Approximating modules by modules of finite projective dimension

W.D. Burgess^{1} and B. Zimmermann Huisgen^{1}

1. Introduction.

Given a finite dimensional algebra Λ, our goal is to find closest possible approximations to the finitely generated Λ-modules by modules of finite projective dimension and to exploit the description of such approximations to gain concrete structural information on the Λmodules of finite projective dimension. The program is carried out in full for left serial algebras, i.e., for algebras with the property that each indecomposable projective left ideal is uniserial. In particular, the resulting structure theory demonstrates the intricacy of the module categories of one-sided serial algebras, which is in sharp contrast to the corresponding theory for two-sided serial algebras. Foundations are laid to tackle the above program in more general situations.

The conceptual framework was introduced by Auslander-Smalø in [3] and Auslander-Reiten in [2], as follows: If A is a full subcategory of Λ-mod and M a finitely generated Λ-module, a right A-approximation of M is a homomorphism f : A → M with A ∈ A such that each homomorphism B → M with B ∈ A factors through f. Provided that such a map f exists, there is one, unique up to isomorphism, for which A has minimal composition length, called the minimal (right) A-approximation of M (by a slight abuse of language we also refer to the module A as the minimal approximation of M). In case each finitely generated (left) Λ-module has an A-approximation, the category A is said to be contravariantly finite in Λ-mod. Our interest here is in the situation where A is P^{∞}, the full subcategory of Λ-mod consisting of all modules of finite projective dimension. While in [2] numerous consequences were derived from contravariant finiteness of P^{∞}, comparatively few classes of algebras are known to enjoy this property, or known not to. There is not even a characterization of the monomial relation algebras for which P^{∞} is contravariantly finite, but the first negative example of [12] has called attention to the problem. Moreover, aside from trivial cases, no structural information on minimal P^{∞}-approximations was available up to now (of course, the identity on M is a minimal P^{∞}-approximation of M in case M has finite projective dimension, and if the left finitistic dimension of Λ is zero, minimal P^{∞}-approximations coincide with projective covers). Existence provided, knowledge of the structure of the minimal P^{∞}-approximations A_{1}, A_{2}, …, A_{n} of the simple Λ-modules is of particular interest since, by [2, Proposition 3.8], the A_{i} are the building blocks of the modules of finite projective dimension, in much the same way as the simple modules are composition factors of arbitrary Λ-modules. More precisely, if P^{∞} is contravariantly finite, a finitely generated module M has finite projective dimension if and only if M is a direct summand of the union of a finite chain of modules with successive factors in {A_{1}, …, A_{n}}.

Our principal objective is to establish contravariant finiteness of P^{∞} in Λ-mod whenever Λ is a finite dimensional split left serial algebra over a field K, and to give an explicit description of the structure of the minimal P^{∞}-approximations of the simple left modules over such algebras. (A split left serial algebra over K is a path algebra modulo relations on a quiver Γ in which at most one arrow emanates from any given vertex.) The resulting approximations display a surprising structural variety on one hand, but turn out to be quite tractable, with respect to syzygies and graphical representations for instance, on the other. In their role as ‘composition factors’ of the modules of finite projective dimension they provide strong clues to the structure of other objects in P^{∞}.

1The work of the authors was partially supported by grants A7539 of the NSERC and DMS 92-04182 of the NSF. Some of this work was done while the first author was enjoying the hospitality of the University of California at Santa Barbara.

The types of modules arising as P^{∞}-approximations of the simple modules in the left serial case also turn out to be useful for approximation purposes over wider classes of algebras (see [11]). Because of the shape of their graphs, we call them saguaros. Roughly speaking, they can be described as follows: Given uniserial modules T_{1}, T_{2}, …, T_{m}, where T_{i} is an epimorphic image of Λe(i) for a primitive idempotent e(i), a saguaro T on this sequence of ‘trunks’ is a factor module of ⨁_{i=1}^{m} T_{i} which identifies b_{i}e(i) with c_{i}e(i + 1) for 1 ≤ i ≤ m − 1, where b_{i}, c_{i} are residue classes, modulo I, of certain paths in Γ; moreover, we require that each T_{i} embed in T via the canonical projection and that all the ‘gluing points’ b_{i}e(i) be non-zero.

We can now give more detail about our main results. Suppose that Λ = KΓ/I is a finite dimensional left serial algebra on a quiver Γ, where KΓ denotes the corresponding path algebra and I is an ideal in KΓ. If J is the Jacobson radical of Λ , the minimal P^{∞}-approximation T (S) of a simple module S = Λe/Je is as follows: Let T_{∗} be the factor module of Λe which has smallest positive composition length with respect to having finite projective dimension. Among the saguaros in P^{∞} with trunk T_{∗} and simple socle, T (S) is then the unique candidate of maximal composition length. The saguaro T (S) can be explicitly constructed from basic information on Γ and I; in particular, it is independent of the base field. Similarly, each of the subcategories P^{(d)} of Λ-mod, having as objects the modules of projective dimension at most d, is contravariantly finite in Λ-mod , and the growth pattern of the corresponding saguaros serving as minimal P^{(d)}-approximations of the simple modules provides an informative picture of the homology of Λ-mod . The graphical rendering of this picture encodes homological data in an easily decipherable form.

Our proofs backing the structure theory for modules of finite projective dimension over left serial algebras are based on techniques which appear to be new and which will subsequently be made available in a broader context [11]. We therefore develop the foundations for more general path algebras modulo relations; the basic theory of saguaros will make up a major portion of Sections 3 and 4. In Section 2, we give a brief introduction to the graphs we use; they are more flexible in adapting to our purposes than those which can be found elsewhere in the literature. In Section 5, we state our main results and prove the first; the second is proved in Section 6. In Section 7, we informally describe an algorithm for finding minimal P^{∞}-approximations in the left serial case and illustrate it with examples.

The authors are indebted to Axel Boldt for carefully proofreading the first version of this paper and making many helpful suggestions towards an improved presentation.

2. Distinguished bases and graphs.

Throughout, Λ = KΓ/I will be a finite dimensional path algebra with relations over a field K, and J will be its Jacobson radical. (In particular, all our algebras will be split.)

As is well known, whenever K is algebraically closed, each finite dimensional K-algebra is Morita equivalent to a split one, whence all the categorical results we prove remain true for arbitrary finite dimensional algebras in that case.

We will identify the primitive idempotents e_{1}, …, e_{n} of Λ corresponding to the vertices of Γ with these vertices, or with the paths of length zero in Γ, as convenient. Set B_{0} = {e_{1}, …, e_{n}}. Given any left Λ-module M, an element m ∈ M is called normed if m = em for some primitive idempotent e ∈ B_{0} (we sometimes say that m is normed of type e), and we call m a top element of M if m is normed and belongs to M \JM. Note that any scalar multiple of a normed element is again normed. Two elements m_{1}, m_{2} ∈ M, normed of the same type e, are called associates if m_{1} = μm_{2} for some unit μ of eΛe. Such a unit is also called a local unit of Λ , and its inverse μ^{−1} in eΛe is its local inverse.

To find what we will call a normed K-basis for Λ, we define B_{1} to be the set of residue classes, modulo I, of all the arrows in Γ; the elements of B_{1} will also be referred to as the arrows in Λ. Since, as usual, we insist that the ideal I of relations be generated by linear combinations of paths of length at least 2, B_{1} is a K-basis for J modulo J^{2}. Next we choose a K-basis B_{2} ⊆ B_{1} · B_{1} for J^{2} modulo J^{3}. Recursively, we choose a K-basis B_{i+1} ⊆ B_{1} ·B_{i} for J^{i+1} modulo J^{i+2}. If we set B = ⋃_{i≥0} B_{i} and B^{∗} = ⋃_{i≥1} B_{i}, then clearly B is a K-basis for Λ consisting of normed elements, and B^{∗} is a basis for J. Any basis for Λ constructed following the above pattern will be called a normed K-basis in the sequel. In general, we of course have considerable freedom in the choice of such bases. Whenever B = ⋃_{i≥0} B_{i} is a normed K-basis for Λ, there is a well-defined concept of length for the elements of B: namely, for b ∈ B,

length(b) = i ⟺ b ∈ B_{i}.

Many authors have found it convenient to represent certain Λ-modules by graphs (see, e.g., [1], [4], [6] and [10]). A special breed of such graphs will make a great deal of information available at a glance and thus clarify our arguments. We introduce these labeled and layered graphs for a certain class of modules informally by means of some illustrative examples.

Example 2.1. Let Γ be the quiver

[figure: 1; β; α; γ; 2; δ; ϵ]

and Λ = KΓ/I, where I is generated by the relations

εδ, δε, δ^{2}, ε^{3}, ε^{2}β, δγ, εγ.

Then the indecomposable projective modules Λe_{1} and Λe_{2} have the following graphs.

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

In the case of Λe_{2}, this is to be interpreted as follows: the top Λe_{2}/Je_{2} is isomorphic to S_{2}; the next radical layer, Je_{2}/J^{2}e_{2}, is equal to A⊕B where A ≅ B ≅ S_{2}, and δe_{2} generates A, while εe_{2} generates B modulo J^{2}e_{2}; moreover, the third radical layer, namely J^{3}e_{2}/J^{4}e_{2} = J^{3}e_{2}, is isomorphic to S_{2} and is generated by ε^{2}e_{2}, while εδe_{2} = δεe_{2} = ε^{3}e_{2} = 0.

A Λ-module M has graph

1_{α} _{β} 2_{ϵ} 2

if the following is true: M/JM ≅ S_{1}; the next two layers, JM/J^{2}M and J^{2}M/J^{3}M = J^{2}M, are each isomorphic to S_{2}; moreover, there exists a top element x of M such that each of the elements αx and βx generates JM modulo J^{2}M, while γx = 0, and each of the elements εαx and εβx generates J^{2}M, while δαx = 0. Clearly, the module M is not determined up to isomorphism by this layered and labeled graph unless K has only two elements. Note, however, that any such module M is necessarily uniserial.

Another graph of a uniserial module N over the same algebra is

1_{α} _{β} _{γ} 2_{δ} _{ϵ} 2

the information carried by the graph being as follows: N/JN ≅ S_{1} and JN/J^{2}N ≅ J^{2}N/J^{3}N = J^{2}N ≅ S_{2}; moreover, there exists a top element y of N such that each of the elements αy and βy generate JN modulo J^{2}N, and finally each of the five elements γy, δαy, εαy, δβy, εβy generates J^{2}N.

On the other hand,

1_{α} _{β} 2_{ϵ} 2_{ϵ} 2

is not the graph of a left Λ-module X, because Kαx = Kβx modulo J^{2}X would imply that Kε^{2}αx = Kε^{2}βx = 0.

Recall that a left serial algebra is just a monomial relation algebra KΓ/I on a quiver Γ which has the property that at most one arrow emanates from any given vertex. In our second example, we present a module over a prototypical left serial algebra to which we will repeatedly refer in later sections.

Example 2.2. 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 module M with the following graph (which determines M up to isomorphism) is an example of a type of module to be called a saguaro in Section 3; here the labels on the edges are uniquely determined and so are omitted.

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

Here the graphical information is to be decoded as follows: M/JM ≅ S_{1} ⊕ S_{2}^{2} ⊕ S_{4}^{2}, and there exist top elements x_{1}, x_{2}, y_{2}, x_{4}, y_{4}, each normed by the idempotent with the same index, such that: JM/J^{2}M = A⊕B with A ≅ B ≅ S_{3}, where A is generated modulo J^{2}M by each of the elements δx_{4}, βx_{2}, and B by either of δy_{4}, βy_{2}. Moreover, J^{2}M/J^{3}M ≅ S_{5}, and this copy of S_{5} is generated modulo J^{3}M by each of the elements αx_{1}, γδy_{4}, γβy_{2}. Finally, J^{3}M/J^{4}M = J^{3}M ≅ S_{5}, and J^{3}M is generated by any of the elements γδx_{4}, γδx_{2}, εαx_{1}, εγδy_{4}, εγβy_{2}. The significance of this particular module for a homological understanding of Λ will become clear in Section 7.

Remarks 2.3 (1) Suppose that Λ = KΓ/I is left serial, i.e., at most one arrow leaves any given vertex of Γ. In that case we have only a single choice of a normed K-basis B as above, B being the set of all residue classes of paths in KΓ which do not belong to I, and we will clearly not lose any information in omitting the labels specifying the edges in the graphs of Λ-modules.

(2) Clearly, layered and labeled graphs of uniserial modules always exist; they are uniquely determined by the isomorphism classes of these modules, once top elements have been fixed. However, in general, the graphs fail to pin down corresponding modules up to isomorphism, as we saw in Example 2.1.

3. Saguaros.

Let B = ⋃_{i≥0} B_{i} and B^{∗} = ⋃_{i≥1} B_{i} be normed K-bases for Λ and J, respectively, as described in Section 2. The basis B will be fixed in what follows.

Definition 3.1 Suppose that T_{1}, …, T_{m} is a sequence of non-zero uniserial left Λ-modules. I. A left Λ-module T is called a saguaro^{2} on (T_{1}, …, T_{m}), relative to the K-basis B, 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

b_{i}t_{i} − c_{i}t_{i+1}, 1 ≤ i ≤ m − 1,

where t_{i} ∈ T_{i} are suitable top elements and b_{i}, c_{i} ∈ B^{∗} have the property that b_{i}t_{i} ≠ 0, c_{i}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.

II. A saguaro is called reduced if it has simple socle.

2The name is that of a cactus found in the Sonoran Desert.

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 displays a canonical sequence of top elements if the simple modules shown in the uppermost layer of the graph are generated by such a canonical sequence. It is clear that layered and labeled graphs displaying a canonical sequence of top elements always exist. They are essentially unique, as long as we keep the distinguished basis B_{1} for J/J^{2} fixed and insist that a canonical sequence of top elements be displayed. 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. Under this relation, the equivalence class of graphs of a saguaro T as above is uniquely determined, and it therefore makes sense to refer to the graph of T. To give an example,

[figure: 1; α; β; 1; α; 2; ϵ; δ; 2; ϵ; 2]

is the graph of a saguaro over the algebra of Example 2.1.

By way of caution: If we do not insist on exhibiting a canonical sequence of top elements, the graph of a saguaro need not be unique in the sense specified above. For example, the left-hand graph below represents a saguaro over the algebra of Example 2.2, and the righthand graph represents the same module, up to isomorphism, but is not equivalent to the former.

[figure: 1; α; 1; α; 1; α; ⊕; 1; 5; 5]

To avoid another layer of terminology, we will tacitly impose the requirement that a canonical sequence of top elements be displayed whenever we refer to the graph of a saguaro in the sequel.

As for the converse: We already remarked in 2.1 that a saguaro is not pinned down up to isomorphism by its graph in general. However, if Λ is a split left serial algebra, the isomorphism type of a saguaro is determined by its graph; one checks this readily, using the fact that in this case the graph of a uniserial left Λ-module is simply a path without double edges which has a ‘top’ and a ‘bottom’. This uniqueness statement remains true for the algebras of the form Λ = O/πO associated with tiled classical orders O over a discrete valuation domain with uniformizing parameter π (see [11]); these algebras, first studied by Jategaonkar and Kirkman-Kuzmanovich, were shown to store a great deal of information about the corresponding orders (see [13], [14], and [7]).

In general, the ordering of the trunks is not uniquely determined by a given saguaro. For example, the module M of Example 2.2 is a saguaro on (T_{1}, T_{2}, T_{3}, T_{4}, T_{5}), on (T_{2}, T_{1}, T_{3}, T_{5}, T_{4}), or on (T_{5}, T_{4}, T_{3}, T_{2}, T_{1}), where T_{1} = Λe_{4}/Λεγδ, T_{2} = Λe_{2}/Λεγβ, T_{3} = Λe_{1}, T_{4} = Λe_{2}/Λε^{2}γβ, and T_{5} = Λe_{4}/Λε^{2}γδ; however M is not a saguaro on the sequence (T_{3}, T_{1}, T_{2}, T_{4}, T_{5}).

The following observations will provide us with some basic information on saguaros and help us to manipulate them.

Observation 3.2 (On the role of scalars.) If T ≅ (⨁_{1≤i≤m} T_{i}) /U is a saguaro, where U is generated by the relations b_{i}t_{i} − c_{i}t_{i+1}, 1 ≤ i ≤ m − 1, as in the definition, and k_{1}, … k_{m−1} are non-zero scalars, then

T ≅ \frac{⨁_{i=1}^{m} T_{i}}{∑_{i≤m−1} Λ(b_{i}t_{i} − k_{i}c_{i}t_{i+1})} . □

Observation 3.3 (On intersections of trunks.) Let T = ∑_{1≤i≤m} T̂_{i} be a saguaro on (T_{1}, …, T_{m})with b_{i}t̂_{i} = c_{i}t̂_{i+1}, 1 ≤ i ≤ m − 1, as in Definition 3.1 (I). Then

T̂_{i} ∩ T̂_{i+1} = Λb_{i}t̂_{i} = Λc_{i}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 follows immediately from the definition. That Λb_{i}t̂_{i} ≠ 0, is a consequence of the facts that b_{i}t_{i} is nonzero in T_{i} by condition (i) of Definition 3.1.I, and that T_{i} is isomorphic to its canonical image T̂_{i} in T by condition (ii) of that definition. This clearly implies 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, k < m and μb_{k}t_{k} is equal to zero in T_{k}, then μc_{k}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}(b_{k}t_{k} − c_{k}t_{k+1}) for certain μ_{k} ∈ Λ. It follows that λt_{i} = μ_{i}b_{i}t_{i}, and hence that λt̂_{i} = μ_{i}c_{i}t̂_{i+1} ∈ T̂_{i+1} ∩ (∑_{l≥j} T̂_{l}), and an obvious induction on i completes the proof. □

The inclusion Soc(T̂_{i}) ⊆ Soc(T) is proper, in general; see Observation 3.8 below for a characterization of reduced saguaros.

Observation 3.4 (Additional information on intersections of trunks when Λ is left serial.) Let Λ be a left serial algebra with normed K-basis B, and let T in Λ-mod be a saguaro on (T_{1}, …, T_{m}) with canonical sequence of top elements t̂_{1}, …, t̂_{m}. For any two indices i < j in {1, …, m} there exist elements a, b ∈ B^{∗} such that at̂_{i} = bt̂_{j} and T̂_{i} ∩ T̂_{j} = Λat̂_{i}.

Proof. We proceed by induction on j − i. The case where j − i = 1 is covered by Observation 3.3, so we may assume j − i ≥ 2. By the induction hypothesis, we can then find u, v ∈ B^{∗} with ut̂_{i+1} = vt̂_{j} such that T̂_{i+1} ∩ T̂_{j} = Λut̂_{i+1}. In view of 3.3, 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}) = Λc_{i}t̂_{i+1} ∩ Λut̂_{i+1}.

If length(c_{i}) ≤ length(u), we obtain u = wc_{i} for some w ∈ B, since Λ is left serial and u, c_{i} are paths starting in the same vertex. In that case, wb_{i}t̂_{i} = wc_{i}t̂_{i+1} = ut̂_{i+1} = vt̂_{j} ≠ 0, which shows in particular that wb_{i} ∈ B^{∗}, and T̂_{i} ∩ T̂_{j} = Λut̂_{i+1} = Λwb_{i}t̂_{i}.

If, on the other hand, length(u) < length(c_{i}), there exists w ∈ B with wu = c_{i}, which implies wvt̂_{j} = wut̂_{i+1} = c_{i}t̂_{i+1} = b_{i}t̂_{i} and T̂_{i} ∩ T̂_{j} = Λc_{i}t̂_{i+1} = Λb_{i}t̂_{i}. □

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 3.5 (On the reordering of trunks.) Let Λ be as in Observation 3.4 and T a saguaro on (T_{1}, …, T_{m}). Moreover, let π ∈ S_{m} be 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 claim can be derived from Observation 3.4 by induction on m. □

In intuitive terms, Observation 3.5 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 3.5 is the following.

Observation 3.6 (Moving two trunks together.) Let Λ be a left serial algebra, T ∈ Λ-mod a saguaro on (T_{1}, …, T_{m}) , and (s, t) 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}}, and
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 3.5. □

The following observation will only play an auxiliary role.

Observation 3.7 (A sufficient condition for a saguaro to be reduced.) Suppose that T is a saguaro on (T_{1}, …, T_{m})with canonical sequence of top elements t̂_{i} ∈ T̂_{i}, and let b_{i}, c_{i} ∈ B be as in Definition 3.1(I). Then T is reduced provided that the following condition is satisfied for 1 ≤ j ≤ m − 1:

(*) Whenever a_{j} ∈ B and α ∈ B_{1} are such that length(a_{j}) < length(b_{j}) and αa_{j}t̂_{j} ≠ 0, then

αa_{j}t̂_{j} ∉ α (∑_{i>j} T̂_{i}) .

Proof. Suppose (*) is satisfied for all j ∈ {1, …, m − 1}, but that, contrary to our claim, Soc(T̂_{1}) = ⋯ = Soc(T̂_{m}) ⊂_{≠} Soc(T), or, equivalently, Soc(T̂_{1} ∩ … ∩ T̂_{m}) ⊂_{≠} Soc(T). Choose a normed element x ∈ Soc(T) \ Soc(T̂_{1} ∩ … ∩ T̂_{m}), say x = ex = ∑_{i=j}^{m} λ_{i}t̂_{i}, with λ_{i} ∈ eΛ for some primitive idempotent e ∈ B_{0}, such that the minimal index j with λ_{j}t̂_{j} ≠ 0 is as large as possible. We have j ≤ m − 1 because Soc(T) ∩ T̂_{m} = Soc(T̂_{1} ∩ … ∩ T̂_{m}). In view of the maximality of j and the fact that T̂_{j} is uniserial, we may choose λ_{j} = μa_{j} for a suitable element a_{j} of B of length less than that of b_{j}, and a local unit μ ∈ Λ; indeed, if t̂_{j} is a top element of type e(j) of T and λ_{j} ∈ eJ^{l}e(j) \ eJ^{l+1}e(j), then l < length(b_{j}) and there exists a path a_{j} from e(j) to e in B_{l} such that λ_{j} = μa_{j} with μ ∈ eΛe \ eJe. We may further assume that μ coincides with the primitive idempotent e ∈ B_{0} such that ea_{j} = a_{j} (by replacing x by μ^{−1}x, where μ^{−1} is the local inverse of μ, if need be). In light of Λλ_{j}t̂_{j} = Λa_{j}t̂_{j} ⊃_{≠} Λb_{j}t̂_{j} ≠ 0, we have Ja_{j}t̂_{j} ≠ 0, and hence there exists an element α ∈ B_{1} with αa_{j}t̂_{j} ≠ 0. But x ∈ Soc(T) and so αx = 0, which means that αa_{j}t̂_{j} = α (− ∑_{i>j} λ_{i}t̂_{i}) ∈ α (∑_{i>j} T̂_{i}). This violates condition (*). □

The next task will be to find a more workable and intuitive criterion for a saguaro to be reduced, provided that Λ is a left serial algebra. As usual, we take B to be the set of residue classes modulo I of the paths in KΓ \ I.

Observation 3.8 (An equivalent condition for reducedness when Λ is left serial.) Let Λ be a left serial algebra and T a saguaro on (T_{1}, …, T_{m}). Then T is reduced if and only if, in the graph of T, no two edges entering a given node from above have the same label.

Suppose, for example, that the indecomposable projectives of Λ are given by

1_{α}  2_{β} 2_{β}  2_{β} 2  2

Then the saguaro on (Λe_{1}, Λe_{2}, Λe_{1}/J^{2}e_{1}) with graph

[figure: 1; α; 2; β; 1; α; 2; β; 2]

is reduced, while the saguaro on the same sequence of trunks, but with the following graph is not.

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

Proof of Observation 3.8. It is clear that the condition on the graph of T is necessary since any subgraph of the graph of T which has the form

[figure: x_{1} •; α; • x_{2}; α; •]

gives rise to an element x_{1} − kx_{2} ∈ Soc(T) \ Soc(T̂_{1}) for some nonzero scalar k ∈ K.

For the converse, suppose that T is a saguaro with canonical sequence of top elements t̂_{1}, …, t̂_{m} such that the graph of T satisfies the condition stated in the observation, i.e., in each subgraph of the form

[figure: x_{1} •; α_{1}; • x_{2}; α_{2}; •]

the labels α_{1}, α_{2} ∈ B_{1} are distinct. A more formal phrasing is as follows: Whenever b_{j}t̂_{j} = p_{l}t̂_{l} for some l > j, where b_{j} is as in the definition of a saguaro and p_{l} ∈ B, then, for any α ∈ B_{1}, the paths b_{j}, p_{l} do not both belong to α · B. In other words, b_{j} and p_{l} do not end with the same arrow.

To have a more convenient induction hypothesis at hand, we will show that T satisfies the following slight strengthening of condition (*) in Observation 3.7: given any path a_{j} ∈ B with length(a_{j}) < length(b_{j}), an arrow α ∈ B_{1}, and a local unit μ of Λ with μαa_{j}t̂_{j} ≠ 0, we have μαa_{j}t̂_{j} ∉ α (∑_{i>j} T̂_{i}).

We will prove this statement by induction on m − j. Assume that T fails to satisfy the condition and let j ≤ m − 1, as well as a_{j}, α, μ, be as above such that μαa_{j}t̂_{j} ≠ 0 while μαa_{j}t̂_{j} ∈ α (∑_{i>j} T̂_{i}). We write μαa_{j}t̂_{j} = α (∑_{i∈L} λ_{i}t̂_{i}), where L ⊆ {j + 1, …, m} and λ_{i} is a non-trivial K-linear combination of paths q^{(i)} with αq^{(i)}t̂_{i} ≠ 0. For i ≤ m − 1, it is moreover legitimate to suppose that all the paths q^{(i)} which appear in λ_{i} have length less than length(b_{i}); indeed, we can shift the term q^{(i)}t̂_{i} to one in T̂_{i+1} = Λt̂_{i+1} if necessary.

If m − j = 1, our assumption amounts to μαa_{m−1}t̂_{m−1} = α(λ_{m}t̂_{m}), whence αa_{m−1} = b_{m−1} and μb_{m−1}t̂_{m−1} = μc_{m−1}t̂_{m} = αλ_{m}t̂_{m} ≠ 0. The length of c_{m−1} and the lengths of the paths occurring in λ_{m} being strictly less than the composition length of T̂_{m}, we deduce μc_{m−1} = αλ_{m}, which means that c_{m−1} = μ^{−1}αλ_{m}. If e ∈ B_{0} is such that ec_{m−1} = c_{m−1}, then μ^{−1} = ke + x, where x is a K-linear combination of paths of positive length starting in e, and k ∈ K is non-zero. Since c_{m−1} ∈ B is a path, we obtain c_{m−1} = lαq^{(m)}, where l ∈ K and q^{(m)} is a suitable path occurring non-trivially in λ_{m}. This shows that both b_{m−1} and c_{m−1} belong to α·B, which, in view of b_{m−1}t̂_{m−1} = c_{m−1}t̂_{m}, contradicts our hypothesis.

Now let m − j ≥ 2, and let l be the least element of L. We expand λ_{l} = ∑_{s=1}^{t} k_{s}q_{s}, where k_{s} ∈ K \ {0} and the q_{s} = q_{s}^{(l)} are distinct paths in B with length(q_{s}) < length(b_{l}) and αq_{s}t̂_{l} ≠ 0 for all s. In particular, all the paths q_{s} have initial vertex e(l), where e(l) denotes the primitive idempotent with e(l)t̂_{l} = t̂_{l}. But distinct paths with the same initial vertex have different lengths, and we may thus choose the indexing in such a way that

length(q_{t}) < ⋯ < length(q_{1}) < length(b_{l}) .

Since the choice of l implies αa_{j}t̂_{j} ∈ T̂_{j} ∩ ∑_{i≥l} T̂_{i} ⊆ T̂_{j} ∩ T̂_{l}, we see that αa_{j} = b_{j} and b_{j}t̂_{j} = p_{l}t̂_{l} for a suitable p_{l} ∈ B (cf. Observations 3.3 and 3.4). By hypothesis, p_{l} ∉ α · B since b_{j} ∈ α · B; in particular, p_{l} is distinct from all the paths αq_{1}, …, αq_{t}. Inserting this information into our assumption, we obtain

(μp_{l} − ∑_{s=1}^{t} k_{s}(αq_{s}))t̂_{l} ∈ α( ∑_{i>l} T̂_{i}) . (‡)

Expanding μp_{l} as a K-linear combination of paths yields μp_{l} = k_{0}p_{l} + y, where k_{0} is a non-zero scalar and y is a linear combination of paths longer than p_{l}. Since p_{l} ∉ α · B, the term k_{0}p_{l} is not canceled by any of the k_{s}αq_{s}. In particular, k_{0}p_{l} is not canceled by the term k_{1}αq_{1}. Note that length(αq_{1}) ≤ length(b_{l}) by construction, while length(αq_{s}) < length(b_{l}) for s ≥ 2. We infer that t = 1, αq_{1} = b_{l} and length(p_{l}) > length(b_{l}), say p_{l} = p̃_{l}b_{l} for some path p̃_{l} ∈ B^{∗}. Next we observe that, if e is the primitive idempotent with eα = α, then the path p̃_{l} ends in e by construction, as does the path b_{l} (since b_{l} = αq_{1}). This shows that the element ν := (μp̃_{l} − k_{1}e) ∈ Λ belongs to eΛe and, in fact, is a local unit in eΛe. Condition (‡) can thus be rewritten in the form

ναq_{1}t̂_{l} ∈ α( ∑_{i>l} T̂_{i}) ,

which is impossible by the induction hypothesis. □

The following observation describes the first syzygy Ω^{1}(T) of a saguaro T in terms of its trunks and ‘gluing points’. Let T ∈ Λ-mod, where Λ is a split left serial algebra, and let (T_{1}, …, T_{m}) be a sequence of trunks for T with b_{i}, c_{i} and t̂_{i} as in Definition 3.1; moreover, suppose that the top element t̂_{i} ∈ T̂_{i} is normed by the primitive idempotent e(i) for 1 ≤ i ≤ m. From Observation 3.3 we know that Soc(T̂_{1}) = ⋯ = Soc(T̂_{m}) ≅ Λe(0)/Je(0) for some primitive idempotent e(0). For 1 ≤ i ≤ m, let p_{i} ∈ B^{∗} be the path from e(i) to e(0) with the property that Λp_{i}t̂_{i} = Soc(T̂_{i}). Note that l ann_{Λ}(p_{i}) ⊇ l ann_{Λ}(p_{j}) if and only if Ω^{1}(T_{i}) is isomorphic to a factor module of Ω^{1}(T_{j}). In particular, l ann_{Λ}(p_{i}) = l ann_{Λ}(p_{j}) implies Ω^{1}(T_{i}) ≅ Ω^{1}(T_{j}).

Observation 3.9 (On the homology of saguaros.) Let T be a saguaro on (T_{1}, …, T_{m}) over a left serial algebra Λ, and let the notation be as introduced above. Choose a permutation σ of {1, …, m} with the property that

l ann_{Λ}(p_{σ(1)}) ⊇ l ann_{Λ}(p_{σ(2)}) ⊇ ⋯ ⊇ l ann_{Λ}(p_{σ(m)}),

and d_{σ(2)}, …, d_{σ(m)} ∈ B^{∗} such that T̂_{σ(i)} ∩ (∑_{j<i} T̂_{σ(j)}) = Λd_{σ(i)}t̂_{σ(i)}. Then

Ω^{1}(T) ≅ Ω^{1}(T_{σ(1)}) ⊕ ⨁_{i=2}^{m} Λd_{σ(i)} .

Remark. In general, T will not be a saguaro on the sequence (T_{σ(1)}, …, T_{σ(m)}) in the sense of Definition 3.1.

Proof. We proceed by induction on the number m of trunks of T. If m = 1, then T = T_{σ(1)}, and our assertion is obvious.

So let m ≥ 2. By the remarks on the reordering of trunks following Observation 3.5, we may assume that σ(m) = m. Then

T′ = ⨁_{i≤m−1} T_{i} / ∑_{i≤m−2} Λ(b_{i}t_{i} − c_{i}t_{i+1})

is a saguaro on (T_{1}, …, T_{m−1}). Since σ induces a permutation of {1, …, m − 1} which correctly orders the annihilators of the trunks of T′, the induction hypothesis yields

Ω^{1}(T′) ≅ Ω^{1}(T_{σ(1)}) ⊕ ⨁_{i≤m−1,i≠σ(1)} Λd_{i} .

Next we consider the projective cover

π : ⨁_{i≤m} Λe(i) → ⨁_{i≤m} T_{i} → ( ⨁_{i≤m} T_{i} / ∑_{i≤m−1} Λ(b_{i}t_{i} − c_{i}t_{i+1})) = T

of T which sends e(i) to t̂_{i} by way of t_{i}. Our claim will readily follow if we can show that

Ker(π) = (Ker(π) ∩ ⨁_{i≤m−1} Λe(i)) ⊕ Λ(b_{m−1}e(m − 1) − c_{m−1}e(m)) ,

because the first summand on the right-hand side is Ω^{1}(T′); moreover, due to the minimality of l ann_{Λ}(p_{m}), we have Λ(b_{m−1}e(m − 1) − c_{m−1}e(m)) ≅ Λc_{m−1}e(m) = Λd_{m}e(m), the last equality following from the fact that T̂_{m} ∩ ∑_{i≤m−1} T̂_{i} = Λc_{m−1}t̂_{m} (Observation 3.3).

To establish the above direct sum decomposition of Ker(π), we first observe that the two terms on the right have trivial intersection; this is another easy consequence of the minimality of l ann_{Λ}(p_{m}). To see that Ker(π) is the sum of the terms on the right, note that Ker(π) = ∑_{i=1}^{m} Jp_{i}e(i)+∑_{i=1}^{m−1} Λ(b_{i}e(i)−c_{i}e(i+1)), whence it suffices to prove that Jp_{m}e(m) is contained in the right hand side. Indeed, the equalities T̂_{m−1} ∩ T̂_{m} = Λb_{m−1}t̂_{m−1} = Λc_{m−1}t̂_{m} yield an element p ∈ B such that p_{m−1} = p b_{m−1} and p_{m} = p c_{m−1} (if e is the primitive idempotent with eb_{m−1} = b_{m−1} and ec_{m−1} = c_{m−1}, then p is the unique path with length(p) = length(p_{m−1}) − length(b_{m−1}) from e to e(0)). Thus, given λ ∈ J, we can write

λp_{m}e(m) = λp c_{m−1}e(m)
= λp_{m−1}e(m − 1) + λp(c_{m−1}e(m) − b_{m−1}e(m − 1)) .

Since λp_{m−1}e(m − 1) ∈ Jp_{m−1}e(m − 1) ⊆ Ker(π) ∩ ⨁_{i≤m−1} Λe(i), the proof is complete. □

Note that Observation 3.9 tells us how to find the graph of Ω^{1}(T) provided that the graph of T and those of the indecomposable projective modules are known. For instance, if Λ is as in Example 2.2 and T ∈ Λ-mod is the saguaro with graph

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

where the trunks are numbered as are the primitive idempotents norming their tops, then

l ann_{Λ}(p_{1}) ⊃_{≠} l ann_{Λ}(p_{3}) = l ann_{Λ}(p_{5}) ⊃_{≠} l ann_{Λ}(p_{2}) = l ann_{Λ}(p_{4}) ,

and accordingly Ω^{1}(T) has the following graph (with Ω^{1}(T_{1}) = 0 omitted).

5  5  5  3 5 ⊕ 5 ⊕ 5 ⊕ 5 5  5  55

Using the same technique, we see that the saguaro with graph

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

has projective dimension 1, whereas the trunks T_{2} and T_{4} have infinite projective dimension. The preceding observation guarantees, however, that any saguaro of finite projective dimension has at least one trunk which shares this property. We record this fact for future reference.

Observation 3.10 (On saguaros of finite projective dimension.) Suppose that Λ is a left serial algebra and T in Λ-mod a saguaro on (T_{1}, …, T_{m}). If T ∈ P^{∞}, there exists an index j such that T_{j} ∈ P^{∞}. □

4. The category of saguaros and approximations of modules by saguaros. Let Λ be a finite dimensional K-algebra with a normed K-basis B, as presented in

Section 2. All saguaros in Λ-mod will be defined relative to this basis.

We denote by S the full subcategory of Λ-mod whose objects are all finite direct sums of saguaros, and by S ∩ P^{∞} the full subcategory of S containing all objects of finite projective dimension. The purpose of this section is to develop the machinery necessary for approximating Λ-modules by objects in S or in S ∩ P^{∞}. The structure of the resulting approximations will then provide us with clues to the structure of the original objects. In the following section, we will see that, not only is the category P^{∞} contravariantly finite in Λ-mod for any left serial algebra Λ, but the minimal right P^{∞}-approximations of the simple Λ-modules belong to S ∩ P^{∞}; as a consequence, saguaros of finite projective dimension form the building blocks of arbitrary Λ-modules of finite projective dimension.

We start with a procedure for gluing saguaros together. Let T^{(1)} and T^{(2)} be saguaros in Λ-mod which share a trunk T_{∗}. To reduce the notation, we identify T_{∗} with an actual submodule of any saguaro T having trunk T_{∗}; in particular, we will assume T_{∗} ⊆ T^{(1)} and T_{∗} ⊆ T^{(2)}.

Definition 4.1 An amalgam of T^{(1)} and T^{(2)} along T_{∗} (written T^{(1)} ∨ T^{(2)}[T_{∗}]) is a saguaro T with trunk T_{∗}, together with Λ-homomorphisms (necessarily embeddings) f_{i} : T^{(i)} → T with f_{i}|_{T_{∗}} = id|_{T_{∗}}, for i = 1, 2, such that for each saguaro T̃ with trunk T_{∗} and each pair of homomorphisms g_{i} : T^{(i)} → T̃, i = 1, 2, with g_{i}|_{T_{∗}} = id|_{T_{∗}}, there exists a unique homomorphism g : T → T̃ such that g_{i} = gf_{i}.

Remark. If we denote by S(T_{∗}) the subcategory of S having as objects the saguaros with trunk T_{∗} and as morphisms the Λ-homomorphisms which restrict to the identity on T_{∗}, then an amalgam T^{(1)} ∨ T^{(2)}[T_{∗}] is nothing but the coproduct of T^{(1)} and T^{(2)} in the category S(T_{∗}). We will not emphasize this point of view, however, since we will also be considering ordinary coproducts, that is, direct sums, of saguaros in Λ-mod.

Once existence is established, uniqueness of amalgams, up to isomorphism, will follow from the universal property. To verify existence, consider the exact sequence

0 → T_{∗} →^{φ} T^{(1)} ⊕ T^{(2)} →^{π} Cok(φ) → 0 ,

where φ sends x to (x, −x), and check that Cok(φ), together with the canonical embeddings

T^{(i)} → T^{(1)} ⊕ T^{(2)} →^{π} Cok(φ) ,

is an amalgam of T^{(1)} and T^{(2)} along T_{∗}; in other words, Cok(ϕ) is in turn a saguaro. To describe the graph of the amalgam T^{(1)} ∨T^{(2)}[T_{∗}], let T^{(1)} be a saguaro on (U_{1}, …, U_{s}) and T^{(2)} one on (V_{1}, …, V_{t}). By Observation 3.5, we may assume that Û_{s} = T_{∗} and V̂_{1} = T_{∗}. Then the graph of T^{(1)} ∨ T^{(2)}[T_{∗}] is obtained from the graphs of T^{(1)} and T^{(2)} by gluing them together in such a way that V̂_{1} is superimposed on Û_{s}.

To exemplify this procedure, let us consider saguaros T^{(1)} and T^{(2)} represented by the graphs

[figure: T_{∗}; T_{∗}; 1; 4; 2; 2; 4; 2; 3; 3; 3; 3]

Then the graph of the amalgam T^{(1)} ∨ T^{(2)}[T_{∗}] is as follows.

[figure: T_{∗}; 1; 4; 2; 4; 2; 3; 3]

By way of caution, we emphasize that, given saguaros T^{(1)} and T^{(2)} with a common trunk T_{∗}, where T^{(1)} has two distinct trunks isomorphic to T_{∗}, say U_{i} and U_{j}, the outcome of the amalgamation process may depend on the chosen identification of T_{∗} with U_{i} or U_{j}.

It will be useful to switch back and forth between the presentation of a saguaro as given in Definition 3.1 and the following alternate presentation.

Proposition 4.2 Let T_{1}, …, T_{m} be non-zero uniserial left Λ-modules, and let t_{i} ∈ T_{i} for i = 1, …, m be top elements. Suppose further that V ⊆ ⨁_{1≤i≤m} JT_{i} is a submodule generated by elements of the form

u_{i}t_{i} − v_{i}t_{σ(i)}, 1 ≤ i ≤ m − 1 ,

where i < σ(i) ≤ m for all i ≤ m − 1 and u_{i}, v_{i} ∈ B^{∗} are normed by the same primitive idempotent. Then T = (⨁_{1≤i≤m} T_{i}) /V is a direct sum of saguaros W^{(1)}, …, W^{(s)}, each being a sum of certain of the factor modules T′_{i} = (T_{i} + V)/V, 1 ≤ i ≤ m. Moreover, each of the factors T′_{i} is non-zero and occurs as trunk of precisely one of the saguaros W^{(j)}.

(We emphasize that we do not require σ to be a permutation. Note moreover that, just as in Observation 3.2, it would not change the conclusion to replace a relation “u_{i}t_{i}−v_{i}t_{σ(i)}” by “u_{i}t_{i} − k_{i}v_{i}t_{σ(i)}” for non-zero scalars k_{i}.)

Proof. Identify the T′_{j} with the images of the canonical maps T_{j} → (⨁_{1≤i≤m} T_{i}) /V, respectively, and observe that T′_{j} ≠ 0 for all j because the elements u_{j}, v_{j} all belong to J.

We proceed by induction on m. If m = 1, there is nothing to be shown. So suppose that m ≥ 2 and that our claim is true for all strictly shorter sequences of uniserial modules T_{i} together with relations as above.

In case T′_{1} ∩ T′_{σ(1)} = 0, i.e., if the canonical image of u_{1}t_{1} in T is zero, we obtain T = T′_{1} ⨁ (⨁_{2≤i≤m} T_{i}) /V′, where V′ is generated by u_{i}t_{i} − v_{i}t_{σ(i)}, 2 ≤ i ≤ m − 1, and we are done by the induction hypothesis.

So suppose that T′_{1} ∩ T′_{σ(1)} ≠ 0. Then X := T′_{1} + T′_{σ(1)} ⊆ T is clearly a saguaro on trunks (T′_{1}, T′_{σ(1)}), and Y := T′_{2} + … + T′_{m} ⊆ T is a finite direct sum of saguaros Y^{(j)} as specified in the proposition; say Y = Y^{(1)} ⨁ … ⊕ Y^{(s)}. Since σ(1) ≥ 2, we know that T′_{σ(1)} occurs as trunk of precisely one of the Y^{(j)}, and, without loss of generality, we may assume that T′_{σ(1)} ⊆ Y^{(1)} is a trunk of Y^{(1)}. From our construction we derive that X ∩ Y = T′_{σ(1)} = X ∩ Y^{(1)}. In particular, T^{(1)} = X + Y^{(1)} is an amalgam X ∨ Y^{(1)}[T′_{σ(1)}] of X and Y^{(1)} along the trunk T′_{σ(1)} and thus in turn is a saguaro. If we set T^{(j)} = Y^{(j)} for 2 ≤ j ≤ s it is now straightforward to see that T^{(1)} ∩ (∑_{2≤i≤s} T^{(i)}) = 0 and to check that the decomposition T = T^{(1)} ⨁ … ⨁ T^{(s)} has all the required properties. □

We say that the category S has finite type if there are finitely many saguaros T^{(1)}, …, T^{(s)} such that each saguaro in S is isomorphic to a direct sum of copies of these. In a subsequent paper ([11]), the second author characterizes – in terms of quivers and relations – the (split) finite dimensional algebras for which S has finite type. In Section 5 of the present paper, we check directly that, whenever Λ is left serial, S has finite type. In particular, this guarantees that S is contravariantly finite in Λ-mod (see [2, Proposition 4.2]).

Proposition 4.3 Suppose that S has finite type and that each Λ-homomorphism f : M → S from a module M ∈ P^{∞} to a simple left Λ-module S factors through some saguaro of finite projective dimension, which may depend on f. Then P^{∞} is contravariantly finite in Λ-mod, and the minimal (right) P^{∞}-approximations of the simple left Λ-modules are objects in S ∩ P^{∞}.

Proof. By [2, Proposition 3.7], P^{∞} is contravariantly finite if and only if all simple left Λ-modules have P^{∞}-approximations. Fix a simple Λ-module S, and let T^{(1)}, …, T^{(r)} be saguaros in S ∩ P^{∞} such that each saguaro in Λ-mod of finite projective dimension is a direct sum of copies of the T^{(i)}. Moreover, for each i, let f_{i1}, …, f_{in_{i}} be a K-basis for Hom_{Λ}(T^{(i)}, S). Then the map

⨁_{i=1}^{r} ⨁_{j=1}^{n_{i}} f_{ij} : ⨁_{i=1}^{r} ⨁_{j=1}^{n_{i}} T^{(i)} ⟶ S

is clearly a P^{∞}-approximation of S by hypothesis, and the source of this map is an object in S ∩ P^{∞}. To complete the argument, use the fact that the minimal P^{∞}-approximation of S is isomorphic to a direct summand of any other P^{∞}-approximation [2]. □

Using [2, Proposition 3.8], we obtain the following corollary.

Corollary 4.4 Suppose that the hypotheses of Proposition 4.3 are satisfied and that T^{(1)}, …, T^{(m)} are the indecomposable objects in S ∩ P^{∞}. Then a finitely generated Λ-module has finite projective dimension if and only if it is a direct summand of an object M of P^{∞} with the following property: M has a series of submodules M = M_{0} ⊇ M_{1} ⊇ ⋯ ⊇ M_{k} = 0 such that each of the consecutive factors is isomorphic to some T^{(i)}. □

The most interesting situations covered by Proposition 4.3 and Corollary 4.4 are those in which S has finite type while P^{∞} does not. Examples can be found in Section 7.

The following somewhat technical lemma will be crucial to the factoring of homomorphisms onto simple modules through saguaros.

Lemma 4.5 Suppose that g ∈ Hom_{Λ}(X, S) is a non-zero homomorphism, where S is a simple left Λ-module and X ∈ P^{∞} has a presentation X = (⨁_{1≤i≤m} Λe(i)) / U with e(i) ∈ B_{0} such that U is generated by elements u_{1}, …, u_{m} subject to the following conditions: If u_{j} = ∑_{1≤i≤m} u_{ji}e(i) with u_{ji}e(i) = u_{ji}, then

(1) l ann_{Λ}(u_{jj}) ⊆ l ann_{Λ}(u_{ji}) for i ≠ j;

(2) u_{ji} = 0 for i < j;

(3) u_{jj} is a (residue class of a) path for each j ≤ m, and u_{jj} ≠ 0 for j ≤ m − 1. Moreover, the composition length of Λu_{ii} is larger than or equal to that of Λu_{jj}, whenever i ≤ j ≤ m.

Finally, suppose that g(e(i) + U) is non-zero only for i = m.

Then g factors through a map g̃ : X̃ → S such that X̃ ∈ P^{∞} has the following simplified presentation: X̃ = (⨁_{1≤i≤m̃} Λe(i)) / Ũ, where Ũ is generated by elements ũ_{1}, …, ũm̃, the components of which satisfy Conditions (1) , (2) and (3) above, as well as the following:

(4) for each j ≤ m̃ − 1 there exists precisely one index σ(j) > j in {1, …, m̃} such that u_{jσ(j)} ≠ 0.

Moreover, g̃(e(i) + Ũ) is non-zero only for i = m̃.

Observe that Conditions (2), (3) and (4) imply that each element ũ_{j}, for j ≤ m̃ − 1, has exactly two non-zero components, namely ũ_{jj} and ũ_{jσ(j)}.

Proof. Note that Conditions (1) and (2) yield

U = ⨁_{1≤j≤m} Λu_{j} ≅ ⨁_{1≤j≤m} Λu_{jj} .

Step 1. We may assume without loss of generality that for all j ≤ m − 1 there exists at least one index i > j such that u_{ji} ≠ 0. For assume that there exists an index j such that u_{ji} = 0 for all i ≠ j. Consider the module X̃ := (⨁_{i≤m,i≠j} Λe(i)) / Ũ with Ũ = ⨁_{i≤m,i≠j} Λũ_{i}, where the ũ_{i} are the canonical projections of the u_{i} onto ⨁_{i≤m,i≠j} Λe(i) along Λe(j). Since g(e(i)+U) = 0 for i ≤ m−1, the homomorphism g factors through the canonical epimorphism π : X → X̃. Moreover, p dim (X̃) = 1 + p dim (Ũ) is finite since Λũ_{l} ≅ Λu_{ll} ≅ Λu_{l} for each l ≠ j and p dim (Λu_{l}) < ∞ by hypothesis.

Step 2. Suppose that there exists an index j ∈ {1, …, m − 1} such that u_{ji} ≠ 0 for more than one index i > j, and let j be the largest such index. We remedy the situation by induction on j.

We list the indices i > j with u_{ji} ≠ 0, say j < i_{0} < i_{1} < ⋯ < i_{t−1}. Allowing for non-integral indices temporarily and letting e(j + s/t) for 1 ≤ s ≤ t − 1 be equal to the primitive idempotent e(j), we consider the Λ-homomorphism

ρ : ⨁_{1≤i≤m} Λe(i) → ⨁_{1≤i≤j−1} Λe(i) ⊕ ⨁_{0≤s≤t−1} Λe(j + s/t) ⊕ ⨁_{j<i≤m} Λe(i) ,

given by ρ(e(i)) = e(i) for i ≠ j and ρ(e(j)) = ∑_{0≤s≤t−1} e(j + s/t). Moreover, we define Ũ to be the submodule of ⨁_{1≤i≤m} Λe(i) ⊕ ⨁_{1≤s≤t−1} Λe(j + s/t) generated by the elements ũ_{i} = ρ(u_{i}) for 1 ≤ i ≤ m, i ≠ j, and ũ_{j+s/t} = u_{jj}e(j + s/t) + u_{ji_{s}}e(i_{s}) for 0 ≤ s ≤ t − 1. We start by observing that the cyclic modules Λũ_{k} add up to the direct sum

⨁_{1≤i≤m} Λũ_{i} ⊕ ⨁_{1≤s≤t−1} Λũ_{j+s/t} ≅ ⨁_{i≠j} Λu_{ii} ⊕ (Λu_{jj})^{t} ,

whence the module X̃ := (⨁_{1≤i≤m} Λe(i) ⊕ ⨁_{1≤s≤t−1} Λe(j + s/t)) / Ũ has the same projective dimension as X. In order to see that ρ induces a homomorphism π : X → X̃, that is, that ρ(U) ⊆ Ũ, we need only check that ρ(u_{j}) ∈ Ũ. Indeed,

ρ(u_{j}) = u_{jj}ρ(e(j)) + ∑_{0≤s≤t−1} u_{ji_{s}}ρ(e(i_{s}))
= u_{jj} ∑_{0≤s≤t−1} e(j + s/t) + ∑_{0≤s≤t−1} u_{ji_{s}}e(i_{s})
= ∑_{0≤s≤t−1} (ũ_{j+s/t} − u_{ji_{s}}e(i_{s})) + ∑_{0≤s≤t−1} u_{ji_{s}}e(i_{s})
= ∑_{0≤s≤t−1} ũ_{j+s/t} ∈ Ũ .

We rename the indices 1, …, j, j + 1/t, …, j + (t − 1)/t, j + 1, …, m, in that order, by the positive integers 1, …, m̃ = m + t − 1, and note that only the indices l > j move to different positions in this process. That the elements ũ_{1}, …, ũm̃ retain properties (1)–(3) is clear. Finally, we observe that we can factor g : X → S in the form g = g̃π, where g̃ ∈ Hom(X̃, S) is defined by g̃(e(l)+Ũ) = 0 for 1 ≤ l < m̃ and g̃(e(m̃)+Ũ) = g(e(m)+U).

At this point, we may have actually increased the number of relations ũ_{l} which deviate from the desired form of having precisely one off-diagonal component which is non-zero. But only relations ũ_{l} with l ≤ j − 1 can be negatively affected. Indeed, the relation ũ_{j} now has the desired form, as do ũ_{j+1/t}, …, ũ_{j+(t−1)/t}, by the definition of ρ. The same is true for the relations ũ_{j+1} = ρ(u_{j+1}), …, ũ_{m} = ρ(u_{m}) since u_{lj} = 0 for l ∈ {j + 1, …, m}. This completes the procedure if j = 1 and permits us to apply the induction hypothesis otherwise. □

5. Modules of finite projective dimension over left serial algebras.

Throughout this section, Λ will be a left serial algebra with underlying quiver Γ and normed K-basis B as described in Section 2; that is, B consists of all non-zero residue classes in Λ of paths in Γ, also called paths in Λ. Recall that, in this case, we have a one-to-one correspondence between isomorphism types of saguaros relative to B and equivalence classes of graphs of saguaros under the equivalence relation which we introduced following Definition 3.1. The algebra Λ being left serial, this relation can be described in somewhat simpler terms: each isomorphism between the undirected graphs of left Λ-modules which preserves the layering and the numbers assigned to the vertices will automatically preserve the labels on the edges.

To see that S has finite type, we need some additional terminology. Let T = ∑_{i=1}^{m} T̂_{i} be a saguaro as in Definition 3.1, 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, written V^{(1)} ≅^{T} V^{(2)}, 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 ⋂_{i∈I_{1}∪I_{2}} T̂_{i}.

c) Finally, 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 3.4. 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. Referring again to Example 2.2 and using the above notation, we see that T̂_{1} + T̂_{2} + T̂_{3} is not a subsaguaro of M, while T̂_{3} + T̂_{4} + T̂_{5} is. Furthermore, by Observation 3.5, we see that, given a subsaguaro V = ∑_{i∈I_{1}} T̂_{i} of T with |I_{1}| = s, there exists a permutation π ∈ S_{m} such that I_{1} = {π(1), …, π(s)}, and T is a saguaro on (T_{π(1)}, …, T_{π(s)}, T_{π(s+1)}, …, T_{π(m)}).

Clearly, a reduced saguaro is irredundant, but not vice versa, in general. For example, the saguaro T with graph

[figure: 3; γ; 4; δ; 2; β; 3; γ; 5]

over the algebra Λ of Example 2.2 is an irredundant saguaro on

(Λe_{3}/Λεγ , Λe_{4}/Λεγδ , Λe_{2}/Λεγβ)

which is not reduced by Observation 3.8.

Observe that a saguaro T in Λ-mod is redundant if and only if the graph of T has a full subgraph of the form

G_{1}  G_{2} a

where G_{1} and G_{2} are identical 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. Moreover, note that there are only finitely many isomorphism types of uniserial modules in Λ-mod, their lengths being bounded by the Loewy length of Λ, and that therefore there are only finitely many graphs of saguaro trunks. This provides us with the prerequisites for a proof of the following theorem.

Theorem 5.1 Let Λ be a split left serial algebra. Then the category S of finite direct sums of saguaros over Λ is of finite type.

Proof. 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. 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, not necessarily distinct, such that α_{j} = α_{k} for j ≠ k implies i_{j} ≠ i_{k}. The number of equivalence classes of graphs of this type is clearly bounded above by 2^{r·|B_{1}|}.

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 3.1 (IV). 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 T̂1i ∩ T̂_{2j} ⊆ D 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 3.4, 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 3.5), 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. □

We next announce the two main theorems of Sections 5 and 6. The proof of the first will be given later in this section, that of the second will follow in Section 6.

Theorem 5.2 Let Λ be a split left serial algebra. Then P^{∞} is contravariantly finite in Λ-mod, and each simple left Λ-module has a right P^{∞}-approximation in S ∩ P^{∞}.

The following theorem gives an explicit description of the minimal P^{∞}-approxima-tions of the simple modules. They turn out to be reduced saguaros. As we will see in Section 7, the description gives rise to a method for constructing these approximations. Before we state the theorem, we point out that, by the proof of Theorem 5.1, there are only finitely many isomorphism types of irredundant saguaros. Since reduced saguaros are irredundant, we thus conclude that each reduced saguaro (in P^{∞}) can be embedded into a reduced saguaro – at least one – of maximal composition length (in P^{∞}).

Theorem 5.3 Let Λ be a left serial algebra. Given a simple left Λ-module S = Λe/Je, let C ⊆ Je have maximal length with respect to the property that T_{∗} = Λe/C has finite projective dimension. Moreover, suppose that T ∈ P^{∞} is a reduced saguaro with trunk T_{∗} having largest possible composition length; say T is a saguaro on (T_{1}, …, T_{m}), where T_{j} = T_{∗}. Then the homomorphism f : T → S which sends t̂_{j} to e + Je and t̂_{i} to 0 for i ≠ j is a minimal right P^{∞}-approximation of S. In particular, T is unique up to isomorphism.

The following restatement of Corollary 4.4 in the present context stresses the importance of the P^{∞}-approximations of the simple Λ-modules as the basic structural constituents of arbitrary Λ-modules of finite projective dimension. This is the first instance – aside from trivial cases – where the corollary takes on a concrete form, inasmuch as the minimal P^{∞}-approximations T (S_{i}) of the simple modules S_{i} can be determined. They can even be constructed algorithmically from the graphs of the Λe_{i} (Section 7).

Corollary 5.4 Let Λ be a left serial algebra. Moreover, suppose that the saguaros T (S_{1}), …, T (S_{n}) are minimal right P^{∞}-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 the union of a chain M_{0} ⊇ M_{1} ⊇ ⋯ ⊇ M_{k} = 0, where each of the consecutive factors is isomorphic to some T (S_{i}). □

The proofs of Theorems 5.2 and 5.3 will remain valid if P^{∞} is replaced by the full subcategory P^{(d)} of P^{∞} whose objects are all finitely generated left Λ-modules of projective dimension less than or equal to d; we leave the verification of the details to the reader. This yields the following supplement to Theorem 5.3.

Supplement 5.5 Let Λ be a left serial algebra. Then, for each d ≥ 0, the category P^{(d)} is contravariantly finite in Λ-mod. A minimal P^{(d)}-approximation of the simple module S = Λe/Je is as follows: Let T_{d} be the non-zero homomorphic image of Λe of smallest positive composition length within P^{(d)}. If T is a reduced saguaro in P^{(d)} with trunk T_{d} which has maximal composition length with respect to these properties, then any homomorphism f : T → S which maps T_{d} onto S and all the other trunks of T onto zero is a minimal P^{(d)}-approximation of S. □

For each simple module S, there is thus a sequence of P^{(d)}-approximations T_{1}(S), T_{2}(S), T_{3}(S), … which terminates in T_{δ}(S), where δ is the left finitistic dimension of Λ (note that this dimension is known to be finite for left serial algebras [10, Theorem 3]). As d increases, the trunks T_{d} may shrink from the bottom up, while the saguaros T_{d}(S) will increasingly ramify on what is left of these trunks, in general. An example of this growth pattern can be found in Section 7.

The first step toward a proof of the theorems is a lemma which is a somewhat strengthened version of [10, Proposition 1]. Recall that an element u of a left Λ-module U is normed if eu = u for some primitive idempotent e.

Lemma 5.6 Let U be a submodule of the radical of a projective module ⨁_{1≤i≤r} Λe(i), where e(i) ∈ B_{0} = {e_{1}, …, e_{n}}. Then U = ⨁_{1≤j≤m} Λu_{j} with m ≤ r such that each u_{j} ∈ U is normed. Moreover, after possibly reordering the e(i), we can choose the u_{j} to be of the form u_{j} = ∑_{1≤i≤r} u_{ji}e(i), where the components u_{ji} = u_{ji}e(i) ∈ Λe(i) satisfy the following conditions:

(a) l ann_{Λ}u_{jj} ⊆ l ann_{Λ}u_{ji} for i ≠ j;

(b) u_{ji} = 0 for i < j;

(c) u_{jj} is (a residue class of ) a path for each j ≤ m, and u_{jj} is non-trivial for j ≤ m−1. Moreover, the composition length of Λu_{ii} is larger than or equal to that of Λu_{jj} whenever i ≤ j ≤ m.

In particular, U ≅ ⨁_{1≤j≤m} Λu_{jj}; in other words, U is isomorphic to the direct sum of the principal left ideals generated by the paths in the diagonal positions of the u_{j}.

Note that in Condition (c), the possibility that u_{mm} = 0 is not excluded and that u_{mm} = 0 implies u_{m} = 0. The reason why we do not suppress the element u_{m} when it is trivial will become clear later.

Proof. We proceed by induction on r. In case r = 1, the lemma is obviously true. Suppose that r ≥ 2, and pick l ≥ 1 minimal with the property that J^{l}U = 0. Now select a normed element u_{1} ∈ U, say u_{1} = eu_{1}, for a suitable e ∈ B_{0}, such that J^{l−1}u_{1} ≠ 0; write u_{1} = ∑_{1≤i≤r} u_{1i}e(i) with u_{1i} ∈ Λe(i). Since a change in indexing is allowed, we may assume without loss of generality that J^{l−1}u_{11} ≠ 0, which yields l ann_{Λ}(u_{1}) = J^{l}e⊕⨁_{i≤n, e_{i}≠e} Λe_{i} = l ann_{Λ}(u_{11}). This in turn means Λu_{1} ≅ Λu_{11} = Λp_{1}, for some path p_{1} starting in e(1). Since u_{11} and p_{1} are associates, i.e., differ at most by a unit in eΛe, it is moreover legitimate to assume that u_{11} = p_{1}. Thus Conditions (a), (b) and (c) are satisfied for u_{1}, and the composition length of Λu_{1} is l.

Next we will check that U = Λu_{1} ⊕ U′, where U′ = U ∩ (⨁_{2≤i≤r} Λe(i)). That Λu_{1} ∩ U′ = 0 is a consequence of the fact that u_{1} satisfies the annihilator condition (a). To see that U ⊆ Λu_{1} + U′, let y = ∑_{1≤i≤r} y_{i}e(i) ∈ U. In view of the facts that J^{l}U = 0 and J^{l−1}u_{11} is a non-zero submodule of the uniserial module Λe(1), we see that the canonical projection of U onto Λe(1) is equal to Λu_{11}. In particular, there is an element λ ∈ Λ such that y_{1}e(1) = λu_{11}, whence y − λu_{1} ∈ U ∩ (⨁_{2≤i≤r} Λe(i)) = U′. The induction hypothesis guarantees a decomposition of U′ = ⨁_{2≤i≤m} Λu_{i} with m ≤ r such that u_{2}, …, u_{m} in turn satisfy (a), (b) and (c). Since J^{l}U′ = 0, the composition length of each of the uniserial modules Λu_{i} ≅ Λu_{ii}, i ≥ 2, is at most l, and therefore the second part of condition (c) is satisfied for the full sequence u_{1}, …, u_{m} as well. □

Proof of Theorem 5.2. Let S be a simple left Λ-module. To show that S has a right P^{∞}-approximation A(S) → S, where A(S) ∈ S ∩ P^{∞}, we check that the hypotheses of Proposition 4.3 are satisfied. In light of Theorem 5.1, only the factorization condition remains to be verified. We let X ∈ P^{∞} and 0 ≠ g ∈ Hom(X, S). First we choose a normed element x_{1} ∈ X with g(x_{1}) ≠ 0. Since dim_{K} S = 1, we obtain a K-vector space decomposition X = Kx_{1} ⊕ ker(g). We can therefore supplement x_{1} by normed elements x_{2}, …, x_{r} ∈ ker(g) so that x_{1}, …, x_{r} is a sequence of top elements of X which generate X over Λ and are K-linearly independent modulo JX. If x_{i} = e(i)x_{i} with e(i) ∈ B_{0}, we consider the following projective cover of X:

⨁_{1≤i≤r} Λe(i) → X, where e(i) ↦ x_{i} .

According to Lemma 5.6, the kernel, U ⊆ ⨁_{1≤i≤r} Je(i), of this projective cover can be decomposed in the form U = ⨁_{1≤j≤m} Λu_{j}, where the u_{j} are normed elements satisfying Conditions (a), (b) and (c) of that lemma, possibly after a renumbering of the idempotents e(i). In particular, we have u_{j} = ∑_{1≤i≤r} u_{ji}e(i), where each component u_{ji} = u_{ji}e(i) is a K-linear combination of paths starting in e(i) and ending in the primitive idempotent e which norms u_{j} from the left; moreover, u_{jj} = p_{j} is a (residue class of a) path from e(j) to e. As a consequence of the adjustment of indexing, we may have given up the equivalence “g(x_{i}) ≠ 0 ⇔ i = 1”, but we still know that g(x_{l}) ≠ 0 for precisely one index l ∈ {1, …, r}. Since p dim X < ∞, we furthermore know that p dim Λu_{j} = p dim Λu_{jj} < ∞ for all j.

Our argument showing that g can be factored through a saguaro of finite projective dimension will involve a number of reduction steps in which we will factor g through increasingly tractable modules from P^{∞}. Step 1. Without jeopardizing the above properties of the u_{j}, we may assume that (d) g(x_{m}) ≠ 0 while g(x_{i}) = 0 for i < m, i.e., l = m, and

(e) r = m, i.e., the number m of relations u_{i} coincides with the number r of primitive idempotents e(i).

While Conditions (a), (b) and (c) allow for the possibility that u_{m} = 0, let us suppress this last element when it is zero in our original configuration. Keeping in mind that m ≤ r, we distinguish two cases.

If l ≤ m, we will establish Conditions (d) and (e) as follows. Consider the canonical epimorphism

π : X = ( ⨁_{1≤i≤r} Λe(i))/ U ⟶ X̃ := ( ⨁_{1≤i≤l} Λe(i))/ Ũ,

where Ũ = ⨁_{j≤l} Λũ_{j} and ũ_{j} = ∑_{i≤l} u_{ji}e(i); in other words, the ũ_{j} for j ≤ l are just the projections of the corresponding u_{j} onto ⨁_{1≤i≤l} Λe(i). We first notice that ker(π) ⊆ ker(g), whence g factors through π. Second, we observe that p dim X̃ = 1 + max_{j≤l}(p dim Λu_{jj}) ≤ p dim X < ∞. Now replace X by X̃, and both r and m by l.

Next suppose that l > m; set I = {1, …, m, l}, X̃ = ( ⨁_{i∈I} Λe(i))/ Ũ, where, this time, Ũ = ⨁_{1≤j≤m} Λũ_{j} and ũ_{j} = ∑_{i∈I} u_{ji}e(i) for j ≤ m. As before, g factors through the canonical epimorphism π : X → X̃, and p dim X̃ ≤ p dim X < ∞. Set ũ_{l} = 0, and renumber {1, …, m, l} by consecutive numbers {1, …, m, m+1}. Finally, replace X by X̃ and m + 1 by m to see that we have placed ourselves in the desired situation; indeed, since our original sequence of u_{j}’s did not contain 0, attaching 0 at the end does not violate Condition (c). This concludes Step 1.

Step 2. In addition to the conditions already imposed on the homomorphism g : X → S and the relations u_{1}, …, u_{m}, we may assume that

(f) for each j ≤ m − 1, there exists precisely one index σ(j) > j with u_{jσ(j)} ≠ 0, and,

(g) u_{jσ(j)} is a scalar multiple of a path in B_{1} for 1 ≤ j ≤ m − 1.

Indeed, Lemma 4.5 permits us to assume Condition (f) without jeopardizing Conditions (a)–(e).

To establish Condition (g), we require some new terminology. We say that a path q ∈ B occurs in a tied position in u_{1}, …, u_{m} if there exists j ≤ m − 1 such that u_{jσ(j)} is a non-trivial K-linear combination of more than one path and q is one of the paths occurring with non-zero coefficient in u_{jσ(j)}. In that case, we also say that u_{jσ(j)} is tied. We will now proceed by induction on the number ν of those paths in B which occur in a tied position in u_{1}, …, u_{m}. If ν = 0, we are done. So suppose ν ≥ 1. Let j be the smallest index with the property that u_{jσ(j)} is tied and let q ∈ B be a path occurring non-trivially in u_{jσ(j)}, say u_{jσ(j)} = k_{0}q + w, where 0 ≠ k_{0} ∈ K and w is a non-zero K-linear combination of paths different from q.

To reduce the number of paths occurring in tied positions by one, we will carry out a subsidiary induction on the number of tied positions in which q occurs (in the sequence u_{1}, …, u_{m}). Set e(j + 1/2) = e(j) and consider the Λ-homomorphism

ρ : ⨁_{1≤i≤m} Λe(i) → ⨁_{1≤i≤j} Λe(i) ⊕ Λe(j + 1/2) ⊕ ⨁_{j<i≤m} Λe(i) ,

given by ρ(e(i)) = e(i) for 1 ≤ i ≤ m, i ≠ j, and ρ(e(j)) = e(j) + e(j + 1/2). Moreover, we define

Ũ = ⨁_{1≤i≤m} Λũ_{i} ⊕ Λũ_{j+1/2} ⊆ ⨁_{1≤i≤m} Λe(i) ⊕ Λe(j + 1/2) ,

as follows:

ũ_{i} = ρ(u_{i}) for 1 ≤ i ≤ m, i ≠ j ,
ũ_{j} = u_{jj}e(j) + k_{0}qe(σ(j)) , and
ũ_{j+1/2} = u_{jj}e(j + 1/2) + w e(σ(j)) .

To see that l ann_{Λ}u_{jj} ⊆ l ann_{Λ}q and l ann_{Λ}u_{jj} ⊆ l ann_{Λ}w, keep in mind that l ann_{Λ}u_{jj} ⊆ l ann_{Λ}u_{jσ(j)} by Condition (a). Thus Conditions (a)–(e) still hold for the ũ_{l} (to achieve formal congruence of the new situation with that of Conditions (d) and (e), we will eventually replace the indices 1, …, j, j + 1/2, j + 1, …, m by successive integers). However, Condition (f) may no longer be satisfied for the elements ũ_{i}, with 1 ≤ i < j; indeed, whenever, σ(i) = j, the element ũ_{i} contains precisely three non-zero components, namely u_{ii}e(i), u_{ij}e(j), and u_{ij}e(j + 1/2). We will attend to this problem later.

In order to see that ρ(U) ⊆ Ũ, it suffices to check that ρ(u_{j}) ∈ Ũ. We compute

ρ(u_{j}) = u_{jj}ρ(e(j)) + (k_{0}q + w)ρ(e(σ(j))
= u_{jj}e(j) + u_{jj}e(j + 1/2) + (k_{0}q + w)e(σ(j))
= ũ_{j} + ũ_{j+1/2} ∈ Ũ .

Thus ρ induces a homomorphism

π : ( ⨁_{1≤i≤m} Λe(i)) / U ⟶ X̃ = ( ⨁_{1≤i≤m} Λe(i) ⊕ Λe(j + 1/2))/ Ũ ,

and g factors as g = g̃π, where g̃(e(m) + Ũ) = g(e(m) + U) and g̃(e(l) + Ũ) = 0 for l < m. Finally, we observe that

p dim X̃ = 1 + p dim( ⨁_{i≠j} Λu_{i} ⊕ (Λu_{j})^{2}) = p dim X < ∞ .

The indices 1, …, j, j +1/2, j +1, …, m are then replaced by consecutive natural numbers in the order indicated. Due to the minimal choice of j, this procedure has not created any new tied positions among the ũ_{l}, and clearly no additional paths from B were introduced into tied positions. Moreover, the number of occurrences of the path q in a tied position among the ũ_{j} is strictly smaller than the number of its occurrences among the u_{i}.

However, as already mentioned above, the new relations ũ_{i} with 1 ≤ i < j may violate Condition (g). To remedy this, we go over the first part of Step 2, in other words, we reapply Lemma 4.5. It is crucial to observe that the procedure leading from the ũ_{k} to new relations, as described in the proof of Lemma 4.5, does not introduce new paths into tied positions and preserves the number of occurrences of q in tied positions since, by construction, none of the positions ũ_{il} for i < j and arbitrary l is tied.

By our subsidiary induction, we can thus eliminate the path q from tied positions in the u_{i}, thereby reducing the number of paths occurring in tied positions by one. The primary induction now completes Step 2.

Step 3. We thus have at our disposal a sequence e(1), …, e(m) of primitive idempotents in B_{0} , together with a submodule ⨁_{1≤i≤m} Λu_{i} ⊆ ⨁_{1≤i≤m} Je(i) of finite projective dimension, such that Conditions (a)–(g) are satisfied for u_{1}, …, u_{m}. We now set T = ⨁_{1≤i≤m} Λe(i)/ ⨁_{1≤i≤m} Λu_{i}. Moreover, we let T′_{i} be the canonical image of Λe(i) in T, and t′_{i} the image of e(i) in T, which gives us u_{jj}t′_{j} = u_{jσ(j)}t′_{σ(j)} for 1 ≤ j ≤ m − 1. In this setting, Proposition 4.2 guarantees that T is a finite direct sum of saguaros W^{(1)}, …, W^{(s)} and that each of the uniserial modules T′_{i} is non-zero and contained, as a trunk, in one of the saguaros W^{(j)}. Since T has finite projective dimension, so do the W^{(j)}. Finally, if T′_{m} ⊆ W^{(s)} is a trunk of the saguaro W^{(s)}, then g clearly factors through W^{(s)}, in view of the fact that g(t′_{i}) = 0 for i < m. This completes the proof of Theorem 5.2. □

6. Proof of Theorem 5.3.

Recall that a left serial ring is a left artinian ring with the property that each indecomposable projective left ideal has a unique composition series.

Lemma 6.1 Let Λ be a left serial ring and X a uniserial left Λ-module of infinite projective dimension. Then X does not contain a chain of submodules of any of the following types:

i) X ⊇ X_{1} ⊇ X_{2} with X_{1} ∈ P^{∞}, but X_{1}/X_{2} ∉ P^{∞}.

ii) X ⊇ X_{1} ⊇ X_{2} with X/X_{2} ∈ P^{∞}, but X_{1}/X_{2} ∉ P^{∞}.

iii) X ⊇ X_{1} ⊇ X_{2} ⊇ X_{3} with X_{1} ∈ P^{∞}, but X_{2}/X_{3} ∉ P^{∞}.

Proof. Parts i) and ii) will be proved simultaneously. Note that from either of these situations we readily deduce X_{2} ∉ P^{∞} and X/X_{1} ∉ P^{∞}.

We will see that the existence of a chain of type i) is equivalent to the existence of one of type ii). So let us start by assuming that we are given a chain as in i),

X = X_{0,0} ⊇ X_{0,1} ⊇ X_{0,2} ⊇ X_{0,3} = 0

with X_{0,1} ∈ P^{∞}, but X_{0,1}/X_{0,2} ∉ P^{∞}. Let π_{1} : P_{1} → X_{0,0} be a projective cover of X_{0,0}, and note that P_{1} is again uniserial by hypothesis. Moreover, we set X_{1,i−1} = π_{1}^{−1}(X_{0,i}) for i = 1, 2, 3, and check that the chain

X_{1,0} ⊇ X_{1,1} ⊇ X_{1,2} ⊇ X_{1,3} = 0

satisfies the hypotheses under ii), if we replace X by X_{1,0}. Indeed, P_{1}/X_{1,0} ≅ X_{0,0}/X_{0,1} ∉ P^{∞} by the above remarks, and therefore X_{1,0} ∉ P^{∞}. Moreover, X_{1,0}/X_{1,2} ≅ X_{0,1} ∈ P^{∞}, and X_{1,1}/X_{1,2} ≅ X_{0,2} ∉ P^{∞} by construction.

Now consider a projective cover π_{2} : P_{2} → X_{1,0}, and set X_{2,i−1} = π_{2}^{−1}(X_{1,i}) for i = 1, 2, 3. Then the chain

X_{2,0} ⊇ X_{2,1} ⊇ X_{2,2} ⊇ X_{2,3} = 0

again satisfies the hypotheses of part i) with X = X_{2,0} ∉ P^{∞}. Indeed, P_{2}/X_{2,0} ≅ X_{1,0}/X_{1,1} ∉ P^{∞}, and hence X_{2,0} ∉ P^{∞}. Moreover, P_{2}/X_{2,1} ≅ X_{1,0}/X_{1,2} ∈ P^{∞}, which shows X_{2,1} ∈ P^{∞}, and X_{2,1}/X_{2,2} ≅ X_{1,2}/X_{1,3} ≅ X_{1,2} ∉ P^{∞} by the initial remarks.

Finally, observe that if p dim (X_{0,1}) = m > 0, then

p dim (X_{1,0}/X_{1,2}) = p dim (P_{2}/X_{2,1}) = m ,

and consequently p dim (X_{2,1}) = m − 1. Thus repetition of the procedure described in the previous two paragraphs will yield chains

X_{2m+1,0} ⊇ X_{2m+1,1} ⊇ X_{2m+1,2}

as in ii) with the property that p dim (X_{2m+1,0}/X_{2m+1,2}) = p dim (X_{2m,1}) = 0. Since X_{2m+1,1}/X_{2m+1,2} ∉ P^{∞} and X_{2m+1,2} ∉ P^{∞} by the above remarks, we obtain that 0 ≠ X_{2m+1,2} is a proper direct summand of X_{2m+1,0}. But this is incompatible with the fact that X_{2m+1,0} is uniserial.

iii) To derive the last assertion from the first, observe that, given a chain as in iii), we conclude that either X_{1}/X_{2} ∉ P^{∞} or else X_{1}/X_{3} ∉ P^{∞}. In the first case, the chain X ⊇ X_{1} ⊇ X_{2} satisfies the hypotheses of part i), in the second case, the chain X ⊇ X_{1} ⊇ X_{3} does. But the non-existence of such chains has already been established. □

Proof of Theorem 5.3. Suppose that S = Λe/Je and T_{∗} = Λe/C are as in the statement of the theorem. In particular, C ⊆ Je is maximal with the property “p dim (C) < ∞” or, equivalently, “p dim (Λe/C) < ∞”. If p dim (S) < ∞, we thus obtain T_{∗} = S, and since S is the only saguaro with trunk S, our claim is obvious in that case. So let S ∉ P^{∞} in the sequel.

Since T_{∗} is a reduced saguaro of finite projective dimension having trunk T_{∗}, and since, by the proof of Theorem 5.1, there are only finitely many isomorphism types of reduced saguaros (recall that reduced saguaros are necessarily irredundant), Observation 3.8 guarantees the existence of a saguaro T as in the statement of Theorem 5.3, i.e., of a saguaro which has maximal composition length among the reduced saguaros in P^{∞} with trunk T_{∗}. By Observation 3.5, the trunk T_{∗} can, moreover, be moved to the last position in some legitimate sequence of the trunks of T.

Furthermore, we know from the proof of Theorem 5.2 that each homomorphism M → S having source M ∈ P^{∞} factors through a map of the form g : V → S, where V ∈ P^{∞} is a saguaro on (V_{1}, …, V_{r}) and g(V̂_{s}) = S for some s, while g maps all the other trunks of V to zero. Under claims 1 and 2 below we will show that

(♣) each such homomorphism g : V → S in turn factors through a homomorphism h : W → S, where W ∈ P^{∞} is a reduced saguaro having trunk T_{∗} such that h(T_{∗}) = S and h is zero on the other trunks of W.

Before we prove (♣), we derive Theorem 5.3 from this statement. To do so, we first argue that each saguaro which is reduced, of finite projective dimension and endowed with trunk T_{∗}, and which has maximal composition length with respect to these three properties, is isomorphic to T. Indeed, let T^{(1)} and T^{(2)} be two saguaros of this type. Then their amalgam T^{(1)} ∨ T^{(2)}[T_{∗}] along T_{∗} again has finite projective dimension, since it is isomorphic to T^{(1)} ⊕T^{(2)}/φ(T_{∗}), where φ is the diagonal embedding of T_{∗} into T^{(1)} ⊕T^{(2)}; see the remarks following Definition 4.1. By condition (♣), any map g : T^{(1)}∨T^{(2)}[T_{∗}] → S, which is non-zero on the top of the trunk T_{∗} and zero on the other trunks of the amalgam, factors through a homomorphism h : W → S as described in (♣); say g = hg′. If k_{i} : T^{(i)} → T^{(1)} ∨ T^{(2)}[T_{∗}] is the canonical embedding for i = 1, 2 and g_{i} : T^{(i)} → S maps the trunk T_{∗} of T^{(i)} onto S and the other trunks to zero, we obtain commutative diagrams

[figure: T^{(i)}; g_{i}; k_{i}; S; T^{(1)} ∨ T^{(2)}[T_{∗}]; g; g′; W; h]

for i = 1, 2, possibly after replacing the g_{i} by non-zero scalar multiples of their original versions. In particular, g′k_{i} induces an isomorphism from the top of the trunk T_{∗} of T^{(i)} to the top of the trunk T_{∗} of W, and consequently g′k_{i} also induces an isomorphism between the copies of T_{∗} sitting inside T^{(i)} and W. The fact that T^{(i)} is reduced, i.e., Soc(T^{(i)}) = Soc(T_{∗}) is simple, thus forces g′k_{i} to be a monomorphism. Since T^{(i)} has maximal composition length with respect to being reduced, of finite projective dimension and endowed with trunk T_{∗}, the same is true for g′k_{i}(T^{(i)}). But W shares these last three properties, and we infer that g′k_{i} : T^{(i)} → W is an isomorphism for i = 1, 2. Thus (g′k_{2})^{−1}(g′k_{1}) : T^{(1)} → T^{(2)} is an isomorphism which induces an isomorphism on the subsaguaros T_{∗} of T^{(1)} and T^{(2)}, respectively. In particular, T^{(1)} ≅ T^{(2)} ≅ T as claimed.

As in Theorem 5.3, let f ∈ Hom_{Λ}(T, S) be such that f(T_{∗}) = S and f is zero on the other trunks of T. Due to uniqueness, as just established, each reduced saguaro W ∈ P^{∞} with trunk T_{∗} embeds into T in such a way that the embedding induces an isomorphism between the distinguished copies of T_{∗} in W and T. It follows that each map h ∈ Hom_{Λ}(W, S) as in (♣) factors through f. In view of the remarks preceding condition (♣), this proves f : T → S to be a right P^{∞}-approximation of S. It is a minimal such approximation because the saguaro T is indecomposable, its socle being simple. Thus we have reduced the proof of the theorem to a verification of (♣).

So suppose that V is a saguaro on (V_{1}, …, V_{r}) and, as before, denote by V̂_{i} the canonical images of the trunks V_{i} in V. Moreover, let g ∈ Hom_{Λ}(V, S) be as in the hypothesis of (♣); in particular, V ∈ P^{∞}. Say g(V̂_{s}) = S, while g is zero on the other trunks of V. In particular, V_{s}/JV_{s} ≅ S, whence the uniserial module V_{s} is an epimorphic image of Λe. We may clearly assume that g does not factor through a map g′ ∈ Hom_{Λ}(V′, S) satisfying all the conditions we imposed on g and having the additional property that the composition length of the saguaro V′ is strictly smaller than that of V. Under this assumption, we may take W = V and h = g, i.e., V has trunk T_{∗} and is reduced, as will be proved in the following.

Claim 1: V_{s} ≅ T_{∗}. Indeed, Observation 3.10 guarantees the existence of an index j ∈ {1, …, r} such that p dim (V_{j}) < ∞.

If j = s, then T_{∗} is an epimorphic image of V_{s} since, by construction, T_{∗} has minimal length among the non-zero epimorphic images of Λe which belong to P^{∞}. Suppose T_{∗} ≅ V̂_{s}/Y. Since p dim (Y) < ∞, the factor module V′ = V/Y again belongs to P^{∞}. Moreover, V′ is a finite direct sum of saguaros on trunks of the form V̂_{i}/(V̂_{i}∩Y). One of these saguaros, Ṽ say, has trunk V̂_{s}/(V̂_{s} ∩ Y) = V̂_{s}/Y ≅ T_{∗}, and clearly f then factors through the map Ṽ → S which sends V̂_{s}/Y onto S and the other trunks of Ṽ to zero. From the minimality condition which we initially imposed on the composition length of V, we infer Y = 0, and thus V̂_{s} ≅ T_{∗}.

Now suppose that j ≠ s. Again the factor module V/V̂_{j} has finite projective dimension and is a direct sum of saguaros, one of which has trunk V̂_{s}/(V̂_{s} ∩ V̂_{j}). And again the map f factors canonically through this latter saguaro. But in view of the fact that V̂_{j} ≠ 0 (even ⋂_{i≤r} V̂_{i} ≠ 0, by definition), this is incompatible with our minimality assumption for V, and the proof of Claim 1 is complete.

Claim 2: V is reduced.

To see this, let v̂_{i} ∈ V̂_{i} for 1 ≤ i ≤ r be a canonical sequence of top elements of V, and let b_{i}, c_{i} ∈ B^{∗} be as in Definition 3.1, i.e., V = ⨁_{1≤i≤r} V_{i}/ ∑_{i≤r−1} Λ(b_{i}v_{i} − c_{i}v_{i+1}), where v_{i} ∈ V_{i} are top elements whose canonical images in V are equal to v̂_{i}, respectively. Moreover, suppose that for 1 ≤ i ≤ r the element v̂_{i} is normed by the primitive idempotent e(i).

Assume that V fails to be reduced. By Observation 3.8 there then exists a pair (j, k) of distinct indices in {1, …, r} such that V̂_{j} ∩ V̂_{k} = Λαa_{j}v̂_{j} = Λαa_{k}v̂_{k}, where αa_{j}, αa_{k} ∈ B^{∗} are paths ending in the same arrow α. We may assume that l ann_{Λ}(p_{j}) ⊇ l ann_{Λ}(p_{k}), where the p_{i} are as in Observation 3.9; that is, the p_{i} ∈ B^{∗} are such that Λp_{i}v̂_{i} = Soc(V̂_{1}) = ⋯ = Soc(V̂_{r}) = Λe(0)/Je(0) for some primitive idempotent e(0). Moreover, we suppose that the pair (j, k) of indices with the above properties is chosen in such a way that the pair of left annihilators (l ann_{Λ}(p_{j}), l ann_{Λ}(p_{k})) is maximal with respect to componentwise inclusion. Let e′ be the starting point of the arrow α. Our next goal is to show that the simple module Λe′/Je′ has finite projective dimension.

First we check that V̂_{i}∩V̂_{k} ⊆ V̂_{j} ∩V̂_{k} for all i with l ann_{Λ}(p_{i}) ⊃_{≠} l ann_{Λ}(p_{k}). Let us assume that this statement is violated for the index i. In other words, we assume the above strict inclusion of left annihilators, together with the strict inclusion V̂_{i} ∩V̂_{k} ⊃_{≠} V̂_{j} ∩V̂_{k}. Obviously, we have i ≠ j. This situation can be visualized as in the figure below which makes it rather obvious that we must have V̂_{j} ∩ V̂_{i} = V̂_{j} ∩ V̂_{k}.

[figure: V̂_{i}; V̂_{k}; V̂_{j}; V̂_{k} ∩ V̂_{i}; •; •; V̂_{k} ∩ V̂_{j}; •]

On one hand, the intersection V̂_{j} ∩ V̂_{i} = V̂_{j} ∩ V̂_{k} equals Λαa_{j}v̂_{j}, on the other, it is equal to Λαa_{k}v̂_{k} = Λαqã_{k}v̂_{k} = Λαqã_{i}v̂_{i} with αqã_{i} ∈ B^{∗}. Depending on whether l ann_{Λ}(p_{j}) contains l ann_{Λ}(p_{i}) or not, either the pair (j, i) or the pair (i, j) thus shares the properties of (j, k). But in view of the proper inclusion l ann_{Λ}(p_{i}) ⊃_{≠} l ann_{Λ}(p_{k}), this contradicts the maximality condition imposed on the pair of annihilators and establishes the statement heading the paragraph.

Next we will use Observation 3.9 to see that the uniserial module X_{1} := Λαa_{k}e(k) is isomorphic to a direct summand of Ω^{1}(V), which will guarantee that p dim X_{1} < ∞. To do so, we let the saguaro V on (V_{1}, …, V_{r}) take over the role played by T in that observation and note that the paths p_{i} as above agree with those of Observation 3.9. Clearly, a permutation σ of {1, …, r} satisfying the hypothesis of 3.9 can be chosen in such a way that σ^{−1}(j) < σ^{−1}(k) and that, for all i < σ^{−1}(k), i ≠ σ^{−1}(j), we have l ann_{Λ}(p_{σ(i)}) ⊃_{≠} l ann_{Λ}(p_{k}); keep in mind that we have l ann_{Λ}(p_{j}) ⊇ l ann_{Λ}(p_{k}) to begin with. In particular, σ^{−1}(j) is the only index l strictly smaller than σ^{−1}(k) which potentially satisfies the equality l ann_{Λ}(p_{σ(l)}) = l ann_{Λ}(p_{k}). The previous paragraph therefore yields

V̂_{k} ∩ ( ∑_{l<σ^{−1}(k)} V̂_{σ(l)}) = V̂_{k} ∩ V̂_{j} = Λαa_{k}v̂_{k} ,

and Observation 3.9 tells us that Λαa_{k}e(k) is isomorphic to a direct summand of Ω^{1}(V), as required.

Finally, we will apply Lemma 6.1(iii) to show that also X := Λa_{k}e(k) has finite projective dimension, which, in view of the isomorphism X/X_{1} ≅ Λe′/Je′, will yield finiteness of the projective dimension of Λe′/Je′ and thus take us to our intermediate goal. To apply the lemma, set X_{2} := Λp_{k}e(k), X_{3} := Jp_{k}e(k), and observe that

X_{2}/X_{3} ≅ Λe(0)/Je(0) ≅ Soc(V̂_{1}) = ⋯ = Soc(V̂_{r}) .

In particular, X_{2}/X_{3} ≅ Soc(T_{∗}). Since the trunk T_{∗} has minimal non-zero length under the stipulation that its projective dimension be finite, we deduce that p dim (X_{2}/X_{3}) = ∞. Part iii) of Lemma 6.1 now guarantees that, indeed, p dim (X) < ∞, and hence that p dim (Λe′/Je′) < ∞.

Observation 3.6 permits us to assume that (j, k) = (j, j + 1). We define a new saguaro V′ = (⨁_{1≤i≤r} V_{i}) / U′, where

U′ = ( ∑_{i≤r−1,i≠j} Λ(b_{i}v_{i} − c_{i}v_{i+1})) + Λ(a_{j}v_{j} − a_{j+1}v_{j+1}) ;

in intuitive terms, V′ is obtained from V through the identification of the highlighted vertices in the figure below.

[figure: V̂_{j}; V̂_{j+1}; e′; •; α; •; α; e′; •; •]

Note that V′ is again a saguaro on (V_{1}, …, V_{r}) if a_{j}, a_{j+1} both have positive length. If at least one of these paths has length zero, V′ will either be a saguaro on (V_{1}, …, V_{j}, V_{j+2}, …, V_{r}) or on (V_{1}, …, V_{j−1}, V_{j+1}, …, V_{r}), which amounts to pruning one of the trunks, V_{j} or V_{j+1}. Moreover, observe that the trunk T_{∗} will never be pruned in this process, since if V_{j} were equal to T_{∗} and a_{j} were a primitive idempotent, we would obtain a_{j} = e′ = e, and the preceding paragraph would show p dim (Λe/Je) = p dim (S) < ∞. But this was excluded at the beginning of the proof. Analogously, we can rule out the situation where T_{j+1} = T_{∗} and a_{j+1} is a primitive idempotent. Thus the trunk T_{∗} of V remains a trunk of V′, and hence it is clear that, via the canonical epimorphism V → V′, the map f : V → S factors through the homomorphism f′ : V′ → S which sends T_{∗} onto S and the other trunks of V′ to zero. Finally, our construction provides us with an exact sequence of the form

0 → Λe′/Je′ →^{ψ} V ⟶^{can} V′ → 0 ,

where ψ sends λe′ + Je′ to λ(a_{j}v̂_{j} − a_{j+1}v̂_{j+1}), which shows that V′ in turn belongs to P^{∞} and that V′ has strictly smaller compostion length than V. But this is incompatible with the minimality assumption for V which we made at the outset and completes the proof that V is reduced. □

7. Algorithm and examples.

In this section we informally describe an algorithm for constructing the graphs of the minimal P^{∞}-approximations of the simple left modules over a left serial algebra Λ.

Let S = Λe/Je be a simple left Λ-module, and let T → S be its minimal P^{∞}- approximation as described in Theorem 5.3. Due to Observations 3.9 and 3.10, we can construct T by starting with a suitable trunk T_{1} of finite projective dimension and by successively attaching trunks, one at a time, so that at all intermediate stages we obtain saguaros of finite projective dimension; in fact, we can proceed in such a way that all the syzygies of the intermediate saguaros are isomorphic to direct summands of the syzygy Ω^{1}(T). However, this procedure involves cumbersome comparisons of annihilators in each step (see Observation 3.9). We will therefore not present the algorithm inherent in our observations, but briefly describe a far simpler one. While it will be obvious that this algorithm always terminates, it will not be immediately clear that it does not terminate prematurely, i.e., come to a halt on a saguaro properly contained in T. For a full justification of this algorithm, we refer to [5], where it will be developed for a larger class of algebras.

Note that, with the aid of the graphical algorithm of [8] and [9], it is a cinch to list all the uniserial left Λ-modules of finite projective dimension. Moreover, in view of Observation 3.8, it can be decided at a glance whether a saguaro is reduced; indeed a saguaro V is reduced if no two distinct edges sitting immediately above one of its nodes carry the same label.

Algorithm 7.1 for the construction of T.

Step 1. Find T_{∗} = Λe/C_{1}, where C_{1} ⊆ Je has maximal length with respect to T_{∗} ∈ P^{∞}. Set T_{1} = T_{∗} = Λt_{1}, where t_{1} is the residue class of e in T_{1}.

Step 2. Attach a trunk T_{2} = Λt_{2} to the highest possible node of JT_{1}, say to the node Λb_{1}t_{1}/Jb_{1}t_{1} where b_{1} ∈ B^{∗}, such that the resulting saguaro T′ = (T_{1} ⊕ T_{2}) / Λ(b_{1}t_{1} − c_{1}t_{2}) with suitable c_{1} ∈ B^{∗} is reduced and such that, moreover, p dim (Λc_{1}) < ∞. In case the set of eligible trunks T_{2} with the property that T′ and c_{1} are as above is non-empty, choose T_{2} so as to give rise to a ‘branch’ c_{1} of maximal length (with respect to our standard concept of ‘length’ for elements of B).

If no such trunk T_{2} exists, then T = T_{1}. Otherwise pass on to Step 3.

Step 3. Let T′ be the saguaro on (T_{1}, T_{2}) obtained in Step 2. If possible, attach a trunk T_{3} = Λt_{3} to the highest possible node of JT_{2}, following the same instructions as in Step 2. In particular, T′′ = ⨁_{i=1}^{3} T_{i} / ∑_{i=1}^{2} Λ(b_{i}t_{i} − c_{i}t_{i+1}) is to be reduced and b_{2}, c_{2} are to belong to B^{∗} such that p dim (Λc_{2}) < ∞. Again our choice will be such that the ‘branch’ c_{2} is as long as possible .

If no such trunk exists, then T = T′. Otherwise, the procedure is repeated in such a way that, at stage i + 1, the trunk T_{i} attached in Step i takes over the role played by T_{1} in Step 2 and by T_{2} in Step 3. That the algorithm terminates is clear, since all the intermediate saguaros are reduced (see proof of Theorem 5.1). □

By way of caution: If T^{(j)} = ⨁_{i=1}^{j} T_{i} / ∑_{i=1}^{j−1} Λ(b_{i}t_{i} − c_{i}t_{i+1}) is the intermediate saguaro obtained in Step j, its syzygy Ω^{1}(T^{(j)}) will in general not be isomorphic to Ω^{1}(T_{1}) ⊕ ⨁_{k=2}^{j} Λc_{k}.

Example 7.2. We again take up Example 2.2. According to the algorithm, the minimal P^{∞}-approximation of S_{1} can be assembled in the following way. Here the first of the graphs represents T_{1} = T_{∗}, the last the minimal P^{∞}-approximation of Λe_{1}/Je_{1}; it coincides with the minimal P^{(1)}-approximation of S_{1}, since Λ has finitistic dimension 1.

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

Note that the algorithm does not completely pin down the sequence in which the trunks are attached. For instance, in the second step we could have adjoined the trunk isomorphic to Λe_{4}/J^{4}e_{4} instead of the one isomorphic to Λe_{2}/J^{4}e_{2}.

This same example shows that, while the category S ∩ P^{∞} containing the minimal P^{∞}-approximations of the simple modules always has finite type, the category P^{∞} is far from sharing this property in general. Indeed, for 0 ≠ k ∈ K, the module

M_{k} := \frac{Λe_{1} ⊕ Λe_{3}}{Λ((α + kεα), γe_{3})}

belongs to P^{(1)}, and it is easy to see that for 0 ≠ k, l ∈ K with k ≠ l the corresponding modules M_{k} and M_{l} are non-isomorphic.

Example 7.3. 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^{∞}-approximation, since the left finitistic dimension of Λ is 3. The P^{(0)}-approximation is simply Λe_{1} and is omitted from the list. The sequence of the vertices in the top rows of the graphs gives the order in which the trunks are added if one follows the algorithm.

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

References

[1] J.L. Alperin, Diagrams for modules, J. Pure and Appl. Algebra 16 (1980), 111-119.

[2] M. Auslander and I. Reiten, Applications of contravariantly finite subcategories, Advances Math. 86(1991), 111-152.

[3] M. Auslander and S.O. Smalø, Preprojective modules over artin algebras, J. Algebra 66 (1980), 61-122.

[4] W.D. Burgess, K.R. Fuller, E. Green and D. Zacharia, Left monomial rings - a generalization of monomial algebras, to appear, Osaka J. Math.

[5] W.D. Burgess and B. Zimmermann Huisgen, An algorithm for approximating modules by saguaros, in preparation.

[6] K.R. Fuller, Algebras from diagrams, J. Pure and Appl. Algebra 48 (1987), 23-37.

[7] E.L. Green, E. Kirkman and J. Kuzmanovich, Finitistic dimension of finite dimensional monomial algebras, J. Algebra 136 (1991), 37-50.

[8] B. Zimmermann Huisgen, Predicting syzygies over finite dimensional monomial relations algebras, manuscripta math. 70 (1991), 157-182.

[9] B. Zimmermann Huisgen, Homological domino effects and the first finitistic dimension conjecture, Invent. Math. 108 (1992), 369-383.

[10] B. Zimmermann Huisgen, Syzygies and homological dimensions over left serial rings, in Methods in Module Theory (G. Abrams, J. Haefner, K.M. Rangaswamy eds.), Lecture Notes in Pure and Applied Mathematics, vol. 140, Marcel Dekker (1992), 161-174.

[11] B. Zimmermann Huisgen, Approximating representations of finite dimensional algebras by structurally simpler ones, in preparation.

[12] K. Igusa, S.O. Smalø and G. Todorov, Finite projectivity and contravariant finiteness, Proc. Amer. Math. Soc. 109 (1990), 937-941.

[13] V.A. Jategaonkar, Global dimension of tiled orders over a discrete valuation ring, Trans. Amer. Math. Soc. 196 (1974), 313-330.

[14] E. Kirkman and J. Kuzmanovich, Global dimensions of a class of tiled orders, J. Algebra 127 (1989), 57-72.

Department of Mathematics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada Electronic mail: wdbsg@acadvm1.uottawa.ca

Department of Mathematics, University of California, Santa Barbara, CA 93106, USA Electronic mail: birge@math.ucsb.edu