On the existence of left and right almost split morphisms

Lidia Angeleri Hügel

Introduction

Left and right almost split morphisms are usually studied for categories of finitely generated modules over artin algebras. In this small note, we consider left and right almost split morphisms for a skeletally small subcategory M of the category of all (right) modules over an arbitrary ring. If the modules M_{i}, i ∈ I, are representatives of the isomorphism classes of M, then the existence of left and right almost split morphisms can be interpreted in terms of finiteness conditions for the modules M = ∐_{i∈I} M_{i} and N = ∏_{i∈I} M_{i} viewed as modules over their endomorphism rings S and T, respectively. An important role in this context is played by the radicals r (M, C)_{S} and _{T} r (A, N) where A and C are R-modules. In case that M is a finite subcategory consisting of modules with local endomorphism ring, we can restrict ourselves to the Jacobson radical J(S) of S, and we obtain for instance that M has left (respectively, right) almost split morphisms if and only if J(S) is a finitely generated left (respectively, right) S-module.

We also study generalized right almost split morphisms, a concept introduced by Brune in [3]. For a right artinian ring R, the existence of such maps means that R is right pure-semisimple, see [11]. This is a consequence of Brune’s work on the so-called Kulikov property, relying on a functorial approach. We now obtain a direct, module-theoretical proof of this result and also a dual characterization of pure-semisimple rings in terms of generalized left almost split morphisms.

The paper is divided into three sections. The first section is devoted to some preliminaries. In Section 2, we relate finiteness conditions for _{S}M and T N to the notion of a finitely (co)generated family of homomorphisms which was introduced by Auslander in [1]. These results are then applied in Section 3 to the study of left and right almost split morphisms.

The author acknowledges a HSPIII-grant of the University of Munich.

1 Preliminaries

Let us first introduce some notation. If R is a ring, we write J(R) for the Jacobson radical of R, and denote by ModR the category of all and by modR the category of all finitely presented right R-modules.

Throughout the paper, we fix a skeletally small subcategory M of ModR which is closed under isomorphic images, and let {M_{i} | i ∈ I} be a complete irredundant set of representatives of the isomorphism classes of M. Further, we set M = ∐_{i∈I} M_{i} with S = End_{R} M, and N = ∏_{i∈I} M_{i} with T = End_{R} N. Moreover, we denote by add M the class consisting of all modules isomorphic to direct summands of finite direct sums of modules of M.

Recall that for two modules X_{R}, Y_{R} the radical r (X, Y ) denotes the collection of all homomorphisms f : X → Y such that there is no isomorphism of the form Z → X ⟶^{f} Y → Z where Z is a module with local endomorphism ring. Then r (X, Y) is an End_{R} Y − End_{R} X - subbimodule of Hom_{R} (X, Y). Let us collect some easy properties of this bimodule.

Lemma 1.1 Let Y_{R} be a module with endomorphism ring E = End_{R} Y, and let X_{R} be an indecomposable module.

(1) J(E) ⊂ r (Y, Y), with equality if Y = ∐_{i=1}^{n} Y_{i} and all Y_{i} have local endomorphism ring.

(2) Hom_{R} (X, Y)/r (X, Y) is either zero or a simple left E-module.

(3) r (X, Y) is a noetherian left E-module if and only if Hom_{R} (X, Y) is a noetherian left E-module.

Proof : The first assertion in (1) is well-known. For the second assertion, assume that Y has a decomposition as stated. This means that E is semiperfect, that is, id_{Y} = ∑_{i=1}^{n} e_{i} for local idempotents e_{1}, …, e_{n} ∈ E. Then every f ∈ r (Y, Y) has the form f = ∑_{i=1}^{n} f e_{i} where Ef e_{i} is properly contained in E e_{i} and therefore lies in J(E) e_{i}, which shows f ∈ J(E).

(2) Assume that there is a nonzero element \overline{f} ∈ Hom_{R} (X, Y)/r (X, Y). Since X is indecomposable, f is a split monomorphism and therefore generates the left E-module Hom_{R} (X, Y). This yields the claim.

(3) Apply statement (2) on the exact sequence of E-modules 0 ⟶ _{E}r (X, Y) ⟶_{E} Hom_{R} (X, Y) ⟶_{E} Hom_{R} (X, Y)/r (X, Y) ⟶ 0. □

2 Families of homomorphisms

Following Auslander [1, §1], we say that a family of homomorphisms (a_{k} : A → X_{k})_{k∈K} is finitely cogenerated if there is a finite subset K_{0} ⊂ K such that the product map a : A → ∐_{k∈K_{0}} X_{k} induced by the a_{k} with k ∈ K_{0} has the property that all a_{k}, k ∈ K, factor through a. We start out with two propositions describing when families of homomorphisms in ⋃_{i∈I} r (A, M_{i}) are finitely cogenerated.

Proposition 2.1 The following statements are equivalent for a module A.

(1) The family of all homomorphisms in ⋃_{i∈I} r (A, M_{i}) is finitely cogenerated.

(2) The left T -module r (A, N) is generated by finitely many maps whose images are contained in a finite subproduct ∏_{i∈I_{0}} M_{i} of N.

(3) There is a map a ∈ r (A, X) such that X ∈ add M and all maps h ∈ r (A, Y) where Y ∈ add M factor through a.

If A is finitely generated, the following statement is further equivalent.

(4) r (A, M) is a finitely generated left S-module.

Proof : (1)⇒(3) : By assumption there are indices i_{1}, …, i_{r} ∈ I and maps a_{k} ∈ r (A, M_{i_{k}}), 1 ≤ k ≤ r, such that all maps in ⋃_{i∈I} r (A, M_{i}) factor through the product map a : A → X = ∐_{k=1}^{r} M_{i_{k}} ∈ add M induced by the a_{k}, and of course a ∈ r (A, X). (3)⇒(2) : By the universal property of products, every map f ∈ r (A, N) factors through a. Moreover, there is a split monomorphism ι : X → ⨁_{k=1}^{n} X_{k} for some X_{1}, …, X_{n} ∈ M, and for any 1 ≤ k ≤ n there is i_{k} ∈ I such that X_{k} ≅ M_{i_{k}}, giving rise to a split monomorphism α_{k} : X_{k} → N. Then the maps

a_{k} : A ⟶^{a} X ⟶^{ι} ⨁_{k=1}^{n} X_{k} ⟶^{pr_{k}} X_{k} ⟶^{α_{k}} N, 1 ≤ k ≤ n, form a generating set

of r (A, N) over T with the required property. (2)⇒(1) : Let a_{k}, 1 ≤ k ≤ n, be a generating set of the left T -module r (A, N) such that there is a finite subset I_{0} ⊂ I with Im a_{k} ⊂ ∏_{i∈I_{0}} M_{i} for all 1 ≤ k ≤ n. Denote by p_{i} : N → M_{i}, i ∈ I, the canonical projections. Then it is easy to check that all maps in ⋃_{i∈I} r (A, M_{i}) factor through the product map induced by the a_{k_{i}} = p_{i}a_{k} ∈ r (A, M_{i}), i ∈ I_{0}, 1 ≤ k ≤ n. Finally, if A is finitely generated, each map in r (A, M) has its image in a finite subcoproduct of M, hence in a direct summand of N. With this in mind, it is easy to see that (4) and (2) are equivalent. □

Proposition 2.2 The following statements are equivalent for a module A.

(1) Every family of homomorphisms in ⋃_{i∈I} r (A, M_{i}) is finitely cogenerated.

(2) r (A, N) is a noetherian left T -module. If A is finitely generated, then (1) and (2) are further equivalent to

(3) r (A, M) is a noetherian left S-module.

Proof : Denote by p_{i} : N → M_{i} and e_{i} : M_{i} → N, i ∈ I, the canonical projections and injections, respectively. (1)⇒(2) : For a submodule _{T} U ⊂ r (A, N) we consider the family of homomorphisms {p_{i} f | f ∈ U, i ∈ I} in ⋃_{i∈I} r (A, M_{i}). By assumption there are indices i_{1}, …, i_{n} ∈ I and maps f_{k} ∈ U, 1 ≤ k ≤ n, such that the product map a : A → ∐_{k=1}^{n} M_{i_{k}} induced by the p_{i_{k}} f_{k} has the property that all other maps of the form p_{i} f with i ∈ I and f ∈ U factor through a. Then also all f ∈ U factor through a, and so the f_{k}, 1 ≤ k ≤ n, form a generating set of _{T} U. (2)⇒(1) : Let now (a_{k} : A → M_{i_{k}})_{k∈K} be a family in ⋃_{i∈I} r (A, M_{i}), and consider the T -submodule U = ∑_{k∈K} T f_{k} of r (A, N) given by the maps

f_{k} : A ⟶^{a_{k}} M_{i_{k}} ⟶^{e_{i_{k}}} N in r (A, N). By assumption _{T} U = ∑_{k∈K_{0}} T f_{k} for

some finite subset K_{0} ⊂ K. This implies that all f_{k}, and therefore also all a_{k}, factor through the product map a : A → ∐_{k∈K_{0}} M_{i_{k}} induced by the a_{k} with k ∈ K_{0}. If A is finitely generated, the equivalence of (1) and (3) is proven with similar arguments, taking into account the fact that each map in r (A, M) has its image in a finite subcoproduct of M. □

Recall that a module is said to be endonoetherian if it is noetherian as a module over its endomorphism ring. As observed by Huisgen-Zimmermann [7], the module N is endonoetherian if and only if for all finitely generated modules A_{R} every family of homomorphisms in ⋃_{i∈I} Hom_{R} (A, M_{i}) is finitely cogenerated. Over a semilocal ring, we can now restrict ourselves to families of homomorphisms in ⋃_{i∈I} r (A, M_{i}) with A indecomposable.

Proposition 2.3 Assume that R is semilocal. Then the following statements are equivalent.

(1) N is endonoetherian.

(2) M is endonoetherian.

(3) For all finitely generated (or equivalently, for all finitely presented) indecomposable modules A_{R} every family of homomorphisms in ⋃_{i∈I} r (A, M_{i}) is finitely cogenerated.

Proof : (2)⇔ (3): M is endonoetherian if and only if Hom_{R} (A, M) is a noetherian left S-module for every finitely generated (or equivalently, for every finitely presented) module A_{R}. Since R is semilocal, every finitely generated module has a finite decomposition in indecomposables [6, 1.14]. So, it suffices to consider finitely generated (or finitely presented) indecomposable modules A, and the statement follows from 2.2 and 1.1. The equivalence (1)⇔ (3) is proven by the same arguments. □

We now consider the dual situation. A family of homomorphisms (b_{k} : X_{k} → C)_{k∈K} is said to be finitely generated [1, §1] if there is a finite subset K_{0} ⊂ K such that the coproduct map b : ∐_{k∈K_{0}} X_{k} → C induced by the b_{k} with k ∈ K_{0} has the property that all b_{k}, k ∈ K, factor through b. Let us describe when families of homomorphisms in ⋃_{i∈I} r (M_{i}, C) are finitely generated. The arguments are dual to those employed above, and we will therefore omit the proofs.

Proposition 2.4 The following statements are equivalent for a module C.

(1) The family of all homomorphisms in ⋃_{i∈I} r (M_{i}, C) is finitely generated.

(2) The right S-module r (M, C) is generated by finitely many maps whose kernels contain a cofinite subcoproduct ∐_{i∈I\I_{0}} M_{i} of M.

(3) There is a map b ∈ r (X, C) such that X ∈ add M and all maps h ∈ r (Y, C) where Y ∈ add M factor through b.

Proposition 2.5 The following statements are equivalent for a module C.

(1) Every family of homomorphisms in ⋃_{i∈I} r (M_{i}, C) is finitely generated.

(2) r (M, C) is a noetherian right S-module.

We now give a dual version of Proposition 2.3. Recall that R is said to be a right Morita ring if it is right artinian and the minimal injective cogenerator of ModR is finitely generated.

Proposition 2.6 Assume that R is semilocal. Further, let W_{R} be a minimal injective cogenerator of ModR, and M_{S}^{∗} = Hom_{R} (M, W)_{S}. The following statements are equivalent.

(1) M_{S}^{∗} is noetherian.

(2) For all finitely cogenerated indecomposable modules C_{R} every family of homomorphisms in ⋃_{i∈I} r (M_{i}, C) is finitely generated. If R is a right Morita ring, then following statement is further equivalent.

(3) For all finitely generated indecomposable modules C_{R} every family of homomorphisms in ⋃_{i∈I} r (M_{i}, C) is finitely generated.

Proof : Since R is semilocal, there are only finitely many simple right R-modules S_{1}, ⋯, S_{n} up to isomorphism. So, W ≅ ∐_{i=1}^{n} C_{i} where C_{i} = E(S_{i}) is an injective envelope of S_{i}. (1)⇒(2) : If C is finitely cogenerated, then Hom_{R} (M, C)_{S} is finitely M^{∗}-cogenerated, and the claim follows from Proposition 2.5. (2)⇒(1) : By Proposition 2.5 we know that r (M, C_{i})_{S} is noetherian for all 1 ≤ i ≤ n, hence by the dual version of statement (3) in Lemma 1.1 also Hom_{R} (M, C_{i})_{S} is noetherian for all 1 ≤ i ≤ n, and M_{S}^{∗} is noetherian. Assume now that R is a right Morita ring. Since R is right artinian, all finitely generated modules are finitely cogenerated, which yields (2)⇒(3). Moreover, since all C_{i} are finitely generated, we deduce (3)⇒(1) as above from 2.5 and the dual version of statement (3) in Lemma 1.1. □

3 Left and right almost split morphisms

Assume now that C is a module in M with local endomorphism ring. Recall that a homomorphism b : X → C with X ∈ add M is said to be right almost split in add M if b is not a split epimorphism and any homomorphism h : Y → C where Y ∈ add M and h is not a split epimorphism factors through b. We have seen in Proposition 2.4 that C has a right almost split morphism in add M if and only if the family of all homomorphisms in ⋃_{i∈I} r (M_{i}, C) is finitely generated. Inspired by Brune’s work [3], we will further say that C has a generalized right almost split morphism in add M if every family of homomorphisms in ⋃_{i∈I} r (M_{i}, C) is finitely generated. The notions of a left almost split morphism and a generalized left almost split morphism in add M are defined dually. Finally, if M consists of modules with local endomorphism ring, we say that M has (generalized) right, respectively left, almost split morphisms if every module belonging to M has a (generalized) right, respectively left, almost split morphism in add M.

The results in Section 2 can now be interpreted as characterizations for the existence of (generalized) left and right almost split morphisms. In particular, we have the following two Corollaries. Observe that statement (3) in Corollary 3.1 has independently been obtained by Dung [4, 2.3], [5, 3.11].

Corollary 3.1 Assume that all M_{i} have local endomorphism ring.

(1) M has generalized right almost split morphisms if and only if r (M, M_{i}) is a noetherian right S-module for all i ∈ I.

(2) M has generalized left almost split morphisms if and only if r (M_{i}, N) is a noetherian left T -module for all i ∈ I.

(3) If M consists of finitely generated modules, then M has (generalized) left almost split morphisms if and only if r (M_{i}, M) is a finitely generated (noetherian) left S-module for all i ∈ I.

Corollary 3.2 Assume that M is a finite subcategory consisting of modules with local endomorphism ring.

(1) M has left (respectively, right) almost split morphisms if and only if J(S) is a finitely generated left (respectively, right) S-module.

(2) M has generalized left (respectively, right) almost split morphisms if and only if J(S) is a noetherian left (respectively, right) S-module.

(3) Assume that all M_{i} have finite length. Then M has left (respectively, right) almost split morphisms if and only if S is left (respectively, right) artinian if and only if M has generalized left (respectively, right) almost split morphisms.

Proof : Note first that we have M = N and S = T. Then (1) and (2) follow from 2.1 and 2.2, or 2.4 and 2.5, respectively, and the fact observed in Lemma 1.1 that J(S) = r (M, M) ≃ ⨁_{i=1}^{n} r (M_{i}, M) ≃ ⨁_{i=1}^{n} r (M, M_{i}).

(3) If all M_{i} have finite length, then so does M, and S is therefore semiprimary. Hence S is left artinian if and only if _{S}J(S) is finitely generated if and only if _{S}J(S) is noetherian, and the symmetric statement holds on the right side. This proves the claim. □

Note that there are categories with left and right almost split morphisms where the module M is not endonoetherian.

Example 3.3 Let R be a hereditary artin algebra, and 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. Set P_{R} = ∐_{j∈J} P_{j}, and _{R}I = ∐_{k∈K} I_{k}. Then add P is closed under submodules, hence it is covariantly finite in modR by [2, 4.7], and it has right and left almost split morphisms. Dually, add I is contravariantly finite in Rmod and has right and left almost split morphisms. However, P_{R} is endonoetherian if and only if R is of tame representation type, and _{R}I is endonoetherian if and only if R is of finite representation type. To prove the latter statements, we argue as in [12, Corollary 12]. Observe first that for the usual artin algebra duality D : modR → Rmod we have D(P_{R}) ≅ ∏_{j∈J} D(P_{j}) ≅ ∏_{k∈K} I_{k}. Moreover, it follows from [12, Proposition 3] that P_{R} satisfies the ascending (descending) chain condition for finite matrix subgroups if and only if the module D(P_{R}) satisfies the descending (ascending) chain condition for finite matrix subgroups, which is further equivalent to the fact that _{R}I satisfies the descending (ascending) chain condition for finite matrix subgroups. But we know from [12, Observation 8] that for the pure-projective modules P_{R} and _{R}I the ascending chain condition for finite matrix subgroups is equivalent to endonoetherianness. So, P_{R}, respectively RI, is endonoetherian if and only if _{R}I, respectively P_{R}, is Σ-pure-injective. Now, it was shown in [9, 4.6] that _{R}I is Σ-pure-injective if and only if the algebra is tame. Furthermore, if P_{R} is Σ-pure-injective, then we know from [12, Proposition 4] that the class of homomorphisms between modules in P is noetherian, and this implies by [1, 5.6] that R is of finite representation type. Conversely, if R is of finite representation type, then every module is endofinite [12, Theorem 6] and hence endonoetherian. So, the proof is complete. □

We close this chapter with an application to pure-semisimple rings. Namely, we can now rediscover a characterization of these rings in terms of generalized right almost split morphisms given by Brune in [3] and later improved by Zimmermann in [11]. Moreover, we give a dual characterization in terms of generalized left almost split morphisms. Recall that a ring R is said to be right pure-semisimple if every right R-module is a direct sum of finitely presented modules.

Theorem 3.4 (1) (cp. [3, §3, Corollary 1] and [11, p. 372]) A right artinian ring R is right pure-semisimple if and only if every finitely presented indecomposable right R-module has a generalized right almost split morphism in modR.

(2) A semilocal ring R is left pure-semisimple if and only if every finitely presented indecomposable right R-module has a generalized left almost split morphism in modR.

Proof : (1) Let {M_{i} | i ∈ I} be a complete irredundant set of representatives of the isomorphism classes of the finitely presented right R-modules, and set M = ∐_{i∈I} M_{i} with S = End_{R} M. Further, let W_{R} be a minimal injective cogenerator of ModR, and E = End_{R} W. As observed by Zimmermann in [11, p. 372], we can assume that R is a right Morita ring. In fact, by [11, Theorem 4], this property follows from the existence of right almost split morphisms in modR for each simple non-projective module C_{R}, and is therefore satisfied whenever every finitely presented indecomposable right R-module has a generalized right almost split morphism in modR or when R is right pure-semisimple. So, the module W_{R} induces a Moritaduality modR → E mod, and we have that M^{∗} = _{E} Hom_{R} (M, W)_{S} is an E-S-bimodule with E ≅ Hom_{R} (Hom_{R} (M, W) ⊗_{S} M, W) ≅ End_{S} M^{∗} and S ≅ (End_{E} M^{∗})^{op}. Moreover, the _{E}Hom_{R} (M_{i}, W) form a complete irredundant set of representatives of the isomorphism classes of the finitely presented left E-modules. Further, we know from [8, Theorem 7] or [10] that R is right pure-semisimple if and only if so is E, which means by [12, Theorem 6] that every pure-projective left E-module is endonoetherian, or in other words, that the left E-module ∐_{i∈I} Hom_{R} (M_{i}, W) is endonoetherian. But by Proposition 2.3 the latter is equivalent to ∏_{i∈I} Hom_{R} (M_{i}, W) ≅ _{E}M^{∗} being endonoetherian. So, we see that R is right pure-semisimple if and only if M_{S}^{∗} is noetherian, and by Proposition 2.6 the proof is complete.

12 L. Angeleri Hügel

(2) Take M as in (1). By 2.3 we know that every finitely presented indecom-

posable module A_{R} has a generalized left almost split morphism if and only if

M is endonoetherian. But the latter means that every pure-projective mod-

ule is endonoetherian, which by [12, Theorem 6] is equivalent to R being left

pure-semisimple. □

References

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[7] B. Huisgen-Zimmermann, unpublished.

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[12] B. Zimmermann-Huisgen, W. Zimmermann: On the sparsity of representations of rings of pure global dimension zero, Trans. Amer. Math. Soc. \underline{320} (1990), 695-711.

Address of the author: Mathematisches Institut der Universität Theresienstraße 39 D-80333 München e-mail: angeleri@rz.mathematik.uni-muenchen.de