j} and such that T̂_{i+1} ∩ T̂_{j} = Λut̂_{i+1}. In view of Observation 6.3, we see moreover that

T̂_{i} ∩ T̂_{j} = T̂_{i} ∩ T̂_{i+1} ∩ T̂_{j} = (T̂_{i} ∩ T̂_{i+1}) ∩ (T̂_{i+1} ∩ T̂_{j}) = Λq′_{i+1}t̂_{i+1} ∩ Λut̂_{i+1}

where q′_{i+1} is a right subpath of p_{i+1} as in Definition 6.1.

If length(q′_{i+1}) ≤ length(u), we obtain u = wq′_{i+1} for a suitable subpath w of p_{i+1}, since both u and q′_{i+1} are right subpaths of p_{i+1}. Observe that, in this case, wq_{i} is a right subpath of p_{i} and wq_{i}t̂_{i} = wq′_{i+1}t̂_{i+1} = ut̂_{i+1} = vt̂_{j} ≠ 0, which shows in particular that T̂_{i} ∩T̂_{j} = Λwq_{i}t̂_{i} = Λvt̂_{j}; thus our claim is satisfied with a = wq_{i} and b = v.

If, on the other hand, length(u) < length(q′_{i+1}), there exists a subpath w of p_{i+1} with wu = q′_{i+1}, which implies wvt̂_{j} = wut̂_{i+1} = q′_{i+1}t̂_{i+1} = q_{i}t̂_{i} and T̂_{i} ∩ T̂_{j} = Λq_{i}t̂_{i} = Λwvt̂_{j}; in other words, our claim is satisfied with a = q_{i} and b = wv. □

It is an obvious consequence of the preceding observation that, given a saguaro T = ∑_{i∈I} T̂_{i} over a left serial algebra and I_{1} ⊆ I, the sum T′ = ∑_{i∈I_{1}} T̂_{i} is in turn a saguaro with trunks {T_{i} | i ∈ I_{1}}.

Observation 6.5. (On the reordering of trunks.) Let T be a saguaro on (T_{1}, …, T_{m}), and π ∈ S_{m} a permutation. If for each i ∈ {1, …, m − 1} we have T̂_{π(i)} ∩ T̂_{π(i+1)} ⊇ T̂_{π(i)} ∩ T̂_{π(j)} for all j > i, then T is also a saguaro on (T_{π(1)}, …, T_{π(m)}).

Proof. The details of the proof can be derived from Observation 6.4 by induction on m. □

In intuitive terms, Observation 6.5 says that, if for all i < m the trunk T̂_{π(i)} is linked to T̂_{π(i+1)} in at least as high a position in the graph of T as it is linked to any of the trunks T̂_{π(i+1)}, …, T̂_{π(m)}, then T is also a saguaro on (T_{π(1)}, …, T_{π(m)}). In particular, this yields: Given any index j ∈ {1, …, m}, there exists a permutation π ∈ S_{m} with π(m) = j (or with π(1) = j) such that T is a saguaro on (T_{π(1)}, …, T_{π(m)}).

A less obvious consequence of Observation 6.5 is the following.

Observation 6.6. (Moving two trunks together.) Again let T ∈ Λ -mod be a saguaro on (T_{1}, …, T_{m}), and let (s, t) be a pair of distinct indices in {1, …, m}. Then there exists a permutation π ∈ S_{m} with the property that T is a saguaro on (T_{π(1)}, …, T_{π(m)}) and (s, t) = (π(l), π(l + 1)) for some l.

Proof. Define I(s), I(t) ⊆ {1, …, m} as follows:

I(s) = {i | i ≠ s, t and T̂_{i} ∩ T̂_{s} ⊋ T̂_{i} ∩ T̂_{t}}
I(t) = {i | i ≠ s, t and T̂_{i} ∩ T̂_{s} ⊆ T̂_{i} ∩ T̂_{t}}.

If |I(s)| = l − 1 for l ≥ 1, set π(l) = s, and define π(l − 1), …, π(1) recursively: If I(s) ≠ ∅, i.e., if l ≥ 2, select π(l − 1) ∈ I(s) so that T̂_{π(l−1)} ∩ T̂_{s} is maximal among the intersections T̂_{i} ∩ T̂_{s} for i ∈ I(s). If I(s) \ {π(l − 1)} ≠ ∅, choose an element π(l − 2) ∈ I(s) \ {π(l − 1)} so that T̂_{π(l−2)} ∩ T̂_{π(l−1)} is maximal among the intersections T̂_{i} ∩ T̂_{π(l−1)}, i ∈ I(s) \ {π(l − 1)}, etc.

Next set π(l +1) = t, and if I(t) ≠ ∅, select π(l +2) ∈ I(t) so that T̂_{π(l+2)} ∩T̂_{t} is maximal among the T̂_{i}∩T̂_{t}, i ∈ I(t). Continue as above: if I(t)\{π(l+2)} ≠ ∅, pick π(l + 3) in this set difference so that T̂_{π(l+3)} ∩ T̂_{π(l+2)} is maximal among the intersections T̂_{i} ∩ T̂_{π(l+2)}, i ∈ I(t) \ {π(l + 2)}, and so forth.

Going back to the definition of a saguaro, one verifies that π satisfies the hypothesis of Observation 6.5. □

The category of finite direct sums of saguaros will serve as a key source of examples to illustrate the usefulness of approximations; see Sections 7 and 8.

7. Second instalment of positive examples: P^{∞}(Λ -mod) where Λ is left serial

The importance of saguaros – or, more generally, modules built on similar patterns – came to our attention through their ‘natural’ occurrence as minimal P^{∞}(Λ -mod)-approximations of the simple modules over a split left serial algebra Λ.

Theorem 7.1. [5, Theorems 5.2, 5.3] Suppose that Λ = KΓ/I is a left serial algebra. Then P^{∞}(Λ -mod) is contravariantly finite, and the minimal P^{∞}(Λ -mod)-approximations 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 C ⊆ Je is maximal with respect to the property that p dim Λe/C < ∞, 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. Then each canonical epimorphism A(S) → S, which maps Λe/C onto S and sends the other trunks of A(S) to zero, is a minimal P^{∞}(Λ -mod)-approx-imation. □

When Λ is a left serial algebra, the minimal P^{∞}(Λ -mod)-approximations of the simple modules can actually be constructed algorithmically from quiver and relations of Λ. In view of Theorem 2.4, they form the basic structural components of arbitrary objects in P^{∞}(Λ -mod).

Corollary 7.2. Let Λ be a left serial algebra. Moreover, suppose that the saguaros A(S_{1}), …, A(S_{n}) are minimal right P^{∞}(Λ -mod)-approximations of the simple left Λ-modules S_{1}, …, S_{n}. Then a finitely generated left Λ-module has finite projective dimension if and only if it is a direct summand of a submodule M having a filtration M = M_{0} ⊇ M_{1} ⊇ ⋯ ⊇ M_{k} = 0, where each of the consecutive factors is isomorphic to some A(S_{i}). □

Example 7.3. Let Λ be a left serial algebra whose quiver has 5 vertices and whose indecomposable projective modules Λe_{i} are given by the following graphs:

1_{α} 2_{β} 3_{γ} 4_{δ} 5_{ϵ} 5_{ϵ}  3_{γ} 5_{ϵ}  3_{γ} 5_{ϵ} 5  5_{ϵ}  5_{ϵ}  5_{ϵ}  5 5_{ϵ}  5  5_{ϵ} 5  5

Then the minimal P^{∞}(Λ -mod)-approximations of the the simple modules S_{i} = Λe_{i}/Je_{i} are as follows (given the fact that they are saguaros, they are in fact determined up to isomorphisms by their graphs):

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

and A(S_{i}) ≅ S_{i} for i = 2, 3, 4. □

The homological information to be gleaned from saguaros can be pushed further. For each d ≥ 0, denote by P^{(d)} the full subcategory of P^{∞}(Λ -mod) whose objects are the modules of projective dimension at most d.

Theorem 7.4. [5] Let Λ again be a left serial algebra. Then, for each d ≥ 0, the category P^{(d)} is contravariantly finite in Λ -mod, and the minimal P^{(d)}-ap-proximations of the simple modules are saguaros.

A minimal P^{(d)}-approximation of the simple module S = Λe/Je is as follows: Let T_{d} be that non-zero homomorphic image of Λe within P^{(d)} which has smallest (positive) composition length. If A_{d}(S) is a saguaro in P^{(d)} having simple socle and trunk T_{d}, which has maximal composition length with respect to these properties, then any homomorphism f : A_{d}(S) → S which maps T_{d} onto S and all the other trunks of A_{d}(S) onto zero is a minimal P^{(d)}-approximation of S. In particular, A_{d}(S) is unique up to isomorphism. □

For each simple module S, there is thus a sequence of minimal P^{(d)}-approxi-mations A_{1}(S), A_{2}(S), A_{3}(S), … which terminates in A_{δ}(S), where δ is the left finitistic dimension of Λ (note that this dimension is known to be finite for left serial algebras [11, Theorem 3]). Clearly A_{δ}(S) coincides with the minimal P^{∞}(Λ -mod)-approximation of S. As d increases, the trunks T_{d} may shrink from the bottom up, while the saguaros A_{d}(S) will increasingly ramify on what is left of these trunks. The following example illustrates this growth pattern; it encodes a great deal of homological information about Λ in a compact form.

Example 7.5. 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. From left to right, we exhibit the minimal P^{(1)}-, P^{(2)}-, P^{(3)}-approximations of S_{1}; the last coincides with the minimal P^{∞}(Λ -mod)-ap-proximation, since the left finitistic dimension of Λ is 3. The P^{(0)}-approximation is simply Λe_{1} and is omitted from the list.

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

8. Approximations over algebras of finite uniserial type

Throughout this section we will assume that Λ = KΓ/I has finite uniserial type. In view of Section 5, this implies in particular that Γ has no double arrows. Let S ⊆ Λ -mod and S^{∞} ⊆ Λ -mod denote the full subcategories having as objects all finite direct sums of saguaros in Λ -mod in the first case, and all finite direct sums of saguaros of finite projective dimension in the second. If the categories S and S^{∞} have finite representation type, they are of course contravariantly finite in Λ -mod and can thus be used to group the objects of Λ -mod according to their minimal S- or S^{∞}-approximations. Whether this yields an effective classification depends on the algebra Λ and on the class of modules to be explored. The objects in S or S^{∞} are, in a way, ‘first approximations’ to ‘useful approximations’. In general, one is led to consider larger contravariantly finite subcategories of Λ -mod, such as CS ⊂ Λ -mod, the category of all finite direct sums of saguaros in Λ -mod and duals of saguaros in mod-Λ, or categories of objects glued together along uniserial submodules from certain cyclic building blocks. An effective method of creating new, more flexible – for approximation purposes – contravariantly finite subcategories from S is provided by the following corollary to Theorem 2.5: If A ⊆ Λ -mod is contravariantly finite, then so is the category E(A) having as objects all direct summands of extensions of modules in A by modules in A.

We believe that the conclusion of the following theorem remains true for arbitrary algebras of finite uniserial type. The somewhat more narrow situation which we address here is technically far less involved, however. As in Section 5, we denote by V_{p} the affine variety describing the uniserial left Λ-modules with mast p. For a characterization of the algebras satisfying the hypothesis of the next theorem, we refer back to Theorem 5.6.

Theorem 8.1. If V_{p} is finite for all paths p in KΓ, the subcategories S and S^{∞} of Λ -mod have finite representation type.

The line of the proof is akin to that of Theorem 5.1 of [5], but the present, far more general, situation calls for non-trivial supplements. We therefore include a fairly detailed argument.

In case the hypothesis of Theorem 8.1 is satisfied, the graphs of the saguaros in Λ -mod are essentially unique, as long as we insist that a canonical sequence of top elements be displayed; see Theorem 5.6(4),(5). To state this uniqueness more concisely, we say that two layered and labeled graphs, G_{1} and G_{2}, are equivalent if there exists an isomorphism of undirected graphs between them which preserves the layering, as well as the numbers attached to the vertices and the labels attached to the edges. (For our use of ‘layered and labeled graphs’, see Section 1.) Under this relation, the equivalence class of graphs of a saguaro T over an algebra Λ as above is uniquely determined, and it therefore makes sense to refer to the graph of T.

Lemma 8.2. If V_{p} is finite for all paths p in KΓ, then any two saguaros in Λ -mod having equivalent graphs (in the sense of Section 7) are isomorphic. More precisely, if T^{(1)} and T^{(2)} are saguaros with identical graphs and t̂_{11}, …, t̂_{1s} (respectively, t̂_{21}, …, t̂_{2s}) are canonical sequences of top elements corresponding to these graphs, then t̂_{1i} ↦ t̂_{2i} induces an isomorphism T^{(1)} → T^{(2)}.

Proof. Recall that Λ = KΓ/I satisfies ‘|V_{p}| < ∞ for all p’ precisely when each graph of a uniserial module is an edge path. In particular, this means that, given a uniserial module U ∈ Λ -mod with top elements x and x′, the left annihilator of x in Λ coincides with that of x′; in other words, the assignment x ↦ x′ determines an automorphism U → U.

Now suppose that T^{(1)} and T^{(2)} in Λ -mod are saguaros with equivalent graphs; without loss of generality, we may assume these graphs to be identical. Then T^{(1)} and T^{(2)} have identical sequences of (isomorphism types) of trunks, (T_{1}, …, T_{s}) say. Write T^{(j)} = ∑_{i=1}^{n} T̂_{ji} for j = 1, 2, where the T̂_{ji} are the canonical images of the T_{i} in T^{(j)}. Moreover, if p_{i} ∈ KΓ is the mast of T_{i}, the fact that the graphs of T^{(1)} and T^{(2)} are the same guarantees the existence of right subpaths q_{1} of p_{1}, q_{i}, q′_{i} of p_{i} for 2 ≤ i ≤ s − 1, q′_{s} of p_{s} such that q_{i}t̂_{ji} = q′_{i+1}t̂_{j,i+1}, j = 1, 2, for canonical sequences of top elements t̂_{ji} corresponding to the coinciding graphs of T^{(1)} and T^{(2)}. By the first paragraph, t̂_{1i} ↦ t̂_{2i} then yields a well-defined isomorphism from T^{(1)} to T^{(2)}. □

For the sake of the proof of Theorem 8.1 we require some additional concepts. The reader solely interested in an overview can safely skip them.

Definition 8.3. Let T = ∑_{i=1}^{m} T̂_{i} be a saguaro as in Definition 6.1, and set I = {1, …, m}.

(a) A submodule V of T of the form V = ∑_{i∈I_{1}} T̂_{i}, where I_{1} ⊆ I, is called a subsaguaro of T if, for all indices j ∈ I \ I_{1},

T̂_{j} ∩ V ⊆ ⋂_{i∈I_{1}} T̂_{i} .

(b) Two subsaguaros V^{(1)} = ∑_{i∈I_{1}} T̂_{i} and V^{(2)} = ∑_{i∈I_{2}} T̂_{i} of T are said to be isomorphic as subsaguaros of T if the following is true: The index sets I_{1} and I_{2} have the same cardinality, s say, and there exists an isomorphism ϕ : V^{(1)} → V^{(2)}, together with orderings {T̂_{11}, …, T̂_{1s}} and {T̂_{21}, …, T̂_{2s}} of the trunks indexed by I_{1} and by I_{2}, respectively, such that ϕ induces an isomorphism T̂_{1j} → T̂_{2j} for 1 ≤ j ≤ s which restricts to the identity on the intersection ⋂_{i∈I_{1}∪I_{2}} T̂_{i}.

(c) The saguaro T is called redundant if it contains nontrivial isomorphic subsaguaros V^{(1)} = ∑_{i∈I_{1}} T̂_{i} and V^{(2)} = ∑_{i∈I_{2}} T̂_{i} with I_{1} ∩ I_{2} = ∅ such that, moreover, T̂_{k} ∩ V^{(1)} = T̂_{k} ∩ V^{(2)} for all k ∈ I \ (I_{1} ∪ I_{2}). Otherwise T is called irredundant.

Observe that each subsaguaro of T is a saguaro in its own right by the remark following Observation 6.4. Intuitively speaking, a subsum V = ∑_{i∈I_{1}} T̂_{i} of trunks of a saguaro T = ∑_{i∈I} T̂_{i} is a subsaguaro if there is no index j ∈ I \ I_{1} such that the trunk T̂_{j} meets V at a properly higher point in the graph of T than it meets the intersection ⋂_{i∈I_{1}} T̂_{i} of the trunks of V.

Lemma 8.4. Again suppose that all the uniserial varieties V_{p} are finite, and let T in Λ -mod be a saguaro with graph G. Then T is redundant if and only if G has a non-trivial full subgraph of the form

G_{1}  G_{2} a

where G_{1} and G_{2} are equivalent trees sharing precisely one vertex a of G such that, moreover, each edge of G which is contiguous with G_{1} ∪G_{2}, without belonging to G_{1} ∪ G_{2}, is incident with a.

Proof. Start by observing that, by Theorem 5.6, our hypothesis forces all graphs of saguaros in Λ -mod to be trees (all graphs of uniserial modules being edge paths). Hence redundancy of T clearly implies the existence of a full subgraph of G as postulated.

Conversely, suppose that G has a subgraph as described in the claim. Without loss of generality, we may assume that G_{1} and G_{2} are identical trees. By Observations 6.5 and 6.6, we are, moreover, free to assume that the graph G has a form as follows

[figure: G_{1}; G_{2}; G_{3}; •; D; G_{4}; _{...}; G_{m}; •]

with a canonical sequence of top elements t̂_{11}, …, t̂_{1s} corresponding to the uppermost layer of vertices in G_{1}, with t̂_{21}, …, t̂_{2s} corresponding to the vertices in the uppermost layer of G_{2}, and t̂_{3}, …, t̂_{l} corresponding to the top vertices of G_{3}, …, G_{m}. For j = 1, 2, define T̂_{(j,i)} = Λt̂_{ji} and V^{(j)} = ∑_{i=1}^{s} ΛT̂_{(j,i)}. Then, clearly, V^{(1)} and V^{(2)} are subsaguaros of T, and the sets I_{j} = {(j, i) | 1 ≤ i ≤ s}, j = 1, 2, indexing their trunks are disjoint. We will show that V^{(1)} and V^{(2)} are isomorphic as subsaguaros of T. Indeed, from Lemma 8.2 we know that the assignment t̂_{1i} ↦ t̂_{2i} induces an isomorphism ϕ : V^{(1)} → V^{(2)}. To see that ϕ induces the identity on D = ⋂_{i≤s} T̂_{(1,i)}∩⋂_{i≤s} T̂_{(2,i)}, it suffices to observe that there are right subpaths a_{i} of mast(T̂_{(1,i)}) = mast(T̂_{(2,i)}), respectively, such that, for any choice of i and h in {1, …, s}, we have T̂_{(1,i)} ∩T̂_{(2,h)} = D = Λa_{i}t̂_{1i} = Λa_{h}t̂_{2h} and a_{i}t̂_{1i} = a_{h}t̂_{2h}; for details, see Observation 6.4. □

Proof of Theorem 8.1. In a first step we show that, up to isomorphism, there are only finitely many irredundant saguaros in Λ -mod. We proceed by induction on the Loewy length L of Λ. The case L = 1 being clear, suppose that L ≥ 2 and that there are r isomorphism types of irredundant saguaros in Λ/J^{L−1}-mod; let G_{1}, …, G_{r} be the corresponding graphs, and recall that the G_{i} are unique up to equivalence. The graphs of the additional irredundant saguaros in Λ -mod are all equivalent to graphs of the form

[figure: G_{i_{1}}; G_{i_{t}}; •; ⋯; •; •; α_{1}; α_{t}]

where t ≥ 1, 1 ≤ i_{j} ≤ r, and the α_{i} are arrows such that α_{j} = α_{k} implies i_{j} ≠ i_{k}. The number of equivalence classes of graphs of this type is clearly bounded above by 2^{ra}, where a is the number of distinct arrows in Γ.

In a second step we prove that each saguaro in Λ -mod is a direct sum of irredundant ones. Suppose that T is a saguaro on (T_{1}, …, T_{m}). This time, we proceed by induction on m. If T is irredundant to begin with, there is nothing to prove; in particular, this is the case when m = 1. So suppose that m ≥ 2 and that T is redundant with non-zero isomorphic subsaguaros V^{(1)} = ∑_{i∈I_{1}} T̂_{i} and V^{(2)} = ∑_{i∈I_{2}} T̂_{i} such that I_{1} ∩ I_{2} = ∅ and T̂_{k} ∩ V^{(1)} = T̂_{k} ∩ V^{(2)} ⊆ V^{(1)} ∩ V^{(2)} for all k ∉ I_{1} ∪ I_{2} as in Definition 8.3. Moreover, let ϕ : V^{(1)} → V^{(2)} be an isomorphism and T̂_{11}, …, T̂_{1s}, resp. T̂_{21}, …, T̂_{2s}, orderings of the trunks indexed by I_{1}, resp. by I_{2}, such that ϕ induces an isomorphism T̂_{1j} → T̂_{2j} for 1 ≤ j ≤ s and restricts to the identity on D := ⋂_{i∈I_{1}} T̂_{1i} ∩ ⋂_{i∈I_{2}} T̂_{2i}. In particular, this implies that, given a top element t̂_{1j} ∈ T̂_{1j}, the image t̂_{2j} := ϕ(t̂_{1j}) is a top element of T̂_{2j} for 1 ≤ j ≤ s.

Observe that T̂_{1i} ∩ T̂_{2j} = D for all i, j ∈ {1, …, s}, since T̂_{1i} ∩ T̂_{2j} ⊆ ⋂_{1≤k≤s} T̂_{1k} ∩ ⋂_{1≤k≤s} T̂_{2k} due to the fact that V^{(1)} and V^{(2)} are subsaguaros of T and I_{1} ∩ I_{2} = ∅. It follows that the submodule V := ∑_{1≤i≤s} Λ(t̂_{1i} − t̂_{2i}) of T is isomorphic to V^{(1)}/D and thus is a direct sum of saguaros, each of which has at most s trunks. By the remark following Observation 6.4, W := ∑_{i∈I\I_{1}} T̂_{i} is in turn a saguaro which clearly has fewer than m trunks because I_{1} ≠ ∅. In view of the fact that we can reorder T_{1}, …, T_{m} into another legitimate sequence of trunks in such a way that the trunks indexed by I_{1} precede those indexed by I \ I_{1} (Observation 6.5), it is now routine to check that V ∩ W = 0. We infer that T = V ⊕ W and apply the induction hypothesis to W and to the saguaros occurring as direct summands of V to complete the proof. □

In a first easy example – Example 4.3 revisited – we present a monomial relation algebra Λ for which P^{∞}(Λ -mod) fails to be contravariantly finite in Λ -mod, while all the varieties V_{p} describing the uniserial left Λ-modules are finite. By Theorem 8.1, this implies that S^{∞} has finite representation type.

Example 8.5. If Λ is the algebra of Example 4.3, then P^{∞}(Λ -mod) fails to be contravariantly finite as we saw earlier. However, Λ satisfies the hypothesis of Theorem 8.1, and so S^{∞} has finite type. A fortiori, S^{∞} is contravariantly finite in Λ -mod. The minimal S^{∞}-approximations A_{i} of the simple left Λ-modules S_{i} are as follows:

1  2  3  4  5  6 4  3  3  5  5  2

Observe that l fin dim Λ = 1 = sup_{1≤i≤6} p dim A_{i}. □

Problem 8.6. Characterize those algebras Λ of finite uniserial type for which the supremum of the minimal S^{∞}-approximations of the simple left Λ-modules equals l fin dim Λ.

While there are algebras for which this equality fails, for instance among the algebras of type Λ = O/πO discussed in Section 4, the realm of validity of this equality even among non-monomial algebras appears to be fairly wide. We conclude with a not so straightforward binomial example. Here the computations leading to the precise shapes of the minimal approximations of the simple modules are a bit more involved; we will suppress them nonetheless.

Example 8.7. Let Λ = KΓ/I, where Γ is the quiver

[figure: 1; α_{1}; β_{1}; 9; α_{9}; 8; α_{8}; 2; β_{2}; α_{2}; γ_{2}; 3; α_{3}; 7; α_{7}; 4; α_{4}; 5; α_{5}; β_{5}; 6; α_{6}]

and the indecomposable projective left Λ-modules are

[figure: 1; α_{1}; β_{1}; 2; 3; 4; 5; 6; 7; 8; 9; 2; γ_{2}; β_{2}; 8; α_{8}; 9; 2; 3; 7; 2; 3; 6; 5; 7; 7; 7; 9; α_{9}; 2; α_{2}; 3; 7; 9; 2; 3; 7; 5; 3; 7; 3; α_{3}; 7; 3; 7; 3; 7; 7; 7; 7]

The only relations which cannot be gleaned from the graphs – the scalars occurring cannot be detected – we take to be α_{9}γ_{2} − α_{3}α_{2}β_{2} and α_{3}α_{2}β_{2}α_{1} − α_{8}β_{1}.

Then Λ has finite uniserial type, without satisfying the stronger hypothesis of Theorem 8.1. However, the categories S and S^{∞} are still of finite representation type. One can compute the minimal S^{∞}-approximation A_{1} of S_{1} to be given by the graph

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

The minimal S^{∞}-approximations A_{i} of the remaining simples S_{i} are:

[figure: 2; 4; 3; 9; 7; 9; 8; 9; A_{2} :; 2; A_{3} :; 7; A_{7} :; 7; A_{8} :; 7]

and A_{i} = S_{i} for i = 4, 5, 6, 9.

Moreover, one obtains sup{p dim A_{i} | 1 ≤ i ≤ 9} = 2. We resolve A_{1}, to indicate how easy it is to find minimal projective resolutions of saguaros. Indeed, the first syzygy Ω^{1}(A_{1}) of A_{1} has graph

[figure: 2; 9; 8; 2; 3; 3; 9; 2; ⨁; 7; ⨁; 9; 2; 3; ⨁; 7; ⨁; 7; 3; 3; 7; 7; 7]

and Ω^{2}(A_{1}) has graph

3  7 7 ⨁ 7

Thus Ω^{2}(A_{1}) is projective. With the ‘repetition method’ of [8], one can finally check that l fin dim Λ = 2, which yields the equality l fin dim Λ = sup{p dim A_{i} | 1 ≤ i ≤ 9}. □

9. Phantoms

When 𝔄 ⊆ Λ -mod fails to be contravariantly finite, we abandon the requirement that the approximating objects, used to compare arbitrary finitely generated modules with the modules in 𝔄, be themselves finitely generated. In [10] it became apparent that this often yields information which is no less effective than that stored in classical (finite dimensional) 𝔄-approximations. Such generalized 𝔄-approximations, ‘phantoms’ as we will call them, of a simple module Λe/Je say, provide a synopsis of the relations present in those objects of 𝔄 which contain a top element of type e. We simply renounce the requirement that this picture should fit into a finitely generated module.

Definitions 9.1. [10] Let ℭ ⊆ 𝔄 be full subcategories of Λ -mod and suppose that 𝔄 is closed under finite direct sums. Moreover, let X be a finitely generated left Λ-module.

(1) A ℭ-approximation of X inside 𝔄 is a homomorphism f : A → X with A ∈ 𝔄 such that each map g ∈ Hom_{Λ}(C, X) with C ∈ ℭ factors through f.

(2) An 𝔄-phantom of X of the first kind is an object H ∈ Λ -mod (not necessarily in 𝔄) with the following property: There exists a finite nonempty subclass ℭ(H) ⊆ 𝔄 such that for each ℭ(H)-approximation f : A → X inside 𝔄, the module H is a subfactor of A. Any direct limit of 𝔄-phantoms of X of the first kind will be called an 𝔄-phantom of X of the second kind.

We will refer to both kinds of phantoms as 𝔄-phantoms of X.

(3) An 𝔄-phantom H of X is called ℭ-effective if H is a direct limit of objects in 𝔄 and there exists a homomorphism f : H → X with the property that each homomorphism g ∈ Hom_{Λ}(C, X) with C ∈ ℭ factors through f.

The effective phantoms are in a sense the best possible substitutes for minimal 𝔄-approximations in the traditional sense. The crucial fact is that, given any module X ∈ Λ -mod, nontrivial phantoms exist. In case X has an 𝔄-approx-imation in the sense of Auslander and Smalø, the minimal 𝔄-approximation A(X) is the ‘best possible’ phantom; indeed, the 𝔄-phantoms of X are precisely the subfactors of A(X) in that situation. The case of interest is addressed by the following theorem, the proof of which indicates a construction pattern for phantoms of infinite K-dimension.

Theorem 9.2. [10] Suppose that 𝔄 ⊆ Λ -mod is closed under finite direct sums and let X ∈ Λ -mod. Then the following conditions are equivalent:

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

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

(3) There exists a countable subclass ℭ ⊆ 𝔄 such that X has ℭ-effective 𝔄phantoms of infinite K-dimension. □

Remarks 9.3. (1) From the proof of Criterion 4.1 it can be gleaned that, under the hypotheses of 4.1, with p_{1}, …, p_{m}, q_{1}, …, q_{m} being paths in KΓ of positive length, the simple module S_{1} has an 𝔄-phantom with graph

[figure: e_{1}; p_{1}; e_{2}; q_{2}; p_{2}; ⋯; e_{m}; q_{m}; p_{m}; e_{1}; q_{1}; p_{1}; e_{2}; q_{2}; p_{2}; ⋯; •; •; ⋯; •; •; •; •; ⋯]

The next remark gives a clue how to start building phantoms.

(2) Let 𝔄 = P^{∞}(Λ -mod) where Λ = KΓ/I is a monomial relation algebra, and suppose that the simple module S = Λe/Je has infinite projective dimension. If α_{1}, …, α_{m} are arrows α_{j} : e → e_{j} ending in distinct vertices e_{1}, …, e_{m} such that p dim Je_{j} = ∞ for 1 ≤ j ≤ m, then

[figure: e; α_{1}; α_{2}; α_{m}; e_{1}; e_{2}; ⋯; e_{m}]

is the graph of a P^{∞}(Λ -mod)-phantom of S.

Moreover, if there exists a module M ∈ P^{∞}(Λ -mod) with graph G having top elements m = em and m′ = e′m′ which correspond to the top vertices of a subgraph of G as follows

[figure: e; α_{m}; e′; β; e_{m}]

then

[figure: e; α_{1}; α_{2}; α_{m}; e′; β; e_{1}; e_{2}; ⋯; e_{m}]

is the graph of a P^{∞}(Λ -mod)-phantom of S. □

Examples 9.4. We revisit the negative examples of Section 4. Throughout, 𝔄 stands for the category P^{∞}(Λ -mod) and S_{1} for the simple left Λ-module corresponding to the vertex ‘1’.

• Example 4.2. The module M = \underrightarrow{lim} M_{n} ∈ P^{∞}(Λ -Mod) with graph

[figure: 1; β; 1; α; β; 1; α; β; ⋯; 2; 2; 2; ⋯]

is an 𝔄-phantom of S_{1} which is ℭ_{0}-effective, where ℭ_{0} = {M_{n} | n ∈ ℕ}; here M_{n} is defined as in 4.2.

On the other hand, the left Λ-module N with graph

[figure: 1; α; β; 1; α; β; 1; α; β; ⋯; 2; γ; 2; 2; 2; ⋯; 1]

– while still an object of P^{∞}(Λ -Mod) with the property that each homomorphism M_{n} → S_{1}, n ∈ ℕ, factors through it – is not an 𝔄-phantom of S_{1}.

Further non-finitely generated phantoms of S_{1} are as follows. For each nonzero scalar k ∈ K and n ∈ ℕ, let x_{1} = ⋯ = x_{n} = e_{1}, and consider the module

L_{nk} = (⨁_{i=1}^{n} Λx_{i}) / (Λ(βx_{1} − kαx_{1}) + ∑_{i=2}^{n} Λ(βx_{i} − kαx_{i} − αx_{i−1}))

in P^{∞}(Λ -mod), which has the following graph relative to the top elements \overline{x}_{1}, …, \overline{x}_{n}:

[figure: 1; β; α; 1; β; α; 1; β; α; 1; β; α; ⋯; 1; α; 1; β; α; 2; 2; 2; 2; 2; 2; 2; ⋯; 2; 2; 2]

Clearly, L_{nk} embeds canonically into L_{mk} for n < m, and it is not difficult to see that the direct limit L_{k} := \underrightarrow{lim} L_{nk} with graph

[figure: 1; β; α; 1; β; α; 1; β; α; 1; β; α; ⋯; 2; 2; 2; 2; 2; 2; 2; ⋯]

is an 𝔄-phantom of S_{1} which is ℭ_{k}-effective, where ℭ_{k} = {L_{nk} | n ∈ ℕ} ⊆ 𝔄.

• Example 4.3. Here the direct limit M = \underrightarrow{lim} M_{n} with graph

[figure: 1; β; 6; δ; γ; 1; α; β; 6; δ; γ; ⋯; 4; 2; 4; 2; ⋯]

is an 𝔄-phantom of S_{1} which is {M_{n} | n ∈ ℕ}-effective. It is not difficult to see that the 𝔄-phantom M is actually 𝔄-effective and hence encodes the full information stored in classical approximations when they exist.

• Example 4.5. Consider the subclasses ℭ = {M_{n} | n ∈ ℕ} and 𝔇 = {N_{n} | n ∈ ℕ} of 𝔄, where M_{n} and N_{n} are the left Λ-modules with graphs

[figure: 1; β; 1; α; β; ⋯; 1; α; β; 2; 2; ⋯; 2; 2]

[figure: 1; β; 3; δ; 3; δ; 3; δ; 1; α; β; 1; α; β; ⋯; 1; α; β; 2; 2; 2; ⋯; 2; 2]

respectively, relative to n top elements which are linearly independent modulo the radical in each case. Then both M = \underrightarrow{lim} M_{n} and N = \underrightarrow{lim} N_{n} are 𝔄phantoms of S_{1}. The phantom M is ℭ-effective, but not 𝔇-effective, while N is (ℭ ∪ 𝔇)-effective; in other words, N is the ‘better’ of the two 𝔄-phantoms of S_{1}, storing more information about 𝔄 than the phantom M. □

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

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