RELATIVE HOMOLOGY

RELATIVE HOMOLOGY M. Auslander∗

Ø. Solberg†

Department of Mathematics

Institutt for matematikk og statistikk

Brandeis University

Universitetet i Trondheim, AVH

Waltham, Mass. 02254–9110

N–7055 Dragvoll

USA

NORWAY

September 15, 2005

Introduction Throughout this paper all rings are assumed to be artin R-algebras over a fixed commutative artin ring R, i.e. R-algebras which are finitely generated R-modules. All modules are assumed to be finitely generated and we denote the category of finitely generated left modules over an artin algebra Λ by mod Λ. Our purpose in this paper is to explain some applications of relative homological algebra to the study of the modules over an artin R-algebra Λ. We start by describing two applications of relative homological algebra, one ring theoretic and the other module theoretic. We hope this will encourage the reader to read the rest of the paper, which consists of a brief summary of relative homological algebra for modules over artin algebras and an explanation of how the two results given in the first section follow from the general theory. Throughout the paper all subcategories are assumed to be additive subcategories. Recall that an additive subcategory of mod Λ is a full subcategory X closed under isomorphisms and all finite direct sums of modules in X and all direct summands of modules in X again are in X . Moreover, for any module A in mod Λ we let add A denote the additive subcategory of mod Λ generated by A, i.e. add A consists of the Λ-modules isomorphic to direct summands of finite direct sums of A. Also, the symbol ⊕ denotes the direct sum in mod Λ.

1

Results

Before stating our first result it is convenient to make the following definition. We say an artin algebra Λ is D Tr-selfinjective if the subcategory add{(Tr D)i Λ}∞ i=0 is of finite type, i.e. there is a module C in mod Λ such that add C = add{(Tr D)i Λ}∞ . Here Tr D0 is defined to be the i=0 identity map. Clearly Λ being D Tr-selfinjective is equivalent to either of the following equivalent conditions: (a) the D Tr-orbit of all the indecomposable injective modules are finite or (b) the union of the Tr D-orbits of all the indecomposable projective modules coincide with the union of the D Tr-orbits of all the indecomposable injective modules. It is obvious that selfinjective algebras as well as algebras of finite representation type are D Tr-selfinjective. The somewhat less obvious result that the Auslander algebras of selfinjective algebras of finite representation type are D Tr-selfinjective gives other examples of D Tr-selfinjective algebras. In this connection it is worth noting that not all Auslander algebras are D Tr-selfinjective. Our reason for introducing D Tr-selfinjective algebras is that these algebras are intimately related to the better known class of algebras of injective dimension 2 which are also of dominant ∗ Partially

supported by NSF Grant No. DMS–8904594 by the Norwegian Research Council for Science and Humanities.

† Supported

1

dimension 2. However, before stating the precise relationship between these type of algebras we recall that an artin algebra Γ is of dominant dimension at least r ≥ 1 (notation: dom.dim Γ ≥ r), if in a minimal injective resolution 0 → Γ → I 0 → I1 → · · · the injective Γ-modules Ij are also projective for j = 0, 1, . . . , r − 1. Theorem 1.1 An artin algebra Γ has dom.dim Γ = 2 and idΓ Γ = 2 if and only if there is a module T over a D Tr-selfinjective algebra Λ satisfying the following: (a) Γ = EndΛ (T ) (b) T = C ⊕ M , where add C = add{(Tr D)i Λ}∞ i=0 and M ' D Tr M . The following is a special case of this result which seems of particular interest. Corollary 1.2 Suppose Λ is a symmetric algebra (e.g. Λ is a modular group ring) and A a Λn−1 i module of finite Ω-period n. Let M = ⊕i=0 ΩΛ (A) and let T = Λ ⊕ M . Then Γ = EndΛ (T ) has the property dom.dim Γ = 2 = idΓ Γ. For those readers unfamiliar with the notation ΩiΛ (M ) used above, we recall that ΩΛ (M ) denotes the kernel of a projective cover of a module M and the ΩiΛ (M ) for all i = 0, 1, . . . are defined i i inductively as follows: Ω0Λ (M ) = M and Ωi+1 Λ (M ) = ΩΛ (ΩΛ (M )) for all i ≥ 0. The ΩΛ (M ) is called the i-th syzygy of M . Before giving our other application we recall the notion of homologically finite subcategories introduced by M. Auslander and S. O. Smalø in [4]. Let Λ be an artin algebra. An additive subcategory X of mod Λ is contravariantly finite in mod Λ, if given any module C in mod Λ there is a morphism f : XC → C with XC in X such that HomΛ (X, XC ) → HomΛ (X, C) → 0 is exact for all X in X . For a given module C such a morphism f : XC → C is called a right X -approximation of C. The approximation is called minimal if in addition a morphism g: XC → XC is an isomorphism whenever f ◦g = f . It is easy to see that if C has a right X -approximation, then there is a minimal right X -approximation which is unique up to isomorphism. Dually, an additive subcategory Y of mod Λ is covariantly finite in mod Λ, if given any module A in mod Λ there is a morphism g: A → Y A with Y in Y such that HomΛ (Y A , Y ) → HomΛ (A, Y ) → 0 is exact for all Y in Y. Dually one also defines left Y-approximations and minimal left Y-approximations. For example, if X = add A for some module A in mod Λ, then X is both contravariantly and covariantly finite in mod Λ. A subcategory X of mod Λ which is both contravariantly and covariantly finite in mod Λ is called functorially finite in mod Λ. Moreover, a subcategory of mod Λ satisfying one of these notions of finiteness is called a homologically finite subcategory of mod Λ. Now we have the necessary background to explain our second application. In [3] M. Auslander and I. Reiten showed that the additive subcategory generated by all n-th syzygy modules, ΩnΛ (mod Λ), is functorially finite for all positive n. We generalize this result as follows. Let X be a functorially finite subcategory of mod Λ containing all the projective Λ-modules. For each module C in mod Λ let f1 (C) f0 (C) · · · → X1 (C) → X0 (C) → C → 0 be exact with each fi (C): Xi (C) → Im fi (C) being a minimal right X -approximation. For each module C in mod Λ denote Im fn (C) by ΩnX (C). Let ΩnX (mod Λ) denote the additive subcategory of mod Λ generated by the modules ΩnX (C) for all modules C in mod Λ. Then we have the following result. Theorem 1.3 The subcategory X ∪ ΩnX (mod Λ) is functorially finite in mod Λ for all positive n.

2

A special case of this result is the following. Let X = add G for a generator G of mod Λ. Then X is functorially finite in mod Λ and contains all the projective Λ-modules. In this case the above result says that the union of X and the additive subcategory of mod Λ generated by all the kernels of the minimal right X -approximations is functorially finite in mod Λ. By [4, Proposition 3.13 (a)] removing a subcategory of finite type from a homologically finite subcategory does not change the homological finiteness. Hence it follows that the additive subcategory of mod Λ generated by all the kernels of the minimal right X -approximations is functorially finite in mod Λ.

2

Relative Homological Algebra

In this section we give a brief summary of relative homological algebra for modules over artin algebras. Let Λ be an artin algebra. A relative homology theory for the category mod Λ consists of a class E of short exact sequences 0 → A → B → C → 0 in mod Λ which is closed under isomorphisms f2 f1 of exact sequences, pushouts, pullbacks and finite sums, i.e. if 0 → A1 → A2 → A3 → 0 and f0

f0

(f1 ,f 0 )

(f2 ,f 0 )

1 2 0 → A01 → A02 → A03 → 0 are in E, then 0 → A1 ⊕ A01 →1 A2 ⊕ →2 A3 ⊕ A03 → 0 is in E. Of particular concern to us are the relative homological theories one obtains as follows. Let X be an arbitrary additive subcategory of mod Λ. Then associated with X is the relative homology theory EX consisting of all exact sequences 0 → A → B → C → 0 such that 0 → HomΛ (X, A) → HomΛ (X, B) → HomΛ (X, C) → 0 is exact for all X in X and also the relative homological theory E X consisting all exact sequences 0 → A → B → C → 0 in mod Λ such that 0 → HomΛ (C, X) → HomΛ (B, X) → HomΛ (A, X) → 0 is exact for all X in X . Now it is not difficult to see that associated with a relative homological theory E of mod Λ is the additive subfunctor F of the bifunctor Ext1Λ ( , ): (mod Λ)op × mod Λ → Ab given by F (C, A) consisting of all exact sequences 0 → A → B → C → 0 in Ext1Λ (C, A) with are in E, where Ab denotes the category of abelian groups and (mod Λ)op denotes the opposite category of mod Λ. Recall that a functor F : (mod Λ)op ×mod Λ → Ab is called a bifunctor. A bifunctor F : (mod Λ)op × mod Λ → Ab is said to be additive if for each C in (mod Λ)op and A in mod Λ the functors F (C, ): mod Λ → Ab and F ( , A): (mod Λ)op → Ab are additive functors. Moreover, a subfunctor F of Ext1Λ ( , ): (mod Λ)op × mod Λ → Ab is called an additive subfunctor if F is an additive bifunctor. In fact the map which assigns to each relative homological theory E the additive subfunctor F of Ext1Λ ( , ) gives a bijection between the set of relative homological theories and the set of additive subfunctors of Ext1Λ ( , ). Since the only subfunctors of Ext1Λ ( , ) which are of interest to us are those associated with a relative homological theory E of mod Λ, we assume from now on that when we say that F is a subfunctor of Ext1Λ ( , ) we mean that F is an additive subfunctor of Ext1Λ ( , ). Suppose now that E is a relative homological theory with corresponding subfunctor F of f2 f1 f0 Ext1Λ ( , ). We say that an exact sequence · · · → C2 → C1 → C0 → 0 is F -exact if the exact sequences 0 → Ker fi → Ci → Ker fi−1 → 0 are in E for all i = 1, 2, . . .. We will often use the notation Ext1F (C, A) for F (C, A) when we want to emphasize that F (C, A) consists of the F -exact sequences 0 → A → B → C → 0. As with standard homological algebra, if 0 → A → B → C → 0 is F -exact, then for all X in mod Λ there are exact sequences

(∗) 0 → HomΛ (X, A) → HomΛ (X, B) → HomΛ (X, C) → Ext1F (X, A), which are functorial in X and the F -exact sequence 0 → A → B → C → 0. As in standard homological algebra we would like to define additive bifunctors ExtiF ( , ): (mod Λ)op ×mod Λ → Ab for i = 0, 1, . . ., such that for an F -exact sequence 0 → A → B → C → 0 we get the usual type of long exact sequences of additive functors 0

→ HomΛ (X, A) → Ext1F (X, A) → Ext2F (X, A) → ...,

→ HomΛ (X, B) → Ext1F (X, B) → Ext2F (X, B)

3

→ HomΛ (X, C) → Ext1F (X, C) → Ext1F (X, C)

which are functorial in X and the F -exact sequence 0 → A → B → C → 0. For this purpose it is convenient to introduce the notion of an F -projective module for a corresponding relative homological theory E. We say that a module P is F -projective if 0 → HomΛ (P, A) → HomΛ (P, B) → HomΛ (P, C) → 0 is exact for all F -exact sequences 0 → A → B → C → 0 in mod Λ. We denote by P(F ) the subcategory of mod Λ consisting of all F -projective modules. Using the exact sequence (∗), it is easy to see as in the standard homological algebra, that a module P is F -projective if and only if Ext1F (P, ) = 0. It is clear that P(F ) contains P(Λ), where P(Λ) is the subcategory of mod Λ consisting of the projective Λ-modules. In particular, from the existence of almost split sequences, it follows that P(FX ) = P(Λ) ∪ X , where FX is the subfunctor of Ext1Λ ( , ) corresponding to EX . Using this one can show that a relative homological theory E has the property that F = F P(F ) if and only if F = FX for some additive subcategory X of mod Λ. Suppose now that F = FP(F ) . Then for each F -exact sequence 0 → A → B → C → 0 and X in mod Λ we have an exact sequence 0 → HomΛ (X, A) → HomΛ (X, B) → HomΛ (X, C) → Ext1F (X, A) → Ext1F (X, B) → Ext1F (X, C), which is functorial in X and F -exact sequences 0 → A → B → C → 0. Thus we see that the hypothesis that F = FP(F ) gives us part of our desired long exact sequence. However in order to get a fully satisfactory theory, it seems necessary to make further assumptions on F . We say that a subfunctor F of Ext1Λ ( , ) has enough projectives if and only if given any Λmodule C there is an F -exact sequence 0 → K → P → C → 0 with P an F -projective module. From this it follows that each Λ-module C has an F -projective resolution, i.e. there is an F -exact sequence · · · → P2 → P1 → P0 → C → 0 with the Pi in P(F ) for all i. In analogy with standard homological algebra we define ExtiF (C, X) to be the i-th homology of the complex 0 → HomΛ (P0 , X) → HomΛ (P1 , X) → HomΛ (P2 , X) → · · · . If one substitutes F -exact sequences and F -projective modules for exact sequences and projective modules, then most of the basic result and concepts of standard homological algebra translate verbatim to the relative context. For instance, it is clear what one means by pd F C, the F projective dimension of a module C as well as gl.dimF Λ, the F -global dimension of Λ. Moreover, for each F -exact sequence 0 → A → B → C → 0 and X in mod Λ we get our desired long exact sequence 0 → HomΛ (X, A) → HomΛ (X, B) → HomΛ (X, C) → Ext1F (X, A) → Ext1F (X, B) → Ext1F (X, C) → Ext2F (X, A) → Ext2F (X, B) → Ext1F (X, C) → ... functorial in the F -exact sequence 0 → A → B → C → 0 and X in mod Λ. In view of these observations it is of particular interest to know when a subfunctor F of Ext1Λ ( , ) has enough projectives. Here we use the notion of homologically finite subcategories defined in section 1. It is relatively straightforward to see that a subfunctor F of Ext1Λ ( , ) has enough projective modules if and only if P(F ) is contravariantly finite in mod Λ and F = FP(F ) . In particular, if X = add A for some module A, then FX has enough projectives and P(FX ) = P(Λ) ∪ X . We also have the dual notion of F -injective modules for a subfunctor F of Ext1Λ ( , ). A module I in mod Λ is F -injective if and only if 0 → HomΛ (C, I) → HomΛ (B, I) → HomΛ (A, I) → 0 is exact for all F -exact sequences 0 → A → B → C → 0 in mod Λ. The subcategory of all F injective modules is denoted by I(F ), which clearly contains the subcategory I(Λ) of all injective Λ-modules. The subfunctor F has enough injectives if and only if given any Λ-module A there is an F -exact sequence 0 → A → I → K → 0 with I in I(F ). Let F X denote the subfunctor of Ext1Λ ( , ) corresponding to the relative homological theory E X for a subcategory X of mod Λ. Then, dual 4

to the characterization of subfunctors of Ext1Λ ( , ) with enough projectives, a subfunctor F of Ext1Λ ( , ) has enough injectives if and only if I(F ) is covariantly finite in mod Λ and F = F I(F ) . In general the F -projective and F -injective modules are related in the following way. Using that for an exact sequence 0 → A → B → C → 0 the sequence HomΛ (X, B) → HomΛ (X, C) → 0 is exact if and only if HomΛ (B, D Tr X) → HomΛ (A, D Tr X) → 0 is exact by [1, III, Corollary 4.2], it follows that I(F ) = I(Λ) ∪ D Tr P(F ) and P(F ) = P(Λ) ∪ Tr DI(F ). This implies that for an additive subcategory X of mod Λ the equality I(FX ) = I(Λ) ∪ D Tr X holds. The duality D and the transpose Tr map contravariantly finite subcategories to covariantly finite subcategories and vice versa. By [4, Proposition 3.13] adding a subcategory of finite type to a homologically finite subcategory does not change the type of homologically finiteness. Using these observations a subfunctor F of Ext1Λ ( , ) has enough projective and injective modules if and only if P(F ) is functorially finite in mod Λ and F = FP(F ) . In particular, if X = add A for some module A in mod Λ, then (a) FX has enough projective nodules and enough injective modules and (b) P(F ) = P(Λ) ∪ X and I(F ) = I(Λ) ∪ D Tr X . The main feature of relative homological algebra is that the class of extensions of modules is restricted while the homomorphisms remain the same. This has the effect of making some Λmodules which are not projective or injective Λ-modules behave like projective or injective objects in the relative homological setting. For example, in the relative homological theory of F = F P(Λ) , the projective Λ-modules also behave like injective objects since an exact sequence 0 → A → B → C → 0 is F -exact if and only if 0 → HomΛ (C, Λ) → HomΛ (B, Λ) → HomΛ (A, Λ) → 0 is exact. In fact, I(F ) = I(Λ) ∪ P(Λ). But this is not the only change that occurs when one goes from standard to relative homology. Some subcategories of mod Λ may be closed under extensions in a suitable relative theory without being extension closed in the standard theory. For example, the functorially finite subcategory Sub(Λ) consisting of all modules isomorphic to submodules of free Λ-modules is extension closed in the relative theory F = F P(Λ) but not in general extension closed in the standard theory. In fact Sub(Λ) is extension closed in the standard theory if and only if Λ satisfies the rather restrictive condition that the Λop -injective envelope of Λop is at most 1 [2, Proposition 3.5]. Many of the results in the theory of homologically finite subcategories require that subcategories are closed under extensions. As we saw above not all homologically finite subcategories are closed under extensions, hence many of the results for homologically finite subcategories do not apply. One example of such a result is the Auslander-Reiten Lemma, which says the following. If Y is an extension closed covariantly finite subcategory of mod Λ, then the subcategory {X ∈ mod Λ | Ext1Λ (X, )|Y = 0} is an extension closed contravariantly finite subcategory of mod Λ. But it turns out that most of the results in the standard theory for homologically finite subcategories generalize to relative homological settings when reformulated properly. For example, the generalization of the Auslander-Reiten Lemma to the relative setting is the following. Theorem 2.1 Let F be a subfunctor of Ext1Λ ( , ) with enough projectives. If Y is an F -extension closed covariantly finite subcategory of mod Λ, then the subcategory {X ∈ mod Λ | Ext1F (X, )|Y = 0} is an F -extension closed contravariantly finite subcategory of mod Λ. We observed above that the subcategory Sub(Λ) is covariantly finite in mod Λ and closed under the relative extensions given by F = F P(Λ) , which has enough projectives. Then Theorem 2.1 implies that the subcategory X = {X ∈ mod Λ | Ext1F (X, )|Sub(Λ) = 0} is an F -extension closed contravariantly finite subcategory of mod Λ. Even though X is given by expressions involving relative homology functors, we have the following description of X not involving any relative homological algebra. The subcategory X is equal to the subcategory {X ∈ mod Λ | HomΛ (X, )|A = 0}, where A = {A ∈ mod Λ | Ext1Λ (A, Λ) = 0} and HomΛ (X, A) is HomΛ (X, A) modulo the morphisms from X to A which factor through a projective Λ-module. For more information on this and additional topics the reader is referred to the papers [5, 6, 7, 8]. The first examples of relative homology theories for categories of modules over a ring were given and developed by G. Hochshild in [12, 1956]. Relative homological algebra was generalized 5

and extended to abstract categories by A. Heller in [11, 1958] and by D. A. Buchsbaum in [9, 1959]. M. C. R. Butler and G. Horrocks showed in [10, 1961] that the relative homology theories in a module category over a ring Λ are in one-one correspondence with the category of additive subfunctors of Ext1Λ ( , ). It is their approach to relative homological algebra we have taken in this paper and also in the papers [5, 6, 7, 8].

3

Outline of proofs

This section is devoted to describing some of the theoretical background for the proofs of the theorems stated in section 1. We start with the ring theoretic application given in Theorem 1.1, which is proven using relative cotilting theory. We begin with a brief review of this theory. Let Λ be an artin algebra. The object of cotilting theory is to compare Λ and EndΛ (T ) for a cotilting module T over Λ. A cotilting module T is a Λ-module with (i) ExtiΛ (T, T ) = 0, for i > 0, (ii) idΛ T < ∞ and (iii) for any injective Λ-module I there is an exact sequence 0 → Tn → Tn−1 → · · · → T1 → T0 → I → 0 with Ti in add T for all i. During the discussion of the theoretical background for Theorem 1.1 we fix F to be Fadd G for some generator G of mod Λ. Recall that F (C, A) = {0 → A → B → C → 0 | HomΛ (G, B) → HomΛ (G, C) → 0 is exact}. Then P(F ) = add G and F has enough projectives and injectives. A relative cotilting module is given by the following. An F -cotilting module is a Λ-module T with (i) ExtiF (T, T ) = 0 for all i > 0, (ii) idF T < ∞ (iii) for all modules I in I(F ) there is an F -exact sequence 0 → Tn → Tn−1 → · · · → T1 → T0 → I → 0 with Ti is in add T for all i. An example of an F -cotilting module that always exists is a module T such that add T = I(F ), which trivially satisfies (i)–(iii). It is this type of relative cotilting modules that we use in proving Theorem 1.1. The standard cotilting modules are obtained by letting G = Λ so F = Ext 1Λ ( , ). The relative extension groups may be zero without the standard extension groups being zero. This enables us to construct more relative cotilting modules than standard ones. For instance, for standard cotilting theory selfinjective algebras have only the trivial cotilting module Λ, while using various relative homological theories one obtains relative cotilting modules other than Λ which give nontrivial results of which Theorem 1.1 is an example. Let T be a Λ-module which is an F -cotilting module. We want to compare Λ and Γ. Since T is a module over Γ, we have the functor HomΛ ( , T ): mod Λ → mod Γ and HomΛ (Λ, Λ T ) ' Γ T . As in the standard case the naturally induced homomorphism Λ → EndΓ (Γ TΛop ) is an isomorphism. Moreover, one can show that the number of nonisomorphic simple Γ-modules is the same as the number of nonisomorphic indecomposable modules in P(F ) = add G. So, if the subcategory P(F ) properly contains P(Λ), then Γ has more nonisomorphic simple modules than Λ. Furthermore, we have the following connections between Λ and Γ. Theorem 3.1 Let F = Fadd G for a generator G of mod Λ and let T be a Λ-module which is an F -cotilting module. Denote EndΛ (T ) by Γ. Then the subcategory HomΛ (P(F ), T ) in mod Γ contains add Γ T and is add T0 for a standard cotilting module T0 over Γ with idΓ T0 ≤ max{idF T, 2}. Moreover, idΓ Γ T ≤ idF Λ T and the modules Λ T and Γ T0 have the property that the natural homomorphism Γ T0 → HomΛ (HomΓ (Γ T0 , Γ T ), Λ T ) is an isomorphism. The last property of the modules T and T0 mentioned in Theorem 3.1 turns out to be important. In this connection it is convenient to make the following definition. Let M be a Λ-module and Σ = EndΛ (M ). If A is a Λ-module such that the natural homomorphism A → HomΣ (HomΛ (A, M ), M ) is an isomorphism, then M is said to dualize A. If A is a direct summand of M , then A is called a 6

dualizing summand of M . Since add Γ T is contained in HomΛ (P(F ), T ) = add T0 , we can choose T0 such that Γ T is a direct summand of Γ T0 . Then Theorem 3.1 says that Γ T is a dualizing summand of the standard cotilting module Γ T0 . The converse of this result is also true, namely all dualizing summands of standard cotilting modules are obtained from relative cotilting modules. More precisely, we have the following result. Theorem 3.2 Let T = T1 ⊕ T2 be a standard cotilting module in mod Γ for an artin algebra Γ. Let Λ = EndΓ (T1 ). Define F to be the subfunctor of Ext1Λ ( , ) given by FP(F ) , where P(F ) = HomΓ (add Γ T , T1 ). Then T1 is a dualizing summand of the Γ-module T if and only if the Λ-module Λ T1 is an F -cotilting module. This shows that relative cotilting modules and dualizing summands of standard cotilting modules are intimately connected and give a way of comparing algebras with a different number of nonisomorphic simple modules. We saw that a module T such that add T = I(F ) always is an F -cotilting module. In this case we have even more information than in the general case. Proposition 3.3 Let F = Fadd G for a generator G of mod Λ, where add G properly contains P(Λ). Let T be a Λ-module such that add T = I(F ). Denote EndΛ (T ) by Γ. Then the Γ-module HomΛ (G, T ) is a standard cotilting module over Γ with idΓ HomΛ (G, T ) = 2. Recall that Theorem 1.1 from section 1 says the following. Theorem 1.1 An artin algebra Γ has dom.dim Γ = 2 and idΓ Γ = 2 if and only if there is a module T over a D Tr-selfinjective algebra Λ satisfying the following: (a) Γ = EndΛ (T ) (b) T = C ⊕ M , where add C = add{(Tr D)i Λ}∞ i=0 and M ' D Tr M . Next we explain part of the proof of this result using what we have discussed so far. Let Λ be an artin algebra. Assume that OΛ = add{(Tr D)i Λ}∞ i=0 is finite. This implies that all the injective Λ-modules are in OΛ . Let C be a Λ-module such that add C = OΛ and M a Λ-module such that M ' D Tr M . Denote C ⊕M by T . Let F = Fadd T . Using that I(F ) = I(Λ)∪Tr DP(F ) and P(F ) = add T , it follows that P(F ) = I(F ) = add T . Hence, T is an F -cotilting module. By the above result we have that idΓ Γ = 2. By Theorem 3.1 we have that idΓ Γ T = 0. Since Λ T is a generator, it follows that Γ T is a projective injective Γ-module. Let P1 → P0 → T → 0 be a projective presentation of T as a Λ-module. Apply the functor HomΛ ( , T ) to this sequence and we obtain the following exact sequence 0 → Γ → HomΛ (P0 , T ) → HomΛ (P1 , T ) → I2 → 0. The modules HomΛ (Pi , T ) for i = 0, 1 are in add Γ T , so that the dominant dimension of Γ is at least 2. Since idΓ Γ = 2, the dominant dimension of Γ must be 2. This proves one direction of Theorem 1.1 we mentioned above. The other direction is more involved and the reader is referred to [8] for further details of the proof. We end this paper by giving some details of the proof of the second result, which we recall next. Let X be a functorially finite subcategory of mod Λ containing the projective Λ-modules. f1 (C)

f0 (C)

For each module C in mod Λ let · · · → X1 (C) → X0 (C) → C → 0 be exact with each fi (C): Xi (C) → Im fi (C) being a minimal right X -approximation. Denote Im fn (C) by ΩnX (C) for each C in mod Λ and n > 0. Let ΩnX (mod Λ) denote the additive subcategory of mod Λ generated by the modules ΩnX (C) for all modules C in mod Λ. Theorem 1.3 The subcategory X ∪ ΩnX (mod Λ) is functorially finite in mod Λ for all positive n.

7

In [3] M. Auslander and I. Reiten proved that ΩnΛ (mod Λ) is functorially finite in mod Λ for all positive n. We claimed in the first section that the above result is an generalization of this result. To see this let F be the subfunctor FX of Ext1Λ ( , ). Then P(F ) = X and F has enough projectives and injectives. Since a relative projective cover of a module C in mod Λ is the same as a minimal right X -approximation, we have that relative syzygies ΩnF (C) coincide with ΩnX (C) for every module C in mod Λ. This gives the analogy with the standard result. To prove the theorem we have to show that for each n the subcategory X ∪ Ω nΛ (mod Λ) is functorially finite. We first prove that it is a contravariantly finite subcategory of mod Λ. The proof goes by induction on n, where we only indicate the steps for n = 1. Since F has enough injectives, we also have negative relative syzygies Ω−n F (C) using a minimal relative injective resolution of C in mod Λ. Let C be an arbitrary module in mod Λ. Then construct the following diagram

0 →

0 0 ↓ ↓ ΩF (I) == ΩF (I) ↓ ↓ Z → PI ↓

0 →

C ↓ 0

↓ →

I ↓ 0

→ Ω−1 0 F (C) →

→ Ω−1 F (C) → 0,

where the diagram is induced by starting with the minimal right X -approximation 0 → ΩF (I) → −1 PI → I → 0. Since 0 → Z → PI → Ω−1 F (C) → 0 is a right X -approximation, Z ' ΩF (ΩF (C))⊕X for some module X in X . Then it is relatively straightforward to see that Z → C is a right X ∪ Ω1X (mod Λ)-approximation. To prove that X ∪ ΩnX (mod Λ) is covariantly finite in mod Λ we make use of the following result by M. Auslander and I. Reiten. Proposition 3.4 ([2, Proposition 1.2 (b)]) Suppose C and D are two categories and G: C → D and H: D → C is an adjoint pair of functors with G a left adjoint and H a right adjoint. Then Im H is covariantly finite in C and the homomorphism I → HG given by the adjointness gives a left Im H-approximation C → HG(C) for each object C in C. Let modF Λ denote the category mod Λ modulo the subcategory P(F ) = X . Then we show that ΩnX : modF Λ → modF Λ has a right adjoint, hence ΩnX (mod Λ) is covariantly finite in modF Λ. Since X is covariantly finite in mod Λ, it follows easily that X ∪ ΩnX (mod Λ) is covariantly finite in mod Λ. Acknowledgements The work leading up to this paper was done while the second author spent the academic years of 1990-91 and 1991-92 in the Department of Mathematics at Brandeis University, Waltham, Mass., USA. He would like to express his gratitude for the hospitality and the support of the department and his coauthor during his stay there.

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[3] M. Auslander, I. Reiten, k-Gorenstein algebras and syzygy modules, Preprint, Mathematics no. 12/1992, The University of Trondheim, Trondheim, Norway. [4] M. Auslander, S. O. Smalø, Preprojective modules over artin algebras, J. Algebra 66 (1980) 61–122. [5] M. Auslander, Ø. Solberg, Relative homology and representation theory I, Relative homology and homologically finite subcategories, Comm. in Alg., to appear. [6] M. Auslander, Ø. Solberg, Relative homology and representation theory II, Relative cotilting theory, Comm. in Alg., to appear. [7] M. Auslander, Ø. Solberg, Relative homology and representation theory III, Cotilting modules and Wedderburn correspondence, Comm. in Alg., to appear. [8] M. Auslander, Ø. Solberg, Gorenstein algebras and algebras with dominant dimension at least 2, Comm. in Alg., to appear. [9] D. A. Buchsbaum, A note on homology in categories, Ann. Math., Princeton 69 (1959) 66–74. [10] M. C. R. Butler, G. Horrocks, Classes of extensions and resolutions, Phil. Trans. Royal Soc., London, Ser. A, 254 (1961) 155–222. [11] A. Heller, Homological algebra in abelian categories, Ann. Math., Princeton, 68 (1958) 484– 525. [12] G. Hochshild, Relative homological algebra, Trans. Amer. Math. Soc. 82 (1956) 246–269.

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