CO– VERSUS CONTRAVARIANT FINITENESS

OF CATEGORIES OF REPRESENTATIONS

B. Huisgen–Zimmermann and S. O. Smalø

1. Introduction and Terminology

Functorial finiteness conditions for certain categories of finitely generated representations of an algebra may have a major impact also on the non-finitely generated representations, as was shown by the authors in [10]. More precisely: Let Λ be an artin algebra, and let P^{∞}(Λ-mod) and P^{∞}(Λ-Mod) be the full subcategories of the categories Λ-mod of finitely generated left Λ-modules and the full module category Λ-Mod, respectively, consisting of the objects of finite projective dimension in either case. Then contravariant finiteness of P^{∞}(Λ-mod) in Λ-mod forces arbitrary modules in P^{∞}(Λ-Mod) to be direct limits of objects in P^{∞}(Λ-mod). When combined with a theorem of Auslander and Reiten [1], this entails that l fin dim Λ = l Fin dim Λ = sup_{1≤i≤n} p dim A_{i} in this case, where A_{1}, …, A_{n} are the minimal right P^{∞}(Λ-mod)-approximations of the simple left Λmodules. (Here l fin dim Λ and l Fin dim Λ are the suprema of the projective dimensions attained on P^{∞}(Λ-mod) and P^{∞}(Λ-Mod), respectively.)

As a byproduct of the described connections, one obtains that \underline{contra}variant finiteness of P^{∞}(Λ-mod) implies \underline{co}variant finiteness of this category in Λ-mod. Indeed, by Crawley-Boevey’s [4, Theorem 4.2], an additive subcategory A of Λ-mod is covariantly finite if and only if the closure \overrightarrow{A} of A under direct limits is closed under direct products as well. As explained above, contravariant finiteness of P^{∞}(Λ-mod) implies P^{∞}(Λ-Mod) = \overrightarrow{P^{∞}}(Λ-mod) and l Fin dim Λ < ∞, which guarantees that \overrightarrow{P^{∞}}(Λ-mod) is closed under direct products.

It is not hard to find examples demonstrating that, in general, contravariant finiteness of P^{∞}(Λ-mod) in Λ-mod is properly stronger than covariant finiteness. In fact, the initial example – due to Igusa, Smalø, and Todorov

Typeset by AMS-TEX

The first author was partially supported by a grant from the NSF while this work was done, while the second author was supported by the US–Norway Fulbright Foundation. Most of this research was carried out while the second author was visiting the University of California at Santa Barbara, and he wishes to thank his coauthor for her kind hospitality.

[11] – of a situation where P^{∞}(Λ-mod) fails to be contravariantly finite already serves to show this. This leaves one wondering whether P^{∞}(Λ-mod) might always be covariantly finite in Λ-mod. The answer is negative, as we show here, but examples are somewhat harder to come by.

The first part of the present paper is devoted to developing a criterion for failure of covariant finiteness of P^{∞}(Λ-mod) and to then applying it to a finite dimensional special biserial algebra (for a definition see under ‘Notation and Terminology’ below). We believe that, in the restricted setting of special biserial algebras Λ, the conditions of this criterion actually provide an equivalent description for failure of covariant finiteness of P^{∞}(Λ-mod). The criterion can be considered as a somewhat weaker twin sibling of the sufficient condition for failure of contravariant finiteness developed by Happel and the first author in [7, Criterion 10]. We remark that, while the concepts of contra- and covariant finiteness are mutually dual, the theories relating them to a prescribed subcategory of Λ-mod are of course not; in particular, the argument backing up the criterion presented here differs substantially from that used to prove [7, Criterion 10].

The homological picture available for algebras Λ having the property that P^{∞}(Λ-mod) is contravariantly finite in Λ-mod naturally raises the question as to how abundant they are. While this condition is known to ‘slice diagonally’ through the standard classes of algebras, the authors conjecture that failure can ‘often’ be fixed in the following sense: Namely, that there exists a sequence Λ_{0} = Λ, Λ_{1}, …, Λ_{m} of Artin algebras such that each Λ_{i} is a one-point extension of Λ_{i−1} and P^{∞}(Λ_{m}-mod) is contravariantly finite in Λ_{m}-mod. Since in each passage from Λ_{i−1} to Λ_{i} both the little and big finitistic dimensions increase by at most 1, this might give a handle on the difference l Fin dim − l fin dim for special classes of algebras. Moreover, the existence of a sequence as above guarantees that l Fin dim Λ < ∞. In general, this process will force one to leave a given ‘nice’ class of algebras, however. The second part of this article is devoted to the explicit construction of such a sequence Λ_{0} = Λ, Λ_{1}, …, Λ_{m} such that Λ is a monomial relation algebra while Λ_{m} cannot be chosen from this class of algebras.

Terminology and Notation.

In the following, Λ will be a split finite dimensional algebra over a field K, i.e., Λ will be of the form KΓ/I for some quiver Γ and an admissible ideal I of the path algebra KΓ. The Jacobson radical of Λ will be denoted by J. For us, the primitive idempotents of Λ will be those which naturally correspond to the vertices of Γ; in fact, we will identify a complete set of primitive idempotents of Λ with the vertices of the quiver.

According to [2], a full subcategory A of Λ-mod is said to be contravariantly finite in Λ-mod if, for each module M in Λ-mod, there exists a homomorphism f : A → M with A ∈ A such that the following sequence of contravariant functors is exact:

Hom_{Λ}(−, A)|_{A} ⟶^{Hom_{Λ}(−,f)} Hom_{Λ}(−, M)|_{A} ⟶ 0;

in other words, exactness of this sequence means that each map g ∈ Hom_{Λ}(B, M) with B ∈ A factors through f. In that case, A is called a right A-approx-imation of M. By [2], the A-approximations of M that have minimal dimension are pairwise isomorphic. This justifies reference to the minimal right A-approximation of M, whenever A is contravariantly finite. The terms covariantly finite and left A-approximation are defined dually.

An algebra Λ is said to be special biserial in case it is isomorphic to a path algebra modulo relations, KΓ/I, with the following properties: Given any vertex e of Γ, at most two arrows enter e and at most two arrows leave e; moreover, for any arrow α in Γ there is at most one arrow β with αβ ∉ I and at most one arrow γ with γα ∉ I.

Recall, moreover, that given two algebras Λ = KΓ/I and Λ′ = KΓ′/I′, the second is called a one-point extension of the first in case Γ′ results from Γ through addition of a single vertex which is a source of Γ′ such that I′ ∩ KΓ = I. For importance and properties of one-point extensions, we refer to [13].

Given paths p and q of Γ, we say that q is a subpath of p if p = p_{2}qp_{1} in KΓ for suitable paths p_{1} and p_{2}. We call q a right (left) subpath of p in case p = p_{2}q (respectively, p = qp_{1}) for suitable paths p_{2} (respectively, p_{1}). The path p is said to start (end) in the arrow α if α is a right (left) subpath of p. Furthermore, we call an element x of a left Λ-module M a top element in case x ∈ M \ JM and ex = x for a primitive idempotent e from our distinguished set.

Finally, we refer the reader to previous work of the authors for their graphing conventions (see, e.g., [7,8,9,10]). The graphs most crucial to the present note are zigzags of the type

[figure: e(1); p_{1}; q_{1}; e(2); p_{2}; q_{2}; ⋯; e(r); p_{r}; ẽ(1); ẽ(2); ẽ(3); ⋯; ẽ(r)]

where the e(i) and ẽ(i) denote vertices and, for each i, the p_{i}, q_{i} denote paths of positive length starting in distinct arrows. That a module M ∈ Λ-mod has the shown graph relative to a sequence x_{1}, …, x_{r} of top elements in particular encodes the following information: x_{i} = e(i)x_{i}, the x_{i} are K-linearly independent modulo JM, each p_{i} has starting point e(i) and endpoint ẽ(i), each q_{i} has starting point e(i) and endpoint { \atop e(i + 1)}, and the multiples q_{i}x_{i} = p_{i+1}x_{i+1}, 1 ≤ i ≤ r − 1, are K-linearly independent elements of the socle of M. The information encoded in the graph guarantees, moreover, that there are no ‘other’ nonzero multiples of the x_{i} apart from those shown; more precisely, the only paths in KΓ not annihilating the element x_{i} ∈ M are the right subpaths of p_{i} and q_{i}.

2. Covariant finiteness of P^{∞}(Λ-mod)

The main goal of this section will be the development and application of conditions which guarantee that a simple left Λ-module fails to have a left P^{∞}(Λ-mod)-approximation. We will start by recalling a result of Auslander and Reiten, including a short alternate proof akin to the arguments of the introduction.

Proposition 1. [1, Proposition 4.2] If l fin dim Λ ≤ 1, then P^{∞}(Λ-mod) is covariantly finite in Λ-mod.

Proof. Suppose l fin dim Λ ≤ 1. By [4, Theorem 4.2], it suffices to prove that each direct product of objects in P^{∞}(Λ-mod) belongs to \overrightarrow{P^{∞}}(Λ-mod). Clearly, each direct product M of objects from P^{∞}(Λ-mod) has projective dimension ≤ 1 in Λ-Mod, and hence [9, Observation 5] shows that M is a direct limit of finitely generated modules of finite projective dimension as required. □

We will see later in this section that the conclusion of Proposition 1 breaks down for algebras of finitistic dimension 2.

Example 2. [11] This is the example exhibited by Igusa, Smalø and Todorov to show that P^{∞}(Λ-mod) may fail to be contravariantly finite, even in the case of a special biserial algebra Λ.

Let Γ be the quiver

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

and I ⊆ KΓ such that, for Λ = KΓ/I, the indecomposable projective left Λ-modules have the following graphs:

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

Here l Fin dim Λ = 1, and hence P^{∞}(Λ-mod) is covariantly finite in Λ-mod by Proposition 1. □

In the sequel, A will denote a full subcategory of Λ-mod.

Criterion 3. Let e(1), …, e(r) be vertices of Γ, and let p_{1}, …, p_{r}, q_{1}, …, q_{r} be 2r paths of positive length in KΓ, none of which is a subpath of any of the others. Moreover, suppose that the following conditions (1) and (2) are satisfied:

(1) For each natural number n, there exists a module M_{n} ∈ A having graph

[figure: x_{n,1}; x_{n,2}; ⋯; x_{n,r}; x_{n,r+1}; ⋯; e(1)′; p_{1}; q_{1}; e(2)′; p_{2}; q_{2}; ⋯; e(r)′; p_{r}; q_{r}; e(1)′; p_{1}; q_{1}; ⋯; e(1); e(2); e(3); ⋯; e(r); e(1); e(2); ⋯; ⋯; x_{n,2r}; x_{n,2r+1}; ⋯; x_{n,nr}; ⋯; e(r)′; p_{r}; q_{r}; e(1)′; p_{1}; q_{1}; ⋯; e(r)′; p_{r}; ⋯; e(r); e(1); e(2); ⋯; e(r)]

relative to a suitable sequence of top elements x_{n,1}, …, x_{n,nr} of M_{n} which are K-linearly independent modulo JM_{n}.

(2) Each module A ∈ A has the following properties:

(i) e(1)(Soc A) ⊆ p_{1}A;

(ii) q_{i}A ∩ (Soc A) ⊆ p_{i+1}A for i < r, and q_{r}A ∩ (Soc A) ⊆ p_{1}A; (iii) If x ∈ A with p_{i}x ∈ Soc A, then q_{i}x ∈ Soc A.

Then S = Λe(1)/Je(1) fails to have a left A-approximation.

Proof. For i > r, let s(i) be the integer in {1, …, r} with i ≡ s(i) (mod r), and define p_{i} := p_{s(i)}, q_{i} := q_{s(i)}.

We assume that, to the contrary of our claim, S = Λe(1)/Je(1) does have a left A-approximation f : S → A with A ∈ A. Choose n > dim_{K} A and write x_{1}, …, x_{nr} for the elements x_{n,1}, …, x_{n,nr} of M_{n}. Moreover, let g : S → M_{n} be the embedding which sends e(1)+Je(1) to p_{1}x_{1}, and choose a homomorphism h : A → M_{n} with g = h ∘ f. Since f(e(1) + Je(1)) ∈ e(1) Soc A, condition 2(i) permits us to pick an element a_{1} ∈ A such that f(e(1)+Je(1)) = p_{1}a_{1}. In view of the equality hf(e(1)+Je(1)) = p_{1}x_{1}, we see that h(a_{1}) = x_{1} + y_{1} with y_{1} ∈ ∑_{j≢1 (mod r)} Λx_{j} + ∑_{j≡1 (mod r)} Jx_{j}. Keep in mind that p_{1} equals p_{r+1} = ⋯ = p_{lr+1}, but is not a subpath of the other p_{i} or any of the q_{i}. Since q_{1} is not a subpath of any of the paths p_{1}, …, p_{r}, q_{2}, …, q_{r} either, we infer that q_{1}h(a_{1}) = q_{1}x_{1}. Due to our choice of a_{1} such that p_{1}a_{1} ∈ Soc A, condition 2(iii) guarantees that q_{1}a_{1} ∈ Soc A as well, in other words, q_{1}a_{1} ∈ q_{1}A ∩ Soc A. Next, condition 2(ii) yields a_{2} ∈ A with p_{2}a_{2} = q_{1}a_{1}. In view of h(p_{2}a_{2}) = q_{1}h(a_{1}) = q_{1}x_{1}, we obtain h(a_{2}) = x_{2} + y_{2} with y_{2} ∈ ∑_{j≢2 (mod r)} Λx_{j} + ∑_{j≡2 (mod r)} Jx_{j}; indeed this follows at once from the nature of the graph of M_{n} and the hypothesis that p_{2} is not a subpath of any of p_{1}, p_{3}, …, p_{r}, q_{1}, …, q_{r}. Now the non-occurrence of q_{2} as a subpath of p_{1}, …, p_{r}, q_{1}, q_{3}, …, q_{r} allows us to deduce q_{2}h(a_{2}) = q_{2}x_{2}, and since p_{2}a_{2} = q_{1}a_{1} ∈ Soc A, we observe that also q_{2}a_{2} ∈ Soc A by 2(iii). Thus 2(ii) in turn provides us with an element a_{3} ∈ A such that p_{3}a_{3} = q_{2}a_{2}. As above, we argue that q_{3}h(a_{3}) = q_{3}x_{3}, and proceeding inductively, we thus obtain a sequence a_{1}, …, a_{nr} of elements in A with the property that q_{i}h(a_{i}) = q_{i}x_{i} for 1 ≤ i ≤ nr. Since the elements q_{1}x_{1}, …, q_{nr}x_{nr} of M_{n} are K-linearly independent by hypothesis, so are a_{1}, …, a_{nr}. But this contradicts our choice of n exceeding dim_{K} A and proves the criterion. □

As our argument makes clear, the requirement that none of the p_{i}, q_{i} is a subpath of any other as called for in the criterion can certainly be weakened. Other than that, the chain reaction exhibited in the proof of Criterion 3 appears prototypical for the failure of a simple module S to have a left P^{∞}(Λ-mod)-approximation. In fact, specializing to special biserial algebras, we believe the answer to the following question to be positive.

Problem 4. Suppose S = Λe(1)Je(1) is a simple left module over a finite dimensional special biserial algebra Λ = KΓ/I. If S does not have a left P^{∞}(Λ-mod)-approximation, do there exist paths p_{1}, …, p_{r}, q_{1}, …, q_{r} in KΓ satisfying the conditions (1) and (2) of Criterion 3?

On the other hand, we point out that it is not known whether failure of covariant finiteness of P^{∞}(Λ-mod) in Λ-mod implies that one of the \underline{simple} left Λ-modules is devoid of a left P^{∞}(Λ-mod)-approximation. This is in contrast with our level of information on contravariant finiteness of P^{∞}(Λ-mod). Indeed, as was shown by Auslander and Reiten in [1], P^{∞}(Λ-mod) is contravariantly finite in Λ-mod provided that all simple Λ-modules have right P^{∞}(Λ-mod)-approximations. We therefore propose

Problem 5. Is P^{∞}(Λ-mod) covariantly finite in Λ-mod in case each simple left Λ-module has a left P^{∞}(Λ-mod)-approximation?

Criterion 3 is particularly suited to the situation where A = P^{∞}(Λ-mod) and Λ is a finite dimensional special biserial algebra (see §1 for a definition). The representation theory of the finitely generated modules over such algebras is exceptionally well understood (see [3,5,6,12,14]). In fact, the indecomposable objects of Λ-mod fall into two classes, ‘bands’ and ‘strings’. All we presently need to know about bands is the following: If B ∈ Λ-mod is a band, e ∈ Λ a primitive idempotent and x ∈ e(Soc B), then x ∈ ∑_{i} u_{i}B ∩ v_{i}B for pairs (u_{i}, v_{i}) of paths of positive length ending in e such that, moreover, u_{i} = α_{i}u′_{i} and v_{i} = β_{i}v′_{i} with distinct arrows α_{i} and β_{i}. The strings, on the other hand, are precisely the objects in Λ-mod having graphs of the form

[figure: •; u_{1}; v_{1}; •; u_{2}; v_{2}; ⋯; •; u_{m}; v_{m}; •; •; •; ⋯; •; •]

relative to a suitable sequence of top elements, such that, for each i, the paths u_{i}, v_{i} start in distinct arrows, and the partners of each pair (v_{i}, u_{i+1}) end in distinct arrows.

Example 6. A finite dimensional special biserial algebra Λ for which P^{∞}(Λ-mod) fails to be covariantly finite in Λ-mod: Let Λ = KΓ/I, where Γ is the quiver

[figure: 2; γ; ρ; 5; χ; ψ; 1; α; β; 4; ϵ; 8; ϵ′; 6; τ; 7; τ′; 3; δ; σ]

and I is generated by

γα − δβ, ρχ − σψ, γχ, ρα, δψ, σβ, ϵγ, ϵ′ϵ, (ϵ′)^{2}, τ′τ, (τ′)^{2}, τσ.

Then the graphs of the indecomposable projective left Λ-modules are

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

We will use Criterion 3 to show that S = Λe_{3}/Je_{3} does not have a left P^{∞}(Λ-mod)-approximation. For that purpose, let r = 2, e(1) = e_{3}, e(2) = e_{5}, p_{1} = β, and p_{2} = χ, q_{1} = α, q_{2} = ψ. Then condition 1 of our criterion is satisfied by the modules M_{n} with graphs

[figure: x_{n,1}; x_{n,2}; x_{n,3}; x_{n,4}; ⋯; x_{n,2n}; 1; β; α; 5; χ; ψ; 1; 5; ⋯; 5; χ; 3; 2; 3; 2; 3; ⋯; 2]

Indeed, one readily verifies that the first syzygy Ω^{1}(M_{n}) of M_{n} has graph

[figure: 2; 3; 2; ⋯; 2; 3; 4; 6; 4; 6; ⋯; 4; 6]

and the second syzygy Ω^{2}(M_{n}) is a direct sum of modules with graphs

6  4 and 7  8

This shows that p dim M_{n} = 2; in particular, M_{n} ∈ P^{∞}(Λ-mod).

It is obvious that for every module A ∈ Λ-mod and every element x ∈ A the implications (βx ∈ Soc A ⟹ αx ∈ Soc A) and (χx ∈ Soc A ⟹ ψx ∈ Soc A) hold. So condition 2(iii) of Criterion 3 is met as well.

To check the remaining conditions under (2), we start by observing that the only nontrivial paths of KΓ ending in e_{3} are the arrows β and ψ, while the only nontrivial paths ending in e_{2} are the arrows α and χ. Given our comments on bands, this implies that, for each band B ∈ Λ-mod, we have e_{3}(Soc B) ⊆ βB ∩ψB and, in particular, ψB ∩(Soc B) ⊆ βB. Analogously, αB ∩(Soc B) ⊆ e_{2}(Soc B) ⊆ χB. Hence, we may focus our attention on the situation where A ∈ P^{∞}(Λ-mod) is a string with S_{3} ⊆ Soc A. Note that A cannot be simple, since p dim S_{3} = ∞. The only possibility for a copy of S_{3} ⊆ Soc A not to belong to βA is that of a string A having graph

[figure: 5; ψ; 1; ⋯; 3; 2; ⋯]

In that case Ω^{1}(A) would have a graph of one of the following types

[figure: 2; 4; •; 2; ⋯; or; or; 6; 6; 4; ⋯]

and Ω^{2}(A) would be a copy of S_{4} ⊕ S_{7} in the first case, a copy of S_{8} in the second, and have a direct summand isomorphic to S_{7} in the third. In all of these cases, we would have p dim Ω^{2}(A) = ∞ contradicting our choice of A in P^{∞}(Λ-mod). This proves e_{3}(Soc A) ⊆ βA and thus 2(i).

For 2(ii) it suffices to observe that any A ∈ P^{∞}(Λ-mod) has the stronger property that e_{2}(Soc A) ⊆ χA. The argument is analogous to the one we just completed for e_{3} and β.

Thus Criterion 3 implies that S = Λe_{3}/Je_{3} does not have a left P^{∞}(Λ-mod)- approximation. □

We remark that l Fin dim Λ = 2 in the preceding example, which shows that Auslander-Reiten’s Proposition 1 cannot be extended to the case of finitistic dimensions exceeding 1.

3. Curing failure of contravariant finiteness of P^{∞}(Λ-mod)

We conjecture that, for any monomial relation algebra and for any special biserial algebra Λ, there exists a sequence of one-point extensions Λ = Λ_{0}, …, Λ_{m} = Δ such that P^{∞}(Δ-mod) is contravariantly finite in Δ-mod. At this point, our conviction is based mainly on a long list of examples. One of our most interesting examples shows that, in general, one cannot expect Δ to retain the ‘good’ properties of Λ in this process, in other words, a ‘cure’ for failure of contravariant finiteness of P^{∞}(Λ-mod) by successive one-point extensions, will usually involve a trade-off. Here we will construct an example of a monomial relation algebra Λ, together with a sequence Λ = Λ_{0}, Λ_{1}, Λ_{2} = Δ as above such that Δ cannot be chosen within the class of monomial relation algebras. Simultaneously, this example will illustrate the potential intricacy of the structure of the minimal right P^{∞}(Δ-mod)- approximations of the simple modules.

Example 7. Let Λ = KΓ/I, where Γ is the quiver

[figure: 7; γ_{1}; 5; β_{1}; 3; ρ; 1; α; 2; δ; ϵ; 6; β_{2}; 4; σ; 8; γ_{2}]

and the ideal I ⊆ KΓ is generated by

δα, ϵα, ϵβ_{i} (i = 1, 2), δγ_{i} (i = 1, 2), ρδ, σϵ, ρ^{2}, σ^{2}.

This yields indecomposable projective left Λ-modules with graphs

[figure: 1; α; 2; δ; ϵ; 3; ρ; 4; σ; 5; β_{1}; 6; β_{2}; 7; γ_{1}; 8; γ_{2}; 2; 3; 4; 3; 4; 2; δ; 2; δ; 2; ϵ; 2; ϵ; 3; 3; 4; 4]

To see that P^{∞}(Λ-mod) fails to be contravariantly finite in Λ-mod, more precisely, that Λe_{1}/Je_{1} fails to have a right P^{∞}(Λ-mod)-approximation, we exhibit a P^{∞}(Λ-mod)-phantom of infinite K-dimension for Λe_{1}/Je_{1} (see [7, Definition 5 and Theorem 9]). The routine check that the following module H is indeed such a phantom is left to the reader: H = \underrightarrow{lim} H_{n} where

H = (Λz ⊕ ⨁_{i=1}^{n} Λx_{i} ⊕ ⨁_{i=1}^{n} Λy_{i}) / U_{n},

with z = e_{1}, x_{2m−1} = e_{5}, x_{2m} = e_{6}, y_{2m−1} = e_{7}, y_{2m} = e_{8} for m ≥ 1, and

U_{n} = Λ(αz − β_{1}x_{1} − γ_{1}y_{1}) + ∑_{i=1}^{n−1} Λ(γ̃_{i}y_{i} − β̃_{i+1}x_{i+1} − γ̃_{i+1}y_{i+1}),

with γ̃_{i} equal to γ_{1} or γ_{2}, depending on whether i is odd or even, and β̃_{i} equal to β_{1} or β_{2}, depending on whether i is odd or even. The modules H_{n} can be pictured via graphs of the form

z x_{1} y_{1} x_{2} y_{2} x_{3} y_{3} x_{4} y_{4} ⋯ y^{n−1} x_{n} y_{n} 1_{α} 5_{β_{1}} 7_{γ_{1}} 6_{β_{2}} 8_{γ_{2}} 5  7  6  8  ⋯  •  •  • 2  2  2  2  2  2  2  2  2  ⋯  2  2  2

relative to the top elements z, x_{1}, …, x_{n}, y_{1}, …, y_{n}, where we extend our graphing conventions as follows: The dotted loop around the vertices labeled ‘2’ which represent αz, β_{1}x_{1}, γ_{1}y_{1} indicates that any two of the three listed vectors are K-linearly independent while dim_{K}(Kαz+Kβ_{1}x_{1}+ Kγ_{1}y_{1}) = 2. The same holds for the additional triples γ̃_{i}y_{i}, β̃_{i}x_{i+1}, γ̃_{i+1}y_{i+1} surrounded by loops. Note that Ω^{1}(H_{n}) ≅ (Λe_{2})^{n} for n ∈ ℕ; in particular, H_{n} ∈ P^{∞}(Λ-mod) for n ∈ ℕ.

Let Γ_{1} be the quiver obtained from Γ by adding a single vertex, labeled 9, and two arrows leaving 9, namely χ_{1} : 9 → 5 and χ_{2} : 9 → 6. Moreover, let Λ_{1} = KΓ_{1}/I_{1}, where the ideal I_{1} ⊆ KΓ_{1} is generated by I and the relation β_{1}χ_{1} − β_{2}χ_{2}.

Next let Γ_{2} be the quiver obtained from Γ_{1} by adding a single vertex, 10, and two arrows leaving 10, namely ψ_{1} : 10 → 7 and ψ_{2} : 10 → 8. Now Δ = Λ_{2} = KΓ_{2}/I_{2}, where I_{2} is generated by I_{1} and the relation γ_{1}ψ_{1} − γ_{2}ψ_{2} ∈ KΓ_{2}.

Clearly, Λ_{1} is a one-point extension of Λ = Λ_{0}, and Δ = Λ_{2} is a one-point extension of Λ_{1}. The ‘new’ indecomposable projective left Δ-modules have graphs

[figure: 9; χ_{1}; χ_{2}; 10; ψ_{1}; ψ_{2}; 5; β_{1}; 6; β_{2}; and; 7; γ_{1}; 8; γ_{2}; 2; δ; 2; ϵ; 3; 4]

Note that Δe_{i} = Λe_{i} for 1 ≤ i ≤ 8.

One can verify that P^{∞}(Δ-mod) is contravariantly finite in Δ-mod by exhibiting right P^{∞}(Δ-mod)-approximations of the simple left Δ-modules S_{i} = Δe_{i}/J(Δ)e_{i} for 1 ≤ i ≤ 10. It is comparatively easy to see that the following are the (minimal) right P^{∞}(Δ-mod)-approximations of S_{2}, …, S_{10}: Namely, Δe_{i} for i = 2, 3, 4, and

[figure: 5; 7; 5; 8; 6; 7; 6; 8; ⨁; ⨁; 2; 2; 2; 2; 7; 5; 7; 6; 8; 5; 8; 6; ⨁; ⨁; 2; 2; 2; 2; 9; 9; 10; 10; ⨁; ⨁; 5; 6; 7; 8]

for i = 5, 6, …, 10, respectively. We will sketch an argument backing the claim that the canonical epimorphism f : A_{1} → S_{1} with

A_{1} = (Δe_{1} ⊕ Δe_{9} ⊕ Δe_{10})/Δ(α, β_{1}χ_{1}, γ_{1}ψ_{1})

is a right P^{∞}(Δ-mod)-approximation of S_{1}. To buttress intuition, start by noting that A_{1} has graph

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

with the above convention for loops. Observe, moreover, that Ω_{Δ}^{1}(A_{1}) ≅ Δe_{2}, whence A_{1} ∈ P^{∞}(Δ-mod).

We will sketch an argument showing that each epimorphism g : M → S_{1} with M ∈ P^{∞}(Δ-mod) factors through f. It is clearly harmless to assume that M is indecomposable. Moreover, it suffices to consider the case where ker(g) does not contain any nonzero submodules in P^{∞}(Δ-mod); indeed, given U ⊆ M with U ∈ P^{∞}(Δ-mod), it is enough to factor the induced map \overline{g} : M/U → S_{1} through f. In particular, this means that M does not contain any submodules isomorphic to Δe_{i}, with i ≥ 2, nor any submodules with graphs of type

10  10  11  11 or  or  or 6  7  8  9

As a consequence, we can zero in on the structure of M as follows: Let A be the class of Δ-modules isomorphic to Δe_{1}, B the class of modules isomorphic to one of the Δ-modules with graphs

[figure: 5; 6; 9; or; or; 5; 6; 2; 2; 2]

and, finally, C the class of those Δ-modules which have one of the graphs

[figure: 7; 8; 10; or; or; 7; 8; 2; 2; 2]

Our assumptions on M guarantee that, up to isomorphism, M = X/Y ∈ P^{∞}(Δ-mod), where

X = ⨁_{1≤i≤m_{1}} A_{i} ⊕ ⨁_{1≤i≤m_{2}} B_{i} ⊕ ⨁_{1≤i≤m_{3}} C_{i}

with A_{i} ∈ A, B_{i} ∈ B, C_{i} ∈ C and Y ⊆ Soc X ≅ S_{2}^{m_{1}+m_{2}+m_{3}}.

Let a_{i}, b_{i}, c_{i} be top elements of A_{i}, B_{i}, C_{i}, respectively, and let a′_{i} = αa_{i}, b′_{i} = (β_{1} + β_{2} + β_{1}χ_{1})b_{i}, c′_{i} = (γ_{1} + γ_{2} + γ_{1}ψ_{1})c_{i}. Moreover, let

⨁_{1≤i≤m_{1}} Δ\overline{a}_{i} ⊕ ⨁_{1≤i≤m_{2}} Δ\overline{b}_{i} ⊕ ⨁_{1≤i≤m_{3}} Δ\overline{c}_{i} ⟶ X/Y

be the obvious projective cover of X/Y mapping \overline{a}_{i}, \overline{b}_{i}, \overline{c}_{i} to a_{i}, b_{i}, c_{i} respectively. Then

Soc X = ⨁_{1≤i≤m_{1}} Δa′_{i} ⊕ ⨁_{1≤i≤m_{2}} Δb′_{i} ⊕ ⨁_{1≤i≤m_{3}} Δc′_{i},

and we can therefore write Y in the form ⨁_{1≤h≤t} Δy_{h} with

y_{h} = ∑_{1≤i≤m_{1}} k_{hi}a′_{i} + ∑_{1≤i≤m_{2}} l_{hi}b′_{i} + ∑_{1≤i≤m_{3}} m_{hi}c′_{i},

where k_{hi}, l_{hi}, m_{hi} ∈ K such that Δy_{h} ≅ Δe_{2} = Λe_{2} for each h. Let k_{h} = (k_{hi})_{1≤i≤m_{1}} ∈ K^{m_{1}}, l_{h} = (l_{hi})_{1≤i≤m_{2}} ∈ K^{m_{2}}, and m_{h} = (m_{hi})_{1≤i≤m_{3}} in K^{m_{3}}. Then the vectors l_{1}, …, l_{t} ∈ K^{m_{2}} are K-linearly independent; indeed if we had ∑_{1≤h≤t} d_{h}l_{h} = 0 with d_{h} ∈ K not all zero, we would obtain a top element z of Ω_{Δ}^{1}(M) with the property that δz = 0, namely z = ∑_{1≤h≤t} d_{h}z_{h}, where

z_{h} = ∑_{1≤i≤m_{1}} k_{hi}α\overline{a}_{i}+ ∑_{1≤i≤m_{2}} l_{hi}(β_{1}+β_{2}+β_{1}χ_{1})\overline{b}_{i}+ ∑_{1≤i≤m_{3}} m_{hi}(γ_{1}+γ_{2}+γ_{1}ψ_{1})\overline{c}_{i}

in Ω_{Δ}^{1}(M). This would place a direct summand isomorphic to S_{3} into Ω_{Δ}^{1}(M), which – in view of p dim_{Δ} Δe_{3}/J(Δe_{3}) = p dim Λe_{3}/Je_{3} = ∞ – is incompatible with p dim_{Δ}(M) < ∞. Similarly m_{1}, …, m_{t} in K^{m_{3}} are linearly independent, since otherwise we would obtain a direct summand isomorphic to S_{4} in Ω_{Δ}^{1}(M). If we set g̃(a_{i}) = \overline{(r_{i}e_{1}, 0, 0)} ∈ A_{1}, where f(a_{i}) = \overline{r_{i}e_{1}} with r_{i} ∈ K, the above independence information allows us to extend the assignment g̃ to a homomorphism g̃ : M → A. Any such homomorphism clearly satisfies f ∘ g̃ = g.

Now let Λ = R_{0}, R_{1}, …, R_{m} = R be successive one-point extensions such that R is a monomial relation algebra. We leave the justification of our claim that P^{∞}(R-mod) fails to be contravariantly finite in R-mod as an exercise, but provide hints of the underlying ideas.

1.) Due to the fact that R is a monomial relation algebra obtained from Λ via one-point extensions, the following holds: Whenever a left R-module M has a submodule N with graph

[figure: e(1); e(2); e(1); e(2); e(3); or; 2; 2; 2; 2]

where e(1), e(2), e(3) are distinct vertices in {e_{1}, e_{5}, e_{6}, e_{7}, e_{8}}, the first syzygy Ω_{R}^{1}(M) of M has a top element x of type e_{2}; moreover, if e(1), e(2), e(3) ∈ {e_{1}, e_{5}, e_{6}}, we can choose x such that Rx has graph

2  2 • or 3

and if e(1), e(2), e(3) ∈ {e_{1}, e_{7}, e_{8}}, we can choose x such that Rx has graph

2  2 • or 4

In each of these two situations, p dim_{R}(M) = ∞.

2.) If there were a P^{∞}(R-mod)-approximation B_{1} of Re_{1}/J(R)e_{1} = Λe_{1}/Je_{1} = S_{1}, then all homomorphisms in Hom_{R}(H_{n}, S_{1}) = Hom_{Λ}(H_{n}, S_{1}) with H_{n} as above would factor through B_{1}, because H_{n} ∈ P^{∞}(R-mod). Using the first part, one would deduce the existence of a submodule of B_{1} with graph

1 5 7 6 8 2 2 2 2 2

and then proceed to show that H = \underrightarrow{lim} H_{n} would still be a P^{∞}(R-mod)- phantom for S_{1}.

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

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

Department of Mathematics, The Norwegian University for Science and Technology, 7055 Dragvoll, Norway

   E-mail address: sverre@matstat.unit.no