Covers and envelopes via endoproperties of modules

Lidia Angeleri-Hügel

Abstract Let R be a ring and M_{R} be an R-module. We characterize the existence of Add Mcovers and Add M-envelopes in terms of finiteness conditions of M over its endomorphism ring. We then present some applications related to the existence of well-behaved direct sum decompositions for direct products of copies of M. Our results can be viewed as natural extensions of classical theorems of Bass and Chase on coherent and perfect rings. ^{1}

Precovers and preenvelopes were introduced in the early eighties by Enochs [23] and, independently, by Auslander and Smalø [12]. Enochs gave a general definition in terms of commutative diagrams for modules over arbitrary rings, whereas Auslander and Smalø, mainly concerned with the case of finitely generated modules over finite-dimensional algebras, stressed the functorial viewpoint and coined the terminology of contravariantly and covariantly finiteness. Both approaches turned out to be extremely fruitful for general module theory as well as for representation theory.

The idea behind these concepts is to exploit interesting features of a special class of modules X for the study of the whole module category. The link between X and the other modules is given by some universal homomorphisms, namely the precovers and the preenvelopes.

The work of Enochs and other authors provided a common frame for a number of classical notions, such as injective envelopes or projective covers, and on the other hand, it also raised many challenging problems that are still object of current research. Similarly, contravariantly and covariantly finite subcategories had a significant impact on the representation theory of finite-dimensional algebras. In fact, Auslander and Reiten established in [10] an intimate relationship between these concepts and the notion of a tilting module, which opened up new applications of tilting theory in different areas, e. g. to quasi-hereditary algebras, algebraic groups, or Cohen-Macaulay rings. Moreover, contravariantly finite subcategories have been employed to study the Homological Conjectures [30], [46], [5] and have provided a better understanding of the interaction between finite-dimensional and infinite-dimensional modules.

1Key words: left/right approximations, endomorphism ring, matrix subgroup, perfect rings, coherent rings. MSC 16D90 16G10 16S50 16D40 16P70 16E50

This interaction is also one of the main aspects of the present paper. We consider an arbitrary ring R and discuss the existence of precovers and preenvelopes with respect to certain classes X of (right) R-modules. Actually, the only constrain on our classes X concerns their size. More precisely, we assume that there is a set of modules {M_{i} | i ∈ I} such that every module in X is isomorphic to a direct summand of a coproduct of copies of the module M = ∐_{i∈I} M_{i}. Such classes are denoted by Add M. We show that the existence of Add M-preenvelopes or Add M-precovers can be translated into module-theoretical properties of M over its endomorphism ring S. This will allow us to handle infinite categories X in terms of finiteness conditions on a single module _{S}M. On the other hand, we will investigate non-finitely generated modules M_{R} and even big products of copies of M_{R} by studying the category X given by the (perhaps finitely generated or indecomposable) direct summands of M_{R}.

One of the main endoproperties we will consider concerns S-submodules of M of some special kind, namely the matrix subgroups introduced by Zimmermann in [51]. We discuss the case that matrix subgroups are finitely generated over S. It turns out that this is closely related to the existence of Add Mpreenvelopes. By specializing to the case M = R, we then rediscover some known features of coherent rings and of the strongly coherent rings studied in [48] and [24].

We then turn to Add M-precovers. Every module C_{R} has an Add Mprecover g : B → C. But is there even an Add M-cover? That is, can we choose g in such a way that it also satisfies a minimality condition ensuring that B is uniquely determined by C up to isomorphism? For the case M = R, the answer is given by Bass’s work on perfect rings. We will see that Bass’s results extend to the general situation. In fact, we prove that Add M-covers exist for instance when Add M consists of modules with semiregular endomorphism ring, or when the endomorphism ring of each M_{i} is local and _{S}M satisfies the descending chain condition on cyclic S-submodules. By using some results on perfect functor categories due to Harada and Simson [27], [44], we then obtain that these perfectness conditions even characterize the existence of Add M-covers in case that all M_{i} are finitely presented.

Finally, we deal with Add M-envelopes, that is, Add M-preenvelopes satisfying a dual minimality condition. It was shown by Krause and Saorín that Add M-envelopes exist if and only if all products of copies of M have wellbehaved direct sum decompositions [34]. We prove that the latter property can be characterized by combining the coherence and perfectness properties mentioned above. This extends a well-known theorem of Chase asserting that all products of projective modules are projective if and only if R is left coherent and right perfect. We then apply our results to endofinite modules over artin algebras.

The paper is organized as follows. In Sections 1 and 2, we review the notions of a precover, a preenvelope, Azumaya’s notion of a locally split epimorphism, and some other related concepts. The existence of Add M-preenvelopes and Add M-covers is discussed in Sections 3 and 4, respectively. In Section 5, we combine the previous results to describe the existence of Add M-envelopes and to study direct products of modules.

The author wishes to thank Wolfgang Zimmermann for many valuable discussions and acknowledges a research grant of the University of Munich inside the HWP-program.

1 Preliminaries

Let us start with some general notation which we will use throughout the paper. We denote by R an arbitrary ring and write ModR and modR for the categories of all, respectively of the finitely presented, right R-modules. By a subcategory we always mean a full subcategory.

Let X be a subcategory of ModR and C be a right R-module. Following Enochs [23], we say that a homomorphism b : X → C is an X -precover if X ∈ X and the abelian group homomorphism Hom_{R}(X′, b) : Hom_{R}(X′, X) → Hom_{R}(X′, C) is surjective for each X′ ∈ X. In the representation theory of artin algebras, the usual terminology is right X -approximation. A homomorphism b : X → C is said to be right minimal if every endomorphism h : X → X such that b h = b is an isomorphism. Right minimal precovers are called covers and are uniquely determined up to isomorphism [47, 1.2.6]. We will say that X is a (pre)covering class, if every right R-module has an X -(pre)cover. X -(pre)envelopes and (pre)enveloping classes are defined dually.

Let us remark that a covering class X which is closed under isomorphic images is also closed under direct summands and coproducts. This follows from the fact that a cover is an isomorphism whenever it is a split epimorphism, see for instance [47, 1.2.8].

We will stick to classes with such closure properties and will make a restriction on their size. We fix for the whole paper a skeletally small subcategory M of ModR, and let {M_{i} | i ∈ I} be a complete irredundant set of representatives of the isomorphism classes of M. We put M = ∐_{i∈I} M_{i} and S = End_{R} M. Our focus will be on the class Add M consisting of all modules isomorphic to direct summands of coproducts of modules of M.

Note that Add M is always a precovering class. In fact, for any module C_{R}, the coproduct map g : M^{(H)} → C induced by all homomorphisms in H = Hom_{R} (M, C) is an Add M-precover. Moreover, Add M is a (pre)enveloping class if and only if it is closed under products [41]. In the latter case, following Krause and Saorín [34], we will call M product-complete.

To study the existence of Add M-preenvelopes, we will often consider a special class of S-submodules of M, the matrix subgroups [51]. Recall that a subgroup U of the abelian group M is said to be a matrix subgroup of M if there is a module A_{R} and an element x ∈ A such that U equals the set H_{A,x}(M)= {f(x) | f ∈ Hom_{R} (A, M)}. Of course, every matrix subgroup is a left S-submodule of M. Moreover, the functor M ↦ H_{A,x}(M) commutes with products and coproducts. A matrix subgroup H_{A,x}(M) is called finite matrix subgroup if A_{R} is finitely presented, see also [31].

Finally, we will also need the category add M of all modules isomorphic to direct summands of finite direct sums of modules of M.

2 Locally split homomorphisms

In this section, we introduce some categories containing Add M which will play an important role in the sequel. Following Azumaya [13], we will say that a homomorphism g : B → A is a locally split epimorphism if for each x ∈ A there is a map φ = φ_{x} : A → B such that x = gφ(x). Of course, every split epimorphism is locally split. Further, given a locally split epimorphism g : B → A and a finite number of elements x_{1}, …, x_{n} ∈ A, we can use an argument of Villamayor (see [19, Proposition 2.2] or [15, Corollary 2]) to construct a map φ : A → B such that x_{i} = gφ(x_{i}) for all 1 ≤ i ≤ n. This shows that every locally split epimorphism g : B → A is a pure epimorphism, and moreover, that g splits whenever A is countably generated. Locally split monomorphisms are defined dually.

A submodule X of a module Y is said to be a locally split (or strongly pure [42], [53]) submodule if the embedding X ⊂ Y is locally split. Examples for locally split submodules are the local direct summands, that is, the submodules X of Y having a decomposition X = ∐_{k∈K} X_{k} with the property that ∐_{k∈K_{0}} X_{k} is a direct summand of Y for every finite subset K_{0} ⊂ K.

We now investigate the category G(M) of all modules A admitting a locally split epimorphism g : M^{(K)} → A for some set K.

The modules in G(R) were already studied by several authors [26], [40], [24], [48]. We adopt the terminology of [48] and call them locally projective. Observe that a module A is locally projective if and only if every epimorphism Y → A is locally split, or equivalently, if A is flat and locally pure-projective. Here A is called locally pure-projective if every pure epimorphism Y → A is locally split. As shown by Azumaya [15, Proposition 8], the locally pureprojective modules coincide with the strict Mittag-Leffler modules studied by Gruson and Raynaud in [26].

In the general case, the modules in G(M) can be characterized as follows.

Lemma 2.1 The following statements are equivalent for a module A_{R}.

(1) A ∈ G(M).

(2) Every Add M-precover of A is a locally split epimorphism.

(3) For all X_{R} and x ∈ X we have H_{X,x}(A) = ∑_{g∈Hom_{R} (M,A)} g(H_{X,x}(M)).

(4) For each x ∈ A there are n ∈ ℕ and homomorphisms f : A → M^{n} and φ : M^{n} → A such that φ f(x) = x.

Proof : (1)⇒(2) : This follows immediately from the fact that the locally split epimorphism g : M^{(K)} → A factors through any Add M-precover of A. Since every module has an Add M-precover, we then conclude (1)⇔(2). (1)⇒(3) : Let h ∈ Hom_{R} (X, A) and g : M^{(K)} → A a locally split epimorphism. Then there is a map φ : A → M^{(K)} such that h(x) = gφh(x) = ∑_{k∈K_{0}}(ge_{k}) (pr_{k}φh)(x) where K_{0} is a finite subset of K and e_{k} : M → M^{(K)} and pr_{k} : M^{(K)} → M, k ∈ K_{0}, are the canonical inclusions and projections, respectively. Since ge_{k} ∈ Hom_{R} (M, A) and (pr_{k}φh)(x) ∈ H_{X,x}(M), we have shown the inclusion ⊂. The reverse inclusion is always true.

(3)⇒(4) : Every x ∈ A can be written as x = id_{A}(x) ∈ H_{A,x}(A) and therefore as x = ∑_{i=1}^{n} g_{i}(y_{i}) for some g_{i} ∈ Hom_{R} (M, A) and y_{i} ∈ H_{A,x}(M), or, more precisely, as x = ∑_{i=1}^{n} g_{i} f_{i}(x) where f_{i} ∈ Hom_{R} (A, M). But this means that x = φ f(x) where f : A → M^{n} is the product map induced by the f_{i} and φ : M^{n} → A is the coproduct map induced by the g_{i}. (4)⇒(1) : For each x ∈ A choose n_{x} ∈ ℕ, f_{x} : A → M^{n_{x}} and φ_{x} : M^{n_{x}} → A such that φ_{x} f_{x}(x) = x. Then it is easy to check that the coproduct map g : ∐_{x∈A} M^{n_{x}} → A induced by the φ_{x} is a locally split epimorphism. □

Observe that a module A ∈ G(M) belongs to Add M provided it is countably generated or pure-projective or has a local endomorphism ring. Moreover, G(M) is closed under coproducts, locally split submodules and locally split epimorphic images.

When M consists of finitely presented modules, G(M) has some further nice properties which we are going to discuss now. To this aim we first need some knowledge on the category M⃗ of all modules which are the direct limit of some direct system of modules in add M.

Lemma 2.2 (Lenzing [35, 2.1]) Assume that M is a subcategory of modR. Then the following statements are equivalent for a module A_{R}.

(1) A ∈ M⃗.

(2) There is a pure epimorphism ∐_{k∈K} X_{k} → A for some modules X_{k} in M.

(3) Every homomorphism h : F → A where F is finitely presented factors through a module in add M.

We then obtain the following result.

Proposition 2.3 Assume that M is a subcategory of modR. Then G(M) consists of all modules in M⃗ which are locally pure-projective. In particular, G(M) is closed under pure submodules.

Proof : If A admits a locally split epimorphism g : M^{(K)} → A for some set K, then it satisfies condition (2) in the above Lemma and therefore lies in M⃗. Further, since M^{(K)} is pure-projective by assumption, the map g factors through any pure epimorphism h : Y → A, which thus has to be locally split as well. This shows that A is locally pure-projective. Conversely, it is straightforward that any locally pure-projective module satisfying condition (2) in the above Lemma belongs to G(M).

It remains to show that G(M) is closed under pure submodules. From condition (3) in Lemma 2.2 it follows that M⃗ is closed under pure submodules, see [33, 3.10]. So, we have only to verify that the class of all locally pureprojective modules is closed under pure submodules. To this end, we apply a result of Azumaya [15, Proposition 4] asserting that a module A is locally pure-projective if and only if for each x ∈ A there are a finitely presented module F and homomorphisms f : A → F and φ : F → A such that φ f(x) = x. So, let us consider a pure submodule A′ ⊂ A of a locally pure-projective module A and an element x ∈ A′. Then there are a finitely presented module F and homomorphisms f : A → F and φ : F → A such that φ f(x) = x. Setting y = f(x) ∈ F, we see that x is contained in the finite matrix subgroup H_{F,y}(A). But A′ ⊂ A being pure implies that H_{F,y}(A) ∩ A′ = H_{F,y}(A′), see [51, p. 1088]. Thus we can find a map φ′ : F → A′ such that x = φ′(y), and we conclude that the maps φ′ and f′ = f|_{A′} : A′ → F satisfy x = φ′ f′(x), as required. □

We will also need a category which can be viewed as the dual counterpart of G(M), namely the category C(M) of all modules which are isomorphic to some locally split submodule of a product of copies M. We denote by Prod M the class of all modules isomorphic to direct summands of products of copies of M. Dually to Lemma 2.1, we can characterize C(M) as follows.

Lemma 2.4 The following statements are equivalent for a module C_{R}.

(1) C ∈ C(M).

(2) Every Prod M-preenvelope of C is a locally split monomorphism.

(3) For all X_{R} and x ∈ X we have H_{X,x}(C) = ⋂_{f∈Hom_{R} (C,M)} f^{−1}(H_{X,x}(M)).

Proof : (1)⇔(2) is proven with arguments dual to those in Lemma 2.1. (1)⇔(3): If C ∈ C(M), then there is a locally split monomorphism τ : C → M^{J} for some set J. Thus every element c ∈ ⋂_{f∈Hom_{R} (C,M)} f^{−1}(H_{X,x}(M)) satisfies τ(c) ∈ ∏_{j∈J} H_{X,x}(M) = H_{X,x}(M^{J}) and admits a map ψ : M^{J} → C such that c = ψτ(c), hence belongs to H_{X,x}(C). Since the inclusion ⊂ is always true, we have thus proven (1)⇒(3). Conversely, if C has property (3), we consider the product map τ : C → M^{J} induced by all homomorphisms in J = Hom_{R} (C, M). Then, if we choose X = M^{J} in (3), it is easy to check that every c ∈ C belongs to H_{M^{J} ,τ(c)}(C) and therefore has the form c = ψτ(c) for some ψ : M^{J} → C. Thus τ is a locally split monomorphism, and C ∈ C(M). □

For further properties of C(M), we refer to [3]. We only point out that C(M) always contains G(M). This can be verified by using condition (4) in 2.1 to construct a locally split monomorphism A → ∏_{x∈A} M^{n_{x}}. We will discuss in 3.4 the case where C(M) and G(M) coincide.

3 Coherence

In this section, we discuss the existence of X -preenvelopes when X is one of the categories considered in the previous section. We will see that this is related to finiteness conditions for matrix subgroups and can be viewed as a form of coherence of M over its endomorphism ring.

We begin with Add M-preenvelopes.

Proposition 3.1 Every finitely presented module has an Add M-preenvelope if and only if all finite matrix subgroups of M_{R} are finitely generated over S.

Proof : We start with the only-if-part. Let A_{R} be a module and assume there is an Add M-preenvelope a : A → X. Since X ∈ Add M, there is a family (X_{k})_{k∈K} in M and a split monomorphism e : X → ∐_{k∈K} X_{k}. Moreover, for any k ∈ K there is a split monomorphism e_{k} : X_{k} → M. Consider the family

(g_{k})_{k∈K} in Hom_{R} (A, M) given by g_{k} : A ⟶^{a} X ⟶^{e} ∐_{k∈K} X_{k} ⟶^{pr_{k}} X_{k} ⟶^{e_{k}} M

where pr_{k} are the canonical projections. For every x ∈ A there is a finite subset K_{x} ⊂ K such that g_{k}(x) = pr_{k} ea(x) = 0 for all k ∈ K \ K_{x}, and so H_{A,x}(M) = ∑_{k∈K_{x}} S · g_{k}(x). For the if-part, we consider a finitely presented module A_{R}, and let A denote the category of all modules X which are isomorphic to A/A′ for some finitely generated submodule A′ ⊂ A. Since A consists of finitely presented modules, we know by assumption that all matrix subgroups of the form H_{X,x}(M) with X ∈ A and x ∈ X are finitely generated over S. As shown by Zimmermann [52], this implies that for all n ∈ ℕ, all X ∈ A and \underline{x} = (x_{1}, …, x_{n}) ∈ X^{n} the image H_{X,\underline{x}}(M) of the S-homomorphism ε\underline{x} : Hom_{R} (X, M) ⟶ M^{n}, f ↦ (f(x_{1}), …, f(x_{n})) is finitely generated. In particular, if we take a generating set a_{1}, …, a_{n} of A_{R} and consider the S-monomorphism ε\underline{a} where \underline{a} = (a_{1}, …, a_{n}) ∈ A^{n}, we obtain that _{S}Hom_{R} (A, M) is finitely generated. Now one can easily check that the map c : A → M^{r} induced by an S-generating set c_{k} : A → M, 1 ≤ k ≤ r, of Hom_{R} (A, M) is an Add M-preenvelope of A. □

Recall that a subcategory M ⊂ modR is said to be covariantly finite in modR if every finitely presented module has an add M-preenvelope. As observed by Crawley-Boevey and Krause, covariantly finiteness can be characterized in terms of the category M⃗. Combining their results with Proposition 3.1, we obtain

Theorem 3.2 The following statements are equivalent for a subcategory M of modR.

(1) All finite matrix subgroups of M_{R} are finitely generated over S.

(2) M is covariantly finite in modR.

(3) [22, 4.2] M⃗ is closed under products.

(4) [33, 3.11] M⃗ is a preenveloping class.

Observe that this theorem contains two known characterizations of coherent rings. In fact, in case that M just consists of the regular module R, condition (3) is nothing else than left coherence of R, and the equivalence of (1) and (3) was already proven by Zimmermann in [51, 1.3], while the equivalence of (3) and (4) is due to Enochs [23, 5.1].

We now give an analogous characterization of the case that all matrix subgroups of M are finitely generated over S. Here the role of M⃗ will be played by G(M).

Lemma 3.3 [2, 2.1] Let A_{R} be a module, x ∈ A, and assume that there are a family (X_{k})_{k∈K} in M and homomorphisms μ : A → ∏_{k∈K} X_{k}, and ε : ∏_{k∈K} X_{k} → A with εμ(x) = x. Further, let λ be a cardinal such that the matrix subgroup of M of the form H_{A,x}(M) is a λ-generated left S-module. Then there is a family (Y_{l})_{l∈L} in M, where the index set L is finite if λ is finite or else has cardinality λ, and there are homomorphisms f : A → ∏_{l∈L} Y_{l} and φ : ∏_{l∈L} Y_{l} → A such that φf(x) = x.

Theorem 3.4 The following statements are equivalent.

(1) All matrix subgroups of M are finitely generated over S.

(2) G(M) = C(M).

(3) G(M) is closed under products.

(4) M^{M} ∈ G(M). If M ⊂ modR, then the following statement is further equivalent.

(5) G(M) is a preenveloping class.

Proof : (1)⇒(2) : By the concluding remarks in Section 2, we have only to prove that C(M) ⊂ G(M). Let A ∈ C(M) and μ : A → M^{K} be a locally split monomorphism. For every x ∈ A there is a map ε : M^{K} → A such that x = ε μ(x). But then condition (1) and Lemma 3.3 tell us that we can choose homomorphisms f : A → M^{n} and φ : M^{n} → A with n ∈ ℕ such that x = φ f(x). In other words, A satifies condition (4) in Lemma 2.1 and is therefore contained in G(M). (2)⇒(3) follows from the fact that C(M) is closed under products; (3)⇒(4) is clear. (4)⇒(1) : It is well known that every matrix subgroup H_{A,x}(M) of M can be written in the form H_{M^{M} , y} (M) by taking the element y = (y_{m})_{m∈M} ∈ M^{M} defined by y_{m} = m if m ∈ H_{A,x}(M) and y_{m} = 0 otherwise, see for instance [48, p. 241]. So, we can always assume that A is in G(M). Then we know from Lemma 2.1 that for any x ∈ A there are n ∈ ℕ and homomorphisms f : A → M^{n} and φ : M^{n} → A such that φ f(x) = x. So, putting m = (m_{1}, …, m_{n}) = f(x), we have for all h ∈ Hom_{R} (A, M) that h(x) = h′(m) with h′ = h φ ∈ Hom_{R} (M^{n}, M). This shows H_{A,x}(M) ⊂ H_{M^{n},m}(M). Since h(m) = hf(x) for all h ∈ Hom_{R} (M^{n}, M), we conclude H_{A,x}(M) = H_{M^{n},m}(M). But this means that m_{1}, …, m_{n} ∈ M generate H_{A,x}(M) over S, and condition (1) is verified. Finally, if M ⊂ modR, then we know from Proposition 2.3 that G(M) is closed under pure submodules. Now, it was shown in [41, 3.5] that a class which is closed under pure submodules is a preenveloping class if and only if it is closed under products. This completes the proof. □

If we restrict to the case M = R, then condition (3) in the above theorem asserts that products of (locally) projective right R-modules are locally projective. Rings with this property are called left strongly coherent, and the equivalence of conditions (1)-(4) in case M = R was independently obtained by Zimmermann-Huisgen [48] and by Garfinkel [24].

Further coherence properties over the endomorphism ring will be considered in a subsequent paper.

4 Perfectness

This section is devoted to the existence of AddM-covers. We will interpret this property as a form of perfectness of M over its endomorphism ring.

Recall that a ring R is said to be semiregular if idempotents lift modulo the Jacobson radical J(R) and R/J(R) is von Neumann regular. These rings were introduced by Oberst and Schneider [39] under the name F-semiperfect and were later studied by Nicholson [38] and other authors.

Proposition 4.1 If End_{R} A is semiregular for all A ∈ Add M, then Add M is a covering class.

Proof : Let C_{R} be a module and g : M^{(H)} → C an Add M-precover. We want to show that g has a right minimal version, i. e. that there is a decomposition M^{(H)} = X ⊕ X′ such that g(X′) = 0 and the restriction g|_{X} : X → C of g on X is right minimal. Of course, g|_{X} is then an Add M-cover. Set E = End_{R} M^{(H)}, and assume that r_{E}(g) = {h ∈ E | g h = 0} is a finitely generated right ideal. Since E is semiregular, we know from [39, Satz 1.2] that there is a right ideal I which satisfies r_{E}(g) + I = E and is minimal with respect to this property. Then r_{E}(g) ∩ I is superfluous in I and therefore also in E. Thus our claim follows from a result of Zimmermann [52] stating that g has a right minimal version provided that (i) idempotents lift modulo J(E); and (ii) there is a right ideal I in E such that r_{E}(g) + I = E and r_{E}(g) ∩ I ⊂ J(E). So, our task now is to show that we can assume r_{E}(g) finitely generated without loss of generality. First of all, putting K = Ker g, we deduce from the exact

sequence 0 → Hom_{R} (M^{(H)}, K)_{E} ⟶ E_{E} ⟶^{Hom_{R} (M^{(H)}, g)} Hom_{R} (M^{(H)}, C)_{E} → 0

that r_{E}(g) ≃ Hom_{R} (M^{(H)}, K). Let us now consider an Add M-precover f : M^{(J)} → K. If the cardinality of J is less or equal the cardinality of H, then the epimorphism Hom_{R} (M^{(H)}, f) : Hom_{R} (M^{(H)}, M^{(J)})_{E} → Hom_{R} (M^{(H)}, K)_{E} shows that r_{E}(g) is even a cyclic ideal. If the cardinality of J is greater than the cardinality of H, then J has a subset L such that M^{(L)} ⊕ M^{(H)} ≅ M^{(J)}, and we can replace g and f by the Add M-precovers g′ = (0, g) : M^{(J)} ≅ M^{(L)} ⊕M^{(H)} → C and f′ = pr⊕f : M^{(J)} ⊕M^{(J)} → M^{(L)} ⊕K = Ker g′, where pr denotes the canonical projection. Then, putting E′ = End_{R} M^{(J)}, we infer from the epimorphism Hom_{R} (M^{(J)}, f′) : E′ ^{2} _{E′} → Hom_{R} (M^{(J)}, Ker g′)_{E′} that r_{E′}(g′) is a finitely generated right ideal of E′, and the proof is complete. □

Our aim is now to prove that the converse of Proposition 4.1 holds true when M consists of finitely presented modules. To this end, we first have to recall Harada’s notion of semi-T-nilpotency and some related concepts.

We say that the family (M_{i})_{i∈I} is locally semi-T-nilpotent if for each sequence of non-isomorphisms M_{i_{1}} ⟶^{f_{1}} M_{i_{2}} ⟶^{f_{2}} M_{i_{3}} ⟶^{f_{3}} … with pairwise different indices (i_{n})_{n∈ℕ} from I, and each element x ∈ M_{i_{1}}, there exists m = m_{x} ∈ ℕ such that f_{m} f_{m−1} … f_{1}(x) = 0. If the same condition is satisfied also when we allow repetitions in the sequence of indices (i_{n})_{n∈ℕ} involved, then the family (M_{i})_{i∈I} is called locally T-nilpotent. If, furthermore, the index m does not depend on the element x, that is, if for each sequence of non-isomorphisms M_{i_{1}} ⟶^{f_{1}} M_{i_{2}} ⟶^{f_{2}} M_{i_{3}} ⟶^{f_{3}} … there exists m ∈ ℕ such that f_{m} f_{m−1} … f_{1} = 0, then the family (M_{i})_{i∈I} is said to be T-nilpotent. In Auslander’s terminology [7], this means that all families of non-isomorphisms between modules from M are noetherian. Of course, if all M_{i} are finitely generated, then (M_{i})_{i∈I} is T-nilpotent whenever it is locally T-nilpotent.

As pointed out by Harada, the concept of T-nilpotency can be interpreted in terms of the functor category. Let X be a skeletally small subcategory of ModR, and let {X_{β} | β ∈ B} be a complete irredundant set of representatives of the isomorphism classes of X. We denote by (X^{op}, Ab) the category of all additive contravariant functors X → Ab from X to the category Ab of abelian groups. It is well known that (X^{op}, Ab) is a locally finitely presented Grothendieck category and that the functors H_{β} = Hom_{R} ( , X_{β}) |X, β ∈ B, form a family of finitely presented projective generators. We say that (X^{op}, Ab) is perfect if each of its objects admits a projective cover.

Recall finally that a left module _{B}X over a ring B is said to be coperfect if it satisfies the descending chain condition for cyclic (or equivalently, finitely generated) B-submodules [18].

We now collect a number of different interpretations for the condition considered in Proposition 4.1.

Proposition 4.2 Assume that the endomorphism ring of each M_{i} is local. Then the following statements are equivalent.

(1) End_{R} A is semiregular for all A ∈ Add M.

(2) The family (M_{i})_{i∈I} is locally T-nilpotent.

(3) Every local direct summand of a module A ∈ Add M is a direct summand.

(4) _{S}M is coperfect. Moreover, the conditions (1)-(4) are satisfied provided that ((add M)^{op}, Ab) is perfect, and the converse implication holds when all M_{i} are finitely generated.

Proof : Harada has proven in [28, 7.3.15] that semiregularity of S is equivalent to local semi-T-nilpotency of the family (M_{i})_{i∈I} and is also equivalent to the property that every local direct summand of M is a direct summand. We now use his result to prove the equivalence of the first three conditions. (1)⇒(2) : Let M_{i_{1}} ⟶^{f_{1}} M_{i_{2}} ⟶^{f_{2}} M_{i_{3}} … be a sequence of non-isomorphisms with possible repetitions in the indices (i_{n})_{n∈ℕ} taken from I. Roughly speaking, we will now force the indices to become pairwise different by considering the (external) direct sum A = ∐_{n∈ℕ} M_{i_{n}}. More precisely, we denote by e_{n} : M_{i_{n}} → A the canonical embeddings and set A_{n} = e_{n}(M_{i_{n}}). Since the endomorphism ring of A ∈ Add M is semiregular, the family (A_{n})_{n∈ℕ} is locally semi-T-nilpotent. Moreover, our sequence of non-isomorphisms corresponds to a sequence of non-isomorphisms between the A_{n} where none of the modules involved occurs more than once. Then for any element x ∈ M_{i_{1}} there exists n ∈ ℕ such that f_{n} f_{n−1} … f_{1}(x) = 0. (2)⇒(1) : If a ring E is semiregular and e ∈ E is idempotent, then eEe is semiregular as well [38, 2.3]. So, it is enough to consider modules A of the form M^{(K)} for some set K. But M^{(K)} = ∐_{(i,k)∈I×K} M_{(i,k)} where M_{(i,k)} ≃ M_{i} for all i ∈ I and k ∈ K, and the family (M_{(i,k)})_{(i,k)∈I×K} is locally semi-T-nilpotent by assumption. Thus the claim follows from Harada’s result. (1)⇔(3) is proven similarly. (2)⇔(4) is due to Huisgen-Zimmermann and Saorín [29, Proposition E]. Finally, the last statement holds by [27, Theorem 5] which asserts that the family (M_{i})_{i∈I} is T-nilpotent if and only if the category ((add M)^{op}, Ab) is perfect. □

Perfect functor categories were also studied by Simson. Flat objects and Mittag-Leffler objects play a prominent role in his characterization, which we are going to recall now.

Let X and H_{β} be as above. A functor F ∈ (X^{op}, Ab) is called flat if it is a direct limit of finitely generated free objects, that is, of finite coproducts of some H_{β}. Furthermore, a direct system (F_{j}, φ_{kj})_{k,j∈K} in (X^{op}, Ab) is said to be factorizable if for every j ∈ J there exists l ≥ j such that φ_{lj} factors through each φ_{kj} with k ≥ j. Extending the concept of a Mittag-Leffler module due to Gruson and Raynaud [26], Simson has studied in [44] the functors F ∈ (X^{op}, Ab) which are colimits of factorizable direct systems consisting of finitely presented objects. He has called them Mittag-Leffler objects. They coincide with the Mittag-Leffler modules of [26] when the category X just consists of the module R_{R}.

Theorem 4.3 (Simson [44, Theorem 5.4 and Corollary 5.2]) The following statements are equivalent for a skeletally small subcategory X ⊂ ModR.

(1) The category (X^{op}, Ab) is perfect.

(2) Every flat object in (X^{op}, Ab) has a projective cover.

(3) Every flat object in (X^{op}, Ab) is projective.

(4) Every flat object in (X^{op}, Ab) is a Mittag-Leffler object. In this case, every X ∈ X has a finite decomposition in modules with local endomorphism ring.

We are now in a position to describe when a category M of finitely presented modules provides for Add M-covers.

Theorem 4.4 Assume that M ⊂ modR. Then the following statements are equivalent.

(1) Add M is a covering class.

(2) ((add M)^{op}, Ab) is perfect.

(3) Add M is closed under direct limits.

(4) G(M) is closed under direct limits.

(5) All modules in M⃗ are Mittag-Leffler modules.

(6) End_{R} A is semiregular for all A ∈ Add M.

(7) M_{R} has a decomposition in modules with local endomorphism ring, and SM is coperfect.

(8) Every pure submodule (or every locally split submodule, or every local direct summand) of a module A ∈ Add M is a direct summand. If M is a finite category, then (1)-(8) are further equivalent to

(9) S is right perfect.

Proof : We will employ the functor H : ModR ⟶ ((add M)^{op}, Ab) which maps every module A_{R} to the functor H_{A} = Hom_{R} ( , A) |_{add M} and every homomorphism g ∈ Hom_{R} (B, A) to the natural transformation Hom_{R} ( , g) : H_{B} → H_{A}. Since M ⊂ modR, we know that H commutes with coproducts and direct limits. As a consequence, H induces an equivalence between M⃗ and the category of the flat objects in ((add M)^{op}, Ab), which restricts to an equivalence between Add M and the category of the projective objects in ((add M)^{op}, Ab), see [35, 2.4]. This shows immediately that condition (3) is satisfied if and only if every flat object in ((add M)^{op}, Ab) is projective, which is equivalent to condition (2) by Theorem 4.3. Further, it is straightforward that a homomorphism g : B → A is an Add Mprecover if and only if Hom_{R} ( , g) : H_{B} → H_{A} is an epimorphism with H_{B} being projective. Also, as an easy consequence of Yoneda’s Lemma, we see that g is right minimal if and only if so is Hom_{R} ( , g). But an epimorphism in ((add M)^{op}, Ab) of the form H_{B} → F is right minimal if and only if it is superfluous, see [9, 2.1]. So, we conclude that under the action of H, the Add M-covers correspond to the projective covers. Thus we have (2)⇒(1). Conversely, condition (1) implies that all flat objects in ((add M)^{op}, Ab) have a projective cover, which again by Theorem 4.3 is equivalent to (2). We remark that (3)⇒(1) can also be obtained as a special case of [47, Theorem 2.2.8]. Next, we recall that G(M) consists of the modules in M⃗ which are locally pure-projective by Proposition 2.3. In particular, this implies that (3)⇒(4) is true. Moreover, if (4) is satisfied, then all modules in M⃗ are locally pureprojective and therefore Mittag-Leffler modules by a result of Azumaya [15, Theorem 5]. This shows (4)⇒(5). Assume now condition (5). Using again Yoneda’s Lemma, we see that a direct system in ModR is factorizable if and only if so is the direct system in ((add M)^{op}, Ab) which corresponds to it under the action of H. Hence, as observed by Simson [44, p. 105], a module A ∈ M⃗ is a Mittag-Leffler module if and only if H_{A} is a Mittag-Leffler object in ((add M)^{op}, Ab). But then we deduce from (5) that all flat objects in ((add M)^{op}, Ab) are Mittag-Leffler objects, which is equivalent to (2) by Theorem 4.3. So, we have proven the equivalence of the first five conditions. Remind further that under condition (2), each M_{i} has a decomposition in modules with local endomorphism ring by Theorem 4.3. From Proposition 4.2 we then obtain (2)⇒(7)⇒(6), while (6)⇒(1) holds by Proposition 4.1. Let us now turn to condition (8). Observe that the cokernel C of a pure monomorphism f : A′ → A with A ∈ Add M belongs to M⃗ by Lemma 2.2. So, if Add M is closed under direct limits, the module C is pure-projective and f splits. This shows that (3) implies the strongest version of (8). Then also the special cases hold true, that is, locally split submodules, and in particular, local direct summands, of modules in Add M split. Conversely, it is enough to require that all local direct summands of modules in Add M are direct summands to obtain condition (6). In fact, under this assumption, M has a decomposition in modules with local endomorphism ring by [25, 2.3], and so Proposition 4.2 applies. Finally, let us consider the case that M is a finite category. Then the wellknown argument of Bass shows that under condition (7) the Jacobson radical J(S) of S is right T-nilpotent. Moreover, S is then obviously semiperfect and therefore right perfect. Conversely, (9)⇒(8) is proven by a standard argument applying the functor Hom_{R} (M, ) : ModR → ModS on pure-exact sequences 0 ⟶ A′ ⟶^{f} A ⟶ C ⟶ 0 in ModR with A ∈ Add M. □

Of course, in the case M = R, Theorem 4.4 contains Theorem P of Bass [17] as well as some other characterizations of perfect rings due to Nicholson [38, 3.9], Huisgen-Zimmermann [49, p. 60] and to Azumaya and Facchini [16, Corollary 2], [14, Theorem 5].

In case that M is an arbitrary projective module, we rediscover and improve a characterization of perfect modules due to Stock. According to [37], a projective module P is called perfect if all modules generated by P have a projective cover.

Corollary 4.5 (cp. [45, 3.9]) Assume that M is projective. Then the following statements are equivalent.

(1) M is perfect.

(2) End_{R} M^{(K)} is semiregular for all sets K. If M is finitely generated, (1) is further equivalent to

(3) S is right perfect.

Proof : (1)⇒(2) was already proven by Mares [37, 2.4 and 4.5]. Conversely, condition (2) implies by our Proposition 4.1 that every module A has a right minimal Add M-precover g : B → A. If A is M-generated, then g is an epimorphism and hence a projective cover, see [47, 1.2.12]. Thus (2)⇒(1). If, moreover, M is finitely generated, the equivalence of (6) and (9) in Theorem 4.4 yields (2)⇔(3). □

Finally, in the case M = modR, closure under direct limits of the category Add M means exactly that R is right pure-semisimple, and our Theorem 4.4 contains known characterizations of pure-semisimple rings due to Azumaya, Facchini, Leszczyǹski and Simson [16, Proposition 5], [15, Theorem 6], [44, Theorem 6.3], [36, Theorem 1].

5 Direct products

Chase has shown in [19] that a ring R is left coherent and right perfect if and only if all products of projective right R-modules are projective, or in other words, iff R_{R} is product-complete. This result can be extended to modules by combining the results of Sections 3 and 4. From Theorems 3.4 and 4.4 we then obtain a whole bunch of conditions characterizing product-complete modules. We only mention the most salient ones.

Theorem 5.1 The following statements are equivalent.

(1) Add M is an enveloping class.

(2) M is product-complete.

(3) Add M = G(M), and all matrix subgroups of M are finitely generated over S. (4)_{S}M is coperfect, and all finite matrix subgroups of M are finitely generated over S. If M ⊂ modR, then (1) is further equivalent to

(5) Add M is a covering class, and M is covariantly finite in modR.

Proof : (1)⇔(2) is due to Krause and Saorín [34], who also proved that product-complete modules are Σ-pure-injective. Note that when M is Σ-pure-injective, every pure epimorphism of the form M^{(K)} → A splits, and hence G(M) = Add M. So (2)⇔(3) follows immediately from Theorem 3.4. Moreover, since every Σ-pure-injective module M satisfies the descending chain condition for matrix subgroups, and in particular for finitely generated Ssubmodules, we obtain (3)⇒(4). Conversely, a module with property (4) obviousy satisfies the descending chain condition for finite matrix subgroups and is therefore Σ-pure-injective. But matrix subgroups of Σ-pure-injective modules are always finite matrix subgroups [50, p. 704], and so we conclude (4)⇒(3). Finally, Σ-pure-injective modules always have a decomposition in modules with local endomorphism ring. Hence (4)⇔(5) is just Theorem 4.4 together with Theorem 3.2. □

Recall that M is said to be endonoetherian, respectively endofinite, if SM is noetherian, respectively a module of finite length. It is well known that endofinite modules are always product-complete. We now prove the converse for pure-projective modules over artin algebras.

Theorem 5.2 Assume that R is an artin algebra with ordinary duality D : modR → Rmod and that M ⊂ modR. Let M^{∗} be the subcategory of Rmod consisting of all duals D(X) of modules X ∈ M, and set M^{∗} = ∐_{i∈I} D(M_{i}). Then the following statements are equivalent.

(1) M is product-complete.

(2) G(M) and G(M^{∗}) are closed under products.

(3) G(M) is closed under products, and M is contravariantly finite in modR.

(4) M^{∗} is product-complete. In particular, M is endofinite under any of these conditions.

Proof : (1)⇒(2) : G(M) is closed under products by Theorem 5.1. As far as G(M^{∗}) is concerned, we argue as in [50, Corollary 12] and use the fact that M is Σ-pure-injective to deduce from [50, Proposition 3] that M^{∗} is endonoetherian. Of course, all matrix subgroups of M^{∗} are then finitely generated over the endomorphism ring, and the claim follows from Theorem 3.4. (2)⇒(3) : When G(M^{∗}) is closed under products, then M^{∗} is covariantly finite in Rmod by Theorems 3.4 and 3.2. Hence, applying D, we obtain that M is contravariantly finite in modR. (3)⇒(1) : We claim that G(M) is closed under direct limits. This will imply that Add M is a covering class by Theorem 4.4, and that M⃗ = G(M) is closed under products, hence M is covariantly finite in modR by Theorem 3.2. So, M will be product-complete by Theorem 5.1. To prove the claim, observe first that M^{∗} is covariantly finite in Rmod, hence M⃗^{∗} is closed under products by Theorem 3.2. Let now A_{R} be a module in M⃗. By Lemma 2.2 there is a pure-exact sequence 0 ⟶ K ⟶ ∐_{k∈K} X_{k} ⟶ A ⟶ 0 with X_{k} in M. Applying D, we obtain a pure-exact sequence 0 ⟶ D(A) ⟶ ∏_{k∈K} D(X_{k}) ⟶ D(K) ⟶ 0 with D(X_{k}) ∈ M^{∗}, and since D(A) is pure-injective, we infer that D(A) lies in Prod M^{∗}, hence in M⃗^{∗}. Again by Lemma 2.2, we deduce that there is a pure-exact sequence 0 ⟶ K′ ⟶ ∐_{j∈J} Y_{j} ⟶ D(A) ⟶ 0 with Y_{j} ∈ M^{∗}, which in turn implies that D^{2}(A) ∈ Prod M. Since G(M) is closed under products, D^{2}(A) is then contained in G(M). But A is a pure submodule of D^{2}(A), and G(M) is closed under pure submodules by Proposition 2.3. So, we conclude that A belongs to G(M), which proves our claim. (2)⇔(4) holds by symmetry. Assume now that the equivalent conditions (1)-(4) are satisfied. In particular, M^{∗} is then Σ-pure-injective, and we have seen in the proof of (1)⇒(2) that it is also endonoetherian. Thus M^{∗} and M are endofinite. □

Recall that an endofinite module only admits finitely many isomorphism classes of indecomposable direct summands [21]. So, if the M_{i}, i ∈ I, are pairwise nonisomorphic indecomposable finitely generated modules over an artin algebra R, then the above theorem tells us that M is product-complete if and only if the index set I is finite. However, we point out that the assumption M ⊂ modR cannot be removed.

Example 5.3 Let R be a finite-dimensional hereditary algebra over a field. Consider the direct sum T of a complete irredundant set of representatives of the isomorphism classes of the Prüfer modules and the (unique) generic Rmodule Q, see [43]. Observe that T is an infinite-dimensional tilting module in the sense of [20], see [4, 1.4]. Thus Gen T is closed under products. Observe that T is a divisible module and that the class of all divisible modules is closed under coproducts and epimorphic images by [43, 4.6]. We infer that every direct product T^{K} is divisible and therefore, by [43, 4.7], a coproduct of indecomposable preinjective modules, Prüfer modules and copies of Q. On the other hand, no indecomposable preinjective module can occur as a direct summand of T^{K}, see [8]. Hence we conclude that T^{K} ∈ Add T. So, T is product-complete, but of course, it is not endofinite. □

Let us now consider a generalization of product-completeness. Following [2], we call M product-rigid if any direct summand of a product of copies of M having a local endomorphism ring is isomorphic to some direct summand of M itself. Then M is product-complete if and only if it is product-rigid and Σ-pure-injective. We close the paper by exhibiting a situation as in Theorem 5.2 where M is product-rigid while M^{∗} is not.

Example 5.4 Let R be a finite-dimensional hereditary algebra over a field. We denote by P = {P_{j} | j ∈ J}, respectively I = {I_{k} | k ∈ K}, a complete irredundant set of representatives of the isomorphism classes of the indecomposable preprojective right modules, respectively of the indecomposable preinjective left modules, and we set P_{R} = ∐_{j∈J} P_{j}, and _{R}I = ∐_{k∈K} I_{k}. Observe that I ≅ ∐_{j∈J} D(P_{j}) where D : modR → Rmod is the usual duality.

(I) First of all, we note that the following statements are equivalent (even over an arbitrary artin algebra).

(1) R is of finite representation type.

(2) P_{R} is product-complete.

(3) Add P is a covering class. In fact, if R is of finite representation type, then every module is endofinite [50, Theorem 6] and hence product-complete. Together with Theorem 5.1, we then obtain (1)⇒(2)⇒(3). For (3)⇒(1), note that by Theorem 4.4 and Proposition 4.2, the family (P_{j})_{j∈J} is T-nilpotent, which implies by [9, 5.6] that R is of finite representation type.

(II) P_{R} is product-rigid. Indeed, one proves as in [1, 4.1, step (4)] that every module A_{R} with a nonzero map A → P has a preprojective summand. On the other hand, we know from [35, 4.6] that _{R}I is Σ-pure-injective if and only if R is tame. So, we infer from Theorem 5.2 and (I) that I cannot be product-rigid whenever R is representation-infinite tame.

(III) Let us remark, however, that I is always coperfect over its endomorphism ring. To verify that, we consider a chain X_{1} → X_{2} → X_{3} → … of nonzero non-isomorphisms between indecomposable preinjective modules and let I_{0}, …, I_{∞} be the preinjective partition [12]. Now, if X_{1} belongs to the layer I_{n}, then X_{2}, X_{3}, … belong to the finite subcategory I_{0} ∪ … ∪ I_{n}, see [1, 4.1], [6]. So, we conclude from the Lemma of Harada and Sai that the family (I_{k})_{k∈K} is T-nilpotent, and Proposition 4.2 completes the proof. □

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[53] W. Zimmermann, On locally pure-injective modules, J. Pure Appl. Algebra 166 (2002), 337–357.

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