Categories of lattices, and their global structure in terms of almost split sequences

A major part of Iyama’s characterization of Auslander-Reiten quivers of representation-finite orders Λ consists of an induction via rejective subcategories of Λ-lattices, which amounts to a resolution of Λ as an isolated singularity. Despite of its useful applications (proof of Solomon’s second...

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Datum:	2004
1. Verfasser:	Rump, W.
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Veröffentlicht:	Інститут прикладної математики і механіки НАН України 2004
Schriftenreihe:	Algebra and Discrete Mathematics
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spelling	irk-123456789-1559522019-06-18T01:30:39Z Categories of lattices, and their global structure in terms of almost split sequences Rump, W. A major part of Iyama’s characterization of Auslander-Reiten quivers of representation-finite orders Λ consists of an induction via rejective subcategories of Λ-lattices, which amounts to a resolution of Λ as an isolated singularity. Despite of its useful applications (proof of Solomon’s second conjecture and the finiteness of representation dimension of any artinian algebra), rejective induction cannot be generalized to higher dimensional Cohen-Macaulay orders Λ. Our previous characterization of finite Auslander-Reiten quivers of Λ in terms of additive functions [22] was proved by means of L-functors, but we still had to rely on rejective induction. In the present article, this dependence will be eliminated. 2004 Article Categories of lattices, and their global structure in terms of almost split sequences / W. Rump // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 1. — С. 87–111. — Бібліогр.: 30 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 16G30, 16G70, 18E10; 16G60. http://dspace.nbuv.gov.ua/handle/123456789/155952 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	A major part of Iyama’s characterization of Auslander-Reiten quivers of representation-finite orders Λ consists of an induction via rejective subcategories of Λ-lattices, which amounts to a resolution of Λ as an isolated singularity. Despite of its useful applications (proof of Solomon’s second conjecture and the finiteness of representation dimension of any artinian algebra), rejective induction cannot be generalized to higher dimensional Cohen-Macaulay orders Λ. Our previous characterization of finite Auslander-Reiten quivers of Λ in terms of additive functions [22] was proved by means of L-functors, but we still had to rely on rejective induction. In the present article, this dependence will be eliminated.
format	Article
author	Rump, W.
spellingShingle	Rump, W. Categories of lattices, and their global structure in terms of almost split sequences Algebra and Discrete Mathematics
author_facet	Rump, W.
author_sort	Rump, W.
title	Categories of lattices, and their global structure in terms of almost split sequences
title_short	Categories of lattices, and their global structure in terms of almost split sequences
title_full	Categories of lattices, and their global structure in terms of almost split sequences
title_fullStr	Categories of lattices, and their global structure in terms of almost split sequences
title_full_unstemmed	Categories of lattices, and their global structure in terms of almost split sequences
title_sort	categories of lattices, and their global structure in terms of almost split sequences
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2004
url	http://dspace.nbuv.gov.ua/handle/123456789/155952
citation_txt	Categories of lattices, and their global structure in terms of almost split sequences / W. Rump // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 1. — С. 87–111. — Бібліогр.: 30 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT rumpw categoriesoflatticesandtheirglobalstructureintermsofalmostsplitsequences
first_indexed	2025-07-14T08:09:18Z
last_indexed	2025-07-14T08:09:18Z
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 1. (2004). pp. 87 – 111 c© Journal “Algebra and Discrete Mathematics” Categories of lattices, and their global structure in terms of almost split sequences Wolfgang Rump Communicated by Yu. A. Drozd Abstract. A major part of Iyama’s characterization of Auslander-Reiten quivers of representation-finite orders Λ consists of an induction via rejective subcategories of Λ-lattices, which amounts to a resolution of Λ as an isolated singularity. Despite of its useful applications (proof of Solomon’s second conjecture and the finiteness of representation dimension of any artinian al- gebra), rejective induction cannot be generalized to higher dimen- sional Cohen-Macaulay orders Λ. Our previous characterization of finite Auslander-Reiten quivers of Λ in terms of additive func- tions [22] was proved by means of L-functors, but we still had to rely on rejective induction. In the present article, this dependence will be eliminated. Introduction Let R be a complete regular local ring of dimension d. An R-algebra Λ is said to be a Cohen-Macaulay order if RΛ is finitely generated and free. A Λ-module M is said to be Cohen-Macaulay if RM is finitely generated and free. The category of Cohen-Macaulay modules over Λ will be denoted by Λ-CM. For example, if d = 0, then R is a field, and Λ-CM is the category of finite dimensional modules over the artinian algebra Λ. For d = 1, Λ is an order over a complete discrete valuation ring R, and Λ-CM is the category of Λ-lattices. 2000 Mathematics Subject Classification: 16G30, 16G70, 18E10; 16G60. Key words and phrases: L-functor, lattice category, τ -category, Auslander- Reiten quiver. Jo u rn al A lg eb ra D is cr et e M at h .88 Categories of lattices, and their global structure... By a theorem of Auslander [2], the category Λ-CM (for Λ Cohen- Macaulay) has almost split sequences if and only if Λ is either non- singular or an isolated singularity, i. e. if gld Λp = d holds for all non- maximal prime ideals p of R. For Λ representation-finite (i. e. the num- ber of isomorphism classes of indecomposables in Λ-CM is finite), it is known [2] that Λ-CM has almost split sequences. Given an isolated singularity Λ, it is natural to ask what are the possi- ble Auslander-Reiten quivers A(Λ-CM) of Λ-CM. In the representation- finite case, this question has been answered for d = 0 by Igusa and Todorov [7, 8, 9] and Brenner [4], for d = 1 by Iyama [10, 11, 12], and for d = 2 by Reiten and Van den Bergh [20]. There is an essential difference between d 6 2 and d > 2. Roughly speaking, the projective and injective objects play a predominant rôle for d > 2. To make this precise, recall that a sequence τA v ֌ ϑA u −→ A of morphisms u, v of a Krull-Schmidt category A is said to be right almost split if u is a right almost split mor- phism, and v = keru is a left almost split morphism. Now A is said to be a strict τ -category [10] if A has right and left almost split sequences for each object A. For d < 2, the projective objects P of Λ-CM can be characterized by τP = 0, but in case d = 2, the projective objects of Λ-mod no longer coincide with the projectives of Λ-CM since in that case, Λ-CM has no projectives at all. This means that for all objects A, the right almost split sequence τA ֌ ϑA → A is left almost split, and vice versa. Thus Λ-CM is a strict τ -category if and only if d 6 2. Among the dimensions d 6 2, the characterization of finite translation quivers of the form A(Λ-CM) has been most difficult in case d = 1. To achieve this, a rather intensive study of τ -categories was necessary [10, 11, 12]. Moreover, the theory of overorders had to be translated into the language of rejective subcategories of Λ-CM (which were invented for that purpose). In this way, the structure of A(Λ-CM) was determined by induction via a decreasing chain of rejective subcategories, a non- commutative analogue to a resolution of singularities. Amazingly, the same induction led Iyama to the proof of two important conjectures in the representation theory of algebras and orders, respectively (see [13]). Another tool for the determination of A(Λ-CM) was an improved theory of ladders, initiated by Igusa and Todorov in 1984. Originally, starting with a suitable morphism, a step of a ladder is given by a com- mutative square Ai+1 ≻ Ai Bi+1 g ≻ Bi g Jo u rn al A lg eb ra D is cr et e M at h .W. Rump 89 such that the mapping cone sequence Ai+1 → Ai ⊕ Bi+1 → Bi is almost split. After a series of modifications, this ultimately led to the concept of ladder functor [22]. For a Krull-Schmidt category A, let M(A) denote the homotopy category of two-termed complexes (see §3). A pair of L- functors is an adjoint pair L+ ⊣ L− of additive functors L±: M(A) → M(A) together with natural transformations λ+: L+ → 1 and λ−: 1 → L−, with additional properties (see §2). More generally, L-functors can be defined for any additive category M instead of M(A). If L-functors exist for M, they are unique, and M carries a structure similar to that of a triangulated category. Therefore, we call such a category M triadic (§2). By definition, a left L-functor L+: M(A) → M(A) applies to any morphism a: A1 → A0 in A. Then λa: L+a → a gives rise to a pullback- pushout square in A, which can be regarded as a step of a (generalized) ladder. In particular, if a ∈ Ob M(A) is of the form 0 → A with A indecomposable and non-projective, then λa gives a commutative square τA ≻ 0 ϑA g ≻ A g corresponding to the almost split sequence τA ֌ ϑA ։ A. Thus L- functors yield almost split sequences in a functorial way, and they also apply to morphisms instead of objects A. Using L-functors, a simplification of the statement and proof of Iyama’s characterization of A(Λ-CM) for representation-finite Λ with d = 1 became possible [22]. In particular, the complicated part of his criterion could be replaced by the existence of an additive function l > 0 on the vertices of A(Λ-CM). In order to extend the characterization of A(Λ-CM) to dimensions d > 2, L-functors might be useful. For arbitrary dimension d, the triadic category M(Λ-CM) has L-functors if and only if Λ-CM has almost split sequences [28]. However, a big obstacle came from the induction via re- jective subcategories for d = 1. This allows no generalization to higher dimensions since a resolution of higher-dimensional non-commutative sin- gularities Λ would not be feasable. Therefore, toward a criterion for d > 2, a fundamental step would be to eliminate that inductive rejection in d = 1. This will be done in the present paper. As a side-effect, The- orem 1 of [22] which was fundamental for the introduction of additive functions, also has been dropped now. Approximately, this latter reduc- tion eliminates half of the use of L-functors in our treatment of d = 1. As already begun in [22], we investigate Λ-CM for d = 1 in a more general setting. We define a Λ-lattice as a finitely presented Λ-module Jo u rn al A lg eb ra D is cr et e M at h .90 Categories of lattices, and their global structure... with no simple submodules. In this way, the classical situation is reduced to what is really needed. In particular, no base ring of Λ has to be specified. In §1, the structure of lattice categories Λ-CM (for d = 1) will be characterized within this general context in category-theoretic terms. L-functors will be introduced and applied to lattice categories in §§2 - 3. For a Krull-Schmidt lattice category A, we show (Theorem 4) that M(A) has L-functors if and only if A is a strict τ -category. From §4 on, we investigate strict τ -categories A. Heuristically, this means that we study the local structure of A, given by its Auslander- Reiten meshes. We will assume that A is L-finite, i. e. for any object a ∈ M(A), the powers L±na stabilize for n ≫ 0. This condition holds, e. g., if A is equivalent to Λ-CM, or its universal cover (see [31, 10]), in case Λ is representation-finite. To characterize A in terms of the Auslander- Reiten quiver, we have to reconstruct its global structure from the local mesh structure. This job will be done by the L-functors. For an object A of A, the repeated application of L+ to 0 → A yields a cokernel P ։ A with P projective, which shows that A has enough projectives in a strict sense. Similarly, the kernel of a cokernel c ∈ A is obtained by applying L+n to c, for n ≫ 0 (Proposition 7). To show that every morphism in A has a kernel, more assumptions are necessary. We call a monomorphism m ∈ A simple if it allows no factorization m = ab into non-invertible monomorphisms a, b. Analoguously, simple epimorphisms are defined. For an L-finite strict τ -category A, a simple monomorphism is either epic or a kernel. If it is epic, it need not be a simple epimorphism. We show that equality of the classes of simple monomorphisms and simple epimorphisms among the monic and epic morphisms establishes a duality between projectives and injectives in A. This condition holds, e. g., when the Auslander-Reiten quiver of A admits an additive function l > 0 (Proposition 9). For a lattice category, such an l is given by the rational rank. If, in addition, A satisfies ⋂ ∞ n=1 RadnA = 0, we prove (Theorem 5) that A has all the relevant global properties of Λ-CM in case d 6 1. In particular, Theorem 5 implies that A is noetherian in a strong sense (Corollary 1). As a further consequence, we get the above mentioned characterization of finite translation quivers arising as Auslander-Reiten quivers of Λ-CM for a Cohen-Macaulay order Λ over a complete discrete valuation ring (Corollary 3). 1. Categories of lattices Let Λ be any ring (associative with 1). By Λ-mod we denote the category of finitely presented left Λ-modules. A module E ∈ Λ-mod is said to be a Λ-lattice [23] if E has no simple submodules. For example, if Λ is an order Jo u rn al A lg eb ra D is cr et e M at h .W. Rump 91 over a Dedekind domain R (see [19]), then Λ-lattices are just what they ought to be. Therefore, we denote the full subcategory of Λ-lattices in Λ-mod (for any ring Λ) by Λ-lat. An additive category A equivalent to a category Λ-lat with Λ left noetherian will be called a left lattice category. We call A a right lattice category if Aop is a left lattice category. If A is a left and right lattice category, we simply speak of a lattice category A. In the particular case where Λ is an R-order, the duality functor E 7→ HomΛ(E, R) provides an equivalence (Λ-lat)op −→∼ Λop-lat, (1) which shows that Λ-lat is in fact a lattice category. In this section, we will give an intrinsic characterization of lattice categories. A morphism in an additive category A is said to be regular if it is monic and epic. We say that A has a quotient category, denoted by Q(A), if the regular morphisms admit a calculus of left and right fractions [5]. Thus if Q(A) exists, it has the same objects as A, and the morphisms of Q(A) are formal fractions fr−1 = s−1g with r, s regular and sf = gr. Moreover, there is a faithful embedding A →֒ Q(A) (2) which respects kernels and cokernels of morphisms. Recall that a short exact sequence A a ֌ B b ։ C in A is defined by the property a = ker b and b = cok a. By ֌ (resp. ։) we indicate that a morphism is a (co-) kernel. An object P of A is said to be projective if for each cokernel c: B ։ C, every morphism P → C factors through c. An object C of A will be called a covering object if every E ∈ ObA admits a cokernel Cn ։ E for some n ∈ N. The dual notions of injective or cocovering objects are defined analoguously. The full subcategories of projective (injective) objects will be denoted by Proj(A) (resp. Inj(A)). An additive category A is said to be preabelian if every morphism of A has a kernel and a cokernel. Proposition 1. For a preabelian category A, the following are equivalent. (a) If a composition fg is a cokernel, then f is a cokernel. (b) For given A c ։ C f ←− B, the morphism (c f): A ⊕ B → C is a cokernel. (c) Every composition fg of cokernels f, g is a cokernel. (d) If f : A c ։ E d → B is a morphism with c = cok(ker f), then d is monic. Jo u rn al A lg eb ra D is cr et e M at h .92 Categories of lattices, and their global structure... Proof. (a) ⇒ (b): Consider the composition c: A ( 1 0 ) −→ A ⊕ B (c f) −→ C. (b) ⇒ (c): Let f : B ։ C be the cokernel of d: D → B, and g: A ։ B. Then property (b) implies that there is a pushout E e ≻ A fg ≻ C PO D g d ≻ B g g g f ≻≻ C. w w w w w w Hence fg = cok e. (c) ⇒ (d): Suppose that dg = 0. Then d factors through e := cok g, and there exists a morphism h with ec = cokh. Hence fh = 0. Therefore, h factors through the kernel of f , and thus ch = 0. Consequently, c = cok h, whence e is invertible. So we get g = 0, which shows that d is monic. (d) ⇒ (a): Consider a factorization f = dc with c = cok(ker f) and d monic. Assume that fg = cok h. Since d is monic, this implies that cgh = 0. So cg factors through fg. Therefore, d is split epic, hence invertible. A preabelian category A which satisfies the equivalent properties of Proposition 1 is called left semi-abelian [21]. By [21], Proposition 1, a preabelian category A is left semi-abelian if and only if for any pullback A a ≻ B C g b d ≻ D g c (3) in A where d is a cokernel, the morphism a is epic. In [21] we called the preabelian categories A where a is even a cokernel for all such pullbacks (3) left almost abelian. Several authors use the term “quasi-abelian” in- stead of “almost abelian” [30, 29, 3]. Following this trend, we replace “left almost abelian” by “left quasi-abelian” in what follows. If A and Aop (resp. Aop) is left quasi-abelian, we call A (right) quasi-abelian. Similarly, we define (right) semi-abelian. We are grateful to Y. Kopylov for pointing out to us that the term “quasi-abelian” dates back to R. Succi Cruciani’s paper [30] of 1973. There is also a Russian tradition [16, 17, 18, 15] that calls quasi-abelian categories “(Răıkov-)semi-abelian”. Proposition 2. Let A be a preabelian category, such that every object A admits a cokernel P ։ A with P projective. Then A is left quasi-abelian. Jo u rn al A lg eb ra D is cr et e M at h .W. Rump 93 Proof. Consider a pullback (3) where d is a cokernel. By assumption, there is a cokernel p: P ։ B with P projective. Hence cp factors through d, and the pullback property implies that p factors through a. Thus a is a cokernel by Proposition 1. By [21], Proposition 6, a semi-abelian category A has a quotient cat- egory if and only if for each pullback (3) with d epic, the morphism a is also epic. Semi-abelian categories with this property (i. e. that epimor- phisms are stable under pullback) are called integral [21]. For an integral category A, the quotient category Q(A) is abelian. Examples of quasi- abelian and integral categories are abundant (see [21], §2). We will see below that lattice categories are integral and quasi-abelian. An additive category A is said to be noetherian if for each object of A, the subobjects satisfy the ascending chain condition. We call A bi-noetherian if A and Aop are noetherian. Assume that every kernel or cokernel can be completed to a short exact sequence, and that condition (a) of Proposition 1 together with its dual is satisfied. (Then (c) of Proposition 1 follows.) This holds, for example, when A is semi-abelian. Define the rational length ρ(A) ∈ N ∪ {∞} of an object A ∈ ObA as the supremum of all n ∈ N for which there exists a chain 0 = A0 a1 ֌ A1 a2 ֌ · · · an ֌ An = A (4) of non-invertible kernels a1, . . . , an ∈ A. According to our assumption, this definition is self-dual, i. e. the sequence of kernels (4) can be replaced by a sequence of cokernels A ։ · · · ։ 0. If A is integral, then [25], Proposition 2, implies that the rational length of an object A ∈ ObA is equal to ρ(A) in the abelian category Q(A). We call A ∈ ObA irreducible if ρ(A) = 1. A chain (4) with Cok ai irreducible will be called a rational composition series of A. Thus if A is integral, every rational composition series (4) is of length n = ρ(A). For a regular morphism r ∈ A, we define the length ρ(r) ∈ N ∪ {∞} as the supremum of all n ∈ N for which r can be written as a composition r = r1 · · · rn into non-invertible regular morphisms ri. We say that r has a composition series if a factorization r = r1 · · · rn with ρ(ri) = 1 exists. If A is integral and quasi-abelian, then every composition series of r is of length ρ(r) by [24], Proposition 2. Proposition 3. An integral quasi-abelian category A is bi-noetherian if and only if its objects have finite rational length, and every regular morphism has finite length. Proof. Assume first that A is bi-noetherian. For any non-zero object A0, there exists a maximal subobject a1: A1 ֌ A0 with a1 non-invertible. Jo u rn al A lg eb ra D is cr et e M at h .94 Categories of lattices, and their global structure... Thus a1 is a kernel with ρ(Cok a1) = 1. By induction, we get a sequence A0 a1 ֋ A1 a2 ֋ A2 a3 ֋ · · · of kernels with ρ(Cok ai) = 1 for all i. So there are commutative diagrams Ai ≻ ≻ A0 ≻≻ Ci Ai−1 g g ai ≻ ≻ A0 w w w w w w ≻≻ Ci−1 g g for all i. Since Aop is noetherian, we infer that An = 0 for some n > 1. Hence ρ(A0) = n < ∞. A similar argument shows that regular morphisms of A have finite length. Conversely, assume that ρ(A) < ∞ and ρ(r) < ∞ for all A ∈ ObA and all regular r ∈ A. Consider a strictly increasing sequence A0 < A1 < A2 < · · · of subobjects of A ∈ ObA. Since ρ(A) < ∞, almost all monomorphisms Ai → Ai+1 must be regular. Therefore, the sequence cannot be infinite. For a quasi-abelian category A, we define the initial category [21] as the full subcategory A◦ of objects D of A such that every monomorphism D′ → D is a kernel. The full subcategory A◦ of A with (A◦)op = (Aop)◦ is called the terminal category of A. By [21], Proposition 8, the categories A◦ and A◦ are abelian. Now we are ready to prove Theorem 1. An additive category A is a lattice category if and only if the following are satisfied. (a) A is preabelian with a projective covering object P and an injective cocovering object I. (b) A has a quotient category. (c) A is bi-noetherian. (d) A◦ = A◦ = 0. Proof. Assume that A = Λ-lat with Λ left noetherian. Then A is pre- abelian and noetherian, and there is a hereditary torsion theory (T , Λ-lat) in Λ-mod, where T is the class of length-finite Λ-modules. By [21], Theo- rem 2, this implies that A is integral, whence (b) holds. Moreover, there is a short exact sequence Λ0 ֌ Λ ։ P in Λ-mod with Λ0 ∈ T and P ∈ Λ-lat. Thus P is a projective covering object. For any non-zero Λ-lattice E, there is a maximal Λ-submodule F . Therefore, if E ∈ A◦, then i: F →֒ E is a kernel in Λ-lat. But the cokernel of i in Λ-lat is zero, a contradiction. Thus A◦ = 0. By symmetry, this proves that every lattice category satisfies (a)-(d). Jo u rn al A lg eb ra D is cr et e M at h .W. Rump 95 Conversely, let (a)-(d) be satisfied. By Proposition 2, this implies that A is quasi-abelian. By [24], Proposition 11, we have Ql(A) ≈ Λ-mod, where Λ := EndA(P )op, and Ql(A) denotes the left abelian cover [21] of A. From [21], Theorem 2, we infer that there is a torsion theory (R(A),F) in Λ-mod with F ≈ A, such that a finitely presented Λ- module M belongs to R(A) if and only if there exists a regular morphism r ∈ F with M = Cok r in Λ-mod. By (d), every simple Λ-module belongs to R(A). Moreover, A is integral by (b). Therefore, the regular morphisms are essentially monic and essentially epic. Hence (c) implies that R(A) is the full subcategory of length-finite modules in Λ-mod. Thus A ≈ F = Λ-lat with Λ left noetherian by (c). As a consequence of Theorem 1 and Proposition 2, we get Corollary. Every lattice category is integral and quasi-abelian. Remark. If (d) in Theorem 1 is replaced by A◦ = A◦ = A, then the conditions (a)-(d) characterize a category A which is equivalent to Λ-mod ≈ (Γ-mod)op with Λ, Γ left artinian. This can be regarded as the 0-dimensional analogue of a lattice category. 2. L-functors In this section we review the basic theory of L-functors, as far as needed for our present purpose. Functors between additive categories are always assumed to be additive. Let M be an additive category. For a full subcategory C of M, a morphism ϕ: a → b of M is said to be C-epic (C-monic) if every morphism c → b (resp. a → c) with c ∈ C factors through ϕ. By [C] we denote the ideal of M generated by the identity morphisms 1c, c ∈ Ob C. If M/[C] has a quotient category, we say that MC := Q(M/[C]) exists. For a class Σ of morphisms, let Pr Σ (resp. In Σ) denote the largest full subcategory C of M such that every ϕ ∈ Σ is C-epic (C-monic). Assume that MC exists. For a morphism α ∈ M, we denote the (co-) kernel of α in MC by kerCα (resp. cokCα) and call this a local (co-)kernel. As a counterpart, we call α ∈ M a global kernel of β ∈ MC if βα = 0 holds in MC , and for each α′ ∈ M with βα′ = 0 in MC there exists a unique γ ∈ M with αγ = α′. By this universal property, the global kernel and its dual, the global cokernel, are unique up to isomorphism. We write α = kerCβ (resp. cokCβ) for the global (co-)kernel of β. We call an object s of M left (right) semisimple if every monomor- phism a → s (epimorphism s → a) splits. The full subcategory of left (right) semisimple objects is denoted by Sl(M) (resp. Sr(M)), and the Jo u rn al A lg eb ra D is cr et e M at h .96 Categories of lattices, and their global structure... objects of S(M) := Sl(M) ∩ Sr(M) will be called semisimple. Note that for a module category M, the semisimple objects coincide with the semisimple modules ([1], Theorem 9.6). Definition 1. Let M be an additive category. By Σ we denote the class of regular morphisms which are Sl(M)-epic and Sr(M)-monic. The morphisms in Σ will be called (absolutely) exact. We call M (absolutely) triadic if the following are satisfied for P := Pr Σ and I := In Σ. (T1) MP and MI exist and are abelian, MP has enough projectives, and MI has enough injectives. (T2) Every morphism in MP (resp. MI) has a global (co-)kernel. (T3) Every global kernel is a global cokernel, and vice versa. Remark. By [28], Corollary of Theorem 1, the global kernels in M coin- cide with the exact morphisms. In [28], we define a triadic category with respect to arbitrary full subcategories P, I of M. Then it can be shown that PrΣ ⊂ P and InΣ ⊂ I. Thus in the absolute case of Definition 1, P and I are as small as possible. In the wider sense of [28], every ad- ditive category M is triadic with respect to the pair P = I = M. The reason why we introduced triadic categories for arbitrary P and I comes from the observation that they naturally arise in the study of orders over a two-dimensional regular ring. By [28], Theorem 1, we have Theorem 2. Let M be a triadic category. There is an equivalence T : MP −→∼ MI such that every exact morphism β can be completed to a triad [28], i. e. a sequence Td α −→ b β −→ c γ −→ d (5) with α = kerIβ, γ = cokPβ, and β = cokIα = kerPγ, such that each commutative diagram ψβ = β′χ with exact β′ induces a morphism of triads Td α ≻ b β ≻ c γ ≻ d Td′ g Tω α′ ≻ b′ g χ β′ ≻ c′ g ψ γ′ ≻ d′. g ω (6) Remark. Using (T2), Theorem 2 implies that every epimorphism γ ∈ MP , and every monomorphism α ∈ MI can be extended to a triad (5). Moreover, each of the commutative squares in (6) extends to a morphism of triads. Jo u rn al A lg eb ra D is cr et e M at h .W. Rump 97 Recall that a pointed functor [14] of M is defined as a functor L−: M → M together with a natural transformation λ−: 1 → L−. Du- ally, we define an augmented functor of M as an endofunctor L+ with a natural transformation λ+: L+ → 1. For an augmented or pointed functor L±, let Pr L± (resp. In L±) denote the largest full subcate- gory C of M such that λ± a is C-epic (resp. C-monic) for every a ∈ ObM. For an adjoint pair of endofunctors L+ ⊣ L− with adjunction Φ: HomM(L+a, b) −→∼ HomM(a, L−b), an augmentation λ+: L+ → 1 of L+ makes L− into a pointed functor via λ− a = Φ(λ+ a ). In other words, the right adjoint of an augmented functor is pointed, and the left adjoint of a pointed functor is augmented. If M is triadic, we define a left triadic functor of M as an augmented functor L+: M → M such that λ+ a is exact for all a ∈ ObM. Thus if L+ is left triadic, every object a of M gives rise to a triad TSa σa−→ L+a λ+ a−→ a πa−→ Sa (7) with a functor S: M → MP . Dually, a pointed functor L− of M with λ− a exact for all a ∈ ObM will be called right triadic. Definition 2. Let M be a triadic category. We define a left L-functor of M as a left triadic functor L+: M → M such that the inclusions Pr L+ ⊂ Pr Σ, In L+ ⊂ In Σ hold, and Sa is semisimple for every a ∈ ObM. Dually, a pointed functor L− of M will be called a right L-functor if it induces a left L-functor Mop → Mop. We say that an additive category has L-functors if it is triadic and admits a left L-functor L+ and a right L-functor L−. By [28], Proposition 15, we have Theorem 3. If an additive category M has L-functors, then L+ is left adjoint to L−. The right adjoint of a left L-functor is a right L-functor, and vice versa. A left or right L-functor of a triadic category is unique, up to isomorphism. 3. L-functors for lattice categories Now we will show how triadic categories arise in the context of lattice categories. Let A be a Krull-Schmidt category, i. e. an additive category such that every object of A is a finite direct sum of objects with local en- domorphism rings. The ideal RadA of A generated by the non-invertible morphisms between indecomposable objects is called the radical of A. Let Mor(A) be the category of two-termed complexes 0 → A1 a → A0 → 0 Jo u rn al A lg eb ra D is cr et e M at h .98 Categories of lattices, and their global structure... in A. So the objects of Mor(A) can be regarded as morphisms a ∈ A, and the morphisms in Mor(A) are tantamount to commutative squares in A. If we identify A ∈ ObA with the identity morphism 1A ∈ Mor(A), then A becomes a full subcategory of Mor(A), such that the ideal [A] of Mor(A) consists of the morphisms ϕ: a → b in Mor(A) which are homotopic to zero. Since every morphism f ∈ A has a decomposition f = e ⊕ r into an isomorphism e and some r ∈ RadA, the factor cat- egory Mor(A)/[A] is equivalent to its full subcategory M(A) of objects A1 a → A0 with a ∈ RadA. For any A ∈ ObA, there are two corre- sponding objects A+: 0 → A and A−: A → 0 of M(A). So we get two equivalences ( )+: A −→∼ A+ and ( )−: A −→∼ A− between A and full subcategories of M(A). A morphism f : A → B in A is said to be right almost split if f ∈ RadA, and every morphism A′ → B in RadA factors through f . If f is right almost split in Aop, then f is called left almost split. A sequence τA vA ֌ ϑA uA−→ A (8) in A is said to be right almost split if uA is right almost split, and vA = keruA is left almost split. Note that a right almost split sequence (8) is uniquely determined by the object A, up to isomorphism. Similarly, a sequence A uA −→ ϑ−A vA ։ τ−A (9) is said to be left almost split if it is right almost split in Aop. A Krull- Schmidt category A with left and right almost split sequences for all A ∈ ObA is called a strict τ -category [10]. To each strict τ -category A, a (valued) translation quiver A(A) can be associated as follows. The class of vertices of A(A) is given by a representative system indA of the isomorphism classes of indecomposable objects in A. For A, B ∈ indA, let dAB be the multiplicity of A in a direct decomposition of ϑB, and d′AB the multiplicity of B in ϑ−A. Then there is an arrow A → B with valuation (dAB, d′AB) whenever dAB 6= 0 (or equivalently, d′AB 6= 0). The translation quiver A(A) is called the Auslander-Reiten quiver [10] of A. Iyama [10] has shown that A(A), together with its natural modulation, determines the associated completely graded τ -category of A up to equivalence. Proposition 4. Let A be a lattice category with the Krull-Schmidt prop- erty. For every indecomposable projective object P , there exists a unique maximal subobject ϑP < P . Jo u rn al A lg eb ra D is cr et e M at h .W. Rump 99 Proof. Since A is noetherian, there exists a maximal subobject E < P . Assume that there is a different maximal subobject F < P . The corresop- nding monomorphisms E → P ← F induce a morphism g: E ⊕ F → P . Since A is semi-abelian, g has a decomposition g = mq with m monic and q a cokernel. Thus m defines a subobject of P which contains E and F . Hence m is invertible. Since P is projective with EndA(P ) local, we infer that either E → P or F → P is split epic, a contradiction. Remark. Let P = P1 ⊕ · · · ⊕ Pn be any projective object in a Krull- Schmidt lattice category, with Pi indecomposable. Then the monomor- phisms uPi : ϑPi → Pi define a monomorphism uP : ϑP → P such that 0 −→ ϑP uP−→ P (10) is a right almost split sequence. By duality, every injective object I of A gives rise to a left almost split sequence I uI −→ ϑ−I → 0. Proposition 5. Let A be a lattice category with the Krull-Schmidt prop- erty. An object of M(A) is left semisimple if and only if it is isomorphic to uP ⊕E− for some projective object P , and an arbitrary object E of A. Proof. Let a: A1 → A0 be a left semisimple object in M(A). Then there is an exact square A ≻≻ A1 P g p f ≻≻ A0 g a with P projective. By [22], Proposition 2 and its dual, this represents a regular morphism ε: p → a in M(A). Hence ε is split monic, and thus invertible. Therefore, f is invertible, which shows that A0 is projective. Now the proof can be completed as in the proof of [27], Proposition 8. Recall that a commutative square (3) is said to be exact if it is a pullback and a pushout. Proposition 5 allows us to determine the exact morphisms (see Definition 1) of M(A). Proposition 6. Let A be a lattice category with the Krull-Schmidt prop- erty. Then M(A) is triadic. A morphism ϕ: b → c in M(A), given by a commutative square (3), is exact if and only if (3) is an exact square. Proof. Assume that b → c is exact. By Proposition 5, this implies that ϕ is A−-epic. Hence (3) is exact by [22], Proposition 2. Conversely, let ϕ be given by an exact square (3). Then ϕ is A−-epic and A+-monic by [22], Proposition 2. Hence ϕ is exact by Proposition 5 and its dual. Now [28], Corollary 2 of Theorem 3, implies that M(A) is triadic. Jo u rn al A lg eb ra D is cr et e M at h .100 Categories of lattices, and their global structure... Theorem 4. Let A be a lattice category with the Krull-Schmidt property. Then M(A) has L-functors if and only if A is a strict τ -category. Proof. This follows by Proposition 4 and the above remark, and Propo- sition 6 together with [28], Theorem 5. 4. L-finiteness For a strict τ -category A, the homotopy category M(A) need not be triadic. Nevertheless, by [22], §3, there exists an augmented functor L+: M(A) → M(A) with a right adjoint L− such that L± become L-functors when A is triadic. An object a of M(A) belongs to Pr L+ (resp. Pr L−) if and only if λ+ a (resp. λ− a ) is invertible. Therefore, we call A left (right) L-finite if for each a ∈ Ob M(A), there is an integer n ∈ N such that L+na ∈ Pr L+ (resp. L−na ∈ In L−). If A is left and right L-finite, we just say that A is L-finite. Thus if A is left L-finite, every a ∈ Ob M(A) gives rise to an exact square B c ≻ A1 P g b p ≻ A0 g a (11) with b = L+na ∈ Pr L+. Hence τP = 0. Proposition 7. Let A be a left L-finite strict τ -category. Then every cokernel has a kernel, and for each A ∈ ObA, there is a cokernel P ։ A with P projective. An object P of A is projective if and only if τP = 0, and a morphism of A is a cokernel if and only if it is Proj(A)-epic. Proof. If we set A1 = 0 in (11), we get a short exact sequence B ֌ P ։ A0 with τP = 0. The proof of [22], Theorem 3, shows that P is projective. Therefore, [22], Proposition 11, implies that the projective objects P are characterized by the property τP = 0. Thus if a in (11) is a cokernel, then p factors through a, which implies that b is split epic. Since b ∈ RadA, we infer that P = 0, whence c = ker a. Finally, let b: B → C be Proj(A)-epic. Consider a short exact sequence C ′ c ֌ P p ։ C and a cokernel q: Q ։ B with P, Q projective. Then bq = pd for some d: Q → P , and it is easily verified that (b p): B ⊕ P → C is a cokernel of ( q 0 −d c ) : Q ⊕ C ′ → B ⊕ P . Hence (b p) has a kernel, which gives an Jo u rn al A lg eb ra D is cr et e M at h .W. Rump 101 exact square E e ≻ P B g f b ≻ C. g g p (12) Hence p factors through b, and the pullback property implies that e is split epic. Since idempotents split in A, we infer that e has a kernel g: K ֌ E. Thus by the pushout property of (12), it follows that b = cok(fg). Proposition 7 shows that the short exact sequences of an L-finite strict τ -category A make A into an Ext-category (see [28]), that is, an exact category with enough projectives and enough injectives such that every split epimorphism has a kernel. Corollary. Let A be an L-finite strict τ -category. Then M(A) has L- functors. Proof. By [22], Proposition 2, and [27], Proposition 8, a morphism in M(A) is exact if and only if it corresponds to an exact square in A. Therefore, the corollary follows by [28], Theorem 5 and Corollary 2 of Theorem 3. Remark. By [22], §6, and [10], §7, it follows that whether a strict τ - category A is L-finite can be read off from the Auslander-Reiten quiver A(A). For the rest of this section, we will derive further consequences of L-finiteness. Since A(A) = A(A/ ⋂ ∞ n=1 RadnA), we eventually assume that ⋂ ∞ n=1 RadnA = 0. Lemma 1. Let A be a Krull-Schmidt category with a commutative dia- gram B e ≻ A J g g b e′ ≻ I. g g a Assume that e is split monic, J injective, and cok b ∈ RadA. Then every retraction of e can be lifted to a retraction of e′. Proof. If fe = 1, then there is a morphism f ′: I → J with f ′a = bf . Hence (1 − f ′e′)b = 0, and thus 1 − f ′e′ ∈ RadA. Therefore, f ′e′ is invertible, and it follows easily that (f ′e′)−1f ′ is the desired lifting. Lemma 2. Let A be an L-finite strict τ -category with ⋂ ∞ n=1 RadnA = 0. Then every object A of A admits a rational composition series. Jo u rn al A lg eb ra D is cr et e M at h .102 Categories of lattices, and their global structure... Proof. By duality, it is enough to show that an infinite strictly ascending sequence A0 < A1 < · · · of subobjects Ai ֌ A cannot exist. We may as- sume, without loss of generality, that A is injective. Suppose first that A is indecomposable. Then all Ai are indecomposable by Lemma 1. There- fore, the inclusions Ai ֌ Ai+1 are in RadA, whence A0 ֌ A belongs to ⋂ ∞ n=1 RadnA, a contradiction. Now let A be decomposable. Since ⋂ ∞ n=1 RadnA = 0, there exists an integer n ∈ N such that An ֌ Ai does not belong to RadA for all i > n. Hence there exists an indecomposable direct summand B of An such that the composition ei: B ֌ An ֌ Ai is split monic for all i > n. So there are commutative diagrams with short exact rows B ≻ en ≻ An ≻≻ Cn B w w w w w w ≻ ei ≻ Ai g g ≻≻ Ci g J g g b ≻ e ≻ A g g ≻≻ C, g ci with J injective and cok b ∈ RadA, where all the ei are split monic. The lifting e of en is split monic by Lemma 1. Moreover, Lemma 1 implies that every retraction of ei lifts to a retraction of e. Therefore, the ci are kernels. So we get an infinite strictly ascending sequence Cn < Cn+1 < · · · of subobjects Ci ֌ C. By induction, this leads to a contradiction. Lemma 3. Let A be a left L-finite strict τ -category. If a pullback is made up of two commutative squares A i ≻ E e ≻ B C g b j ≻ F g g f ≻ D, g c (13) where the left-hand square is exact, then the right-hand square is a pull- back. Proof. Let z: Z → E be a morphism with ez = gz = 0. Since the left- hand square is a pullback, there is an x: Z → A with ix = z and bx = 0. Hence ( b ei ) x = 0, and thus x = 0. So we get z = 0. Next let p: P → B and q: P → F be morphisms with cp = fq. Assume first that P is projective. Since the left-hand square is a pushout, there are morphisms p′: P → E and q′: P → C with q = gp′+jq′. Hence Jo u rn al A lg eb ra D is cr et e M at h .W. Rump 103 c(p − ep′) = fj · q′. So we get a morphism h: P → A with p − ep′ = eih and q′ = bh. Thus h′ := p′ + ih satisfies p = eh′ and q = gh′. Now let P be non-projective. Then there is a cokernel r: Q ։ P with Q projective, and we get a morphism h′′: Q → E with pr = eh′′ and qr = gh′′. Hence eh′′ and gh′′ annihilate the kernel k of r. Consequently, h′′k = 0, and thus h′′ = h′r for some h′: P → E. So we get p = eh′ and q = gh′. Let A be an L-finite strict τ -category. For a monomorphism A → B, let B/A denote the poset of subobjects E of B with A 6 E 6 B. Then every exact square (3) with monomorphisms a, d gives rise to an isomorphism of posets B/A ∼= D/C. In fact, if E ∈ B/A is given, then the corresponding F ∈ D/C is obtained via (13) by taking the pushout of i and b. By Lemma 3 and its dual, this correspondence E 7→ F is bijective. Let us call a monomorphism A → B simple if B/A has exactly two el- ements. Dually, we call an epimorphism simple if it is a simple monomor- phism in Aop. Proposition 8. Let A be an L-finite strict τ -category. A morphism a: A1 → A0 is a cokernel if and only if there exists no factorization a = me with a simple monomorphism m. Proof. Assume that a is not a cokernel. By Proposition 7, there ex- ists an exact square (11) with P projective and b = uP d for some d: B → ϑP . Since uQ is a simple monomorphism for any indecomposable direct summand Q of P , there exists a factorization b = st with a simple monomorphism s. By Lemma 3, the pushout of t and c yields an exact square D ≻ E P g s ≻ A0 g m with a simple monomorphism m such that a factors through m. Con- versely, let a = me be a cokernel with m monic. Then e factors through a. Thus m is split epic, hence invertible. Corollary. Let A be an L-finite strict τ -category. A simple monomor- phism m: A → B is either epic or a kernel. Proof. Suppose that fm = 0 with f 6= 0. We show that m = ker f . Assume that fg = 0. Since (g m) is not a cokernel, it factors through a simple monomorphism. Hence g factors through m. Jo u rn al A lg eb ra D is cr et e M at h .104 Categories of lattices, and their global structure... 5. The dualizing property Let A be an L-finite strict τ -category. For any simple monomorphism a: A1 → A0 in A, there is an exact square (11) with b = uP for some indecomposable projective P . If a ∈ RadA, we may regard (11) as an exact morphism in M(A). Since uP ∈ Pr L+, we necessarily have uP ∼= L+na for n ≫ 0. Therefore, up to isomorphism, P is uniquely determined by a. Definition 3. Let A be a strict τ -category. A function l: ObA → N is said to be additive if for A, B ∈ ObA, l(A ⊕ B) = l(A) + l(B) (14) l(A) = l(ϑA) − l(τA) = l(ϑ−A) − l(τ−A). (15) If l(A) > 0 for A 6= 0, then we write l > 0. We say that A is dualizing if there is a one-to-one correspondence between the isomorphism classes of indecomposable projective P with uP epic and the isomorphism classes of indecomposable injective I with uI monic, given by an exact square ϑP ≻ I P g uP ≻ ϑ−I. g uI (16) Assume that A is L-finite. Then the correspondence (16) is explic- itly given by uP = L+nuI and uI = L−nuP for n ≫ 0. Therefore, the dualizing property merely depends on the Auslander-Reiten quiver A(A) (cf. [10], §7). An additive function l: ObA → N admits a natural exten- sion to Ob M(A). Namely, for an object a: A1 → A0 of M(A), we define l(a) := l(A0) − l(A1). Thus l(A+) > 0 and l(A−) 6 0. Moreover, we have l(L+a) = l(L−a) = l(a) for all a ∈ Ob M(A). For an indecomposable projective P ∈ ObA, the monomorphism uP is obviously simple. Assume that uP is epic (which happens, e. g., if A is a lattice category), and let us try to prove that uP is a simple epimorphism. For a factorization uP = ab, there are two possibilities. If a ∈ RadA, then b is invertible. Otherwise, a is split epic. So b is a regular morphism of the form b: ϑP → P ⊕ C. For lattices over an order, of course, this is not possible, unless C = 0. The reason is that the rational rank of lattices is an additive function. Proposition 9. For an L-finite strict τ -category A, the following are equivalent. Jo u rn al A lg eb ra D is cr et e M at h .W. Rump 105 (a) A is dualizing. (b) A regular r ∈ A is a simple monomorphism if and only if it is a simple epimorphism. If A has an additive function l > 0, then A is dualizing. Proof. The equivalence (a) ⇔ (b) follows by the above. Assume that A has an additive function l > 0. We show that uP is a simple epimorphism for any indecomposable projective P . By the dual of Proposition 8, there is a factorization uP = ab with a simple epimorphism b. If a ∈ RadA, then a has a factorization a = uP c. Hence uP (1 − cb) = 0, and thus cb = 1, a contradiction. So we infer that a is of the form a: P ⊕C ։ P . Since l(b) = l(L−nb) = l(uI) = 0 for n ≫ 0 and some injective I, we get l(C−) = l(a) = l(uP ) − l(b) = 0. Hence a is invertible. Lemma 4. Let A be an L-finite dualizing strict τ -category. If a morphism c: A → C does not factor through a regular morphism r: A → B of length 1, then there is an exact square A c ≻ C B g r d ≻ D g r′ (17) with ρ(r′) = 1. A regular f ∈ A is a simple epimorphism if and only if ρ(f) = 1. Proof. Assume first that r is a simple epimorphism. If c does not factor though r, then the dual of Proposition 8 implies that ( c r ) is a kernel. Hence there exists an exact square (17) with a simple epimorphism r′. Now let f : E → B be a regular morphism with ρ(f) = 1. By Proposition 9, it remains to show that f is a simple monomorphism. By Proposition 8, there exists a factorization f = rs with a simple monomorphism r. We show that s is epic. Thus let c be a morphism with cs = 0. If c factors through r, then c = 0 since rs is regular. Otherwise, by Proposition 9, the above argument yields a commutative diagram (17) with a simple monomorphism r′. Thus df = drs = r′cs = 0, and therefore, d = 0. Hence r′c = 0, which gives c = 0. This shows that r, s are regular, whence s is invertible. Lemma 5. Let A be an L-finite dualizing strict τ -category. If B is an irreducible object and r: A → B a simple monomorphism, then ρ(A) 6 1. Jo u rn al A lg eb ra D is cr et e M at h .106 Categories of lattices, and their global structure... Proof. If r is not regular, then r is a kernel by the Corollary of Proposi- tion 8. Then A = 0. So let r be regular, and let c: A ։ C be a cokernel. Then Lemma 4 yields a commutative diagram (17) with r′ regular and ρ(r′) 6 1. If ρ(r′) = 0, then d is a cokernel, whence C ∼= D = 0. Oth- erwise, we may assume that (17) is exact. Then d = cok(r · ker c), and thus d is invertible. Hence c is invertible. Lemma 6. Let A be an L-finite dualizing strict τ -category with ⋂ ∞ n=1 RadnA = 0. Every non-zero morphism f : A → B with B irre- ducible admits a factorization f = rc with a cokernel c and a regular morphism r having a composition series. Proof. If f is not a cokernel, then Proposition 8 yields a factorization f = r1f ′ with a simple monomorphism r1: B1 → B. By Lemma 5, ρ(B1) = 1. So we can apply the same argument to f ′, which leads to a strictly decreasing sequence B > B1 > B2 · · · of subobjects. As the Bi are irreducible, the inclusions Bi+1 → Bi belong to RadA. Therefore, we end up with a factorization f : A c ։ Bn r → B, where r is regular with a composition series of length n. Now we are ready to prove our main theorem. Theorem 5. Let A be an L-finite dualizing strict τ -category with ⋂ ∞ n=1 RadnA = 0. Then A is a bi-noetherian integral quasi-abelian cate- gory. Proof. Let A0 < A1 < · · · be a strictly increasing infinite sequence of subobjects of A. If A is irreducible, then Lemma 6 implies that the inclusions Ai → A are regular with a composition series. Therefore, the Ai with i > 1 are irreducible by Lemma 5. So the inclusions Ai → Ai+1 belong to RadA, whence A0 → A is in ⋂ ∞ n=1 RadnA, a contradiction. Now we proceed by induction. By Lemma 2, there exists a rational composition series 0 → · · · ֌ B ֌ A. This gives a short exact sequence B ֌ A ։ C with C irreducible. If Ai 6 B for all i, we are done. Otherwise, the composition pi: Ai → A ։ C is non-zero for some n ∈ N. By Lemma 6, pi = rici with a cokernel ci and regular ri. So there are commutative diagrams Bi ≻ ≻ Ai ci ≻≻ Ci Bj g ≻ ≻ Aj g cj ≻≻ Cj g (18) Jo u rn al A lg eb ra D is cr et e M at h .W. Rump 107 with short exact rows and monic vertical morphisms for n 6 i 6 j. By the inductive hypothesis, the ascending sequences of subobjects Bn 6 Bn+1 6 · · · 6 B and Cn 6 Cn+1 6 · · · 6 C become stationary, i. e. Bi = Bi+1 and Ci = Ci+1 for some i > n. Taking a cokernel P ։ Ai+1 with P projective, it follows easily that the left-hand square in (18) with j = i + 1 is a pushout. Hence Ai = Ai+1, a contradiction. By duality, this proves that A is bi-noetherian. Next let f : A → B be any non-zero morphism in A. Consider a rational composition series 0 → · · · ֌ D d ֌ B, and let c: B ։ C be the cokernel of d. Then C is irreducible. We shall prove, by induction, that f has a kernel. By Lemma 6, there is a factorization cf = rc′ with a cokernel c′ and a regular morphism r. This gives a commutative diagram D′ ≻ d′ ≻ A c′ ≻≻ C ′ D g g ≻ d ≻ B g f c ≻≻ C g r with exact rows. By our inductive hypothesis and Lemma 6, there is a kernel k: K ֌ D′ of g. Now it is easily verified that d′k = ker f . By duality, this shows that A is preabelian, hence quasi-abelian by Proposi- tion 2. Since A is bi-noetherian, it follows that regular morphisms have a composition series. By Lemma 4 and [21], Proposition 6, this implies that A is integral. Let us call an additive category A strongly noetherian if the category mod(A) of coherent functors Aop → Ab is abelian and noetherian. (For equivalent descriptions of mod(A), see [21], and [28], Proposition 5.) More explicitly, this property can be expressed as follows. A non-empty class Σ of morphisms f : Af → A in A is said to be an (additive) sieve [6] of A if for f, g ∈ Σ, the morphism (f g): Af⊕Ag → A and each composite morphism fh with h ∈ A belongs to Σ. Now A is strongly noetherian if and only if every sieve Σ of any object of A is principal, i. e. every morphism in Σ factors through a fixed f ∈ Σ. We call A strongly bi- noetherian if A and Aop is strongly noetherian. For example, a ring R is left noetherian if and only if the category R-proj of finitely generated projective left R-modules is strongly noetherian. A will be called a strong lattice category if A ≈ Λ-mod ≈ (Γ-mod)op with Λ, Γ noetherian. Corollary 1. Let A be an L-finite dualizing strict τ -category with ⋂ ∞ n=1 RadnA = 0. Then A is strongly bi-noetherian. Jo u rn al A lg eb ra D is cr et e M at h .108 Categories of lattices, and their global structure... Proof. Let Σ be a sieve of A ∈ ObA. Since A is semi-abelian, every f : Af → A in Σ admits a factorization f = mfcf with a cokernel cf and a monomorphism mf . Since A is noetherian, there exists an h ∈ Σ such that every f ∈ Σ factors through mh: B → A. Therefore, replacing A by B, we may assume, without loss of generality, that there is a cokernel f0 ∈ Σ. By [22], Corollary of Proposition 9, there exists an integer n ∈ N such that every morphism f : Af → A in RadnA belongs to [Proj(A)], and thus factors through f0. Now we construct a finite sequence f0, f1, . . . , fm in Σ such that every f ∈ Σ factors through (f0, f1, . . . , fm): Af0 ⊕ · · · ⊕ Afm → A. Define Ri := RadiA r Radi+1A and ui: ϑiA → · · · uϑA−→ ϑA uA−→ A. Let i be the greatest integer < n with Σ ∩ Ri 6= ∅. Then there is a morphism f1 = uid1 ∈ Σ with d1: D1 ֌ ϑiA split monic and D1 indecomposable. So we have ϑiA = D1 ⊕ C. Denote the injection ( 0 1 ) : C ֌ D1 ⊕ C by d′1. If there exists a morphism in Σ ∩ Ri which factors through uid ′ 1, then there is an f2 = uid ′ 1d2 ∈ Σ with a split monomorphism d2: D2 ֌ C and D2 indecomposable. After finitely many steps, we get a sequence f0, f1, . . . , fj such that every f ∈ Σ ∩ Ri factors through (f0, f1, . . . , fj). Therefore, modulo (f0, f1, . . . , fj), we can replace i by a smaller integer. By induction, this proves the corollary. Corollary 2. Let A be a Krull-Schmidt category with finitely many iso- morphism classes of indecomposable objects. The following are equivalent. (a) A is a strong lattice category. (b) A is an L-finite strict τ -category with ⋂ ∞ n=1 RadnA = 0, having an additive function l > 0. Proof. (a) ⇒ (b): By [26], A is a strict τ -category. The remaining as- sertions follow by [26], and the implication (b) ⇒ (c) of [22], Theorem 4 (see also [22], Proposition 10). (b) ⇒ (a): Proposition 9, Theorem 5, and Proposition 7 imply that A satisfies (a)-(c) of Theorem 1. Suppose that there is a non-zero object A in A◦. Since A is noetherian, there exists a simple monomorphism m: B → A. Since A is L-finite, l(m) = l(uP ) = 0 for some projective object P . On the other hand, A ∈ ObA◦ implies that m is a kernel, which gives a contradiction. Hence A is a lattice category. By Corollary 1, Proj(A) and Inj(A) are strongly bi-noetherian. Thus A is a strong lattice category. By the remark of §4, L-finiteness of a strict τ -category is a property of its Auslander-Reiten quiver. Therefore, we may speak of an L-finite translation quiver Q. By [10], Theorem 7.1, this property of Q can be Jo u rn al A lg eb ra D is cr et e M at h .W. Rump 109 checked easily. A translation quiver Q with valuation (d, d′) is said to be admissible [12] if there exists a function c: Q → N r {0} with cX = cτX for non-projective vertices X, and cXdXY = d′XY cY for all X, Y ∈ Q. Corollary 3. For a finite admissible translation quiver Q, the following are equivalent. (a) There exists an order Λ over a complete discrete valuation ring R with Q = A(Λ-CM). (b) Q is L-finite and admits an additive function l > 0. Proof. By [12], 4.2.1, there exists a modulation for Q, and the mesh cate- gory A is R-linear for some complete discrete valuation ring R. Moreover, ⋂ ∞ n=1 RadnA = 0. By [22], Proposition 8, the existence of an additive function l > 0 implies that A is a strict τ -category. Therefore, the equiv- alence (a) ⇔ (b) follows by Corollary 2. Remark. There are 0-dimensional analogues of Corollary 2 and Corol- lary 3 that also follow by Theorem 5. Here the additive function l has to be replaced by a function l > 0 with l(P ) = l(ϑP )+1 and l(I) = l(ϑ−I)+1 for indecomposable P, I with P projective and I injective. Furthermore, the condition of L-finiteness can be dropped since RadA is nilpotent in this case. References [1] F. W. Anderson, K. R. Fuller: Rings and Categories of Modules, Springer, New York - Heidelberg - Berlin 1974 [2] M. Auslander: Isolated singularities and existence of almost split sequences, Springer Lecture Notes in Mathematics, Vol. 1178 (1986), 194-242 [3] A. Bondal, M. Van den Bergh: Generators and Representability of Functors in Commutative and Noncommutative Geometry, math.AG/0204218 v2 [4] S. Brenner: A combinatorial characterization of finite Auslander-Reiten quivers, Representation Theory I, Finite Dimensional Algebras, Proceedings, Springer LNM 1177 (1984), 13-49 [5] P. Gabriel, M. Zisman: Calculus of Fractions and Homotopy Theory, Springer, Berlin - Heidelberg - New York 1967 [6] A. Grothendieck, J. L. Verdier: Séminaire de Géométrie Algébrique (SGA IV, Exp. I.4), Springer Lecture Notes in Math. 269, 1972 [7] K. Igusa, G. Todorov: Radical Layers of Representable Functors, J. Algebra 89 (1984), 105-147 Jo u rn al A lg eb ra D is cr et e M at h .110 Categories of lattices, and their global structure... [8] K. Igusa, G. Todorov: A Characterization of Finite Auslander-Reiten Quivers, J. Algebra 89 (1984), 148-177 [9] K. Igusa, G. Todorov: A numerical characterization of finite Auslander-Reiten quivers, Representation Theory I, Finite Dimensional Algebras, Proceedings, Springer LNM 1177 (1984), 181-198 [10] O. 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Rump: Regular over-orders of lattice-finite rings and the Krull-Schmidt prop- erty, J. Pure Appl. Algebra 179 (2003), 169-173 [27] W. Rump: The triangular structure of ladder functors, 21pp., preprint [28] W. Rump: Triads, 26pp., preprint [29] J.-P. Schneiders: Quasi-abelian categories and sheaves, Mém. Soc. Math. France (N. S.), no. 76 (1999), vi+134 [30] R. Succi Christiani: Sulle categorie quasi abeliane, Rev. Roumaine Math. Pures Appl. 18 (1973), 105-119 Jo u rn al A lg eb ra D is cr et e M at h .W. Rump 111 [31] A. Wiedemann: The existence of covering functors for orders and consequences for stable components with oriented cycles, J. Algebra 93 (1985), 292-309 Contact information W. Rump Institut für Algebra und Zahlentheorie, Uni- versität Stuttgart, Pfaffenwaldring 57, D- 70550 Stuttgart, Germany E-Mail: rump@mathematik.uni-stuttgart.de Received by the editors: 16.10.2003 and final form in 26.01.2004.

Categories of lattices, and their global structure in terms of almost split sequences

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