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A Homological Bridge Between Finite and Infinite Dimensional Representations of Algebras

B. Huisgen-Zimmermann and S. O. Smalø

Abstract. Given a finite dimensional algebra Λ, we show that a frequently satisfied finiteness condition for the category P^{∞}(Λ-mod) of all finitely generated (left) Λ-modules of finite projective dimension, namely contravariant finiteness of P^{∞}(Λ-mod) in Λ-mod, forces arbitrary modules of finite projective dimension to be direct limits of objects in P^{∞}(Λ-mod). Among numerous applications, this yields an encompassing sufficient condition for the validity of the first finitistic dimension conjecture, that is, for the little finitistic dimension of Λ to coincide with the big (this is well-known to fail over finite dimensional algebras in general).

A Homological Bridge Between Finite and Infinite Dimensional Representations of Algebras

B. Huisgen-Zimmermann and S. O. Smalø

1. Introduction

Let Λ be an Artin algebra – the reader may, for example, think of a finite dimensional algebra over a field – and let Λ-mod be the category of all finitely generated left Λ-modules, Λ-Mod the category of all left Λ-modules. The full subcategories consisting of the objects of finite projective dimension in Λ-mod and Λ-Mod are denoted by P^{∞}(Λ-mod) and P^{∞}(Λ-Mod), respectively. Our interest is focused on the following two homological dimensions of Λ: the left little finitistic dimension, ℓ fin.dim Λ, which is the supremum of the projective dimensions attained on the objects in P^{∞}(Λ-mod), and the left big finitistic dimension, ℓ Fin.dim Λ, which is defined correspondingly based on P^{∞}(Λ-Mod). For several decades, the smoothest possible connection between P^{∞}(Λ-mod) and P^{∞}(Λ-Mod), namely equality of the little and big finitistic dimensions, had been conjectured to hold, at least when Λ is finite dimensional over a field. However, in 1991 this equality was shown to fail, even over finite dimensional monomial relation algebras [13]. Our goal here is to prove that a certain, very frequently satisfied, finiteness condition on P^{∞}(Λ-mod) entails that each object in the big category P^{∞}(Λ-Mod) is a direct limit of objects in the small category P^{∞}(Λ-mod); in particular, this condition implies that the two finitistic dimensions coincide.

Before we state this result with greater precision, we give a brief overview over previously established connections between Λ-mod and Λ-Mod. In the late seventies, Ringel used the thorough understanding of Λ-mod for tame hereditary algebras Λ which was available at that point to give detailed descriptions of representations in Λ-Mod in this situation [19]. We quote from his introduction: “The recent progress in the representation theory of finite dimensional algebras was limited mainly to the modules of finite length and one would be interested to know in which way the structure of the modules of finite length determines the behavior of arbitrary modules. Two results of this type are known...”. The results he subsequently refers to are as follows: If Λ has finite representation type, then all objects in Λ-Mod are direct sums of objects from Λ-mod (see [1] and [20]); if, on the other hand, Λ fails to have finite representation type, Λ-Mod \ Λ-mod contains indecomposable objects [2]. In the meantime, Crawley-Boevey added several striking results to this list by showing that, for a finite dimensional algebra Λ over an algebraically closed base field, certain non-finitely generated modules, called generic modules, completely determine the representation type of Λ; they are the non-finitely generated indecomposable modules which have finite length over their endomorphism rings (see [6] and [7]). In particular, he proved that Λ has infinite representation type if and only if there exist generic objects in Λ-Mod, tame representation type if and only if, for each dimension d, there exist only finitely many generic Λ-modules having length d over their endomorphism rings; finally, Λ has domestic type precisely when there are only finitely many isomorphism classes of generic modules in Λ-Mod altogether.

The work of the first author was partially supported by an NSF grant, while that of the second author was partially supported by a Fulbright grant.

In 1979, Auslander and Smalø introduced the concept of contravariant finiteness of a full subcategory A of Λ-mod [4]. A is said to be contravariantly finite in Λ-mod in case each object M in Λ-mod has an A-approximation in the following sense: there exists a homomorphism f : A → M with A ∈ A such that each map in Hom_{Λ}(B, M) with B ∈ A factors through f. One of the reasons why this concept is of major importance to the homology of a finite dimensional algebra lies in the fact that, if P^{∞}(Λ-mod) is contravariantly finite in Λ-mod, then the minimal P^{∞}(Λ-mod)-approximations of the simple left Λ-modules – A_{1}, …, A_{n} say – constitute the basic building blocks for arbitrary representations in P^{∞}(Λ-mod). In fact, due to Auslander and Reiten [3], a module M belongs to P^{∞}(Λ-mod) if and only if M is a direct summand of a module N having a filtration N = N_{0} ⊃ N_{1} ⊃ … ⊃ N_{m} = 0 such that all consecutive factors N_{i}/N_{i+1} belong to {A_{1}, …, A_{n}}. In particular, this structure result implies of course that ℓ fin.dim Λ = sup{p dim A_{i} | i = 1, …, n}.

Our main result (Theorem 4.4) states that contravariant finiteness of P^{∞}(Λ-mod) in Λ-mod forces each left Λ-module of finite projective dimension to be a direct limit of finitely generated modules of finite projective dimension. More strongly, each object in P^{∞}(Λ-Mod) is then a direct limit of finitely generated modules having filtrations with consecutive factors in {A_{1}, …, A_{n}}. As a consequence, contravariant finiteness of P^{∞}(Λ-mod) entails the equality ℓ fin.dim Λ = ℓ Fin.dim Λ. This answers in the positive a question left open in [10]. As an extra bonus, the result tells us where to look if we wish to determine the big finitistic dimension, a notorious quandary in general. In view of the preceding paragraph, we can obviously conclude that ℓ Fin.dim Λ = sup{p dim A_{i} | i = 1, …, n} in the contravariantly finite case. We illustrate a ‘typical’ computation of this ilk with an example. Another application of our main theorem, combining it with a result of Crawley-Boevey, was pointed out to us by H. Krause. Namely, contravariant finiteness of P^{∞}(Λ-mod) in Λ-mod implies covariant finiteness and thus secures the existence of almost split sequences in P^{∞}(Λ-mod). On the other hand, the conclusion of our result is not left-right symmetric. In the final section of this paper, we present a finite dimensional monomial relation algebra Λ such that P^{∞} is contravariantly finite on one side, while the big and little finitistic dimensions of Λ differ on the other.

Some of our tools should be of independent interest. In Section 2, we show that, over a left noetherian ring, each left module M is the directed union of those countably generated submodules which have projective dimensions bounded above by that of M. As a consequence, the left big finitistic dimension of a left artinian ring equals the supremum of those finite projective dimensions which are attained on countably generated left modules. In Section 3, we collect some closure properties of the category Add M consisting of all direct sums of direct summands of a fixed module M. In particular, we explore closure of such categories under direct limits.

Acknowledgements. This paper was written while the second author was visiting the University of California at Santa Barbara, and he would like to thank the members of the mathematics department of UCSB, and in particular his coauthor, for the hospitality he received during his stay there. Both authors are indebted to Hans Sverre Smalø for an excellent job of typesetting and graphing.

2. The big finitistic dimension equals the countably generated finitistic dimension

The primary goal of this section is to show that, for each left artinian ring R, the big finitistic dimension, ℓ Fin.dim R, coincides with the supremum of those finite projective dimensions which are attained on countably generated left R-modules. The argument is an off-spring of Kaplansky’s classical theorem [16] stating that every projective R-module is a direct sum of countably generated components.

Recall that a module X is said to be countably presented if there exists an exact sequence

R^{(K)} → R^{(L)} → X → 0

with |K|, |L| ≤ ℵ_{0}. Clearly, every left noetherian ring has the property that each of its countably generated left modules is countably presented; the same is true if the base ring R is countable or a countably generated algebra over a field.

Proposition 2.1 Let R be a ring with the property that each countably generated left Rmodule is countably presented. Then each left R-module M is the directed union of those countably generated submodules which have projective dimensions bounded above by that of M.

Remarks. 1. This proposition is akin to Corollary 3.2.5 of [18], but gives more structural information about the module M. We thank H. Krause for bringing this reference to our attention.

2. Observe that, in the conclusion of Proposition 2.1, the attribute ‘countably generated’ cannot be replaced by ‘finitely generated’. In fact, not even over a finite dimensional algebra, need modules of finite projective dimension be direct limits of finitely generated modules of finite projective dimension. This is an immediate consequence of the fact that the big finitistic dimension may exceed the little.

Proof of the proposition: We may assume that p dim M < ∞, say p dim M = m. It clearly suffices to show that each countable subset of M is contained in a countably generated submodule M′ with p dim M′ ≤ m. Let

\xymatrix{0 \ar[r] & P_{m} \ar[r]^{f_{m}} & P_{m−1} \ar[r]^{f_{m−1}} & ⋯ \ar[r] & P_{1} \ar[r]^{f_{1}} & P_{0} \ar[r]^{f_{0}} & M \ar[r] & 0}

be a projective resolution of M. By Kaplansky’s theorem [16], each P_{i} is of the form P_{i} = ⨁_{j∈A_{i}} P_{ij}, where all of the P_{ij}’s are countably generated.

Given a countable subset U of M, choose a countable subset B_{0}^{(1)} ⊆ A_{0} such that f_{0}(⨁_{j∈B_{0}^{(1)}}P_{0j}) contains U. Then the kernel of the restriction of f_{0} to ⨁_{j∈B_{0}^{(1)}} P_{0j} is in turn countably generated by hypothesis, which permits us to pick a countable set B_{1}^{(1)} ⊆ A_{1}, with the property that

Ker(f_{0}) ∩ ( ⨁_{j∈B_{0}^{(1)}} P_{0j}) ⊆ f_{1}( ⨁_{j∈B_{1}^{(1)}} P_{ij}).

An obvious induction further yields countable subsets B_{i}^{(1)} ⊆ A_{i} for 1 ≤ i ≤ m such that

Ker(f_{i−1}) ∩ ( ⨁_{j∈B_{i−1}^{(1)}} P_{i−1,j}) ⊆ f_{i}( ⨁_{j∈B_{i}^{(1)}} P_{ij})

for 1 ≤ i ≤ m.

Now set B_{m}^{(2)} = B_{m}^{(1)}. In view of the fact that the image

f_{m}( ⨁_{j∈B_{m}^{(2)}} P_{mj}) ⊆ Ker f_{m−1}

is countably generated, we can find a countable subset B_{m−1}^{(2)} ⊆ A_{m−1} containing B_{m−1}^{(1)} such that

f_{m}( ⨁_{j∈B_{m}^{(2)}} P_{mj}) ⊆ Ker(f_{m−1}) ∩ ( ⨁_{j∈B_{m−1}^{(2)}} P_{m−1,j}).

An induction analogous to the preceding one then yields countable subsets B_{i}^{(2)} ⊆ A_{i}, 0 ≤ i ≤ m such that B_{i}^{(1)} ⊆ B_{i}^{(2)} and

f_{i+1}( ⨁_{j∈B_{i+1}^{(2)}} P_{ij}) ⊆ Ker(f_{i}) ∩ ( ⨁_{j∈B_{i}^{(2)}} P_{ij})

for 0 ≤ i ≤ m − 1.

Next set B_{0}^{(3)} = B_{0}^{(2)}, and continue. We iterate these m-step inductions moving back and forth along the given projective resolution of M, and an induction on this level will supply us, for each k ∈ N, with countable sets B_{0}^{(k)}, ⋯, B_{m}^{(k)} such that B_{i}^{(k−1)} ⊆ B_{i}^{(k)} ⊆ A_{i} for 0 ≤ i ≤ m, having the additional properties that

Ker(f_{i−1}) ∩ ( ⨁_{j∈B_{i−1}^{(k)}} P_{i−1,j}) ⊆ f_{i}( ⨁_{j∈B_{i}^{(k)}} P_{ij})

for 1 ≤ i ≤ m and odd k, as well as

f_{i+1}( ⨁_{j∈B_{i+1}^{(k)}} P_{i+1,j}) ⊆ Ker(f_{i}) ∩ ( ⨁_{j∈B_{i}^{(k)}} P_{i,j})

for 0 ≤ i ≤ m − 1 and even k.

Finally, we set B_{i} = ∪_{k∈N}B_{i}^{(k)} ⊆ A_{i} for 0 ≤ i ≤ m, define M′ to be the countably generated submodule f_{0}(⨁_{j∈B_{0}} P_{0j}) of M, and observe that

\xymatrix{0 \ar[r] & ⨁_{j∈B_{m}} P_{mj} \ar[r]^{f_{m}} & ⋯ \ar[r] & ⨁_{j∈B_{1}} P_{1j} \ar[r]^{f_{1}} & ⨁_{j∈B_{0}} P_{0j} \ar[r]^{f_{0}} & M′ \ar[r] & 0}

is a projective resolution of M′. This guarantees that p dim M′ ≤ m as required. □

Corollary 2.2 If R is a left artinian ring, then ℓ Fin.dim R equals

sup{p dim M | M a countably generated left R−module with p dim M < ∞}.

Proof: Let N be any left R-module of finite projective dimension. Since R satisfies the hypothesis of Proposition 2.1, we have N = \underrightarrow{lim}_{i∈I}N_{i}, where (N_{i})_{i∈I} is a directed family of countably generated modules with p dim N_{i} ≤ p dim N for i ∈ I. In view of the fact that the functor Tor commutes with direct limits, this implies that the flat dimension of N is bounded above by the supremum of the flat dimensions of the N_{i}. But our hypothesis on R makes the flat dimension equal to the projective dimension and thus completes the argument. □

3. Remarks on the category Add(M)

Given a module M, we denote by Add(M) the full subcategory of R-Mod having as objects the modules which are isomorphic to arbitrary direct sums of direct summands of M. The main purpose of this section is to assemble a number of properties of Add(M) for a Σ-pure injective module M, to be used towards our main theorem. (That M be Σ-pure injective means that all direct sums of copies of M are pure injective.) However, due to the fact that this category holds interest in its own right, we will go a little beyond the requirements of the following section.

Start by recalling that every finitely generated module over an Artin algebra is Σ-pure injective, since of finite length over its endomorphism ring. On the other hand, Σ-pure injectives are far from being finitely generated over the base ring, in general; in particular, any generic module has this property. More generally, recall that the results in [22] and [9] give an equivalent characterization of Σ-pure injectivity of an R-module M in terms of a descending chain condition for certain submodules of M over its endomorphism ring, a condition which is obviously inherited by pure R-submodules of M; as a consequence, Σ-pure injectivity is passed on to pure submodules.

Our first observations address closure of the category Add(M) under direct limits (we follow the convention of reserving the term ‘direct limit’ for colimits over directed index sets). A twin sibling of the following fact was proved by Lenzing in [17], where it is shown that, for any finitely presented module M of finite length, the category Add(M) has the mentioned closure property.

Observation 3.1 Let R be any ring and M a Σ-pure injective left R-module. Then every directed system of objects in Add(M) attains its direct limit in Add(M).

Proof: Let (A_{i}, f_{ij})_{i,j∈I,i≤j} be any directed system of objects A_{i} in Add(M), and denote by A its direct limit. Moreover, let ι_{j} : A_{j} → ⨁_{i∈I} A_{i} be the canonical embeddings. It is well known that the short exact sequence

0 → ∑_{i,j∈I,i<j}(ι_{j}f_{ij} − ι_{i})(A_{i}) → ⨁_{i∈I} A_{i} → A → 0

is pure. Its middle term being Σ-pure injective by hypothesis, the sequence therefore splits. Now each Σ-pure injective module is a direct summand of components, each of which has a local endomorphism ring [22], p. 1100, and consequently the Krull-Remak-Schmidt-Azumaya Theorem guarantees that all terms of the above exact sequence in turn belong to Add(M). □

Interestingly, the question whether Add(M) is closed under direct limits can be decided by testing solely well-ordered chains in general. We record this fact as

Observation 3.2 Given any left R-module M, the category Add(M) is closed under arbitrary direct limits, provided that it is closed under direct limits over well-ordered index sets.

Proof: Again we let (A_{i}, f_{ij})_{i,j∈I,i≤j} be a directed system of objects in Add(M) and denote its direct limit by A. If the index set I is finite, our claim is trivial, so let us suppose that |I| = ℵ_{α} for some ordinal number α. We will prove the observation by a transfinite induction on α. In case α = 0, we may clearly assume that I = {i_{1}, i_{2}, i_{3}, …} and that i_{j} < i_{k} in the partial order of I whenever j < k. Our hypothesis therefore guarantees that A is in turn an object of Add(M).

Now we suppose that α > 0 and that for all ordinal numbers β < α our claim is true; in other words, we assume that all direct limits of directed systems of objects in Add(M) extending over index sets of cardinalities less than ℵ_{α} belong to Add(M). We write Ω_{α} for the first ordinal number of cardinality ℵ_{α} and index the set I by the ordinals below Ω_{α}, i.e., we write I = {i_{β} | β < Ω_{α}}. It will be convenient to identify an ordinal number γ with the set of all ordinal numbers strictly smaller than γ. Our aim is to show that A is the direct limit of a directed system (C_{β}, g_{β,γ})_{β,γ<Ω_{α},β≤γ} with the property that all of the objects C_{β} belong to Add(M). Since the new index set, Ω_{α}, is totally ordered, we can then again invoke the hypothesis to complete the proof.

For that purpose, we observe that, given any infinite subset I′ ⊆ I, there exists a directed subset I′′ of I which contains I′ and has the same cardinality as I′. This allows us to express I as the directed union of a chain of subsets I_{γ}, γ < Ω_{α}, such that I_{γ} ⊆ I_{δ} whenever γ < δ, that I_{γ} is finite whenever γ is finite, and I_{γ} has cardinality less than or equal to that of γ whenever γ is an infinite ordinal number. Indeed, set I_{0} = {i_{0}}, let δ > 0, and suppose that, for all γ < δ, directed subsets I_{γ} satisfying the above requirements and having the property that {i_{β} | β < γ} ⊆ I_{γ} have already been constructed. Set L_{δ} = ∪_{γ<δ}I_{γ} ∪ {i_{δ}}. By construction L_{δ} is either finite or else has a cardinality bounded above by that of δ. If L_{δ} is finite, we can clearly choose a finite directed subset I_{δ} of I containing L_{δ}; if, on the other hand, L_{δ} is infinite, the initial remark of this paragraph permits us to choose a directed subset I_{δ} of I which contains the set L_{δ} and has the same cardinality as the latter. A subsidiary induction thus gives us the desired family of subsets I_{γ} for γ < Ω_{α}. The hypothesis of the principal induction now yields that for each γ, the direct limit C_{γ} of the directed subsystem (A_{i}, f_{ij})_{i,j∈I_{γ},i≤j} of our original system is an object of Add(M). Letting g_{β,γ} : C_{β} → C_{γ} for β < γ be the natural map resulting from the fact that I_{β} ⊆ I_{γ}, we are thus in a position to apply the hypothesis to the system (C_{β}, g_{β,γ})_{β,γ<Ω_{α},β≤γ} to conclude that A = \underrightarrow{lim} C_{β} in turn belongs to Add(M). □

An additional remark on the category Add(M) for a Σ-pure injective module M will turn out helpful in Section 4, namely

Observation 3.3 Let M be Σ-pure injective, and suppose that all monomorphisms in add(M) split. Then the same is true for monomorphisms in Add(M); in other words, given any monomorphism f : A → B with A, B in Add(M), there exists a submodule C of B such that B = Im(f) ⨁ C, and both Im(f) and C again belong to Add(M).

Proof: Let f be as specified in the claim, and write B = ⨁_{i∈I} B_{i}, where each B_{i} is isomorphic to a direct summand of M. Then f is necessarily pure, since the image f(F) of any finitely generated submodule F of A is contained in a finite direct sum of the B_{i} and hence is a direct summand of B by hypothesis. As mentioned in the beginning of this section, Σ-pure injectivity is passed on to pure submodules, and therefore Im(f) is a direct summand of B. Finally we use once more the fact that each Σ-pure injective module is a direct sum of submodules with local endomorphism rings, in order to deduce that both Im(f) and any complement of Im(f) in B are in turn direct sums of direct summands of M. □

4. Fin.dim versus fin.dim in case P^{∞}(Λ-mod) is contravariantly finite

This section is devoted to proving our main result, namely that, for any Artin algebra Λ, the left big and little finitistic dimensions coincide, provided that the full subcategory P^{∞}(Λ-mod) of finitely generated left Λ-modules of finite projective dimension is contravariantly finite in the category of all finitely generated left Λ-modules.

Recall that P^{∞}(Λ-mod) is contravariantly finite in Λ-mod if, for each finitely generated left Λ-module M, there exists an object A ∈ P^{∞}(Λ-mod) and a homomorphism f : A → M such that the induced sequence of functors from P^{∞}(Λ-mod) to the category of abelian groups,

Hom_{Λ}(−, A)|P_{∞}(Λ-mod) → Hom_{Λ}(−, M)|P_{∞}(Λ-mod) → 0,

is exact; see [4]. In that case, A is called a (right) P^{∞}(Λ-mod)-approxiamtion of M. It is well-known that, existence provided, the P^{∞}(Λ-mod)-approximations of minimal length of a given module M ∈ Λ-mod are all isomorphic, and hence it makes sense to refer to the minimal P^{∞}(Λ-mod)-approximation of M in that case.

It is known that for any left serial algebra Λ, the category P^{∞}(Λ-mod) is contravariantly finite in Λ-mod [5]; moreover, the minimal P^{∞}(Λ-mod)-approximations of the simple left Λmodules can be explicitly described in that case. Initial criteria for contravariant finiteness in more general situations were developed in [10].

Throughout this section, we will abbreviate ℓ fin.dim Λ by fin.dim Λ and ℓ Fin.dim Λ by Fin.dim Λ.

Theorem 4.1 Suppose that P^{∞}(Λ-mod) is contravariantly finite in Λ-mod. Then

fin.dim Λ = Fin.dim Λ.

Proof: Since Fin.dim Λ is the supremum of those projective dimensions which are attained on countably generated left Λ-modules of finite projective dimension by Corollary 2.2, it is enough to focus on a countably generated module M with p dim M < ∞, say p dim M = n. Let M_{1} ⊆ M_{2} ⊆ M_{3} ⊆ ⋯ ⊆ M_{r} ⊆ ⋯ ⊆ M be a chain of finitely generated submodules such that M = ∪_{i≥1}M_{i}. For each i, we fix the beginning of a finitely generated projective resolution of M_{i}, say

0 → Ω_{n,i} → P_{n−1,i} → ⋯ → P_{1,i} → P_{0,i} → M_{i} → 0

and, over each of the inclusions M_{i} → M_{i+1}, we choose a chain morphism consisting of maps f_{k,i,i+1} : P_{k,i} → P_{k,i+1} for 0 ≤ k ≤ n − 1, and f_{n,i,i+1} : Ω_{n,i} → Ω_{n,i+1}. Defining f_{k,i,j} = f_{k,j−1,j} ∘⋯∘f_{k,i,i+1} for all j > i and 0 ≤ k ≤ n, we thus obtain a directed system of exact sequences indexed by the natural numbers. Passing to the direct limit of this system gives us a projective resolution of the module M which we label

0 → \underrightarrow{lim} Ω_{n,i} → P_{n−1} → ⋯ → P_{1} → P_{0} → M → 0

In particular, P_{n} := \underrightarrow{lim} Ω_{n,i} is projective due to the fact that p dim M = n. Clearly, all P_{k} are countably generated since our directed system extends over ℕ. Therefore, we can decompose P_{n} in the form P_{n} = ⨁_{j∈ℕ} Q_{n,j}, where all the Q_{n,j} are finitely generated. Denoting by g_{i} : Ω_{n,i} → P_{n} the canonical morphisms, we obtain an increasing sequence i_{1}, i_{2}, i_{3}, … of natural numbers such that ⨁_{j≤m} Q_{n,j} ⊆ Im g_{i_{m}} for m ∈ ℕ. It is clearly harmless to pass to a suitable cofinal subsystem of the Ω_{n,i}, which permits us to assume that i_{k} = k for all k. Next we observe that each Ω_{n,i} contains a submodule P_{n,i} = ⨁_{j≤i} P_{n,j,i} isomorphic to the finite direct sum ⨁_{j≤i} Q_{n,j} with the property that g_{i} restricts to an isomorphism P_{n,j,i} → Q_{n,j} for all j ≤ i; just keep in mind that the restriction of the map g_{i} to the preimage of ⨁_{j≤i} Q_{n,j} splits. Since g_{i+1}f_{n,i,i+1}(P_{n,i}) = g_{i}(P_{n,i}), the map f_{n,i,i+1} induces a split monomorphism P_{n,i} → g_{i+1}^{−1}(⨁_{j≤i+1} Q_{n,j}). Consequently, an obvious induction on i allows us to choose the P_{n,j,i} in such a way that the squares

\xymatrix{P_{n,i+1} \ar[r]^{g_{i+1}} & ⨁_{j≤i+1} Q_{nj} \\ P_{n,i} \ar[u]^{f_{n,i,i+1}} \ar[r]^{g_{i}}_{≅} & ⨁_{j≤i} Q_{nj} \ar@{^{(}->}[u]}

commute. In other words, this process yields a directed subsystem

\xymatrix{⋮ & ⋮ \\ P_{n,i+1} \ar[u] \ar@{^{(}->}[r] & Ω_{n,i+1} \ar[u] \\ P_{n,i} \ar@{^{(}->}[u] \ar@{^{(}->}[r] & Ω_{n,i} \ar[u] \\ ⋮ \ar[u] & ⋮ \ar[u]}

of the system (Ω_{n,i}, f_{n,i,j})_{i,j∈ℕ,i≤j} such that \underrightarrow{lim} P_{n,i} = \underrightarrow{lim} Ω_{n,i} = P_{n}.

At this point, we interrupt the argument with a lemma which will allow us to supplement our original directed system of resolutions

(S_{i}) P_{n−1,i} → P_{n−2,i} → … → P_{0,i} → M_{i} → 0

by a system

(T_{i}) P′_{n−1,i} → P′_{n−2,i} → … → P′_{0,i} → N_{i} → 0,

where N_{i} ∈ P^{∞}(Λ-mod), together with an epimorphism (T_{i}) → (S_{i}) of directed systems, to the effect that N = \underrightarrow{lim} N_{i} has a projective dimension bounded above by fin.dim Λ and the kernel of the induced epimorphism N → M is ‘under control’.

The following lemma is based on the fact that contravariant finiteness of P^{∞}(Λ-mod) in Λ-mod implies the existence of an injective cogenerator inside the category P^{∞}(Λ-mod) (see [4]); indeed, if I is the minimal right P^{∞}(Λ-mod)-approximation of the minimal injective cogenerator for Λ-mod, then each object of P^{∞}(Λ-mod) embeds into an object of add I and every inclusion I′ ↪ X with I′ in add I and X in P^{∞}(Λ-mod) splits. In particular, this entails that every monomorphism in add(I) splits, i.e., the hypotheses of Observation 3.3 are satisfied for M = I.

Lemma 4.2 Let I be a relative injective cogenerator for P^{∞}(Λ-mod) as above, and suppose that we are given an exact commutative diagram of the form

[figure: 0; X; β; P; α; Y; 0; Q; g; 0; X′; β′; f_{X}; P′; α′; f_{P}; Y′; f_{Y}; 0; Q′; h_{Q}; g′]

with Q, Q′, P and P′ ∈ P^{∞}(Λ-mod). Then there exist modules I_{0} and I′_{0} in add I, together with homomorphisms γ : Q → I_{0} and γ′ : Q′ → I′_{0}, as well as a homomorphism h : I′_{0} → I_{0} such that the following diagram has exact rows and commutes.

[figure: 0; X; β; P; α; Y; 0; 0; Q; g; _{\binom{βg}{γ}}; P ⨁ I_{0}; (1,0); Z; 0; 0; X′; f_{X}; β′; P′; f_{P}; α′; Y; f_{Y}; 0; 0; Q′; h_{Q}; g′; _{\binom{β′g′}{γ′}}; P′ ⨁ I′_{0}; _{(_{{f_{P} \atop 0}} _{{0 \atop h}})}; (1,0); Z′; 0]

Here Z = Coker \binom{βg}{γ}, Z′ = Coker \binom{β′g′}{γ′} are in P^{∞}(Λ-mod), and the remaining maps are induced by the cokernels.

Proof of the lemma: Since I is a cogenerator for P^{∞}(Λ-mod), we can choose monomorphisms γ : Q → I_{0} and γ′ : Q′ → I′_{0}. So if we introduce maps Q → P ⊕ I_{0} and Q′ → P ⊕ I′_{0} as in the above diagram and denote by Z and Z′ their cokernels, respectively, the two squares in each of the top and bottom planes commute. Moreover, Z, Z′ in turn belong to P^{∞}(Λ-mod) by the hypothesis on Q and Q′. Next we use the relative injectivity of I to obtain h : I′_{0} → I_{0} such that γh_{Q} = hγ′. It is now straightforward to check that the entire diagram commutes, which completes the proof of the lemma. □

We return to the proof of Theorem 4.1. At this point, we label the maps in the projective resolutions of the M_{i}, say g_{k,i} : P_{k,i} → P_{k−1,i}. By applying the lemma, first to the diagrams

[figure: 0; P_{n−1,i+1}/Ω_{n,i+1}; g_{n−1,i+1}; P_{n−2,i+1}; g_{n−2,i+1}; Im(g_{n−2,i+1}); 0; P_{n−1,i+1}/P_{n,i+1}; 0; P_{n−1,i}/Ω_{n,i}; g_{n−1,i}; f_{n−1,i,i+1}; P_{n−2,i}; g_{n−2,i}; f_{n−2,i,i+1}; Im(g_{n−2,i}); 0; P_{n−1,i}/P_{n,i}]

for i ∈ ℕ, and by then moving inductively along the sequences

P_{n−1,i} → P_{n−2,i} → … → P_{0,i} → M_{i} → 0,

we obtain the following directed system of diagrams in Λ-mod:

[figure: 0; 0; 0; 0; 0; 0; Ω_{n,i}; P_{n−1,i}; g_{n−1,i}; P_{n−2,i}; g_{n−2,i}; ⋯; P_{1,i}; P_{0,i}; M_{i}; 0; 0; P_{n,i}; P_{n−1,i}; P_{n−2,i} ⨁ I_{n−2,i}; (1,0); ⋯; P_{1,i} ⨁ I_{1,i}; (1,0); P_{0,i} ⨁ I_{0,i}; (1,0); N_{i}; 0; 0; I_{n−2,i}; _{\binom{0}{1}}; ⋯; I_{1,i}; _{\binom{0}{1}}; I_{0,i}; _{\binom{0}{1}}; K_{i}; 0; 0; 0; 0; 0]

Here the objects I_{k,i} all belong to add(I). The upper two horizontal sequences of each of these diagrams are exact by construction, whereas the induced kernel sequence in the third row will not be exact in general; in fact, the homology in the term labeled n − 2 of that sequence is isomorphic to Ω_{n,i}/P_{n,i}. On the other hand, the direct limit of the inclusions P_{n,i} ↪ Ω_{n,i} is the identity P_{n} → P_{n}, and hence the snake lemma ensures that, post-limits, we arrive at an exact commutative diagram of the form

[figure: 0; 0; 0; 0; 0; 0; P_{n}; P_{n−1}; P_{n−2}; ⋯; P_{1}; P_{0}; M; 0; 0; P_{n,}; P_{n−1}; P_{n−2} ⨁ I_{n−2}; (1,0); ⋯; P_{1} ⨁ I_{1}; (1,0); P_{0} ⨁ I_{0}; (1,0); N; 0; 0; I_{n−2}; _{\binom{0}{1}}; ⋯; I_{1}; _{\binom{0}{1}}; I_{0}; _{\binom{0}{1}}; K; 0; 0; 0; 0; 0]

Now Observation 3.1 yields I_{j} ∈ Add(I) and, as a consequence, Observation 3.3 gives us K ∈ Add(I). Since the N_{i} belong to P^{∞}(Λ-mod) by construction, the relative cogenerating property of I furthermore permits us to choose a directed system of embeddings N_{i} ↪ I′_{i} with I′_{i} ∈ add(I), which shows that N = \underrightarrow{lim} N_{i} in turn embeds into an object of Add(I) by 3.1. In view of 3.3, this implies that the exact sequence 0 → K → N → M → 0 splits, and therefore p dim M ≤ p dim N ≤ sup{p dim N_{i} | i ∈ ℕ} ≤ fin.dim Λ. □

In the light of Theorem 4.1, the following is an immediate consequence of the result of Auslander and Reiten quoted in the introduction; just keep in mind that each P^{∞}(Λ-mod)- approximation of a module M contains a minimal one as a direct summand [3].

Corollary 4.3 If P^{∞}(Λ-mod) is contravariantly finite in Λ-mod, and C_{1}, …, C_{n} are arbitrary P^{∞}(Λ-mod)-approximations of the simple left Λ-modules, then

Fin.dim Λ = sup{p dim C_{i} | 1 ≤ n}.

□

We can actually strengthen the conclusion of Theorem 4.1, so as to provide information on the structure of the non-finitely generated objects in P^{∞}(Λ-Mod) as follows: Given modules M_{1}, ⋯, M_{n} in Λ-mod we denote by filt(M_{1}, ⋯, M_{n}) the full subcategory of Λ-mod having as objects all finitely generated modules X that possess filtrations with consecutive factors in {M_{1}, ⋯, M_{n}}; in other words, filt(M_{1}, ⋯, M_{n}) consists of those modules X which contain chains of the form X = X_{0} ⊇ X_{1} ⊇ ⋯ ⊇ X_{r} = 0 such that each factor X_{i}/X_{i+1} is isomorphic to some M_{j}. Moreover, \overrightarrow{filt}(M_{1}, ⋯, M_{n}) will stand for the full subcategory of Λ-Mod the objects of which are the direct limits of modules in filt(M_{1}, ⋯, M_{n}). Due to the above-mentioned result of Auslander and Reiten, we know: In case P^{∞}(Λ-mod) is contravariantly finite in Λ-mod and A_{1}, ⋯, A_{n} are the minimal P^{∞}(Λ-mod)-approximations of the simple left Λ-modules, P^{∞}(Λ-mod) consists of the direct summands of modules in filt(A_{1}, ⋯, A_{n}). In view of the proof of Theorem 4.1, this description of the finitely generated modules of finite projective dimension extends to nonfinitely generated candidates as follows.

Theorem 4.4 If P^{∞}(Λ-mod) is contravariantly finite in Λ-mod and A_{1}, ⋯, A_{n} are as above, then

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

In particular, each object of P^{∞}(Λ-Mod) is a direct limit of modules in P^{∞}(Λ-mod).

Proof : That \overrightarrow{filt}(A_{1}, …, A_{n}) is contained in P^{∞}(Λ-Mod) is clear. For the other inclusion, start by noting that each full subcategory C of Λ-Mod which is closed under direct limits is also closed under direct summands. This is well known (see e.g. [8], Lemma 1), but we include the easy argument: If B is a direct summand of an object C in C and π : C → C a projection onto B, then B is the direct limit of the system

\xymatrix{C \ar[r]^{π} & C \ar[r]^{π} & C \ar[r]^{π} & ⋯}

By Proposition 2.1, each object in P^{∞}(Λ-Mod) is the direct union of countably generated objects in P^{∞}(Λ-Mod), whence it suffices to show that each countably generated module M of finite projective dimension belongs to \overrightarrow{filt}(A_{1}, …, A_{n}). But the proof of Theorem 4.1 shows that M is a direct summand of a direct limit of modules in P^{∞}(Λ-mod) and thus belongs to the closure of P^{∞}(Λ-mod) under direct limits as explained above. That the latter category is contained in \overrightarrow{filt}(A_{1}, …, A_{n}), finally, follows from [3] and another application of our initial remark. □

Note that our arguments for Theorems 4.1 and 4.4 only use the existence of a P^{∞}(Λ-mod) approximation of D(Λ), where D is the standard duality mod-Λ → Λ-mod. This observation does not lead to any significant generalization of our results however. Indeed, by [11], in the presence of the inequality fin.dim Λ < ∞, the existence of a P^{∞}(Λ-mod)-approximation of D(Λ) forces P^{∞}(Λ-mod) to be contravariantly finite.

The final consequence of our main results was observed by Henning Krause, who pointed out to us that a result of Crawley-Boevey applies to our context.

Corollary 4.5 If the subcategory P^{∞}(Λ-mod) of Λ-mod is contravariantly finite, then it is also covariantly finite. In particular, P^{∞}(Λ-mod) has almost split sequences in that case.

Proof : Suppose that P^{∞}(Λ-mod) is contravariantly finite in Λ-mod. As we know, ℓFin.dim Λ is finite in that case, equal to d say. In particular, P^{∞}(Λ-Mod) is closed under direct products – just keep in mind that direct products of projectives are projective. By Theorem 4.4, this is the same as to say that the closure of P^{∞}(Λ-mod) under direct limits is also closed under direct products. Hence Theorem 4.2 of [8] tells us that P^{∞}(Λ-mod) is indeed covariantly finite. □

On the other hand, it is not true in general that covariant finiteness of P^{∞}(Λ-mod) in Λ-mod implies contravariant finiteness. Indeed, since the category of modules of projective dimension ≤ 1 is always covariantly finite in Λ-mod by [3], the examples in [15] and [21] exhibit covariantly finite categories P^{∞}(Λ-mod) which fail to be contravariantly finite. A further discussion of the two properties can be found in [14].

5. Examples

Throughout this section, Λ = KΓ/I will be a finite dimensional path algebra modulo relations over a field K. We will denote by J its Jacobson radical, by e_{i} the primitive idempotents of Λ going with the vertices of the quiver Γ, and by S_{i} = Λe_{i}/Je_{i} representatives of the simple left Λ-modules. Moreover, given a left Λ-module M and a primitive idempotent e_{i}, we will call an element x ∈ M a top element of type e_{i} in case x ∈ M \ JM and e_{i}x = x. We start with an example which illustrates the applicability of corollary 4.3.

Example 5.1. Let Λ be the monomial relation algebra KΓ/I where Γ is the quiver

[figure: 12; χ; 11; ε; 4; ψ_{4}; μ; 6; ψ_{6}; σ; 7; τ; 8; 1; δ; α; 9; β; 2; ν; π; ρ; 5; ψ_{5}; 10; γ; 3; ψ_{3}]

and I ⊆ KΓ is the unique ideal of the path algebra with the property that the indecomposable projective left Λ-modules have the following graphs. For our graphing conventions, we refer the reader to [12].

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

We will sketch a proof for the fact that P^{∞}(Λ-mod) is contravariantly finite in Λ-mod by exhibiting (minimal) P^{∞}(Λ-mod)-approximations of the simple left Λ-modules and by partly verifying that they have the claimed properties. Due to [3], securing approximations for the simple modules guarantees contravariant finiteness. The following module A_{1} is a P^{∞}(Λ-mod)-approximation of S_{1}. Namely, A_{1} = B_{1} ⊕ C_{1}, where B_{1} corresponds to the representation

[figure: K; _{\binom{0}{1}}; K; _{\binom{1}{0}}; K^{2}; 0; μ; 0; 0; 0; K; _{\binom{1}{1}}; 1; K; 1; K; ν; 0; 0; 0]

of Γ modulo I. The module C_{1} is defined similarly, with the vertex 10 taking over the role of 9 and the arrow γ taking over that of β.

Note first that Ω^{1}(A_{1}) = (Λe_{2}/Λν)^{2} ⊕ (Λe_{4})^{2}, hence p dim A_{1} = 3. In particular, A_{1} ∈ P^{∞}(Λ-mod). Moreover, denote by K the kernel of the homomorphism (ψ, ρ) : B_{1} ⊕C_{1} → S_{1}, where ψ : B_{1} → S_{1} and ρ : C_{1} → S_{1} are the canonical epimorphisms (unique up to a nonzero scalar factor). It can be checked that Ext_{Λ}^{1}(M, K) = 0 for each object M ∈ P^{∞}(Λ-mod), which shows that the map Hom_{Λ}(M, A_{1}) → Hom_{Λ}(M, S_{1}) induced by (ψ, ρ) is onto, as required. Along the same line, one can verify that the minimal P^{∞}(Λ-mod)- approximations A_{i} of the simple modules S_{i} with i ≥ 2 are as follows: A_{2} = Λe_{2}/Λν; A_{3} = Λe_{3}; A_{4} corresponds to the representation

[figure: K; \binom{0_{1}}{0}; K; _{\binom{0_{0}}{1}}; K^{3}; f_{4}; (1,0,0); K; 0; 0; 0; 0; 0; 0; 0; 0; 0]

where f_{4} has matrix ( _{0}_{0}_{0}_{1}_{0}_{0}_{1}_{0}_{0} ); moreover, A_{5} = Λe_{5}; A_{6} = Λe_{6}/Λσ; A_{7} = S_{7}; A_{8} = S_{8}; A_{9} is

B_{1} as defined above; A_{10} is C_{1} as defined above; A_{11} = A_{12} = Λe_{1}1 ⊕ Λe_{1}2/Λ(ε, χ). Thus Corollary 4.3 is applicable. It yields

ℓfin.dim Λ = ℓFin.dim Λ = sup{p dim A_{i} | 1 ≤ i ≤ 12} = p dim A_{1} = 3.

Our final example exhibits a class of finite dimensional monomial relation algebras Λ such that ℓfin.dim Λ < ℓFin.dim Λ, while the subcategory P^{∞}(mod-Λ) ⊆ mod-Λ of the category of all finitely generated right Λ-modules is contravariantly finite.

Example 5.2. Given a positive integer n, let Λ = KΓ/I be the monomial relation algebra defined in [13], proof of Theorem E, where Γ is the quiver

[figure: 2; γ_{2}; γ_{3}; γ_{1}; 1; σ; τ; χ_{1}; χ_{2}; α_{0}; a_{0}; α_{1}; β_{1}; a_{1}; α_{2}; ⋯; α_{n+1}; a_{n+1}; c_{1}; χ_{1}; c_{2}; χ_{2}; b; β_{2}]

The relations are such that the indecomposable projective right Λ-modules have the following graphs (in particular, e_{2}Λ ≅ S_{2}).

[figure: 1; σ; τ; γ_{1}; γ_{2}; γ_{3}; 2; •; a_{0}; 1; τ; 1; γ_{1}; γ_{2}; γ_{3}; 2; 2; 2; 1; γ_{1}; γ_{2}; 1; γ_{1}; γ_{2}; 2; 2; 2; 2; 2; 2; 2; a_{1}; a_{2}; a_{3}; a_{4}; ⋯; a_{n+1}; b; c_{1}; c_{2}; a_{0}; a_{1}; a_{2}; a_{3}; ⋯; a_{n}; a_{0}; b; c_{1}; 1; τ; c_{2}; 1; τ; 1; γ_{2}; a_{0}; 1; γ_{1}; γ_{2}; a_{0}; 1; 1; γ_{1}; 1; 1; γ_{1}; γ_{2}; 2; 2; 2; 2; 2; 2]

In view of the fact that there are unique arrows 1 → a_{0}, a_{i} → a_{i+1} for i ≥ 0, and b → b, b → a_{0}, c_{i} → c_{i}, 1 → c_{i} for i = 1, 2, the above graphs determine Λ up to isomorphism.

In [13], it was shown that ℓfin.dim Λ = n+1, whereas ℓFin.dim Λ = n+2. On the other hand, the subcategory P^{∞}(mod-Λ) of finitely generated right Λ-modules of finite projective dimension is contravariantly finite in mod-Λ. The minimal P^{∞}(mod-Λ)-approximations A(i) of the simple right Λ-modules e_{i}Λ/e_{i}J for i ∈ Γ_{0} are listed below.

[figure: A(1); A(2); A(a_{0}); A(a_{1}); A(a_{2}); A(a_{3}); ⋯; A(a_{n+1}); 1; σ; τ; 2; •; a_{0}; a_{1}; •; a_{2}; a_{3}; ⋯; a_{n+1}; 1; τ; 1; 1; a_{1}; a_{2}; ⋯; a_{n}; 1; a_{0}]

[figure: A(b); A(c_{1}); A(c_{2}); b; a_{1}; c_{1}; c_{2}; b; c_{1}; 1; τ; c_{2}; 1; τ; a_{0}; 1; 1; 1; 1]

We give a bit of detail to back our claim for A(1) and A(b). Clearly, p dim A(1) = 1. Let M be any object in P^{∞}(mod-Λ) containing a top element x of type e_{1}. Since p dim e_{2}Λ/e_{2}J < ∞ and since the vertex 2 is a source of Γ, we may assume that Me_{2} = 0. Note moreover that Ω^{1}(M)e_{1}J^{2}e_{1} = 0, because Ω^{1}(M) is contained in the radical of a projective module. This implies that Ω^{1}(M) cannot contain a top element of type e_{1}, for otherwise all higher syzygies Ω^{j}(M) would contain such a top element as well. Consequently, the graph of Λx coincides with that of A(1) and Λx is a direct summand of M. This guarantees that each homomorphism M → S_{1} factors through A(1).

We are still briefer in justifying our claim for A(b). We just point to the fact that all homomorphisms M → e_{b}Λ/e_{b}J where M ∈ P^{∞}(mod-Λ) has a graph belonging to the series

[figure: b; b; b; ⋯; b; b; b; b; a_{0}; a_{0}; ⋯; a_{0}; a_{0}]

factor through A(b). Another application of Corollary 4.3 therefore yields that

r Fin.dim Λ = sup{p dim A_{i} | i ∈ Γ_{0}} = p dim A(a_{0}) = p dim A(b) = 2.

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

Department of Mathematics, The Norwegian University for Science and Technology, 7055 Dragvoll, Norway Electronic mail address: sverresm@math.ntnu.no