VIEWING FINITE DIMENSIONAL REPRESENTATIONS

THROUGH INFINITE DIMENSIONAL ONES

Dieter Happel and Birge Huisgen-Zimmermann

Dedicated to the memory of Maurice Auslander whose questions triggered this work

Abstract. We develop criteria for deciding the contravariant finiteness status of a subcategory 𝔄 ⊆ Λ -mod, where Λ is a finite dimensional algebra. In particular, given a finite dimensional Λ-module X, we introduce a certain class of modules – we call them 𝔄-phantoms of X – which indicate whether or not X has a right 𝔄-approximation: We prove that X fails to have such an approximation if and only if X has infinite-dimensional 𝔄-phantoms. Moreover, we demonstrate that large phantoms encode a great deal of additional information about X and 𝔄 and that they are highly accessible, due to the fact that the class of all 𝔄-phantoms of X is closed under subfactors and direct limits.

1. Introduction and preliminaries

Given a finite dimensional algebra Λ and a resolving contravariantly finite subcategory 𝔄 of the category Λ -mod of finitely generated left Λ-modules, the minimal right 𝔄-ap-proximations of the simple left Λ-modules provide significant structural information about arbitrary objects of 𝔄: Indeed, if these approximations are labeled A_{1}, …, A_{n}, then a finitely generated Λ-module M belongs to 𝔄 if and only if M is a direct summand of a module that has a filtration with consecutive factors in {A_{1}, …, A_{n}}; this was proved by Auslander and Reiten [2]. (For the basic definitions, see ‘Prerequisites’ below.) Among other consequences, this result has an obvious homological pay-off: Namely, if all of the A_{i} have finite projective dimension, then the maximum of these dimensions coincides with the supremum of the projective dimensions attained on objects of 𝔄.

Our goal here is twofold. On one hand, we describe typical scenarios preventing the existence of right 𝔄-approximations of a given Λ-module X; as a result, we establish ‘negative criteria’ for the contravariant finiteness of subcategories 𝔄 of Λ -mod. On the other hand – maybe more importantly – our investigation reveals that, for non-contravariantly finite subcategories 𝔄 of Λ -mod, certain modules of infinite dimension over the base field take over the role which is played by the minimal right 𝔄-approximations in the contravariantly finite case. We call these modules ‘𝔄-phantoms’ and define them as follows: First, given a subclass ℭ of 𝔄 and a finitely generated left Λ-module X, we label as ‘ℭ-approximation of X inside 𝔄’ any homomorphism f : A → X with A ∈ 𝔄 having the property that all

Typeset by AMS-TEX

The research of the second author was partially supported by a grant from the National Science Foundation. Part of this work was done while the first author was visiting UCSB. He would like to thank his coauthor for her kind hospitality.

homomorphisms g : C → X with C ∈ ℭ factor through f; in other words, with the prop-

erty that the induced map of functors Hom(−, A)|_{ℭ} ⟶^{Hom(−,f)} Hom(−, X)|_{ℭ} is surjective. Following common practice, we will also refer to the module A as a ℭ-approximation of X inside 𝔄 in that case. Next we will call a Λ-module H, not necessarily in 𝔄, an 𝔄-phantom of X in case, for each finitely generated submodule H′ of H, there exists a finite subclass ℭ = ℭ(H′) of 𝔄 with the property that H′ occurs as a subfactor of each ℭ-approximation of X inside 𝔄. Clearly, the class of all 𝔄-phantoms of X is closed under subfactors; so, in case X has a right 𝔄-approximation, the 𝔄-phantoms of X coincide with the subfactors of the minimal approximation. Our main result (Theorem 9) ensures existence of phantoms in general, under the mild hypothesis that 𝔄 be closed under finite direct sums: Namely, we prove that X has 𝔄-phantoms of infinite dimension over the base field if and only if X fails to have a right 𝔄-approximation. For an intuitive idea of the information stored in such phantoms, suppose for the moment that X = S is simple; comparable to the minimal right 𝔄-approximation of S (in case of existence), the 𝔄-phantoms of S represent, in as highly compressed a form as possible, the relations characterizing those objects of 𝔄 which carry S in their tops. In essence, our criteria for failure of contravariant finiteness provide instructions for the uncovering of 𝔄-phantoms which are too big (namely, non-finitely generated) to be compatible with the existence of traditional 𝔄-approximations.

Our primary interest will be in the category 𝔄 = P^{∞}(Λ -mod) of finitely generated Λ-modules of finite projective dimension, even though most of our results will address the situation of an arbitrary subcategory of Λ -mod which is closed under finite direct sums. Loosely speaking, the category P^{∞}(Λ -mod) is contravariantly finite in Λ -mod when it is either very large (e.g., when Λ has finite global dimension) or very small (e.g., when Λ has vanishing finitistic dimension); in fact, by [3], every representation-finite subcategory of Λ -mod is contravariantly finite. There are several other classes of algebras for which the problem of whether P^{∞}(Λ -mod) is contravariantly finite in Λ -mod is settled, e.g., for left serial algebras, it is settled in the positive [4]. However, in general, it is difficult to decide for an algebra Λ, given via quiver and relations for instance, whether P^{∞}(Λ -mod) has this property. In particular, until recently there was only a single instance for which failure of contravariant finiteness of P^{∞}(Λ -mod) had been established; this is an example of a monomial relation algebra, due to Igusa, Smalø, and Todorov [10], which is so closely related to the Kronecker algebra that a proof for failure of contravariant finiteness of P^{∞}(Λ -mod) can be gleaned from the representation theory of this latter algebra. Our curiosity in this direction was triggered by a question of M. Auslander as to whether, for the monomial relation algebra with differing big and little finitistic dimensions which was exhibited by the second author in [8], P^{∞}(Λ -mod) is contravariantly finite. In the meantime, Smalø and the second author have shown that contravariant finiteness of P^{∞}(Λ -mod) forces the big and little finitistic dimensions of Λ to coincide [9]; hence the answer to Auslander’s question is negative. However, the problem of exhibiting a simple module which fails to have a right P^{∞}(Λ -mod)-approximation still proved rather intractable without a systematic theory providing direction.

The core of the paper is Section 3. The concepts introduced at the beginning of that section appear cogent in the light of the preliminary criterion for failure of contravariant finiteness presented in Section 2 and its applications to special cases. We present several examples to illustrate how phantoms mark the dividing line between contravariant finiteness and failure thereof, and to show concretely what type of information they store. In particular, we apply our techniques to the example mentioned at the outset (Section 4). These sample applications also indicate how sensitive the property of contravariant finiteness of P^{∞}(Λ -mod) in Λ -mod is, with respect to modifications of the relations of the underlying algebra.

Acknowledgment: The authors would like to thank Sverre Smalø for his careful reading of a preliminary manuscript and several helpful suggestions.

Prerequisites: Throughout, Λ will denote a basic finite dimensional algebra over a field K, with a fixed set of primitive idempotents, e_{1}, …, e_{n}, and J will be the Jacobson radical of Λ. The simple left Λ-modules Λe_{i}/Je_{i} will be abbreviated by S_{i}. Moreover, Λ -mod will stand for the category of all finitely generated left Λ-modules, and Λ -Mod for the category of all left Λ-modules.

Let 𝔄 be a full subcategory of Λ -mod. Following Auslander/Smalø [3] and Auslander/Reiten [2], we say that a module M ∈ Λ -mod has a (right) 𝔄-approximation in case there exists a homomorphism φ : A → M with A ∈ 𝔄 such that each homomorphism B → M with B ∈ 𝔄 factors through φ. By [2], existence of any 𝔄-approximation of M entails existence of a minimal 𝔄-approximation of M, i.e., one of minimal K-dimension, which is unique up to isomorphism. If each object in Λ -mod has an 𝔄-approximation, then 𝔄 is said to be contravariantly finite in Λ -mod (see [2,3]). Our favorite choice of a subcategory 𝔄 ⊆ Λ -mod will be the subcategory of all finitely generated left Λ-modules of finite projective dimension; we label it P^{∞}(Λ -mod). Finally, we call a module category 𝔄 ⊆ Λ -mod resolving in case 𝔄 is closed under extensions, as well as kernels of epimorphisms, and contains all indecomposable projective left Λ-modules. Clearly, P^{∞}(Λ -mod) is an instance of a resolving subcategory of Λ -mod.

Given a module M ∈ Λ -Mod, we call an element m ∈ M a top element of M if m ∈ M \ JM and e_{i}m = m for some i; in this case, we also say that m is a top element of type e_{i}. In all of our examples, Λ will be a split finite dimensional algebra over K, that is, Λ will be of the form KΓ/I where Γ is a quiver and I an admissible ideal in the path algebra KΓ. We will briefly and informally review the second author’s conventions for the graphical communication of information about countably generated Λ-modules. For additional detail, see [7,8]. (We point out that our labeled graphs are related to the module diagrams studied by Alperin [1] and Fuller [5].)

Let Γ be the quiver

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

and Λ = KΓ/⟨ρ^{2}⟩. To say that a left Λ-module M has the layered and labeled graph shown below,

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

with respect to a sequence m_{1}, m_{2}, m_{3}, m_{4} of top elements of M which are K-linearly independent modulo JM, is to convey the following information:

• M/JM ≅ S_{1}^{2} ⊕ S_{2} ⊕ S_{3}, the top elements m_{i} of M have type e_{i} for i = 1, 2, 3, and m_{4} has type e_{1};

• JM/J^{2}M ≅ S_{4}^{3}, the three copies of S_{4} modulo J^{2}M being generated by αm_{1}, βm_{1}, ϵm_{3}, and such that γm_{2} is congruent to a nonzero scalar multiple of βm_{1} modulo J^{2}M, and αm_{4} congruent to a nonzero scalar multiple of ϵm_{3};

• J^{2}M/J^{3}M = J^{2}M ≅ S_{4} is generated by ραm_{1}, and is also generated by any of the elements ρβm_{1}, ργm_{2}, or δm_{3}.

Finally, we will consider the following two finitistic dimensions of Λ: the left little finitistic dimension, l fin dim Λ, which is the supremum of the finite projective dimensions attained on Λ -mod, and the left big finitistic dimension, l Fin dim Λ, which stands for the analogous supremum attained on all of Λ -Mod.

2. A few motivating examples

In this section, we specialize to the situation where the algebra Λ is split, i.e., we assume throughout that Λ = KΓ/I is a path algebra modulo relations. We start by exhibiting an elementary sufficient condition for failure of contravariant finiteness of P^{∞}(Λ -mod) in Λ -mod. In our first example we will apply it to the algebra constructed by Igusa/Smalø/Todorov [10], which provided the first known instance of such failure. While this criterion is quite easy to handle, its scope is rather limited, and it will later be supplemented by a criterion of far wider applicability.

For convenience of exposition, we will often view left Λ-modules M as representations of the quiver Γ satisfying the commutativity relations dictated by I; given a path p : e_{1} → e_{2} in KΓ, we will in that case, write f_{p} : e_{1}M → e_{2}M for the K-linear map corresponding to p.

Elementary Criterion 1. Let Λ = KΓ/I. Suppose that e_{1} and e_{2} are vertices of the quiver Γ (not necessarily distinct) and p, q ∈ KΓ \ I paths from e_{1} to e_{2} with Λp ∩ Λq = 0 (we view p and q as elements of Λ whenever indicated by the context). Moreover, suppose that

(i) the cyclic module Λ(p, q) generated by the element (p, q) ∈ Λ^{2} has finite projective dimension, and that one of the following two conditions is satisfied: either,

(ii) whenever M ∈ P^{∞}(Λ -mod), then f_{p}(e_{1}M \ JM) ∩ f_{q}(e_{1}JM) = ∅; or,

(ii’) whenever M ∈ P^{∞}(Λ -mod), then ker(f_{p}) ⊆ ker(f_{q}), and ker(f_{p}) ⊆ e_{1}JM. Then the simple module S_{1} = Λe_{1}/Je_{1} does not have a right P^{∞}(Λ -mod)-approximation.

Before we justify the criterion, we point out that the second part of Criterion 1(ii’) can often be verified without effort; we label it as follows:

(iii) Whenever M in P^{∞}(Λ -mod), then ker(f_{p}) ⊆ e_{1}JM. Indeed, suppose that p is an arrow such that Λp splits off in Je_{1} (this is obviously true when Λ is a monomial relation algebra). Then Condition (iii) holds if and only if p dim Λp = ∞. For, if p dim Λp = ∞ and M ∈ Λ -mod contains a top element of type e_{1} which is annihilated by p, then Λp is isomorphic to a direct summand of Ω^{1}(M), which entails that p dim M = ∞. If, on the other hand, p dim Λp < ∞, then the module M = Λ/Λp violates Condition (iii).

A readily recognizable situation in which the blanket hypothesis of the criterion, as well as conditions (i) and (ii) are satisfied is as follows: p is an arrow e_{1} → e_{2}, q ∈ KΓ \ I a path from e_{1} to e_{2} of positive length which is different from p such that Jp = qJ = 0, and p dim Λq < ∞, while p dim Λe_{2}/Je_{2} = ∞.

Proof of Criterion 1. We start by assuming the blanket hypothesis of the criterion and condition (i) to construct an infinite family of objects (N_{n})_{n∈ℕ} in P^{∞}(Λ -mod). Namely, for n ∈ ℕ, we let b_{1} = ⋯ = b_{n} = e_{1}, define a left Λ-module

N_{n} := (⨁_{i=1}^{n} Λb_{i}) / (∑_{i=1}^{n−1} Λ(pb_{i} − qb_{i+1})) ,

and write \overline{b_{i}} for the residue class of b_{i} in N_{n}. To compute the first syzygy Ω^{1}(N_{n}) of N_{n}, consider the projective cover π : ⨁_{i=1}^{n} Λb_{i} → N_{n} with π(b_{i}) = \overline{b_{i}}, and set C_{i} = Λ(pb_{i} − qb_{i+1}) for 1 ≤ i ≤ n − 1. Note that C_{i} ≃ Λ(p, q), whence p dim C_{i} < ∞ by condition (i). We will conclude that p dim N_{n} < ∞ by showing that ker π = ⨁_{i=1}^{n−1} C_{i}. Suppose that

0 = ∑_{i=1}^{n−1} λ_{i}(pb_{i} − qb_{i+1}) = λ_{1}pb_{1} + (λ_{2}p − λ_{1}q)b_{2} + ⋯ + (λ_{n−1}p − λ_{n−2}q)b_{n−1} − λ_{n−1}qb_{n}

for certain coefficients λ_{i} ∈ Λ. This implies that λ_{1}p = 0, λ_{n−1}q = 0 and λ_{i}p − λ_{i−1}q = 0 for 2 ≤ i ≤ n − 1, and in view of the hypothesis that Λp ∩ Λq = 0, the latter equations entail λ_{i}p = λ_{i−1}q = 0. Thus we obtain λ_{i}(pb_{i} − qb_{i+1}) = 0 for 1 ≤ i ≤ n − 1 as required.

Case I. Suppose that, in addition, condition (ii) holds, but that there nonetheless exists a right P^{∞}(Λ -mod)-approximation φ : A → S_{1} for S_{1}. Pick n > length(A), define f : N_{n} → S_{1} via f(\overline{b_{1}}) = e_{1} + Je_{1}, f(\overline{b_{i}}) = 0 for i ≥ 2, and let g ∈ Hom_{Λ}(N_{n}, A) be such that f = φg, i.e., such that the following diagram commutes:

\xymatrix{ & N_{n} \ar[dl]_{g} \ar[d]^{f} \\ A \ar[r]^{φ} & S_{1}}

Next pick m ≥ 1 minimal with the property that g(\overline{b_{1}}), …, g(\overline{b_{m}}) are K-linearly dependent modulo JA; such an integer m exists because length(N_{n}/JN_{n}) = n > length(A). Say ∑_{i=1}^{m} k_{i}g(\overline{b_{i}}) ∈ JA with k_{i} ∈ K, not all zero. Clearly, k_{m} ≠ 0. Moreover, k_{1} = 0, since 0 = f(∑_{i=2}^{m} k_{i}\overline{b_{i}}) = φg(∑_{i=2}^{m} k_{i}\overline{b_{i}}) = −k_{1}φg(\overline{b_{1}}) = −k_{1}(e_{1} + Je_{1}). In particular, this shows that m ≥ 2. Now set x = ∑_{i=2}^{m} k_{i}\overline{b}_{i−1}. We will check that g(x) = ∑_{i=1}^{m−1} k_{i+1}g(\overline{b_{i}}) again belongs to JA, a contradiction to the minimal choice of m. Indeed, pg(x) = g(∑_{i=2}^{m} k_{i}p\overline{b}_{i−1}) = g(∑_{i=2}^{m} k_{i}q\overline{b_{i}}) = qg(∑_{i=2}^{m} k_{i}\overline{b_{i}}) ∈ qJA, which by condition (ii) entails that g(x) is not a top element of A. This shows g(x) ∈ JA and thus completes the argument for Case I.

Case II. Now suppose that conditions (i) and (ii’) hold. Again assume that S_{1} has a right P^{∞}(Λ -mod)-approximation φ : A → S_{1}, and for n > length(A), define f : N_{n} → S_{1} as in case I. In turn choose g ∈ Hom_{Λ}(N_{n}, A) such that f = φg. But this time, pick m ∈ ℕ minimal with the property that ∑_{i=1}^{m} k_{i}g(\overline{b_{i}}) ∈ JA and q(∑_{i=1}^{m} k_{i}g(\overline{b_{i}})) = 0, for some scalars k_{i} ∈ K which are not all zero. Such an m exists because ker(g) intersects the K-subspace of N_{n} generated by \overline{b}_{1}, …, \overline{b}_{n} non-trivially. As before we obtain k_{1} = 0 and m ≥ 2, and again, we set x = ∑_{i=2}^{m} k_{i}\overline{b_{i−1}} and compute pg(x) = q(∑_{i=2}^{m} k_{i}g(\overline{b_{i}})) = 0. Now condition (ii’) guarantees that g(x) = ∑_{i=1}^{m−1} k_{i+1}g(\overline{b_{i}}) is not a top element of A and that qg(x) = q(∑_{i=1}^{m−1} k_{i+1}g(\overline{b_{i}})) = 0. This is, once more, incompatible with the minimal choice of m. □

Example 2. [10] Let Λ = KΓ/I be the monomial relation algebra with quiver

\xymatrix{Γ : & 1 \ar[r]^{β} \ar[r]^{α} & 2 \ar[l]^{γ}}

and ideal I = ⟨αγ, βγ, γβ⟩ of relations. Then the indecomposable projective left Λ-modules have the following graphs:

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

Set p = β and q = α, and verify the simplified versions of conditions (i) and (ii), as spelled out before the proof of Criterion 1: Clearly, αJ = Jβ = 0 and p dim Λα = 0 < ∞, while p dim Λe_{2}/Je_{2} = ∞. Thus the criterion guarantees that S_{1} does not have a right P^{∞}(Λ -mod)-approximation. □

Note that a typical class of modules defeating attempts to find a right P^{∞}(Λ -mod)-ap-proximation of S_{1} in Example 2 is the following class (M_{n})_{n∈ℕ} of strings of composition length 2n, uniquely determined by their graphs:

[figure: M_{n} :; 1; β; 1; α; β; ⋯; 1; α; β; 2; 2; 2; 2]

(Observe that this family of modules M_{n} represents just a minor simplification of the family of test modules N_{n} exhibited in the proof of Criterion 1 for the choices p = β and q = α; namely M_{n} ≅ N_{n}/Λα\overline{b_{1}}.) A stumbling block for contravariant finiteness can be more succinctly communicated via the infinite dimensional direct limit of the strings M_{n}:

[figure: \underrightarrow{lim} M_{n} :; 1; β; 1; α; β; 1; β; α; ⋯; 2; 2; 2]

At the same time, this limit is a ‘minimal’ module M ∈ P^{∞}(Λ -Mod) with the property that all homomorphisms M_{n} → S_{1}, n ∈ ℕ, factor through M.

By the preceding example one could be led to believe that – under the assumption that Λβ is a direct summand of Je_{1} having infinite projective dimension – the existence of a family of modules (M_{n})_{n∈ℕ} as above should imply non-existence of a P^{∞}-approximation of the simple module associated with the vertex ‘1’. This is not the case, however. In fact, the existence of modules (M_{n})_{n∈ℕ} of the indicated shape inside P^{∞}(Λ -mod) is only potentially troublesome. Continuing to denote the algebra of Example 2 by Λ, we next give an example of an algebra Δ with Λ -mod ⊆ Δ -mod such that Δe_{i} = Λe_{i} for i = 1, 2 and (Λ -mod)∩P^{∞}(Δ -mod) = P^{∞}(Λ -mod); in particular, all of the Λ-modules M_{n} belong also to P^{∞}(Δ -mod). On the other hand, we will see that P^{∞}(Δ -mod) is contravariantly finite in Δ -mod in this example.

Example 3. Let Δ = KΓ′/I′, where Γ′ is

\xymatrix{3 \ar[r]^{δ} & 1 \ar[r]^{β} \ar[r]^{α} & 2 \ar[l]^{γ}}

and the ideal I′ is generated by monomial relations such that the indecomposable projective left Δ-modules have graphs

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

This time the ‘zipper effect’ of the previous example (where the requirement that the homomorphisms f ∈ Hom_{Λ}(M_{n}, S) be factorizable through a module M ∈ P^{∞}(Λ -mod) forces the K-dimension of M to grow with increasing n) can be stopped. This is due to the new projective module Λe_{3}.

We will prove the contravariant finiteness of P^{∞}(Δ -mod) by explicitly describing minimal P^{∞}(Δ -mod)-approximations of the simple left Δ-modules. We claim that the following canonical epimorphism is a right P^{∞}(Δ -mod)-approximation of S_{1}:

[figure: 1; β; 3; δ; φ_{1} :; A_{1}; =; 1; α; S_{1}; 2]

Note that A_{1} = (Δe_{1}⊕Δe_{3})/(Δαe_{1}+Δ(βe_{1}−αδe_{3})) is the injective envelope of the simple module S_{2} = Δe_{2}/Je_{2}. We again use J to denote the Jacobson radical of Δ. and if x_{1} and x_{3} stand for the residue classes of e_{1} and e_{3} in A_{1}, respectively, we let φ_{1}(x_{1}) = e_{1} + Je_{1} and φ_{1}(x_{3}) = 0.

Note first that Ω^{1}(A_{1}) = (Δe_{2})^{2}, whence p dim A_{1} = 1. Now let f : M → S_{1} be an epimorphism with M ∈ P^{∞}(Δ -mod), m_{1} ∈ M a top element of type e_{1} such that f(m_{1}) = e_{1} + Je_{1}, and let m_{2}, …, m_{r} ∈ ker(f) be such that m_{1} + JM, …, m_{r} + JM form a K-basis for e_{1}(M/JM). Then each βm_{i} is a nonzero element in the socle of M, since otherwise Ω^{1}(M) would contain a direct summand isomorphic to S_{2}, which is impossible in view of the fact that p dim S_{2} = ∞, while p dim M < ∞. More strongly, this argument shows that βm_{1}, …, βm_{r} are K-linearly independent, whence in particular β\overline{m}_{1} remains nonzero in \overline{M} = M/ (∑_{i=2}^{2} Δβm_{i}). It is enough to factor the map \overline{f} : \overline{M} → S_{1} induced by f through φ_{1}. In doing this, it is clearly harmless to assume that α\overline{m}_{1} = kβ\overline{m}_{1} for some scalar k ∈ K which may be zero; if this is not a priori the case, we factor out the submodule Δ(α − β)\overline{m}_{1} in addition. Furthermore, we may assume that the elements m_{2}, …, m_{r} ∈ ker(f) are chosen in such a way that there exists an integer s between 2 and r with the property that β\overline{m}_{1} = α\overline{m}_{i} for 2 ≤ i ≤ s and Δβ\overline{m}_{1} ∩ (∑_{i=s+1}^{r} Δα\overline{m}_{i}) = 0.

Set B = ∑_{i=1}^{r} Δ\overline{m}_{i}, and let ι : B → \overline{M} be the canonical embedding. In view of the prededing adjustments, we can define a map σ ∈ Hom_{Δ}(B, A_{1}) by setting σ(\overline{m}_{1}) = x_{1}+kδx_{3}, σ(\overline{m}_{i}) = δx_{3} for 2 ≤ i ≤ s, and σ(\overline{m}_{i}) = 0 for s+1 ≤ i ≤ r. Since A_{1} is injective, σ can be extended to a homomorphism τ ∈ Hom_{Δ}(\overline{M}, A_{1}) which makes the lower triangle in the diagram below commute.

\xymatrix{A_{1} \ar[r]^{φ_{1}} & S_{1} \\ B \ar[u]^{σ} \ar[r]^{ι} & \overline{M} \ar[u]_{\overline{f}} \ar@{-->}[ul]_{τ}}

Our construction entails that the upper triangle then commutes as well, which shows that φ_{1} is indeed a (minimal) P^{∞}(Δ -mod)-approximation of S_{1}.

It is less involved to see that the canonical maps

[figure: 2; γ; 3; δ; φ_{2} :; S_{2}; and; φ_{3} :; S_{3}; 1; 1]

are right P^{∞}(Δ -mod)-approximations of S_{2} and S_{3}, respectively. By [2], this shows that P^{∞}(Δ -mod) is contravariantly finite in Δ -mod. □

Example 3 also shows that the hypotheses of Criterion 1 cannot be simplified to the combination of conditions (i) and (iii), where (iii) is as in the remark following the statement of the criterion. Indeed, if in Example 3, we take p = β and q = α, then both (i) and (iii) are satisfied, but S_{1} does admit a right P^{∞}-approximation.

Finally, we modify the algebra Δ of Example 3 very slightly to an algebra Ξ, with the effect that P^{∞}(Ξ -mod) again fails to be contravariantly finite. Here condition (ii) fails for any choice of p and q, but (i) and (ii’) are satisfied. This sequence of modifications illustrates a phenomenon which will become more obvious in the sequel: Namely, that contravariant finiteness of P^{∞}(Λ -mod) in Λ -mod – as well as failure of this condition – is highly unstable.

Example 4. The quiver of Ξ is that of the algebra Δ in Example 3, but we delete one of the relations, with the effect that the K-dimension of Ξ exceeds that of Δ by 1, and the indecomposable projective left Ξ-modules take on the form

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

We again apply Criterion 1 with the choice p = β and q = α. Clearly, Λα ∩ Λβ = 0.

Moreover, p dim Ξ(β, α) = p dim ( ^{2}^{γ}_{1} ) = 0, whence condition (i) of the criterion is

satisfied.

Next, we check that condition (ii’) of our criterion is satisfied. Let M be in P^{∞}(Ξ -mod). As in Example 3, it is readily checked that, given any top element x ∈ M of type e_{1}, we have βx ≠ 0. So, in proving that for an arbitrary element x ∈ M, the vanishing of βx implies the vanishing of αx, we may assume that x = e_{1}x ∈ JM with βx = 0; since βJ = Kβδ and αJ = Kαδ, we may moreover assume that x ∈ δM. Suppose that αx ≠ 0. In view of the equality αδJ = 0, this implies that x ∈ δM\δJM, i.e., x = δy for some top element y ∈ M of type e_{3}. Let π : P = Ξe_{3} ⊕ Q → M be a projective cover with π(e_{3}) = y. Then βδe_{3} is a nonzero element of ker π = Ω^{1}(M), and since Ξβδe_{3} ≃ S_{2} has infinite projective dimension and is thus not a direct summand of Ω^{1}(M), we see that βδe_{3} ∈ JΩ^{1}(M) ∩ βJP = βΩ^{1}(M); the last equality follows from [5, Lemma 1]. Thus βδe_{3} = βz, where z = e_{1}z ∈ e_{1}Ω^{1}(M)e_{3} ⊆ e_{1}JP e_{3}. Since, clearly, the desired implication ‘βu = 0 ⟹ αu = 0’ does hold for arbitrary elements u of a projective left Ξ-module, we deduce that αδe_{3} = αz and conclude that αx = απ(z) = 0 as required.

Finally, we note that condition (ii) fails in this example. Indeed, if it would hold, it would be true for p = β and q = α. However,the left Ξ-module

M = (Ξe_{1} ⊕ Ξe_{3})/(Ξαe_{1} + Ξ(βe_{1} − αδe_{3}))

with graph

[figure: 1; β; 3; δ; 1; α; β; 2; 2]

has syzygy Ω^{1}(M) = ^{2}^{γ}_{1} ⊕ ^{2}^{γ}_{1} , and thus p dim M = 1, but if x_{i} is the residue class of e_{i} in M for i = 1, 3, then x_{1} is a top element of type e_{1} with βx_{1} = αδx_{3} ∈ αJM. □

As in Example 2, we can again – in the preceding example – pin down classes of modules of finite projective dimension which are responsible for failure of contravariant finiteness of P^{∞}(Ξ -mod). For instance, there is no homomorphism φ : A → S_{1} with A ∈ P^{∞}(Ξ -mod) such that all the canonical epimorphisms from the modules

[figure: 1; β; 3; δ; 3; δ; 3; δ; E_{n} :; 1; α; β; 1; α; β; ⋯; 1; α; β; 2; 2; 2; 2; 2]

onto S_{1} can be factored through φ. Observe, however, that they can all be factored through the canonical surjection from

[figure: 1; β; 3; δ; 3; δ; 3; δ; E = \underrightarrow{lim} E_{n} :; 1; α; β; 1; α; β; 1; α; β; ⋯; 2; 2; 2; 2]

onto S_{1}.

3. Relative approximations and phantoms

If 𝔄 ⊂ Λ -mod is a resolving contravariantly finite subcategory of Λ -mod, then, according to [2], the minimal right 𝔄-approximations of the simple left Λ-modules hold a substantial amount of information on arbitrary objects of 𝔄. The gap in the available information when 𝔄 is not contravariantly finite is to be filled by direct limits of ‘partial minimal approximations’ as indicated informally in Section 2. (We follow the convention that ‘direct limits’ are colimits extending over directed index sets.)

Definitions 5. Let ℭ ⊂ 𝔄 be full subcategories of Λ -mod such that 𝔄 is closed under finite direct sums, and let \overrightarrow{𝔄} be the closure of 𝔄 under direct limits in Λ -Mod. Moreover, let X be a finitely generated left Λ-module.

(1) A (right) ℭ-approximation of X inside 𝔄 (resp. inside \overrightarrow{𝔄}) is a homomorphism f : A → X with A in 𝔄 (resp. A in \overrightarrow{𝔄}) such that

Hom(−, A)|_{ℭ} ⟶^{Hom(−,f)} Hom(−, X)|_{ℭ} ⟶ 0

is an exact sequence of functors from ℭ to the category of abelian groups, i.e., with the property that each map g ∈ Hom_{Λ}(C, X) with C in ℭ factors through f.

If ℭ = 𝔄, a ℭ-approximation of X inside 𝔄 will simply be called a (right) 𝔄-approxima-tion of X, in accordance with the existing terminology.

(2) A ℭ-phantom of X relative to 𝔄 is an object H in Λ -Mod (not necessarily in \overrightarrow{𝔄}) with the following property: For every finitely generated submodule H′ ⊆ H, there exists a finite non-empty set ℭ′ ⊂ ℭ such that H′ is isomorphic to a subfactor of \underline{each} ℭ′-approx-imation of X inside 𝔄.

When ℭ = 𝔄, we will more briefly refer to 𝔄-phantoms of X.

(3) A ℭ-phantom H of X relative to 𝔄 is called effective if there exists a homomorphism f : H → X which is a ℭ-approximation of X inside \overrightarrow{𝔄} (in particular, H ∈ \overrightarrow{𝔄} in that case).

The objects that will draw our particular attention in the sequel will be the 𝔄-phantoms of X. Start by noting that, given any subclass ℭ ⊆ 𝔄, every ℭ-phantom of X relative to 𝔄 is actually an 𝔄-phantom of X. Moreover, observe that the class of all 𝔄-phantoms of X is closed under subfactors and hence also under direct limits. In fact, it is readily verified that a left Λ-module H is an 𝔄-phantom of X if and only if H is the direct limit of a direct system of finitely generated 𝔄-phantoms of X. In case X fails to have an 𝔄-approxima-tion, there will be a plethora of 𝔄-phantoms as we will shortly see. This will facilitate the construction of infinite dimensional phantoms, which in turn lie at the heart of our criteria for failure of contravariant finiteness of 𝔄.

The effective phantoms, on the other hand, are in a sense the best possible substitutes for minimal approximations in the sense of Auslander/Smalø [3] and Auslander/Reiten [2]. For example, if X = Λe/Je is simple, the effective 𝔄-phantoms of X compress information about the relations of those modules in 𝔄 which have a top element of type e into the tightest possible format. Compared with classical approximations, we simply renounce the requirement that this picture should fit into a finitely generated module.

Basic Observations 6. Let ℭ, 𝔄, \overrightarrow{𝔄} and X be as in Definition 5.

(1) Since \overrightarrow{𝔄} is closed under arbitrary direct sums, it is clear that X always has ℭ-ap-proximations inside \overrightarrow{𝔄}. Just add up a sufficient number of copies of each object C in ℭ to cover all homomorphisms C → X. Similarly, X has a ℭ-approximation inside 𝔄 whenever ℭ ⊆ 𝔄 is a finite subclass, because Hom_{Λ}(C, X) has finite K-dimension for each C ∈ ℭ.

(2) If X has phantoms relative to 𝔄 which have unbounded lengths, then X fails to have an 𝔄-approximation. In particular, this is true in case X has a non-finitely generated 𝔄-phantom.

(3) If ℭ ⊆ 𝔇 ⊆ 𝔄, each ℭ-phantom relative to 𝔄 is also a 𝔇-phantom relative to 𝔄. Note the stringency of our definition in this connection: a finitely generated module H is a ℭ-phantom of X relative to 𝔄 if and only if it is a ℭ′-phantom for some finite subclass ℭ′ of ℭ.

(4) Suppose that, in addition to being closed under finite direct sums, 𝔄 is closed under direct summands. Then the existence of an 𝔄-approximation of X implies the existence of a unique minimal such approximation by [2], say A(X). In that case, A(X) is the only effective 𝔄-phantom of X, and all other 𝔄-phantoms of X are subfactors of A(X).

□

7. The Examples of Section 2 Revisited. Let X = S_{1} = Λe_{1}/Je_{1}, and 𝔄 = P^{∞}(Λ -mod).

In Example 2, the module

[figure: M = \underrightarrow{lim} M_{n} :; 1; β; 1; α; β; 1; β; α; ⋯; 2; 2; 2]

is an effective ℭ-phantom of S_{1} inside P^{∞}(Λ -Mod), where ℭ = {M_{n} | n ∈ ℕ}, but

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

is neither the source of a ℭ-approximation of S_{1} inside P^{∞}(Λ -Mod), nor a ℭ-phantom of S_{1} relative to P^{∞}(Λ -mod).

As for Example 3: Start by observing that the above graphs uniquely define left modules over the modified algebra Δ, again denoted M_{n}, M and N, and the class of Δ-modules ℭ = {M_{n} | n ∈ ℕ} in turn belongs to P^{∞}(Δ -mod). Moreover, the homomorphism f : M → S_{1} = Δe_{1}/Je_{1} which sends the top element represented by the left-most ‘1’ in the graph of M to e_{1} + Je_{1} and sends the top elements displayed farther to the right to zero is still a ℭ-approximation of S_{1} inside P^{∞}(Δ -mod). However, in the present setup, both M and N fail to be ℭ-phantoms of S_{1} relative to P^{∞}(Δ -mod); indeed, as we saw earlier, S_{1} has a P^{∞}(Δ -mod)-approximation in that example.

Finally, let us focus on Example 4. Viewing M_{n} and M as left Ξ-modules, and keeping in mind that the M_{n} belong to P^{∞}(Ξ -mod), we find that M is an effective ℭ-phantom of S_{1} relative to P^{∞}(Ξ -mod), as in Example 2. Moreover, if 𝔈 = {E_{n} | n ∈ ℕ} with E_{n} as defined after Example 4, then M is also an 𝔈-phantom of S_{1} relative to P^{∞}(Ξ -mod), but not an effective one because the canonical epimorphism

[figure: 1; β; 3; δ; E_{2}; =; 1; α; S_{1}; 2]

does not factor through M. The better 𝔈-phantom here is E = \underrightarrow{lim} E_{n}, which is actually an effective (ℭ ∪ 𝔈)-phantom of S_{1} relative to P^{∞}(Ξ -mod). □

Next we prepare for a general existence result. In a nutshell: Whenever 𝔄 ⊆ Λ -mod fails to be contravariantly finite, there exist 𝔄-phantoms of infinite K-dimension.

Proposition 8. Suppose that 𝔄 ⊆ Λ -mod is closed under finite direct sums and that X ∈ Λ -mod does not have a (right) 𝔄-approximation. Then there exists a countable subclass ℭ of 𝔄 such that X fails to have a ℭ-approximation inside 𝔄.

Proof. By repeatedly applying the first of the observations under 6, we show that, for each d ≥ 1, there exists a finite subset ℭ_{d} of 𝔄 such that X does not have a ℭ_{d}-approximation of K-dimension ≤ d inside 𝔄.

Assuming the contrary for some d ≥ 1, we pick a module Y_{1} in 𝔄 with Hom_{Λ}(Y_{1}, X) ≠ 0 – such a module Y_{1} exists by hypothesis – and let f_{1} : A_{1} → X be a {Y_{1}}-approximation of X inside 𝔄 such that dim_{k}A_{1} ≤ d. In particular, f_{1} is nonzero. Since X fails to have an 𝔄-approximation, there exists an object Y_{2} in 𝔄 such that some homomorphism in Hom_{Λ}(Y_{2}, X) fails to factor through f_{1}. Let f_{2} : A_{2} → X be an {A_{1}, Y_{2}}-approximation of X inside 𝔄; by assumption, A_{2} can be chosen to have K-dimension at most d. Inductively, our assumption thus yields a family (A_{n})_{n≥1} of objects of 𝔄 with dim_{k}A_{n} ≤ d for all n, together with finitely generated left Λ-modules Y_{n} and homomorphisms f_{n} : A_{n} → X such that f_{n} is an {A_{n−1}, Y_{n}}-approximation of X inside 𝔄, but fails to be a {Y_{n+1}}-approxima-tion. Accordingly, we can pick g_{n} ∈ Hom_{Λ}(A_{n}, A_{n+1}) such that f_{n} = f_{n+1}g_{n} and none of the g_{n} is an isomorphism. But since f_{1} = f_{2}g_{1} = f_{3}g_{2}g_{1} = ⋯ = f_{n+1}g_{n} ⋯ g_{1} is nonzero, we deduce g_{n} ⋯ g_{1} ≠ 0 for all n, which contradicts the Harada-Sai Lemma [6] and proves our assumption to be absurd.

Letting ℭ_{d} for d ≥ 1 be as in our initial claim, the countable subset ℭ = ⋃_{d≥1} ℭ_{d} of 𝔄 is clearly as desired. □

We apply this proposition to obtain the announced existence result.

Theorem 9. Suppose that 𝔄 ⊆ Λ -mod is closed under finite direct sums, and let X ∈ Λ -mod. Then the following conditions are equivalent:

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

(2) There exists a countable subclass ℭ ⊆ 𝔄 such that X has an effective ℭ-phantom of countably infinite K-dimension relative to 𝔄.

(3) X has an 𝔄-phantom of infinite K-dimension.

Proof. ‘(1) ⟹ (2)’. Assume that (1) holds. Then Proposition 8 yields a countable subclass 𝔇 = {D_{1}, D_{2}, D_{3}, … } of 𝔄 such that X does not have a 𝔇-approximation inside 𝔄. However, by the first of the observations under 6, there exists a {D_{1}}-approximation of X inside 𝔄, say f_{1} : A_{1} → X.

Next we pick an {A_{1}}-approximation f_{2} : A_{2} → X of X inside 𝔄, together with a map g_{1,2} ∈ Hom_{Λ}(A_{1}, A_{2}) satisfying f_{1} = f_{2} ∘ g_{1,2}, such that dim_{K}(g_{1,2}(A_{1})) is as small as possible. Consequently, the following is true: Whenever f′_{2} : A′_{2} → X is an {A_{2}}-ap-proximation of X inside 𝔄 and g′ ∈ Hom_{Λ}(A_{2}, A′_{2}) is such that f_{2} = f′_{2} ∘ g′, we have g′(g_{1,2}(A_{1})) ≃ g_{1,2}(A_{1}).

We now choose any {D_{2}, A_{2}}-approximation f_{3} : A_{3} → X of X inside 𝔄, and subsequently an {A_{3}}-approximation f_{4} : A_{4} → X inside 𝔄, together with a map g_{3,4} ∈ Hom_{Λ}(A_{3}, A_{4}) such that f_{3} = f_{4} ∘ g_{3,4} and dim_{K}(g_{3,4}(A_{3})) is minimal.

Continuing along this line, we obtain a sequence of objects (A_{n})_{n≥1} and maps f_{n} : A_{n} → X such that, for n ≥ 2, f_{2n−1} is a {D_{n}, A_{2n−2}}-approximation of X inside 𝔄 and f_{2n} is an {A_{2n−1}}-approximation which is coupled with a map g_{2n−1,2n} ∈ Hom_{Λ}(A_{2n−1}, A_{2n}) such that f_{2n−1} = f_{2n} ∘ g_{2n−1,2n} and dim_{K}(g_{2n−1,2n}(A_{2n−1})) is minimal.

Set ℭ = {A_{1}, A_{2}, A_{3}, … } and supplement the above maps g_{n,n+1} for odd n by homomorphisms g_{n,n+1} ∈ Hom_{Λ}(A_{n}, A_{n+1}) with f_{n} = f_{n+1} ∘ g_{n,n+1} for n even. If, for n < m, we moreover define g_{n,m} = g_{m−1,m} ∘ ⋯ ∘ g_{n,n+1} : A_{n} → A_{m}, then (A_{n}, g_{n,m})_{n,m∈ℕ,n<m} is an inductive system with f_{n} = f_{m} ∘ g_{n,m}. Set

A = \underrightarrow{lim} A_{n}, f = \underrightarrow{lim} f_{n} ∈ Hom_{Λ}(A, X),

and let h_{n} : A_{n} → A be the canonical maps. Clearly, A belongs to \overrightarrow{𝔄} (see Definition 5). Moreover, each homomorphism in Hom_{Λ}(C, X) with C ∈ ℭ factors through f and, a fortiori, so does each homomorphism in Hom_{Λ}(D_{n}, X). In other words, f : A → X is a ℭ ∪ 𝔇-approximation of X inside \overrightarrow{𝔄}.

Next we want to identify A as a ℭ-phantom of X relative to 𝔄. Our construction entails that, for m > 2n, we have g_{2n,m} ∘ g_{2n−1,2n}(A_{2n−1}) ≃ g_{2n−1,2n}(A_{2n−1}), and consequently we have h_{2n}(U_{2n}) ≃ U_{2n} if we define U_{2n} = g_{2n−1,2n}(A_{2n−1}). Since A is the directed union of the submodules h_{2n}(U_{2n}), n ∈ ℕ, it suffices to show that each of the finitely generated modules U_{2n} is a ℭ-phantom of X relative to 𝔄. For that purpose, consider the finite subset ℭ(U_{2n}) = {A_{2n}} of ℭ, and let f′ : A′ → X be a ℭ(U_{2n})-approximation of X inside 𝔄. If g′ ∈ Hom_{Λ}(A_{2n}, A′) is such that f_{2n} = f′ ∘ g′, our construction yields U_{2n} ≃ g′(U_{2n}) ⊆ A′, which shows that U_{2n} is indeed a ℭ-phantom of X relative to 𝔄, and hence so is A. By the preceding paragraph, A is, in fact, even an effective ℭ-phantom of X relative to 𝔄.

Finally, we note that dim_{K}A ≤ ℵ_{0} by construction. To prove the reverse inequality, we assume, to the contrary, that dim_{k}A < ∞. But this means that f is a 𝔇-approximation of X inside 𝔄, which contradicts our choice of 𝔇 and completes the proof of ‘(1) ⟹ (2)’.

The implication ‘(2) ⟹ (3)’ follows from the fact that each ℭ-phantom relative to 𝔄, where ℭ ⊆ 𝔄, is also a 𝔄-phantom, and ‘(3) ⟹ (1)’ is a consequence of the basic observation 6(2). □

The following is an upgraded version of the elementary Criterion 1 for non-existence of a P^{∞}(Λ -mod)-approximation of a given simple module S. The idea underlying the proof is the same, even though we impose no restrictions on the subcategory 𝔄 ⊆ Λ -mod this time. In particular, this criterion again points to a countable subclass ℭ of 𝔄 which obstructs the approximability of S by a finitely generated module of finite projective dimension. In view of the proof of Theorem 9, it can hence be used towards the explicit construction of 𝔄-phantoms of S. While this criterion will be instrumental in resolving the problem of contravariant finiteness in our key example (Section 4), for complex non-monomial algebras, it may still be nontrivial to verify or refute Condition (2) below. We therefore add an illlustration of how the underlying idea can still be used towards deciding questions of contravariant finiteness, even when the criterion is not readily applicable verbatim.

Criterion 10. Suppose that Λ is a split finite dimensional algebra and 𝔄 a full subcategory of Λ -mod. Moreover, let e_{1}, …, e_{m} be pairwise orthogonal primitive idempotents of Λ, and p_{1}, …, p_{m}, q_{1}, …, q_{m} ∈ J with p_{i} = p_{i}e_{i} and q_{i} = q_{i}e_{i} such that the following conditions are satisfied:

(1) For each n ∈ ℕ, there is a module M_{n} ∈ 𝔄, together with a sequence x_{n1}, …, x_{n,mn} of mn top elements of M_{n} which are K-linearly independent modulo JM_{n} such that 0 ≠ p_{r(i)}x_{ni} = q_{r(i+1)}x_{n,i+1} for 1 ≤ i < mn, where r(i) ∈ {1, …, m} is congruent to i modulo m.

(2) For any object C in 𝔄, the following are true:

(i) if x ∈ C is a top element of type e_{1} then p_{1}x ≠ 0; (ii) if y, z ∈ C with 0 ≠ p_{r(i)}y = q_{r(i+1)}z, then p_{r(i+1)}z ≠ 0.

Then S_{1} = Λe_{1}/Je_{1} does not have an {M_{n} | n ∈ ℕ}-approximation inside 𝔄. In particular, 𝔄 is not contravariantly finite in Λ -mod in that case.

Proof. Assume to the contrary that f : A → S_{1} is an {M_{n} | n ∈ ℕ}-approximation of S_{1}, and choose n ∈ ℕ such that dim_{K}A < mn. Fixing n, we will briefly write x_{i} for x_{ni}, 1 ≤ i ≤ mn. Consider the homomorphism g : M_{n} → S_{1}, defined by g(x_{1}) = e_{1} + Je_{1} and g(x_{i}) = 0 for 2 ≤ i ≤ mn; this definition is meaningful because x_{1} is a top element of M_{n} of type e_{1}. Choose h : M_{n} → A such that g = fh. Moreover, note that, due to the linear dependence of the elements h(x_{1}), …, h(x_{mn}) of A, there exists a natural number t, together with scalars k_{t}, …, k_{n} such that k_{t} ≠ 0 whereas 0 = p_{r(t)} ∑_{i=t}^{mn} k_{i}h(x_{i}) = p_{r(t)}z, where z = ∑_{i=t}^{mn} h(e_{r(t)}k_{i}x_{i}).

Choose t minimal with the property that the next-to-last equation is satisfied for some nonzero scalar k_{t} and some scalars k_{t+1}, …, k_{mn}. We claim that t > 1. Indeed, if t = 1, then f(z) = g(∑ e_{1}k_{i}x_{i}) = k_{1}g(x_{1}) ≠ 0 and hence z is a top element of A of type e_{1}. By (2)(i) this implies that p_{1}z ≠ 0, a contradiction. Now set y = ∑_{i=t}^{mn} h(e_{r(t−1)}k_{i}x_{i−1}). By the minimal choice of t, we obtain p_{r(t−1)}y ≠ 0. Using condition (1) and the choice of the r(i), we further compute that p_{r(t−1)}y = q_{r(t)}z, and – invoking condition (2)(ii) – we conclude p_{r(t)}z ≠ 0. But this is incompatible with our choice of t. □

As is backed up by the proof of Criterion 10, the hypotheses of this criterion entail the existence of an infinite dimensional phantom of S_{1} relative to 𝔄, a graph of which contains a subgraph

[figure: 1; p_{1}; 2; q_{2}; p_{2}; 3; p_{3}; q_{3}; ⋯; m; p_{m}; q_{m}; 1; q_{1}; p_{1}; 2; q_{2}; p_{2}; 3; p_{3}; q_{3}; ⋯]

As mentioned earlier, the criterion works well for 𝔄 = P^{∞}(Λ -mod) when Λ is a monomial relation algebra. One of the reasons for this can be found in the following observation which shows how easy it is to get a chain of P^{∞}(Λ -mod)-phantoms started in the monomial situation.

Remarks 11. Let Λ = KΓ/I be a monomial relation algebra, and suppose that the simple module S_{1} = Λe_{1}/Je_{1} has infinite projective dimension. If α_{1}, …, α_{r} are arrows α_{j} : e_{1} → e_{j} ending in distinct vertices e_{1}, …, e_{r}, such that p dim Λα_{j} = ∞ for 1 ≤ j ≤ r, then

[figure: e_{1}; α_{1}; α_{2}; α_{r}; e_{1}; e_{2}; ···; e_{r}]

is a subgraph of the graph of a P^{∞}(Λ -mod)-phantom C of S_{1}. More precisely, there exists a top element c ∈ C of type e_{1} such that α_{i}c ≠ 0 for 1 ≤ i ≤ r.

If, moreover, there exists a module M ∈ P^{∞}(Λ -mod) with a graph containing a subgraph of the form

[figure: e_{1}; α_{r}; x; e; q; y; e_{r}]

where x, y ∈ M are top elements of types e_{1} and e respectively and q denotes a path in KΓ \ I, then there exists a P^{∞}(Λ -mod)-phantom of S_{1} whose graph contains a subgraph of the form

[figure: e_{1}; α_{1}; α_{j}; α_{r}; e; q; e_{1}; ···; e_{j}; ···; e_{r}]

with respect to suitable top elements.

Proof. The second statement clearly follows from the first. To justify the first, we start by noting that ⨁_{1≤j≤r} Λα_{j} is a direct summand of Je_{1}, due to the fact that Λ is a monomial relation algebra. Let C ∈ P^{∞}(Λ -mod) have a top element c of type e_{1}. To see that α_{j}c ≠ 0 for 1 ≤ j ≤ r, consider a projective cover

π : P = Λx_{0} ⊕ ⨁_{i∈I} Λx_{i} → C

such that x_{0} = e_{1} and π(x_{0}) = c. If we had α_{j}c = 0, we could conclude that Λα_{j}x_{0} is a direct summand of ker π = Ω^{1}(C) ≤ JP, because Λα_{j}x_{0} is a direct summand of JP, which is incompatible with our setup. □

While most of our applications demonstrate the use of phantoms towards a proof that P^{∞}(Λ -mod) fails to be contravariantly finite, phantoms may also be helpful in finding P^{∞}(Λ -mod)-approximations.

Example 12. [2, Example on p. 137] Let Λ = KΓ/I be based on the quiver

[figure: 1; α; β; 2; γ; 3; δ; 4]

such that the Λe_{i} have graphs:

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

Clearly, S_{1} is the only simple left Λ-module of infinite projective dimension. By Remark 11, S_{1} has a P^{∞}(Λ -mod)-phantom A_{1}, with graph ^{1}^{α}_{1} , and it is readily checked that there is no object in P^{∞}(Λ -mod) having a submodule with graph [figure: 1; α; e; 1] . One deduces that A_{1} → S_{1} is a (minimal) P^{∞}(Λ -mod)-approximation of S_{1}. □

For non-monomial relation algebras, one often needs to slightly vary the idea of Criterion 10. We illustrate the construction of phantoms in such a non-monomial situation. Example 13. Let Λ = KΓ/I, where Γ is the quiver

[figure: 1; α; β; 7; ρ_{3}; ρ_{4}; 5; ρ_{2}; ρ_{1}; 2; γ; 3; δ; 6; σ_{1}; σ_{2}; 8; σ_{4}; σ_{3}; 4; ϵ]

and I ⊂ KΓ is the unique ideal containing γα − δβ and having the property that the indecomposable projective left Λ-modules have the graphs

[figure: 1; α; β; 2; γ; 3; δ; 4; ϵ; 5; ρ_{1}; ρ_{2}; 6; σ_{1}; σ_{2}; 7; ρ_{3}; ρ_{4}; 8; σ_{3}; σ_{4}; 2; γ; 3; δ; 4; 4; 4; 2; γ; 5; 3; δ; 6; 5; ρ_{1}; ρ_{2}; 5; 6; σ_{1}; σ_{2}; 6; 4; 4; 4; 2; γ; 5; 3; δ; 6; 4; 4]

We will see that S_{1} = Λe_{1}/Je_{1} does not have a right P^{∞}(Λ -mod)-approximation by constructing P^{∞}(Λ -mod)-phantoms of infinite K-dimension. We start by observing that for each module M in P^{∞}(Λ -mod) with top element m of type e_{1}, either αm ≠ 0 or βm ≠ 0. Thus the module

1  1 C_{1} =  _{⊕} 2  3

of finite projective dimension is an (effective) add(C_{1})-phantom of S_{1} inside P^{∞}(Λ -mod); a fortiori, C_{1} is a P^{∞}(Λ -mod)-phantom of S_{1}. Since there exist modules in P^{∞}(Λ -mod) having subgraphs

[figure: x_{1}; 1; α; 5; ρ_{1}; 2; resp.; y_{1}; 1; β; 6; σ_{1}; 3]

where x_{1} and y_{1} stand for top elements, namely

[figure: 1; α; 5; ρ_{1}; ρ_{2}; 2; 5; resp.; 1; β; 6; σ_{1}; σ_{2}; 3; 6]

and since each M ∈ P^{∞}(Λ -mod) with a top element m of type e_{5} (resp. e_{6}) satisfies ρ_{2}m ≠ 0 (resp. σ_{2}m ≠ 0), the module

C_{2} = [figure: 1; α; 5; ρ_{1}; ρ_{2}; 2; 5]  ⊕  [figure: 1; β; 6; σ_{1}; σ_{2}; 3; 6]

is an add(C_{2})-phantom of S_{1} inside P^{∞}(Λ -mod).

In the next step, we observe that P^{∞}(Λ -mod) contains objects with subgraphs

[figure: 1; α; 5; ρ_{1}; ρ_{2}; 7; ρ_{3}; 2; 5; resp.; 1; β; 6; σ_{1}; σ_{2}; 8; σ_{3}; 3; 6]

where again x_{1} and x_{2}, resp. y_{1} and y_{2}, denote top elements, and since each module M ∈ P^{∞}(Λ -mod) with top element m of type e_{7} (resp. e_{8}) satisfies ρ_{4}m ≠ 0 (resp. σ_{4}m ≠ 0) the module

C_{3} = [figure: 1; α; 5; ρ_{1}; ρ_{2}; 7; ρ_{3}; ρ_{4}; 2; 5; 5]  ⊕  [figure: 1; β; 6; σ_{1}; σ_{2}; 8; σ_{3}; σ_{4}; 3; 6; 6]

is an effective add(C_{3})-phantom of S_{1} inside P^{∞}(Λ -mod). A fortiori, C_{3} is a P^{∞}(Λ -mod)- phantom inside P^{∞}(Λ -mod).

Proceeding in this fashion, we obtain modules C_{n} ∈ P^{∞}(Λ -mod) of length 4n, namely

C_{n} = [figure: 1; 5; 7; ⋯; 7; 2; 5; 5; 5; 5] ⊕ [figure: 1; 6; 8; ⋯; 8; 3; 6; 6; 6; 6]

all of which are P^{∞}(Λ -mod)-phantoms of S_{1}. This yields the P^{∞}(Λ -mod)-phantom \underrightarrow{lim} C_{n} of infinite K-dimension, and shows that S_{1} fails to have a P^{∞}(Λ -mod)-approximation inside P^{∞}(Λ -mod). □

Problem 14. Characterize the simple modules over monomial relation algebras which fail to have right P^{∞}(Λ -mod)-approximations in terms of their infinite dimensional phantoms.

4. A less elementary example

We apply Criterion 10 to a less elementary example which, in fact, motivated a major portion of this article. Namely, for the finite dimensional monomial relation algebra Λ of [8] with l fin dim Λ < l Fin dim Λ, we exhibit a simple left Λ-module which fails to have a P^{∞}(Λ -mod)-approximation.

Example 15. We refer the reader to [8, p. 378] for a definition of Λ = KΓ/I. We will apply Criterion 10 to show that the simple module S_{2} = Λe_{2}/Je_{2} fails to have a right P^{∞}(Λ -mod)-approximation. For that purpose, we let 𝔄 = P^{∞}(Λ -mod), set m = 1, and focus on the single primitive idempotent e_{2}. Moreover, we make the choices p = γ_{1} and q = γ_{2} + τγ_{2}, let n ∈ ℕ, and set x_{i} = e_{2} for i = 1, …, n.

First we exhibit modules M_{n} as in part (1) of the criterion. Namely, we define

M_{n} = (⨁_{i=1}^{n} Λx_{i}) / (∑_{i=1}^{n−1} Λz_{i}) ,

where z_{i} = px_{i} − qx_{i+1} for 1 ≤ i ≤ n − 1. Observe that M_{n} ∈ P^{∞}(Λ -mod) for each n. Indeed, the sum ∑_{i=1}^{n−1} Λz_{i} is direct and can be seen to have finite projective dimension as follows: The graph of Λz_{i} relative to the top element z_{i} is

[figure: 1; α_{0}; σ; τ; χ_{2}; a_{0}; α_{1}; β_{1}; 1; 1; σ; χ_{1}; χ_{2}; a_{1}; b_{1}; 1; c_{1}; c_{2}]

whence the graphical method of [7, Section 5] yields Ω^{1}(Λz_{i}) = ( ^{c_{1}}^{χ′_{1}}_{c_{1}}  )⊕( ^{c_{2}}^{χ′_{2}}_{c_{2}}  ). Thus

Ω^{1} (∑_{i=1}^{n−1} Λz_{i}) ≅ (Λe(c_{1}))^{n−1} ⊕ (Λe(c_{2}))^{n−1}

is projective as required.

To check condition (2)(i) of Criterion 10, suppose that C belongs to P^{∞}(Λ -mod) and has a top element x of type e_{2}. Then px = γ_{1}x ≠ 0, since otherwise Ω^{1}(C) would have a direct summand isomorphic to the left ideal Λγ_{1}. But this left ideal has infinite projective dimension, as can again be checked with the aid of its graph

[figure: 1; α_{0}; τ; a_{0}; β_{1}; 1; σ; χ_{1}; b; 1; c_{1}]

and the method of [7]. This shows that condition (2)(i) is indeed met.

Finally, let us check that condition (2)(ii) of Criterion 10 is satisfied. Again let C ∈ P^{∞}(Λ -mod), and suppose that y, z ∈ C are such that 0 ≠ py = qz, where p and q are as above. From the fact that qz = γ_{2}z + τγ_{2}z does not vanish, we deduce γ_{2}z ≠ 0, which in turn implies that e_{2}z is a top element of type e_{2} of C; this implication is an immediate consequence of the fact that the vertex e_{2} is a source of Γ. Consequently, the preceding paragraph yields pz ≠ 0 as required. Thus Criterion 10 applies to complete that proof that S_{2} does not have a right P^{∞}(Λ -mod)-approximation.

An infinite dimensional P^{∞}(Λ -mod)-phantom of S_{2} resulting from the preceding argument can be visualized as follows:

[figure: 2; γ_{1}; 2; γ_{1}; q; 2; γ_{1}; q; ⋯; 1; 1; 1]

where again q = γ_{2} + τγ_{2}. □

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Fakultät für Mathematik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany E-mail address: happel@mathematik.tu-chemnitz.de

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