Length categories of infinite height
aa r X i v : . [ m a t h . R T ] F e b LENGTH CATEGORIES OF INFINITE HEIGHT
HENNING KRAUSE AND DIETER VOSSIECK
Dedicated to Dave Benson on the occasion of his 60th birthday.
Abstract.
For abelian length categories the borderline between finite andinfinite representation type is discussed. Characterisations of finite represen-tation type are extended to length categories of infinite height, and the minimallength categories of infinite height are described.
Contents
1. Introduction 12. Length categories 13. Grothendieck groups and almost split sequences 34. Effaceable functors and pure-injectives 55. Uniserial categories 96. Minimal length categories of infinite height 12References 161.
Introduction
An abelian category is a length category if it is essentially small and every objecthas a finite composition series [16]. The height of a length category is the supremumof the Loewy lengths of all objects.The aim of this note is to explore the structure of length categories of infiniteheight. Length categories of finite height arise from artinian rings by taking thecategory of finite length modules. Also, length categories of infinite height areubiquitous, and typical examples are the uniserial categories which are not of finiteheight. Recall that a length category is uniserial if every indecomposable objecthas a unique composition series [1]. For instance, the category of nilpotent finitedimensional representations of a cyclic quiver over any field is uniserial and ofinfinite height.The paper is divided into three parts. First we extend known characterisations offinite representation type for module categories to more general length categories,including those of infinite height (Theorem 4.10). Then we show that uniserialcategories satisfy these finiteness conditions (Corollary 5.4 and Theorem 5.10). Inthe final part, we describe the minimal length categories of infinite height, and itturns out that only uniserial categories occur (Theorem 6.1).2.
Length categories
In this section we collect some basic concepts that are relevant for the study ofabelian length categories.For a module M over a ring Λ let ℓ Λ ( M ) denote its composition length. February 17, 2017.
Ext-finite categories.
A length category C is (left) Ext-finite if for every pair ofsimple objects S and T ℓ
End C ( T ) (Ext C ( S, T )) < ∞ . A length category C is equivalent to a module category (consisting of the finitelygenerated modules over a right artinian ring) if and only if the following holds [16]:(1) The category C has only finitely many simple objects.(2) The category C is Ext-finite.(3) The supremum of the Loewy lengths of the objects in C is finite. Hom-finite categories.
Let C be an essentially small additive category. Let uscall C (left) Hom-finite if for all objects X, Y in C the End C ( Y )-module Hom C ( X, Y )has finite length. Clearly, this property implies that C is a Krull-Schmidt category,assuming that C is idempotent complete. Lemma 2.1.
Let C be a Krull-Schmidt category. Then C is Hom-finite providedthat for all pairs of indecomposable objects X, Y the
End C ( Y ) -module Hom C ( X, Y ) has finite length.Proof. Choose decompositions X = L i X i and Y = L j Y n j j , n j >
0, such that the Y j are indecomposable and pairwise non-isomorphic. Set Y ′ = L j Y j . Then ℓ End C ( Y ) (Hom C ( X, Y )) = X i ℓ End C ( Y ) (Hom C ( X i , Y ))and ℓ End C ( Y ) (Hom C ( X, Y )) = ℓ End C ( Y ′ ) (Hom C ( X, Y ′ ))= X j ℓ End C ( Y j ) (Hom C ( X, Y j ))since End C ( Y ′ ) / rad End C ( Y ′ ) ∼ = Y j End C ( Y j ) / rad End C ( Y j ) . Now the assertion follows. (cid:3)
Example 2.2.
Let k be a commutative ring and C a k -linear category. If the k -module Hom C ( X, Y ) has finite length for all
X, Y in C , then C and C op are Hom-finite. Finitely presented and effaceable functors.
Let C be an abelian category. Anadditive functor F : C →
Ab is finitely presented if there is a presentation(2.1) Hom C ( Y, − ) −→ Hom C ( X, − ) −→ F −→ , and we call F effaceable if there is such a presentation such that the morphism X → Y in C is a monomorphism. We denote by Fp( C , Ab) the category of finitelypresented functors F : C →
Ab and by Eff( C , Ab) the full subcategory of effaceablefunctors. Note that Fp( C , Ab) is an abelian category, and Eff( C , Ab) is a Serresubcategory.We recall the following duality. The assignment F F ∨ given by F ∨ ( X ) = Ext ( F, Hom C ( X, − ))yields an equivalence(2.2) Eff( C , Ab) op ∼ −−→ Eff( C op , Ab) , where Ext ( − , − ) is computed in the abelian category Fp( C , Ab); see Theorem 3.4in Chap. II of [5]. If 0 → X α −→ Y β −→ Z → C and F = Coker Hom C ( α, − ), then F ∨ ∼ = Coker Hom C ( − , β ) and F ∨∨ ∼ = F . ENGTH CATEGORIES OF INFINITE HEIGHT 3
The
Yoneda functor
C −→
Fp( C op , Ab) , X Hom C ( − , X )admits an exact left adjoint that sends Hom C ( − , X ) to X ; it annihilates the efface-able functors and induces an exact functor(2.3) Fp( C op , Ab)Eff( C op , Ab) −→ C which is an equivalence; see [2, p. 205] and [15, III, Prop. 5].3.
Grothendieck groups and almost split sequences
Let C be an essentially small abelian category. The Grothendieck group K ( C ) isthe abelian group generated by the isomorphism classes [ C ] of objects C ∈ C subjectto the relations [ C ′ ] − [ C ] + [ C ′′ ], one for each exact sequence 0 → C ′ → C → C ′′ → C . Analogously, we write K ( C ,
0) for the abelian group generated by theisomorphism classes [ C ] of objects C ∈ C subject to the relations [ C ′ ] − [ C ] + [ C ′′ ],one for each split exact sequence 0 → C ′ → C → C ′′ → C . Thus there is acanonical epimorphism π : K ( C , −→ K ( C ) . Our aim is to find out when the kernel of π is generated by elements [ X ] − [ Y ]+[ Z ]that are given by almost split sequences 0 → X → Y → Z → C . Almost split sequences.
Let C be a Krull-Schmidt category. Recall from [7]that a morphism α : X → Y in C is left almost split if it is not a split mono andevery morphism X → Y ′ that is not a split mono factors through α . Dually, amorphism β : Y → Z in C is right almost split if it is not a split epi and everymorphism Y ′ → Z that is not a split epi factors through β . An exact sequence0 → X α −→ Y β −→ Z → almost split if α is left almost split and β is right almostsplit.We say that C has almost split sequences if for every indecomposable object X ∈ C , there is an almost split sequence starting at X when X is non-injective,and there is an almost split sequence ending at X when X is non-projective. Lemma 3.1.
A morphism α : X → Y in C is left almost split if and only X isindecomposable and α induces an exact sequence Hom C ( Y, − ) −→ Hom C ( X, − ) −→ F −→ in Fp( C , Ab) such that F is a simple object.Proof. See Proposition 2.4 in Chap. II of [5]. (cid:3)
Lemma 3.2.
For an indecomposable object X ∈ C are equivalent: (1) There is an almost split sequence → X → Y → Z → in C . (2) There is a simple object S ∈ Eff( C , Ab) such that S ( X ) = 0 .Proof. (1) ⇒ (2): Use Lemma 3.1.(2) ⇒ (1): The functor S admits a minimal projective presentation0 −→ Hom C ( Z, − ) −→ Hom C ( Y, − ) −→ Hom C ( X, − ) −→ S −→ C , Ab) since C is Krull-Schmidt. It follows from Proposition 4.4 in Chap. IIof [5] that the underlying sequence 0 → X → Y → Z → C . (cid:3) HENNING KRAUSE AND DIETER VOSSIECK
Length and support.
Let C be an essentially small additive category and supposethat C is Krull-Schmidt. Let ind C denote a representative set of the isoclasses ofindecomposable objects. For an additive functor F : C →
Ab setsupp( F ) = { X ∈ ind C |
F X = 0 } and let ℓ ( F ) denote the composition length of F in the category of additive functors C →
Ab.
Lemma 3.3.
For an additive functor F : C → Ab we have ℓ ( F ) = X X ∈ ind C ℓ End C ( X ) ( F X ) . Proof.
Let F : C →
Ab be a simple functor and
F X = 0 for some X ∈ ind C . Thenwe have supp( F ) = { X } and ℓ ( F ) = 1 = ℓ End C ( X ) ( F X ). From this the assertionfollows by induction on ℓ ( F ). (cid:3) Lemma 3.4.
Let C be Hom-finite and F : C → Ab a finitely generated functor.Then ℓ ( F ) is finite if and only if supp( F ) is finite.Proof. Apply Lemma 3.3. Clearly, supp( F ) is finite when ℓ ( F ) is finite. For theconverse observe that ℓ End C ( X ) ( F X ) is finite for all X ∈ C since C is Hom-finite and F is the quotient of a representable functor. (cid:3) Remark . Let Fp( C , Ab) be abelian and F ∈ Fp( C , Ab). Then ℓ ( F ) does notdepend on the ambient category since every simple object in Fp( C , Ab) is simple inthe category of all additive functors
C →
Ab.Let C be Hom-finite and fix an object X ∈ C . The assignment χ X : Fp( C , Ab) −→ Z , F ℓ End C ( X ) ( F X )induces a homomorphism K (Fp( C , Ab)) −→ Z . Lemma 3.6.
Let C be Hom-finite. Given functors F and ( F i ) i ∈ I in Fp( C , Ab) , [ F ] ∈ h [ F i ] | i ∈ I i ⊆ K (Fp( C , Ab)) implies supp( F ) ⊆ [ i ∈ I supp( F i ) . Proof.
Fix X ∈ ind C . We have X supp( F ) if and only if χ X ( F ) = 0. Thus X S i ∈ I supp( F i ) implies χ X ( F i ) = 0 for all i ∈ I . If [ F ] is generated by the [ F i ],then this implies χ X ( F ) = 0. Thus X supp( F ). (cid:3) Relations for Grothendieck groups.
Let C be an essentially small abelian cat-egory and consider the Yoneda functor C −→
Fp( C , Ab) , X h X := Hom C ( X, − ) . Lemma 3.7.
The Yoneda functor induces an isomorphism of abelian groups K ( C , ∼ −→ K (Fp( C , Ab)) . Proof.
The Yoneda functors identifies C with the full subcategory of projectiveobjects in Fp( C , Ab). From this the assertion follows since every object in Fp( C , Ab)admits a finite projective resolution. (cid:3)
Given an almost split sequence 0 → X → Y → Z → C , let S X denote thecorresponding simple functor in Fp( C , Ab) with supp( S X ) = { X } ; see Lemma 3.2. Lemma 3.8.
Let C be an abelian category. Then the following are equivalent: (1) The kernel of π : K ( C , → K ( C ) is generated by elements [ X ] − [ Y ] + [ Z ] that are given by almost split sequences → X → Y → Z → in C . (2) [ F ] ∈ h [ S X ] | → X → Y → Z → almost split i for all F ∈ Eff( C , Ab) . ENGTH CATEGORIES OF INFINITE HEIGHT 5
Proof.
An exact sequence η : 0 → X → Y → Z → C gives rise to an exactsequence0 −→ Hom C ( Z, − ) −→ Hom C ( Y, − ) −→ Hom C ( X, − ) −→ F η −→ C , Ab) with [ F η ] = [ h X ] − [ h Y ]+[ h Z ]. The assertion then follows by identifying[ X ] with [ h X ] for all X ∈ C , keeping in mind Lemma 3.7. (cid:3) Proposition 3.9.
Let C be an essentially small abelian Krull-Schmidt category.Consider the following conditions: (1) Every effaceable finitely presented functor
C → Ab has finite length. (2) The kernel of π : K ( C , → K ( C ) is generated by elements [ X ] − [ Y ] + [ Z ] that are given by almost split sequences → X → Y → Z → in C .Then (1) implies (2) and the converse holds when C is Hom-finite.Proof. (1) ⇒ (2): Let S be a simple composition factor of an effaceable functor.Choosing a minimal projective presentation of S in Fp( C , Ab) gives rise to an almostsplit sequence 0 → X → Y → Z → C so that S = S X ; see Lemma 3.2. Thuscondition (2) in Lemma 3.8 holds.(2) ⇒ (1): Fix F ∈ Eff( C , Ab). Then Lemmas 3.6 and 3.8 imply that supp( F )is finite. From Lemma 3.4 it follows that F has finite length. (cid:3) Remark . The property that every effaceable functor
C →
Ab has finite lengthis self-dual, thanks to the duality (2.2).4.
Effaceable functors and pure-injectives
The aim of this section is a characterisation of the length categories C such thatevery effaceable finitely presented functor C →
Ab has finite length. This involvesthe study of pure-injective objects, and we need to embed C into a Grothendieckcategory. Locally finitely presented categories.
Let A be a Grothendieck category. Anobject X ∈ A is finitely presented if Hom A ( X, − ) preserves filtered colimits, and wedenote by fp A the full subcategory of finitely presented objects in A . The category A is called locally finitely presented if A has a generating set of finitely presentedobjects [10].Let C be an essentially small abelian category. We denote by Lex( C op , Ab) thecategory of left exact functors C op → Ab and set A = Lex( C op , Ab). Observe that A is a locally finitely presented Grothendieck category. The category C identifieswith fp A via the functor C −→ A , X Hom C ( − , X ) , and every object in A is a filtered colimit of objects in C . Locally noetherian categories.
A Grothendieck category A is called locally noe-therian if A has a generating set of noetherian objects. In that case finitely pre-sented and noetherian objects in A coincide.A Grothendieck category A is called locally finite if A has a generating set offinite length objects. When A is locally finite, then every noetherian object hasfinite length, since any object is the directed union of finite length subobjects. Thusfinitely presented and finite length objects in A coincide HENNING KRAUSE AND DIETER VOSSIECK
Purity.
Let C be an essentially small abelian category and A = Lex( C op , Ab).We recall the following construction from [12, § C = Fp( C , Ab) op and ¯ A =Lex( ˇ C op , Ab). Observe that ˇ C is abelian and identifies with fp ¯ A . The functor h : C −→ ˇ C , X Hom C ( X, − )is right exact and extends to a colimit preserving and fully faithful functor h ! : A −→ ¯ A , X ¯ X that makes the following square commutative: C ˇ CA ¯ A hh ! Note that h ! is the left adjoint of h ∗ : ¯ A → A given by h ∗ ( X ) = X ◦ h .There is a notion of purity for A . A sequence 0 → X → Y → Z → A is pure-exact if Hom A ( C, − ) takes it to an exact sequence of abelian groups forall finitely presented C ∈ A . An object M ∈ A is pure-injective if every puremonomorphism X → Y induces a surjective map Hom A ( Y, M ) → Hom A ( X, M ). Lemma 4.1. (1)
A sequence → X → Y → Z → in A is pure-exact if andonly if the induced sequence → ¯ X → ¯ Y → ¯ Z → in ¯ A is exact. (2) The functor X ¯ X identifies the pure-injective objects in A with theinjective objects in ¯ A .Proof. See Lemma 4 in § § (cid:3) The category ¯ A has enough injective objects. Thus every object in A admitsa pure monomorphism into a pure-injective object. We call such a morphism a pure-injective envelope if it becomes an injective envelope in ¯ A . Example 4.2.
Suppose that C is Hom-finite. Then every finitely presented objectin A is pure-injective. This follows from Theorem 1 in § A for a representative set of the indecomposable pure-injectiveobjects in A , containing exactly one object for each isomorphism class. For a class X ⊆ ¯ A set X ⊥ = { M ∈ Ind
A |
Hom ¯ A ( X, ¯ M ) = 0 for all X ∈ X } . We recall the following detection result; see [18, Thm. 3.8] and [20, Thm. 4.2].
Proposition 4.3.
Let X , Y be Serre subcategories of ˇ C . Then X ⊆ Y ⇐⇒ X ⊥ ⊇ Y ⊥ . (cid:3) Effaceable functors.
We compute the orthogonal complement of the category ofeffaceable functors.
Lemma 4.4.
We have
Eff( C , Ab) ⊥ = { M ∈ Ind
A | M is injective } .Proof. Fix F ∈ Eff( C , Ab) with presentation (2.1) and M ∈ Ind A . Then we haveHom ¯ A ( F, ¯ M ) = 0 if and only if every morphism X → M factors through X → Y .It follows that Hom ¯ A ( F, ¯ M ) = 0 when M is injective. If M is not injective, thenthere is a monomorphism α : M → N in A that does not split. Moreover, α isnot a pure monomorphism since M is pure-injective. Thus we may assume that C = Coker α is in C . Write N = colim i N i as a filtered colimit of objects in C . Thenfor some i the induced morphism β i : N i → C is an epimorphism. Let α i : M i → N i be the kernel of β i and set F i = Coker Hom C ( α i , − ). Then F i is in Eff( C , Ab) andHom ¯ A ( F i , ¯ M ) = 0 by construction. (cid:3) ENGTH CATEGORIES OF INFINITE HEIGHT 7
Let Fp( C , Ab) denote the full subcategory of finite length objects in Fp( C , Ab).
Lemma 4.5.
Let C be a Krull-Schmidt category. An object in Ind A belongs to (Fp( C , Ab) ) ⊥ if and only if it is not the pure-injective envelope of the source of aleft almost split morphism in C .Proof. An object M ∈ Ind A belongs to (Fp( C , Ab) ) ⊥ if and only if Hom ¯ A ( S, ¯ M ) =0 for every simple objects S in ˇ C . By Lemma 3.1, any simple object S in ˇ C arises asthe kernel of a morphism ¯ X → ¯ Y that corresponds to a left almost split morphism X → Y in C . Moreover, the morphism S → ¯ X is an injective envelope in ˇ C sinceEnd C ( X ) is local. It remains to observe that a morphism X → M in A is apure-injective envelope if and only if ¯ X → ¯ M is an injective envelope in ¯ A . (cid:3) Proposition 4.6.
Let A be a locally finitely presented Grothendieck category andset C = fp A . Suppose that C is an abelian Krull-Schmidt category. Then thefollowing conditions are equivalent: (1) Every effaceable finitely presented functor
C → Ab has finite length. (2) Every indecomposable pure-injective object in A is injective or the pure-injective envelope of the source of a left almost split morphism in C .Proof. Effaceable functors and finite length functors form Serre subcategories inFp( C , Ab). Their orthogonal complements in Ind A are described in Lemmas 4.4and 4.5. It remains to apply Proposition 4.3. (cid:3) Fp-injective objects.
Let A be a locally finitely presented Grothendieck category.An object X ∈ A is called fp-injective if Ext A ( C, X ) = 0 for every finitely presentedobject C ∈ A .Let C be an essentially small abelian category and set A = Lex( C op , Ab).
Lemma 4.7.
A functor X ∈ A is exact if and only if X is an fp-injective object.Proof. Using the identification C ∼ −→ fp A , the functor X is exact iff for every exactsequence η : 0 → A → B → C → A the induced sequenceHom A ( η, X ) : 0 → Hom A ( C, X ) → Hom A ( B, X ) → Hom A ( A, X ) → A ( C, X ) = 0. Clearly, this implies exactness of Hom A ( η, X )for any exact η : 0 → A → B → C → A . Conversely, let µ : 0 → X → Y → C → A and write Y = colim i Y i as filtered colimit of finitely presentedobjects. This yields an exact sequence µ j : 0 → X j → Y j → C → A for some j . Now exactness of Hom A ( µ j , X ) imlies that µ splits. (cid:3) Lemma 4.8.
Let X ∈ A . Then ¯ X is fp-injective in ¯ A = Lex( ˇ C op , Ab) .Proof.
We apply Lemma 4.7. Recall that ˇ C = Fp( C , Ab) op . Thus ¯ X is exactfor X = Hom C ( − , C ) with C ∈ C . Any object X ∈ A is a filtered colimit ofrepresentable functors. Thus it remains to observe that a filtered colimit of exactfunctors is exact. (cid:3) Lemma 4.9.
Let A be a locally noetherian Grothendieck category. Then everyfp-injective object in A is injective and decomposes into indecomposable objects.Proof. When A is locally noetherian, then finitely presented and noetherian objectsin A coincide and are therefore closed under quotients. Now apply Baer’s criterionto show that fp-injective implies injective. The decomposition into indecomposablesfollows from an application of Zorn’s lemma, using that fp-injective objects areclosed under filtered colimits. (cid:3) HENNING KRAUSE AND DIETER VOSSIECK
Finite type.
We are now ready to extend some known characterisations of fi-nite representation type for module categories to more general abelian categories,including the length categories of infinite height.Recall that every essentially small abelian category C with all objects in C noe-therian embeds into a locally noetherian Grothendieck category A with C ∼ −→ fp A . Theorem 4.10.
Let A be a locally noetherian Grothendieck category and set C =fp A . Suppose that C is Hom-finite. Then the following are equivalent: (1) Every effaceable finitely presented functor
C → Ab has finite length. (2) The kernel of π : K ( C , → K ( C ) is generated by elements [ X ] − [ Y ] + [ Z ] that are given by almost split sequences → X → Y → Z → in C . (3) The category C has almost split sequences, and every non-zero object in A has an indecomposable direct summand that is finitely presented or injective. (4) The category C has almost split sequences, and every indecomposable objectin A is finitely presented or injective.Proof. We identify ˇ C ∼ −→ fp ¯ A .(1) ⇔ (2): See Proposition 3.9.(1) ⇒ (3): Let X = 0 be an object in A . Suppose first that Hom ¯ A ( S, ¯ X ) = 0for a simple and effaceable S ∈ ˇ C . Choose an injective envelope α : S → ¯ C in ˇ C .Then any non-zero morphism φ : S → ¯ X factors through α , since ¯ X is fp-injectiveby Lemma 4.8. On the other hand, α factors through φ since C is pure-injective byExample 4.2. Thus C is isomorphic to a direct summand of X . Now suppose thatHom ¯ A ( F, ¯ X ) = 0 for all effaceable F ∈ ˇ C . Then ¯ X : Fp( C , Ab) → Ab vanishes onEff( C , Ab) and identifies with an exact functor C op → Ab via the functor (2.3). Thus X is injective and has an indecomposable summand since A is locally noetherian;see Lemma 4.9.Let X ∈ C be an indecomposable non-injective object. Then there is an epimor-phism Hom C ( X, − ) → F with F = 0 effaceable. The object F has finite lengthand we may assume that F is simple. It follows from Lemma 3.2 that there is analmost split sequence 0 → X → Y → Z → C .For every indecomposable non-projective Z ∈ C , there is an almost split se-quence 0 → X → Y → Z → A , by Theorem 1.1 in [21]. The object X isindecomposable, and therefore in C , by the first part of the proof.(3) ⇒ (4): Clear.(4) ⇒ (1): Use Proposition 4.6. (cid:3) Remark . Suppose the conditions in Theorem 4.10 hold. Then effaceable andfinite length functors agree if and only if every injective object in C is zero. Thisfollows from the proof of Proposition 4.6. Remark . Theorem 4.10 generalises various known characterisations of finiterepresentation type for module categories.For a ring Λ, let mod Λ denote the category of finitely presented Λ-modules.Recall that Λ has finite representation type if mod Λ is a length category with onlyfinitely many isomorphism classes of indecomposable objects.(1) A ring Λ has finite representation type if and only if every finitely presentedfunctor mod Λ → Ab has finite length [3].(2) An artin algebra Λ has finite representation type if and only if for C = mod Λthe kernel of π : K ( C , → K ( C ) is generated by elements [ X ] − [ Y ] + [ Z ] that aregiven by almost split sequences 0 → X → Y → Z → C [6, 11]. The proof of Theorem 4.10 is close to Butler’s original proof. Auslander’s proof is based onthe use of a bilinear form on K ( C , ENGTH CATEGORIES OF INFINITE HEIGHT 9 (3) An artin algebra Λ has finite representation type if and only if every inde-composable Λ-module is finitely presented [3, 4, 24].
Example 4.13.
Fix a field and consider the category of nilpotent finite dimensionalrepresentations of the following quiver with relations: ◦ ◦ ( αβ = 0) α β We denote this length category by C and observe that it is discrete : the fibresof the map K ( C , → K ( C ) are finite. However, the finiteness conditions inTheorem 4.10 are not satisfied. For instance, the unique injective and simple objectis not the end term of an almost split sequence in C .5. Uniserial categories
In this section we establish for uniserial categories the finiteness conditions inTheorem 4.10. Moreover, we show that uniserial categories of infinite height areprecisely the length categories such that effaceable and finite length functors agree.Let C be an essentially small abelian category. Set A = Lex( C op , Ab) and identify C with the full subcategory of finitely presented objects in A . Length and height.
Fix an object X ∈ A . The composition length of X isdenoted by ℓ ( X ). The socle series of X is the chain of subobjects0 = soc ( X ) ⊆ soc ( X ) ⊆ soc ( X ) ⊆ . . . of X such that soc ( X ) is the socle of X (the largest semisimple subobject of X ) and soc n +1 ( X ) is given by soc n +1 ( X ) / soc n ( X ) = soc ( X/ soc n ( X )). We setht( X ) = n when n is the smallest integer such that soc n ( X ) = X , and ht( X ) = ∞ when such n does not exist.Let X = colim X i be written as a filtered colimit in A . Then soc n ( X ) =colim soc n ( X i ) for all n ≥
0. Thus X = S n ≥ soc n ( X ) when every object in C has finite length. Uniserial categories.
Recall that C is uniserial if C is a length category and eachindecomposable object has a unique composition series. Lemma 5.1.
Let C be an abelian length category. Then C uniserial if and only if ht( X ) = ℓ ( X ) for every indecomposable X ∈ C .Proof. Let X ∈ C be indecomposable. If ht( X ) = ℓ ( X ), then the socle series of X is the unique composition series of X .Now assume ht( X ) = ℓ ( X ). Then there exists some n ≥ n +1 ( X ) / soc n ( X ) = S ⊕ . . . ⊕ S r with all S i simple and r >
1. Choose n minimal and let soc n ( X ) ⊆ U i ⊆ X begiven by U i / soc n ( X ) = S i Then we have at least r different composition series0 = soc ( X ) ⊆ soc ( X ) ⊆ . . . ⊆ soc n ( X ) ⊆ U i ⊆ . . . ⊆ soc n +1 ( X ) ⊆ . . . of X . (cid:3) Lemma 5.2.
Let C be a uniserial category. Then C is Hom-finite.Proof. We need to show for all
X, Y in C that the End C ( Y )-module Hom C ( X, Y )has finite length. It suffices to assume that Y is indecomposable; see Lemma 2.1.We claim that ℓ End C ( Y ) (Hom C ( X, Y )) ≤ ℓ ( X ) . Using induction on ℓ ( X ) the claim reduces to the case that X is simple. So let S =soc( Y ) and write E = E ( Y ) for its injective envelope. Note that soc n ( E ) = Y for n = ℓ ( Y ), by Lemma 5.1. Thus any endomorphism E → E restricts to a morphism Y → Y . Write i : S → Y for the inclusion. Then any morphism j : S → Y inducesan endomorphism f : E → E such that f | Y ◦ i = j . Thus the End C ( Y )-moduleHom C ( S, Y ) is cyclic, and it is annihilated by the radical of End C ( Y ). ThereforeHom C ( S, Y ) is simple. (cid:3)
Proposition 5.3.
Let C be a uniserial category. Then every non-zero object in A has an indecomposable direct summand that belongs to C or is injective.Proof. From Lemma 5.1 it follows that for every indecomposable injective object E in A the subobjects form a linear chain0 = E ⊆ E ⊆ E ⊆ . . . with E n = soc n ( E ) in C for all n ≥ E = S n ≥ E n . Note that E = E ℓ ( E ) when ℓ ( E ) < ∞ .Fix X = 0 in A and choose a simple subobject S ⊆ X . Let U ⊆ X be a maximalsubobject containing S such that S ⊆ U is essential; this exists by Zorn’s lemma.Then U is injective or belongs to C . In the first case we are done. So assume U ∈ C .We claim that the inclusion U → X is a pure monomorphism. To see this, choose amorphism C → X/U with C ∈ C . This yields the following commutative diagramwith exact rows. 0 U V C U X X/U V = L i V i as a direct sum of indecomposable objects. Then there exists anindex i such that the composite S ֒ → U → V i → X is non-zero. Thus S → V i isessential and V i → X is a monomorphism. It follows from the maximality of U that U → V i is an isomorphism. Therefore the top row splits, and this yields the claim.It remains to observe that every object in C is pure-injective since C is Hom-finite;see Example 4.2 and Lemma 5.2 (cid:3) Let Fp( C , Ab) denote the full subcategory of finite length objects in Fp( C , Ab).
Corollary 5.4.
Let C be a uniserial category. Then every effaceable finitely pre-sented functor C → Ab has finite length. Moreover effaceable and finite lengthfunctors agree if and only if all injective objects in C are zero. In that case we havean equivalence (5.1) Fp( C op , Ab)Fp( C op , Ab) ∼ −→ C . Proof.
The first assertion follows from Proposition 5.3 and Theorem 4.10, keepingin mind that C is Hom-finite by Lemma 5.2. Effaceable and finite length functorsagree if and only if all injective objects in C are zero, by Remark 4.11. Having thisproperty, the equivalence is (2.3). (cid:3) Remark . An interesting instance of the equivalence (5.1) arises from the studyof Greenberg modules; see [13, V, §
4, 1.8] and [25, §§ Remark . It would be interesting to have a more direct proof of Corollary 5.4,avoiding the embedding of C into a Grothendieck category. ENGTH CATEGORIES OF INFINITE HEIGHT 11
Serre duality.
Our next aim is a characterisation of uniserial categories that in-volves Serre duality. To this end recall the following characterisation in terms ofExt-quivers [1].
Proposition 5.7.
A length category C is uniserial if and only it satisfies the fol-lowing condition and its dual: For each simple object S there exists, up to isomor-phism, at most one simple object T such that Ext C ( S, T ) = 0 , and in this case ℓ End C ( T ) (Ext C ( S, T )) = 1 . (cid:3) We fix a field k and write D = Hom k ( − , k ) for the usual duality.Let C be a k -linear abelian category such that Hom C ( X, Y ) is finite dimensionalfor all
X, Y ∈ C . Following [8], the category C satisfies Serre duality if there existsan equivalence τ : C ∼ −→ C with a functorial k -linear isomorphism D Ext C ( X, Y ) ∼ −→ Hom C ( Y, τ X ) for all
X, Y ∈ C . The functor τ is called Serre functor or Auslander-Reiten translation . Note that aSerre functor is k -linear and essentially unique provided it exists; this follows fromYoneda’s lemma.The following result is well-known and describes the structure of a length cate-gory with Serre duality. Let us recall the shape of the relevant diagrams.˜ A n : 1 2 3 · · · n n + 1 A ∞∞ : · · · ◦ ◦ ◦ ◦ · · · Proposition 5.8.
Let C be a Hom-finite k -linear length category and suppose C admits a Serre functor τ . Then C is uniserial. The category C admits a uniquedecomposition C = ` i ∈ I C i into connected uniserial categories with Serre duality,where the index set equals the set of τ -orbits of simple objects in C . The Ext-quiverof each C i is either of type A ∞∞ (with linear orientation) or of type ˜ A n (with cyclicorientation).Proof. We apply the criterion of Proposition 5.7 to show that C is uniserial. Tothis end fix a simple object S . Then Ext C ( S, T ) ∼ = D Hom C ( T, τ S ) = 0 for somesimple object T if and only if T ∼ = τ S . Moreover, ℓ End C ( τS ) (Ext C ( S, τ S )) = 1. Aquasi-inverse of τ provides a Serre functor for C op . Thus the category C is uniserial.The structure of the Ext-quiver of C follows from Proposition 5.7. The Serrefunctor acts on the set of vertices and the τ -orbits provide the index set of thedecomposition C = ` i ∈ I C i into connected components. The Ext-quiver of C i is oftype A ∞∞ if the corresponding τ -orbit is infinite. Otherwise, the Ext-quiver of C i isof type ˜ A n where n + 1 equals the cardinality of the τ -orbit. (cid:3) Auslander-Reiten duality.
For an abelian category A let St A denote the stablecategory modulo injectives , which is obtained from A by identifying two morphisms φ, φ ′ : X → Y if Ext A ( − , φ ) = Ext A ( − , φ ′ ) . When A has enough injective objects this means φ − φ ′ factors through an injectiveobject. We write Hom A ( − , − ) for the morphisms in St A .Let us recall from [22, Corollary 2.13] the following version of Auslander-Reitenduality for Grothendieck categories, generalising the usual duality for modules overartin algebras [7]. Proposition 5.9.
Let A be a k -linear and locally finitely presented Grothendieckcategory. There exist a functor τ : fp A → St A with a natural isomorphism (cid:3) (5.2) D Ext A ( X, − ) ∼ = Hom A ( − , τ X ) for all X ∈ fp A . Uniserial categories of infinite height.
We are now ready to characterise uni-serial categories of infinite height in terms of finitely presented functors.
Theorem 5.10.
Let C be a k -linear length category such that Hom C ( X, Y ) is finitedimensional for all X, Y ∈ C . Then the following are equivalent: (1)
A finitely presented functor
C → Ab is effaceable if and only if it has finitelength. (2) The category C has Serre duality. (3) The category C is uniserial and all connected components have infiniteheight.Proof. Set A = Lex( C op , Ab) and identify C ∼ −→ fp A .(1) ⇒ (2): We claim that the functor τ : C → St A in Proposition 5.9 yields aSerre functor for C .First observe that Hom A ( − , X ) = Hom A ( − , X ) for every X ∈ C . Condition(1) implies that every injective object in C is zero; see Remark 4.11. Now let φ : I → X be a morphism in A with indecomposable injective I . Then Ker φ isindecomposable, and therefore injective or finitely presented, by Theorem 4.10.Thus φ = 0.The assumption on C in (1) is also satisfied by C op , thanks to the duality (2.2).Thus Hom C ( X, − ) = Hom C ( X, − ) for every X ∈ C .The functor τ : C → St A in Proposition 5.9 lands in C , because all indecompos-able objects in St A belong to C by Theorem 4.10. In fact, the formula (5.2) yieldsan almost split sequence 0 → τ Z → Y → Z → C for every indecomposableobject Z . Thus τ : C → C is essentially surjective on objects, since the category C has almost split sequences by Theorem 4.10. The defining isomorphism for τ showsthat τ is fully faithful, sinceHom C ( − , − ) = Hom C ( − , − ) = Hom C ( − , − ) . Indeed, the induced map τ X,Y : Hom C ( X, Y ) → Hom C ( τ X, τ Y ) is a monomor-phism. On the other hand, τ induces mapsHom C ( X, Y ) ∼ −→ D Hom C ( X, Y ) → D Ext C ( Y, τ X ) ∼ −→ Hom C ( τ X, τ Y )where the middle one is the dual of the monomorphismExt A ( − , τ X ) −→ D Hom A ( X, − )from [22, Theorem 2.15]. Thus τ X,Y is bijective.(2) ⇒ (3): Use Proposition 5.8.(3) ⇒ (1): Use Corollary 5.4. (cid:3) Minimal length categories of infinite height
Throughout this section we fix an algebraically closed field k .Our aim is an explicit description of the length categories of infinite height thatare minimal in the sense that every proper closed subcategory has only finitelymany isoclasses of indecomposable objects. Here, a full subcategory of an abeliancategory is closed if it is closed under subobjects and quotients. Theorem 6.1.
Let C be a k -linear length category with the following properties: (1) The category C has only finitely many isoclasses of simple objects. (2) The spaces
Hom C ( X, Y ) and Ext C ( X, Y ) are finite dimensional for all ob-jects X, Y in C . (3) The category C has infinite height. (4) Every proper closed subcategory of C has only finitely many isoclasses ofindecomposable objects. ENGTH CATEGORIES OF INFINITE HEIGHT 13
Then C is equivalent to the category of nilpotent finite dimensional representationsof some cycle Z n : 1 2 3 · · · n − n ( n ≥ . We do not prove the theorem as it stands. Instead we switch to an equivalentformulation involving representations of admissible algebras.
Admissible algebras.
Let A be a k -algebra and denote by R its radical. We call A admissible if it satisfies the following two conditions:(a) The algebras A/R and
R/R are finite dimensional.(b) The algebra A is separated and complete for the R -adic topology: A ∼ −→ lim ←− l ≥ A/R l , canonically.In the sequel, every admissible algebra A will be considered together with its R -adic topology. Accordingly, an ideal I of A is open if I contains some power R l iff I is of finite codimension. In other words, our admissible algebras are exactly theprofinite algebras satisfying (a). For more on profinite algebras, one may consult[26]. The closure of I equals the intersection of all I + R l .Given an admissible algebra A , we are really interested in the category mod A of(left) A -modules of finite length. Condition (a) ensures that mod A has only finitelymany isoclasses of simples and that the spaces Hom A ( M, N ) and Ext A ( M, N ) arefinite dimensional for
M, N in mod A . Condition (b) comes up naturally when onetries to recover A from mod A , up to Morita equivalence; see [15]. Complete path algebras.
Let Q be a finite quiver. The complete path algebra k J Q K consists of the formal series P u a u u where u runs through the paths of Q and a u ∈ k . Here, the paths ǫ i of length 0, corresponding to the vertices i , are included.The multiplication is defined by X u a u u ! X v b v v ! = X w X uv = w a u b v ! w. For any integer l ≥
0, the elements P u a u u , where u is restricted to the paths oflength ≥ l , form an ideal k J Q K ≥ l of k J Q K ; this ideal is precisely the l -th power of theradical of k J Q K . Consequently k J Q K is admissible and, according to our convention,will always be considered together with its k J Q K ≥ -adic topology. Description of admissible algebras by quiver and relations.
Any admissible k -algebra A that is basic , i.e., the quotient A/R is a finite direct product of copiesof k , can be presented as the complete path algebra of a finite quiver modulo aclosed ideal: choose a decomposition 1 A = e + · · · + e n of the unit element intopairwise orthogonal primitive idempotents; such a decomposition can be obtainedby lifting the unique one for A/R . The quiver Q A then has vertices 1 , . . . , n ,and there are arrows α mji : i → j , 1 ≤ m ≤ n ji , where n ji is the dimension of e j ( R/R ) e i . Choose further elements a mji ∈ e j Re i , 1 ≤ m ≤ n ji , whose classes forma basis of e j ( R/R ) e i . The choices uniquely determine a continuous homomorphism k J Q A K → A that maps each ǫ i to e i and each α mji to a mji . This homomorphismis surjective; its kernel I is contained in k J Q A K ≥ and necessarily closed. Thepresentation A ∼ ←− k J Q A K /I allows one to interprete a module in mod A as a finitedimensional representation of Q A that satisfies the relations of some sufficientlysmall ideal I + R l , depending on the module. Representation types.
Let A be an admissible k -algebra.The algebra A is representation-finite if mod A has only finitely many isoclassesof indecomposables. In this case, A is necessarily finite dimensional. Otherwise,there is some infinite dimensional indecomposable projective P , having a simpletop, and P admits indecomposable quotients of any finite length ≥ A is mild if any quotient A/I by some closed ideal I = 0 isrepresentation-finite. Remark . In the definition of ‘mild’, the restriction to closed ideals is not nec-essary: if I is an arbitrary ideal with closure ¯ I , the algebras A/ ¯ I and A/I have thesame finite dimensional modules. On the other hand, if
A/I is representation-finite, I is even open. The main result.
Very special but frequently encountered examples of admissiblealgebras are the complete path algebras k J Z n K of the cyclic quivers Z n : 1 2 3 · · · n − n ( n ≥ . α α α α n − α n − α n They admit an alternative description: consider the discrete valuation k -algebra = k J T K with maximal ideal m = T k J T K and fraction field K . Then k J Z n K isisomorphic to the following hereditary -order in the simple K -algebra M n ( K ): · m m0 0 m · m m0 0 0 · m m · · · · · · · · The Auslander-Reiten quiver of mod k J Z n K is identified with Z A ∞ /τ n . If theorientation · · · → → → A ∞ is used, then the vertex ( − i, l ) corresponds tothe uniserial indecomposable of length l with top located at i .It is well-known that k J Z n K is mild. Theorem 6.3.
If a k -algebra is admissible, basic, infinite dimensional, and mild,then it is isomorphic to k J Z n K for some n ≥ . Our result is close to well-known characterisations of hereditary orders; see [17].The only novelty is that we can do without assuming a ‘large’ center or some ‘purity’condition [14].
Proof.
Let the k -algebra A be admissible, basic, infinite dimensional, and mild.We denote by R the radical of A and fix a decomposition 1 A = e + · · · + e n intopairwise orthogonal primitive idempotents.Step 1: For any i , the algebra e i Ae i is a quotient of k J T K , and for any i, j , thespace e j Ae i is cyclic as a right module over e i Ae i or as a left module over e j Ae j . Indeed, by well-known observations due to Jans [19] and Kupisch [23], a similarstatement is true for each of the representation-finite algebras
A/R l . Our claimfollows by passage to the limit.Step 2: The e j Ae j - e i Ae i -bimodule e j Ae i carries two natural filtrations. Thefirst one is its radical filtration given by the subbimodules( e j Ae i ) ≥ l = X r + s = l ( e j Re j ) s A ( e i Re i ) r . By step 1 it coincides with the radical filtration of e j Ae i over e i Ae i or over e j Ae j ,it contains each non-zero subbimodule of e j Ae i exactly once, and each quotient( e j Ae i ) ≥ l / ( e j Ae i ) ≥ l +1 has dimension ≤ ENGTH CATEGORIES OF INFINITE HEIGHT 15
The second one is the filtration ( e j R m e i ) m ≥ induced by the R -adic filtrationon A . Each term, being a subbimodule of e j Ae i , appears in the first filtration, andeach quotient e j R m e i /e j R m +1 e i is semisimple over e i Ae i and over e j Ae j .Remembering that the second filtration is separated, we conclude that it is ob-tained from the first one by possibly introducing repetitions and that both filtrationsdefine the same topology.Step 3: Let e be an idempotent of A that is the sum of some of e , . . . , e n .We claim that eAe is admissible : indeed, we just saw in step 2 that each space e j ( R/ReR ) e i (with e i and e j occurring in e ) has dimension at most 1. Arguing asabove, the topologies on e j Ae i induced by the eRe -adic filtration on eAe and the R -adic filtration on A coincide. Therefore eAe ∼ −→ lim ←− l ≥ eAe/ ( eRe ) l , canonically.Of course, our claim would be wrong without the assumption of mildness of A .Step 4: Let e be as in step 3. We claim that eAe is mild . Indeed, let J be anon-zero ideal of eAe and I = AJA its extension to A . Since A/I is in particularfinite dimensional, the fully faithful left adjoint N ( A/I ) e ⊗ eAe/J N of therestriction functor M eM then maps mod eAe/J into mod A/I . Since
A/I isrepresentation-finite, so is eAe/J . Consequently, eAe is mild.Step 5:
For at least one i , the algebra e i Ae i is infinite dimensional, i.e., isomor-phic to k J T K . Otherwise all e j Ae i and therefore A itself are finite dimensional bystep 1.Step 6: Consider the case n = 2 (thus 1 A = e + e ) and assume for definitenessthat e Ae is isomorphic to k J T K (step 5). Then e Ae Ae = 0: otherwise A/Ae A is still isomorphic to k J T K and representation-infinite, contradicting the mildnessof A . Therefore e Ae Ae = ( e Re ) r for some uniquely determined integer r ≥ e Re ) l into ideals of e Ae :(6.2) e A ( e Re ) l Ae and then back into ideals of e Ae :(6.3) e Ae A ( e Re ) l Ae Ae = ( e Re ) r ( e Re ) l ( e Re ) r = ( e Re ) l +2 r . Since the sequence of ideals in (6.3) is strictly decreasing, so is the one in (6.2).In particular, also e Ae is isomorphic to k J T K (step 5) and e Ae Ae = 0. Inter-changing e and e , we have e Ae Ae = ( e Re ) s for some uniquely determinedinteger s ≥
1. We get( e Re ) s = e Ae Ae Ae Ae = e A ( e Re ) r Ae and s = dim( e Re ) s / ( e Re ) s = dim e Ae Ae /e A ( e Re ) r Ae ≥ r. By symmetry we even have r = s . If r = s ≥
2, the quiver of A is ◦ ◦ and already A/R is representation-infinite: contradiction! Thus r = s = 1 and A is isomorphic to k J Z K .Step 7: Since A clearly is connected, one immediately deduces from step 6: forany two distinct idempotents e i and e j , ( e i + e j ) A ( e i + e j ) is isomorphic to k J Z K . Step 8: Consider the case n = 3 (thus 1 A = e + e + e ) and put R i = e i Ae i , M i = e i +1 Ae i , N i = e i − Ae i . In this step the indices are taken modulo 3. By theprevious step we have the relations R i +1 M i = M i R i , R i − N i = N i R i , N i +1 M i = R i , M i − N i = R i . Now M i +1 M i = 0, since N i +2 M i +1 M i = R i +1 M i = 0, for instance, and therefore M i +1 M i = N i R r ( i ) i = R r ( i ) i − N i for some uniquely determined integer r ( i ) ≥
0. Calculating in two ways: M i +2 ( M i +1 M i ) = M i +2 N i R r ( i ) i = R r ( i )+1 i , ( M i +2 M i +1 ) M i = R r ( i +1) i N i +1 M i = R r ( i +1)+1 i , we see that r ( i ) is independent of i ; denote this integer again by r . Similarly, wehave N i − N i = M i R si = R si +1 M i for all i and some uniquely determined integer s ≥
0. Calculating again in twoways: N i +2 ( M i +1 M i ) = N i +2 N i R ri = M i R si R ri = M i R r + si , ( N i +2 M i +1 ) M i = R i +1 M i = M i R i , we see that r + s = 1 and hence that ( r, s ) equals (1 ,
0) or (0 , A is isomorphic to k J Z K .Step 9: In the general case, taking into account steps 3 and 4 and applyingsteps 6 and 8, one immediately sees that the quiver of A is a cycle. Since A isinfinite dimensional, there cannot be any relation. (cid:3) References [1] I. Kr. Amdal and F. Ringdal, Cat´egories unis´erielles, C. R. Acad. Sci. Paris S´er. A-B (1968), A85–A87 and A247–A249.[2] M. Auslander, Coherent functors, in
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E-mail address : [email protected] Dieter Vossieck, Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, D-33501 Biele-feld, Germany.
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