Dominant dimension and tilting modules
aa r X i v : . [ m a t h . R T ] O c t DOMINANT DIMENSION AND TILTING MODULES
VAN C. NGUYEN, IDUN REITEN, GORDANA TODOROV, AND SHIJIE ZHU
Abstract.
We study which algebras have tilting modules that are both generated and co-generated by projective-injective modules. Crawley-Boevey and Sauter have shown that Aus-lander algebras have such tilting modules; and for algebras of global dimension 2, Auslanderalgebras are classified by the existence of such tilting modules.In this paper, we show that the existence of such a tilting module is equivalent to thealgebra having dominant dimension at least 2, independent of its global dimension. In generalsuch a tilting module is not necessarily cotilting. Here, we show that the algebras which havea tilting-cotilting module generated-cogenerated by projective-injective modules are precisely1-Auslander-Gorenstein algebras.When considering such a tilting module, without the assumption that it is cotilting, westudy the global dimension of its endomorphism algebra, and discuss a connection with theFinitistic Dimension Conjecture. Furthermore, as special cases, we show that triangular matrixalgebras obtained from Auslander algebras and certain injective modules, have such a tiltingmodule. We also give a description of which Nakayama algebras have such a tilting module.
Contents
Introduction 21. Projective-injectives and the subcategory C Λ C Λ C Λ
52. Dominant dimension and tilting modules (or cotilting modules) 62.1. Numerical condition 62.2. Maps from X to projective non-injective modules 72.3. Existence of tilting modules in C Λ in terms of dominant dimension 82.4. Existence of cotilting modules in C Λ C Λ C Λ for Nakayama algebras 225.1. Nakayama algebras 225.2. Cyclic-Nakayama algebras with dominant dimension at least 2 225.3. Linear-Nakayama algebras with dominant dimension at least 2 235.4. The tilting module T C for Nakayama algebras 24References 24 Date : October 4, 2017.2010
Mathematics Subject Classification.
Key words and phrases. tilting modules, dominant dimension, Auslander algebras, Nakayama algebras.
Introduction
Let Λ be an artin algebra and let mod Λ be the category of finitely generated left Λ-modules. Throughout the paper, gldim Λ denotes the global dimension of Λ, domdim Λ denotesits dominant dimension (c.f. Definition 2.3.1), and Gdim Λ denotes its Gorenstein dimension(c.f. Remark 2.4.6). For any Λ-module M , projdim M denotes its projective dimension andinjdim M denotes its injective dimension.Let e Q be the direct sum of representatives of the isomorphism classes of all indecomposableprojective-injective Λ-modules. Let C Λ := (Gen e Q ) ∩ (Cogen e Q ) be the full subcategory of mod Λconsisting of all modules generated and cogenerated by e Q . When gldim Λ = 2, Crawley-Boeveyand Sauter showed in [10, Lemma 1.1] that the algebra Λ is an Auslander algebra if and only ifthere exists a tilting Λ-module T C in C Λ . In fact, T C is the direct sum of representatives of theisomorphism classes of indecomposable modules in C Λ . Furthermore T C is the unique tiltingmodule in C Λ and it is also a cotilting module.There is another characterization of Auslander algebras as algebras Λ such that gldim Λ ≤ ≥
2. From the above result in [10], it follows that in global dimension 2, theexistence of a tilting module in C Λ is equivalent to domdim Λ ≥
2. In this paper, we show thatthe existence of such a tilting module is equivalent to domdim Λ ≥ Theorem 1.
Let Λ be an artin algebra. Let e Q be the projective-injective Λ -module as above. (1) The following statements are equivalent: (a) domdim Λ ≥ , (b) C Λ contains a tilting Λ -module T C , (c) C Λ contains a cotilting Λ -module C C . (2) If a tilting module T C exists, then T C ≃ e Q ⊕ (cid:0)L i Ω − P i (cid:1) , where Ω − P i is the cosyzygyof P i and the direct sum is taken over representatives of the isomorphism classes of allindecomposable projective non-injective Λ -modules P i . (3) If a cotilting module C C exists, then C C ≃ e Q ⊕ ( L i Ω I i ) , where Ω I i is the syzygy of I i and the direct sum is taken over representatives of the isomorphism classes of allindecomposable injective non-projective Λ -modules I i . Dominant dimensions of algebras under derived equivalences induced by tilting moduleswere studied by Chen and Xi; in particular they looked at a special class of the so-called canonical tilting modules [9, p.385] (or canonical k -tilting modules to specify the projectivedimension being k , c.f. Remark 1.2.3). Recently, the same tilting modules, also called k -shifted modules , are studied by Pressland and Sauter in [23]. They show that the existence ofa k -shifted module is equivalent to the dominant dimension of the algebra being at least k .We remark that our tilting module T C in C Λ is a canonical 1-tilting module. However whendomdim Λ ≤
1, a canonical 1-tilting module never belongs to C Λ .In this paper, we concentrate on the existence and properties of the classical tilting module T C in the subcategory C Λ ; in addition to its description, we also consider classes of algebras Λwhich have such tilting modules in C Λ .Theorem 1 is proved and discussed in detail in Sections 2.3 and 2.4. As a generalizationof [10, Lemma 1.1], we describe Auslander algebras as algebras Λ with finite global dimensionsuch that there exists a tilting-cotilting module in C Λ (see Corollary 2.4.13). More generally,we characterize a larger class, of 1-Auslander-Gorenstein algebras (c.f. Definition 2.4.10) as: Theorem 2.
Let Λ be an artin algebra. Then the subcategory C Λ contains a tilting-cotiltingmodule if and only if Λ is a -Auslander-Gorenstein algebra. OMINANT DIMENSION AND TILTING MODULES 3
The (sub)structures of classes of such algebras with their homological properties are de-scribed in the following diagram (see Definition 2.4.10 and Remark 2.4.11 for some definitions):
Existence of tilting-cotilting module in C Λ -Auslander-Gorenstein algebras injdim Λ Λ ≤ ≤ domdim Λ,gldim Λ ∈ { , , ∞} DT r -selfinjective algebras
Gdim Λ = 2 = domdim Λ,gldim Λ ∈ { , ∞} Selfinjective algebras
Gdim Λ = 0, domdim Λ = ∞ ,gldim Λ ∈ { , ∞} Auslander algebras
Gdim Λ = 2 = domdim Λ,gldim Λ = 2(Non-semisimple)
Non-Auslander
DT r -selfinjective algebras
Gdim Λ = 2 = domdim Λ,gldim Λ = ∞ In Section 3.1, we gather further properties of algebras with dominant dimension at least 2.From the results in [20, 21, 25], it follows that such algebras are isomorphic to End Λ ( X ) op forsome algebra Λ and a Λ-module X which is a generator and a cogenerator; we recall what thesealgebras Λ and modules X should look like and also give a precise description of the tiltingmodule T C in terms of Λ and X in Proposition 3.1.6. In Section 3.2, given an artin algebraΛ with gldim Λ = d and a tilting module T C ∈ C Λ (if it exists), we study the endomorphismalgebra B C := End Λ ( T C ) op . We show that d − ≤ gldim B C ≤ d , and gldim B C = d − τ T C ) < d , (see Corollary 3.2.5 and Theorem 3.2.9). Applying this together withthe description of algebras of dominant dimension at least 2 in Section 3.1, we obtain a resultabout the Finitistic Dimension Conjecture for a certain class of artin algebras of representationdimension at most 4 in Corollary 3.3.9.In Section 4, we construct classes of algebras closely related to Auslander algebras whichhave tilting modules in the subcategory C Λ of mod Λ. More precisely we have: Theorem 3.
Let A be an Auslander algebra. Let E be an injective A -module such that End A ( E ) is a semisimple algebra and Hom A ( E, Q ) = 0 , for all projective-injective A -modules Q . Then A [ E ] , the triangular matrix algebra of A and the A - End A ( E ) op -bimodule E , has atilting module in the subcategory C A [ E ] . In Section 5, we use a numerical condition to give a characterization of Nakayama algebrasΛ which have a tilting module in C Λ . This class of algebras has been classified by Fuller in [11,Lemma 4.3]; we give a combinatorial approach using Auslander-Reiten theory: Theorem 4.
Let Λ be a Nakayama algebra with n simple modules. Let c be an admissiblesequence of a given Kupisch series. Let the set P c label all indecomposable projective non-injective Λ -modules, the set Q c label all indecomposable projective-injective Λ -modules, and c j be the length of the indecomposable module P j .Then there exists a tilting module in C Λ if and only if P c ⊆ { j − c j ∈ Z n | j ∈ Q c } . The description of such a tilting module is given in Section 5.4.
Acknowledgement:
We would like to thank Rene Marczinzik, Matthew Pressland, andJulia Sauter for helpful conversations and remarks, especially Matthew for pointing out a
VAN C. NGUYEN, IDUN REITEN, GORDANA TODOROV, AND SHIJIE ZHU mistake in the original proof of Theorem 3.2.9. The third author would like to thank NTNUfor their hospitality during her several visits while working on this project. This work wasdone when the first author was a Zelevinsky Research Instructor at Northeastern University,she thanks the Mathematics Department for their support.1.
Projective-injectives and the subcategory C Λ Let Λ be an artin algebra and let mod Λ be the category of finitely generated left Λ-modules.
Definition 1.0.1.
A Λ-module X is called a generator if for any Λ-module M , there isan epimorphism X m → M , for some m . A Λ-module X is called a cogenerator if for anyΛ-module M , there is a monomorphism M → X m , for some m .We denote by Gen( X ), respectively Cogen( X ), the full subcategories of mod Λ consistingof modules generated by X , respectively cogenerated by X . Notice that X is a generator-cogenerator if and only if each indecomposable projective Λ-module and indecomposableinjective Λ-module is isomorphic to a direct summand of X . Definition 1.0.2.
Let e Q := L ti =1 Q i , where the Q i are representatives of the isomorphismclasses of all indecomposable projective-injective Λ-modules. Let C Λ := (Gen e Q ) ∩ (Cogen e Q )be the full subcategory of mod Λ consisting of all modules generated and cogenerated by e Q .We are going to investigate when there exists a tilting module in C Λ .1.1. General properties of the subcategory C Λ . We now describe some basic homologicalproperties of the modules in the subcategory C Λ for artin algebras Λ. Proposition 1.1.1.
Let Λ be an artin algebra with gldim Λ = d . Let C Λ = (Gen e Q ) ∩ (Cogen e Q ) ,where e Q is the above projective-injective Λ -module. Let X be any module in C Λ . Then projdim X ≤ d − and injdim X ≤ d − .Proof. Since X is in C Λ , there exist short exact sequences:0 → N → Q → X → → X → Q ′ → L → , with Q and Q ′ projective-injective Λ-modules. Then there are induced long exact sequences: · · · → Ext d Λ ( , N ) → Ext d Λ ( , Q ) → Ext d Λ ( , X ) → Ext d +1Λ ( , N ) → · · · , and · · · → Ext d Λ ( L, ) → Ext d Λ ( Q ′ , ) → Ext d Λ ( X, ) → Ext d +1Λ ( L, ) → · · · , which show that Ext d Λ ( , X ) = 0 and Ext d Λ ( X, ) = 0, since Q is injective, Q ′ is projective andgldim Λ = d . Also Ext j Λ ( , X ) = 0 and Ext j Λ ( X, ) = 0, for all j ≥ d . Hence, projdim X ≤ d − X ≤ d − (cid:3) As an Auslander algebra A has gldim A ≤
2, we obtain the following consequence:
Corollary 1.1.2.
Let A be an Auslander algebra. Let X be in C A . Then projdim X ≤ and injdim X ≤ . Proposition 1.1.3.
Let Λ be an artin algebra. Then: (1) If P is projective and P is in C Λ , then P is projective-injective.If I is injective and I is in C Λ , then I is projective-injective. (2) Let X be in C Λ . Then the projective cover P ( X ) of X is injective, and the injectiveenvelope I ( X ) of X is projective. Hence, P ( X ) and I ( X ) are in C Λ . OMINANT DIMENSION AND TILTING MODULES 5
Proof. (1) Let P be projective in C Λ . Then P is a quotient of a projective-injective module Q .Since P is projective, it is a summand of Q , and therefore it is injective as well. Dually theinjective module I ∈ C Λ is also projective.(2) Since X is in C Λ , there is a projective-injective module Q which maps onto X . Thus,the projective cover P ( X ) is a direct summand of Q and so it is injective. Similarly, I ( X ) isprojective by a dual argument. (cid:3) Lemma 1.1.4.
Let Λ be an artin algebra. Let X be in C Λ . Let Y be a Λ -module with projdim Y = 1 . Then Ext ( Y, X ) = 0 .Proof.
Let 0 → K → P → X → P the projective cover of X .Consider the induced exact sequence · · · → Ext ( Y, P ) → Ext ( Y, X ) → Ext ( Y, K ) · · · → . Here Ext ( Y, P ) = 0 since P is injective, and Ext ( Y, K ) = 0 since projdim Y = 1. Therefore,we have Ext ( Y, X ) = 0 as claimed. (cid:3)
Tilting modules in C Λ . Usually, there will be only partial tilting modules in C Λ , andin general there could be no tilting module in C Λ . In this subsection, we show some of theproperties of a tilting module in C Λ , if it exists. We recall here the definition of tilting andcotilting modules, since both will be studied extensively in this paper: Definition 1.2.1.
Let Λ be an artin algebra. A basic Λ-module T is called partial tilting ifit satisfies conditions (1) and (2). It is called tilting module if it satisfies (1), (2), and (3).(1) projdim Λ T ≤ ( T, T ) = 0(3) There is an exact sequence 0 → Λ → T → T →
0, where T , T ∈ add T .A Λ-module C is called cotilting if it satisfies conditions (1 o ), (2 o ) and (3 o ).(1 o ) injdim Λ C ≤ o ) Ext ( C, C ) = 0(3 o ) There is an exact sequence 0 → C → C → D Λ →
0, where C , C ∈ add C . Remark 1.2.2 ([1], Corollary VI. 4.4) . Let n be the number of non-isomorphic simple Λ-modules. Let T be a partial tilting module. Then the condition (3) is equivalent to:(3 ′ ) The number of non-isomorphic indecomposable summands of T is n . Remark 1.2.3.
To avoid confusion, we clarify the use of terminology “tilting module” here. • In our definition, “tilting” means the classical tilting module as in Definition 1.2.1,with projdim Λ T ≤
1. In particular, we denote a classical tilting module by T C if it liesin C Λ . We will prove later that T C is unique, if it exists. • In the literature, some authors use the terminology “tilting modules” for generalizedtilting modules (e.g. Happel [14]): (1) projdim
T < ∞ , (2) Ext i ( T, T ) = 0, for all i > → Λ → T → T → · · · → T m → m > T i ∈ add T for all 0 ≤ i ≤ m . A generalized tilting module T with projdim T = k is also called k -tilting module in [23, Definition 2.3]. • For an algebra Λ with dominant dimension at least k , Chen and Xi defined in [9] the canonical k -tilting module as follows: Let e Q be the direct sum of representatives of theisomorphism classes of all indecomposable projective-injective Λ-modules, and0 → Λ d → I d → I d → I → · · · VAN C. NGUYEN, IDUN REITEN, GORDANA TODOROV, AND SHIJIE ZHU be a minimal injective resolution of Λ. Then the module T ( k ) := e Q ⊕ Im d k is a basic k -tilting module and it is called the canonical k -tilting module . A canonical k -cotiltingmodule is defined dually. Lemma 1.2.4.
Let Λ be an artin algebra. Let n be the number of non-isomorphic simple Λ -modules. Let { X i } i ∈ I be a set of indecomposable modules such that X i ≇ X j for all i = j and Ext ( X i , X j ) = 0 for all i, j . Assume that projdim X i = 1 for all i ∈ I . Then the set I isfinite and has at most n elements.Proof. Let X , . . . , X s be any s modules in this set. Then L si =1 X i is a partial tilting module.Every partial tilting module can be completed to a tilting module (see [8]). A tilting modulehas n non-isomorphic indecomposable summands. Therefore, s ≤ n . So there are at most n modules in the set { X i } i ∈ I . (cid:3) Proposition 1.2.5.
Let Λ be an artin algebra. Let e Q be the projective-injective module definedabove and let C Λ = (Gen e Q ) ∩ (Cogen e Q ) . Let { X i } i ∈ I be the set of representatives of theindecomposable modules in C Λ such that projdim X i = 1 . Then: (1) The set { X i } i ∈ I is finite, that is, I = { , , . . . , s } for some s < ∞ . (2) Let X = L si =1 X i . Then e Q ⊕ X is a partial tilting module. (3) If there is a tilting module T C in C Λ , then T C = e Q ⊕ X . (4) If there is a tilting module T C in C Λ , then T C is unique.Proof. (1) It follows from Lemma 1.1.4 that Ext ( X i , X j ) = 0, for all i = j . Since projdim X i =1, it follows from Lemma 1.2.4 that there are at most n modules X i , where n is the number ofnon-isomorphic simple Λ-modules.(2) Follows from the definition of partial tilting module.(3) This follows since all other modules in C Λ have projective dimension ≥ T C = e Q ⊕ X , hence it is unique. (cid:3) The following proposition is about the add T C -approximations of projective modules. Proposition 1.2.6.
Let Λ be an artin algebra and P be a projective Λ -module. Suppose thereexists a tilting module T C in C Λ . Let f P : P → T P be a minimal left add T C -approximation of P . Then T P is projective-injective.Proof. Let f P : P → T P be a minimal add T C -approximation of P . Then T P = Q P ⊕ M P ,where Q P is projective-injective and projdim M P = 1 and f P = ( s, ρ ) : P → Q P ⊕ M P . Let σ : Q ′ P → M P be the projective cover of M P ; here Q ′ P is projective-injective by Proposition1.1.3(2). Then ρ factors through σ , i.e. ρ = σa for some a : P → Q ′ P . It is easy to check that( s, a ) : P → Q P ⊕ Q ′ P is an add T C -approximation. Hence T P is a direct summand of Q P ⊕ Q ′ P and therefore it is projective-injective. (cid:3) Dominant dimension and tilting modules (or cotilting modules)
In this section we show that the existence of a tilting module (or a cotilting module) in thesubcategory C Λ is equivalent to the dominant dimension of Λ being at least 2.2.1. Numerical condition.
We now state a numerical condition which will be necessary andsufficient for the existence of a tilting module in C Λ .Let Q := add e Q be the subcategory of C Λ consisting of projective-injective modules where e Q = ⊕ ti =1 Q i as in Definition 1 . .
2. Let X := add X be the subcategory of C Λ consisting ofmodules with projective dimension 1, where X = ⊕ si =1 X i as in Proposition 1 . .
5. We denote
OMINANT DIMENSION AND TILTING MODULES 7 by n Q the number of non-isomorphic indecomposable modules in Q and by n X the number ofnon-isomorphic indecomposable modules in X . Hence, n Q = t and n X = s . Remark 2.1.1.
Let n be the number of non-isomorphic simple Λ-modules. Since by Propo-sition 1.2.5(2), e Q ⊕ X is a partial tilting module, it follows that n Q + n X ≤ n .Combining this remark and Proposition 1.2.5 we obtain the following important numericalcondition for the existence of a tilting module in C Λ . Corollary 2.1.2.
Let Λ be an artin algebra with n non-isomorphic simple modules. Let Q and X be the above subcategories. Then there is a tilting module in C Λ if and only if n Q + n X = n .Proof. If there is a tilting module T C in C Λ then it has n Q summands from Q and n X summandsfrom X by Proposition 1.2.5(3). (cid:3) Maps from X to projective non-injective modules. In this part we define a mapping Ω : ind X → ind P , where P is the subcategory of projective non-injective Λ-modules. Thismapping will be a bijection exactly when there is a tilting module in C Λ , which will be shownin Corollary 2.2.8. This will be used in a very essential way in the proof of the main Theorem2.3.4. We need some preparation: Lemma 2.2.1. [3, II, Lemma 4.3]
Let → A g → B f → C → be a non-split exact sequence inan additive category C . Then: (1) If End C ( A ) is local, then f : B → C is right minimal in C . (2) If End C ( C ) is local, then g : A → B is left minimal in C . Lemma 2.2.2.
Let → Y g → Q f → X → be a non-split exact sequence. (1) Suppose Y is indecomposable, g is left minimal and Q is projective. Then X is inde-composable and f is right minimal. (2) Suppose X is indecomposable, f is right minimal and Q is injective. Then Y is inde-composable and g left minimal.Proof. (1) By Lemma 2 . . Y being indecomposable implies that f is right minimal. Hence f is a projective cover of X . To show that X is indecomposable, suppose X = X ⊕ X where X and X are both non-zero. Consider the projective covers Q and Q of X and X respectivelyand the associated exact sequences:0 → Y → Q → X → , → Y → Q → X → . Then Q ⊕ Q is the projective cover of X ⊕ X ∼ = X . Because the projective cover of X isunique up to isomorphism it follows that Q ≃ Q ⊕ Q . Therefore we have Y ≃ Y ⊕ Y . Since Y is indecomposable, either Y = 0 or Y = 0. If Y = 0, then X ≃ Q = 0, which contradictsthe fact that g is left minimal. A similar contradiction is drawn if we assume Y = 0. Therefore, X is indecomposable.(2) This is the dual statement of (1) . (cid:3) Corollary 2.2.3.
Let → Y g → Q f → X → be a non-split exact sequence, where Q is aprojective-injective module. Then the following statements are equivalent: (1) X is indecomposable and f is a projective cover, (2) Y is indecomposable and g is an injective envelope. Applying Corollary 2.2.3 recursively, we have the following result:
VAN C. NGUYEN, IDUN REITEN, GORDANA TODOROV, AND SHIJIE ZHU
Corollary 2.2.4.
Suppose → X → I d → I d → · · · is a minimal injective resolution of anindecomposable module X . If I , I , · · · , I k are also projective, then Im d i are indecomposablefor all ≤ i ≤ k . Lemma 2.2.5.
Let X be an indecomposable module in X and let → P i → Q p → X → be the minimal projective resolution of X . Then the following statements hold: (1) The module Q is projective-injective. (2) The syzygy Ω X = P is indecomposable, projective and non-injective. (3) The map i : P → Q is an injective envelope of P .Proof. (1) By Proposition 1 . . Q is also injective.(2) It is clear that P is projective non-injective. By Corollary 2.2.3, P is indecomposable.(3) The fact that the map i is an injective envelope also follows from Corollary 2.2.3. (cid:3) Definition 2.2.6.
Let P be the subcategory of projective non-injective modules in mod Λ.Denote by [ M ] the isomorphism class of a Λ-module M . Then by Lemma 2 . .
5, we know that Ω ([ X ]) := [Ω X ] = [ P ] defines a set-theoretic map: Ω : ind X → ind P .Now we show the main Lemma: Lemma 2.2.7.
Let Λ be an artin algebra. Let n be the number of non-isomorphic simple Λ -modules. Then (1) Ω : ind X → ind P is an injection of sets, (2) n Q + n X ≤ n , (3) n Q + n X = n if and only if Ω is a bijection.Proof. (1) Injectivity of Ω : Suppose X ≇ X in ind X . We will show that Ω ([ X ]) = Ω ([ X ]).In fact, taking the minimal projective resolution of X and X , we get0 → P → Q → X → , → P → Q → X → . Assume P ∼ = P . By Corollary 2 . . Q and Q are injective envelopes of P and P respec-tively. Hence Q ∼ = Q , and then X ∼ = X which is a contradiction.(2) It follows from (1) that n X ≤ n P . Therefore n Q + n X ≤ n Q + n P = n. Then (3) is clear. (cid:3)
Corollary 2.2.8.
Let Λ be an artin algebra with n simple modules. Then there is a tiltingmodule T C in C Λ if and only if Ω is a bijection. Corollary 2.2.9.
If a tilting module T C exists, then T C ≃ e Q ⊕ (cid:0)L i Ω − P i (cid:1) , where Ω − P i isthe cosyzygy of P i and the direct sum is taken over representatives of the isomorphism classesof all indecomposable projective non-injective Λ -modules P i . Existence of tilting modules in C Λ in terms of dominant dimension. We nowprove the main theorem. Recall that the dominant dimension of a (left) Λ-module M isdefined as follows. Definition 2.3.1.
Let 0 → Λ M → I → I → · · · → I m → · · · be a minimal injectiveresolution of M . Then domdim Λ M = sup { k | I i is projective, for all 0 ≤ i < k } . The leftdominant dimension of the algebra Λ is defined to be domdim Λ Λ. OMINANT DIMENSION AND TILTING MODULES 9
Remark 2.3.2.
The dominant dimension of a right module and the right dominant dimensionof the algebra are defined similarly. It is well known that domdim Λ Λ = domdim Λ Λ for anyalgebra Λ (see e.g. [21, Theorem 4]). So for the rest of this paper, we will denote both left andright dominant dimension of Λ by domdim Λ and call it the dominant dimension of Λ. Remark 2.3.3.
Here are some basic properties of dominant dimension:(1) Let Q be a projective-injective module. Then domdim Q = ∞ .(2) domdim Λ = min { domdim Λ P | Λ P is indecomposable projective } .(3) If Λ is selfinjective, then domdim Λ = ∞ . Theorem 2.3.4.
Let Λ be an artin algebra. Then the following statements are equivalent: (1) The subcategory C Λ contains a tilting Λ -module T C , (2) domdim Λ ≥ .Proof. (1) = ⇒ (2). Let P be an indecomposable projective Λ-module.If P is projective-injective then domdim P = ∞ by Remark 2.3.3.If P is projective non-injective we will show that in the minimal injective copresentation of P → P → I → I → I → I → . . . , both I and I are projective-injective. To show this we use the assumption that there isa tilting module T C in C Λ . By property (3) in Definition 1.2.1 of tilting modules, for eachprojective P there is an exact sequence0 → P g −→ T f −→ T → , where T and T are in add T C and the map g is a minimal left add T C -approximation of P . Itfollows by Proposition 1.2.6 that T is projective-injective. Call it Q . Let T i −→ Q be theinjective envelope of T . Since T is in add T C ⊂ C Λ , the injective envelope Q is also projective-injective by Proposition 1.1.3(2). Combining these two statements, we get the minimal injectivecopresentation of P → P g −→ Q if −→ Q , where Q and Q are projective-injective modules. Hence, domdim P ≥
2. By Remark 2.3.3,we have domdim Λ ≥ ⇒ (1). We use the fact that domdim Λ ≥ Ω : ind X → ind P is a bijection. Then apply Corollary 2.2.8 to conclude (1). It follows from Lemma 2.2.7 that Ω is an injective map. To show that it is surjective, we consider P ∈ ind P and find X ∈ ind X sothat Ω X ∼ = P . Let P a −→ Q be the injective envelope of P . The module Q is projective-injectivesince domdim P ≥
2. Consider the induced short exact sequence0 → P a −→ Q b −→ X −→ . Let X c −→ I be the injective envelope of X . We have a minimal injective copresentation of P :0 → P a −→ Q cb −→ I. The assumption that domdim P ≥ I is projective-injective. Therefore X is asubmodule of a projective-injective module. Since X is also a quotient of Q and projdim X = 1,it follows that X is in X . Furthermore, X is indecomposable since P is indecomposable, byCorollary 2.2.3. Therefore X is in ind X and Ω X ∼ = P . Thus Ω is a surjection and therefore abijection. By Corollary 2.2.8, it follows that there is a tilting module T C in C Λ . (cid:3) Using this theorem, we can deduce the following result of Crawley-Boevey and Sauter [10].
Corollary 2.3.5. If gldim Λ = 2 , then C Λ contains a tilting Λ -module if and only if Λ is anAuslander algebra.Proof. An Auslander algebra is an algebra A with gldim A = 2 and domdim A = 2. (cid:3) More directly by Theorem 2 . .
4, for m -Auslander algebras Λ, we can always guarantee theexistence of such a tilting module in C Λ . Recall that Iyama introduced the notion of higherAuslander algebras (see [17, 2.2]): an artin algebra Λ is called m -Auslander if gldim Λ ≤ m + 1 ≤ domdim Λ. It is easy to see that m -Auslander algebras are either semisimple or satisfygldim Λ = domdim Λ. Notice that Auslander algebras are precisely 1-Auslander algebras. Corollary 2.3.6.
For any integer m ≥ and any m -Auslander algebra Λ , its subcategory C Λ always contains a tilting Λ -module. Example 2.3.7.
In this example, we illustrate Corollary 2.3.6 for a 2-Auslander algebra. LetΛ be the Nakayama algebra given by the following quiver and relations αγ = γβ = 0.12 3 α β γ with the AR-quiver The subcategory C Λ is add { , , , } , where there is a tilting Λ-module T C = ⊕ ⊕ .2.4. Existence of cotilting modules in C Λ . Notice that a left Λ-module Λ T is tilting if andonly if D ( T ) Λ as a right Λ-module is cotilting. As a dual statement, we provide a result onthe existence of a cotilting module here. Theorem 2.4.1.
Let Λ be an artin algebra. Then the following statements are equivalent: (1) C Λ contains a cotilting Λ -module, (2) C Λ op contains a tilting Λ op -module, (3) domdim Λ op ≥ . On the other hand, by definition, we have domdim Λ op = domdim Λ Λ which is the same asdomdim Λ as we mentioned before (see Remark 2.3.2). So combining our results, we have: Corollary 2.4.2.
Let Λ be an artin algebra. Then the following statements are equivalent: (1) domdim Λ ≥ , (2) C Λ contains a tilting Λ -module T C , (3) C Λ contains a cotilting Λ -module C C . Remark 2.4.3.
If a cotilting module C C exists, then C C ≃ e Q ⊕ ( L i Ω I i ), where Ω I i is thesyzygy of I i and the direct sum is taken over the representatives of the isomorphism classes ofall indecomposable injective non-projective Λ-modules I i . Remark 2.4.4.
By [23, Proposition 2.6], for k ≥
1, the existence of the canonical k -tilting(or k -cotilting) modules is equivalent to the dominant dimension of the algebra being at least k . One can see that the implications (1) = ⇒ (2) and (1) = ⇒ (3) in Corollary 2.4.2 followimmediately from the existence of the canonical 1-tilting (or 1-cotilting) modules. However,our proof of the equivalence of statements (1) , (2) , (3) is done using a different approach. OMINANT DIMENSION AND TILTING MODULES 11
In general, the tilting module and the cotilting module in C Λ from Corollary 2.4.2 are notnecessarily the same module. Now we discuss when C Λ contains a module which is both tiltingand cotilting. We call this module the tilting-cotilting module in C Λ . Definition 2.4.5. [4] An artin algebra Λ is called
Gorenstein if both injdim Λ Λ < ∞ andinjdim Λ Λ < ∞ . Remark 2.4.6.
It is conjectured that for an artin algebra Λ, injdim Λ Λ < ∞ is equivalentto injdim Λ Λ < ∞ (Gorenstein Symmetry Conjecture). But we know that if injdim Λ Λ andinjdim Λ Λ are both finite, then injdim Λ Λ = injdim Λ Λ (e.g. see [27]); in this case, we callthis number Gorenstein dimension , denoted as Gdim Λ := injdim Λ Λ = injdim Λ Λ . Anartin Gorenstein algebra Λ is called Iwanaga-Gorenstein of Gorenstein dimension m , ifGdim Λ = m . To avoid confusion, we point out that in the literature, there is an original notionof m -Gorenstein algebra (e.g. see [5]) which is different from the notion of Iwanaga-Gorensteinalgebra of Gorenstein dimension m .It is well known that selfinjective algebras and algebras of finite global dimensions areGorenstein. For the convenience of the readers, we show: Proposition 2.4.7.
Let Λ be an artin algebra with gldim Λ = d < ∞ . Then there exists anindecomposable projective Λ -module P with injdim P = d .That is, if gldim Λ = d < ∞ then Λ is Iwanaga-Gorenstein with Gdim Λ = d .Proof. Since gldim Λ = d , injdim P ≤ d for all projective Λ-module P . Here, we claim that atleast one P satisfies injdim P = d . Otherwise, since any Λ-module M has a finite projectiveresolution 0 → P d → · · · → P → M →
0, with each injdim P i < d , then injdim M < d andhence gldim Λ < d , a contradiction. So gldim Λ = d implies injdim Λ Λ = d . Similarly, we haveprojdim D (Λ Λ ) = d = injdim Λ Λ . Therefore, Λ is Iwanaga-Gorenstein with Gdim Λ = d . (cid:3) Recently Iyama and Solberg defined m -Auslander-Gorenstein algebra in [18]. They alsoshowed that the notion of m -Auslander-Gorenstein algebra is left and right symmetric. Definition 2.4.8. [18] An artin algebra Λ is called m -Auslander-Gorenstein ifinjdim Λ Λ ≤ m + 1 ≤ domdim Λ . Proposition 2.4.9. [18, Proposition 4.1]
Let Λ be an artin algebra. (1) If Λ is an m -Auslander-Gorenstein algebra, then either injdim Λ Λ = m +1 = domdim Λ holds or Λ is selfinjective. (2) An algebra Λ is m -Auslander-Gorenstein if and only if Λ op is m -Auslander-Gorenstein. Remark 2.4.10.
Let Λ be an artin algebra. We have the following equivalent characterizationsof m -Auslander-Gorenstein algebras:(1) Λ is m -Auslander-Gorenstein,(2) Λ is Iwanaga-Gorenstein with Gdim Λ ≤ m + 1 ≤ domdim Λ,(3) Λ is selfinjective or injdim Λ Λ = injdim Λ Λ = m + 1 = domdim Λ,(4) Λ is selfinjective or injdim Λ Λ = m + 1 = domdim Λ,(5) Λ is selfinjective or injdim Λ Λ = m + 1 = domdim Λ,(6) injdim Λ Λ ≤ m + 1 ≤ domdim Λ,(7) injdim Λ Λ ≤ m + 1 ≤ domdim Λ.In particular, a 1-Auslander-Gorenstein algebra is either a selfinjective algebra or a Goren-stein algebra satisfying injdim Λ Λ = 2 = domdim Λ. The authors called it “minimal m -Auslander-Gorenstein” in the introduction of [18]. Later in the paper,they called it “ m -Auslander-Gorenstein” for simplicity. Remark 2.4.11.
The algebras satisfying the condition injdim Λ Λ = domdim Λ = 2 are called
DT r -selfinjective algebras and they were classified by Auslander and Solberg in [7].We have a characterization of 1-Auslander-Gorenstein algebras in terms of the existence ofthe tilting-cotilting module in C Λ : Theorem 2.4.12.
Let Λ be an artin algebra. Then the following statements are equivalent: (1) Λ is -Auslander-Gorenstein, (2) C Λ contains a tilting-cotilting module.Proof. (1) = ⇒ (2). Assume that (1) holds, then domdim Λ ≥
2. By Corollary 2 . .
2, thesubcategory C Λ contains a tilting module T C . It suffices to show that injdim T C ≤ T be any non-injective indecomposable summand of T C (if it exists). Then byProposition 1 . . T is neither projective nor injective with projdim T = 1. Moreover, T hasa minimal projective resolution: 0 → P → P → T → , where P is projective-injective. Because domdim Λ ≥ T is a submodule of a projective-injective module I . But I /T must be injective since injdim Λ Λ ≤
2. Hence injdim T = 1.Therefore injdim T C ≤ T C is a cotilting module.(2) = ⇒ (1). Assume that (2) holds, then Corollary 2 . . ≥
2. Let P be an indecomposable projective non-injective module (if it exists). Let f : P → I ( P ) bean injective envelope of P . Then we know that X ∼ = Coker f is a non-injective summand ofthe tilting module T C . Since T C is also cotilting, we have that injdim X = 1, which impliesinjdim P = 2. Hence, we show injdim Λ Λ ≤ Λ Λ ≤ ≤ domdim Λ which means that Λ is 1-Auslander-Gorenstein. (cid:3) We have the following statement which generalizes Crawley-Boevey and Sauter’s result [10,Lemma 1.1] from an algebra Λ with gldim Λ = 2 to an algebra Λ with any finite gldim Λ.
Corollary 2.4.13.
Let Λ be an artin algebra with gldim Λ < ∞ . Then the subcategory C Λ contains a tilting-cotilting module if and only if Λ is an Auslander algebra.Proof. Assume C Λ contains a tilting-cotilting module, then Λ is 1-Auslander-Gorenstein byTheorem 2.4.12. It follows from Remark 2.4.10 that Λ is Iwanaga-Gorenstein with Gdim Λ = 2.By Proposition 2.4.7, this forces gldim Λ = 2. Also domdim Λ ≥ . . . .
12 and the factthat an Auslander algebra is 1-Auslander-Gorenstein. (cid:3)
Remark 2.4.14. • Theorem 2.4.12 and Corollary 2 . .
13 suggest that 1-Auslander-Gorenstein algebras aregeneralizations of Auslander algebras in the sense of the existence of a tilting-cotiltingmodule in C Λ . • A more general situation is considered by Pressland and Sauter (c.f. [23, Proposition3.7, Theorem 3.9]). They show that Λ is an m -Auslander-Gorenstein algebra if andonly if canonical k -tilting modules coincide with canonical ( m +1 − k )-cotilting modules,for all 0 ≤ k ≤ m + 1. Example 2.4.15.
In this example, we present a 1-Auslander-Gorenstein algebra which con-tains a tilting-cotilting module in C Λ but is not an Auslander algebra, since its gldim Λ = ∞ .Let Λ be the Nakayama algebra given by the following quiver and relations γβα = δγβ = αǫ = ǫδ = 0. We omit the modules when drawing the AR-quiver. OMINANT DIMENSION AND TILTING MODULES 13 α βγδ ǫ with the AR-quiver • • • • •• • •
There exists a tilting-cotilting module T C = P ⊕ P ⊕ P ⊕ P ⊕ S in C Λ .3. Homological applications of the tilting module in C Λ More on dominant dimension.
Algebras with dominant dimension at least 2 havebeen studied since the 1960’s by Morita [20], Tachikawa [25], Mueller [21], Ringel [24] andmany others. Morita and Tachikawa showed that any artin algebra of dominant dimension atleast 2 is an endomorphism algebra of a generator-cogenerator of another artin algebra. Thisgives us a full machinery for producing algebras whose dominant dimension is at least 2, andhence, algebras which have a tilting module in C Λ . Theorem 3.1.1. [20] , [25] , [21, Theorem 2] For an artin algebra Γ , the following statementsare equivalent: (1) domdim Γ ≥ , (2) Γ ≃ End Λ ( X ) op , where X is a generator-cogenerator of an artin algebra Λ . Furthermore, there is a more precise result on the dominant dimension:
Lemma 3.1.2. [21, Lemma 3]
Let Γ ≃ End Λ ( X ) op , where X is a generator-cogenerator of anartin algebra Λ . Then domdim Γ ≥ m + 2 if and only if Ext i Λ ( X, X ) = 0 , for all ≤ i ≤ m , m ∈ { , , , . . . } . From this lemma, the following well-known results can be deduced.
Corollary 3.1.3.
Let X be a generator-cogenerator of an artin algebra Λ and Γ ≃ End Λ ( X ) op . (1) If X is a summand of a module Y , then domdim End Λ ( X ) op ≥ domdim End Λ ( Y ) op . (2) Suppose Λ is non-selfinjective. If injdim Λ Λ ≤ m or injdim Λ Λ ≤ m , then domdim Γ ≤ m + 1 . (3) If gldim Λ ≤ m , then domdim Γ ≤ m + 1 . (4) If Λ is non-semisimple hereditary, then domdim Γ = 2 . Next, we recall how to construct the algebra Λ and Λ-module X such that Γ ≃ End Λ ( X ) op is of dominant dimension at least 2, under the assumption that both Γ and Λ are basic. Ingeneral, we emphasize that such an algebra Λ is only unique up to Morita equivalence, see also[2, IV], [24] for details. Proposition 3.1.4.
Let X be a module over an artin algebra Λ and Γ := End Λ ( X ) op . Then: (1) Hom Λ ( X, X i ) are all the indecomposable projective Γ -modules, where X i runs throughthe non-isomorphic indecomposable direct summands of X . (2) If X is a cogenerator of Λ , then the Hom Λ ( X, I i ) are all the indecomposable projective-injective Γ -modules, where I i runs through the non-isomorphic indecomposable injective Λ -modules. Proposition 3.1.5.
Let Γ be a basic algebra of dominant dimension at least . Let e Q bethe direct sum of representatives of the isomorphism classes of all indecomposable projective-injective Γ -modules. Then the algebra Λ and Λ -module X can be chosen to be Λ = End Γ ( e Q ) op and X = Hom Γ ( e Q, D Γ) so that Γ ≃ End Λ ( X ) op . We now describe the tilting module T C (whose existence is given by Theorem 2.3.4) in termsof the algebra Λ and a generator-cogenerator X . Proposition 3.1.6.
Let Γ be a basic algebra with domdim Γ ≥ . Let T C be the tilting modulein C Γ . Let Γ ≃ End Λ ( X ) op , for an artin algebra Λ and a Λ -generator-cogenerator X . Then: (1) T C ≃ ( L i Hom Λ ( X, I i )) ⊕ (cid:16)L j Hom Λ ( X, I ( X j ) /X j ) (cid:17) , where { I i } are the indecompos-able injective Λ -modules, { X j } are the non-injective indecomposable direct summandsof X , and { I ( X j ) } are the corresponding injective envelopes of { X j } . (2) In part (1), the first summand is a projective-injective Γ -module isomorphic to e Q , andthe second summand is a non-projective-injective Γ -module isomorphic to (cid:0)L i Ω − P i (cid:1) ,as described in Corollary 2.2.9. The endomorphism algebra of the tilting module.
Using the Morita-Tachikawacorrespondence as in Theorem 3.1.1, for any artin algebra Λ, taking a generator-cogenerator X ∈ mod Λ, the algebra Γ = End Λ ( X ) op is an artin algebra of dominant dimension at least 2.So by Theorem 2.3.4, there exists a unique tilting module T C ∈ C Γ . We are going to study theglobal dimension of the endomorphism algebra B C := End Γ ( T C ) op and in the next section, weobtain its relationship with the Finitistic Dimension Conjecture.We recall some facts about tilting modules and torsion classes due to Brenner and Butler(see [1, VI, § §
4] for more details and proofs). Let Λ be an artin algebra and T be any tiltingΛ-module. Then there is a torsion pair ( T ( T ) , F ( T )) in mod Λ: T ( T ) := Gen Λ ( T ) = { M ∈ mod Λ | Ext ( T, M ) = 0 } and ( ∗ ) F ( T ) := Sub Λ ( τ T ) = { M ∈ mod Λ | Hom Λ ( T, M ) = 0 } . ( ∗∗ )Let B := End Λ ( T ) op , then T is also a tilting B op -module. We have a torsion pair ( X ( T ) , Y ( T )): X ( T ) := D Gen B op ( T ) = { M ∈ mod B | Hom B ( M, DT ) = 0 } and Y ( T ) := D Sub B op ( τ T ) = { M ∈ mod B | Ext B ( M, DT ) = 0 } . In all four descriptions above, the first equalities may be considered as definitions and thesecond equalities are consequences of the results stated in [1, VI, § § Theorem 3.2.1 (Brenner-Butler Tilting Theorem) . [1, VI, § a ) The functors
Hom Λ ( T, − ) and − ⊗ B T induce quasi-inverse equivalences: T ( T ) Hom Λ ( T, − ) −−−−−−−−→←−−−−−−−−− ⊗ B T Y ( T ) . ( b ) The functors
Ext ( T, − ) and Tor B ( − , T ) induce quasi-inverse equivalences: F ( T ) Ext ( T, − ) −−−−−−−−−→←−−−−−−−−− Tor B ( − ,T ) X ( T ) . We will first state and prove several general lemmas about tilting modules.
Lemma 3.2.2.
Let Λ be an artin algebra, T be any tilting Λ -module, and B := End Λ ( T ) op . (1) Let U be a B -module. Then the first syzygy Ω U of U is in Y ( T ) . (2) There exists a Λ -module M ∈ T ( T ) such that Ω U ∼ = Hom Λ ( T, M ) .Proof. Let U be a B -module and Hom Λ ( T, T ) → U be its projective cover. In the exactsequence 0 → Ω U → Hom Λ ( T, T ) → U →
0, the middle term Hom Λ ( T, T ) is in Y ( T ) byTheorem 3.2.1(a), and Y ( T ) is a torsion-free class so it is closed under submodules. Therefore,the first syzygy Ω U of U is in Y ( T ). So Ω U ∼ = Hom Λ ( T, M ), for some M ∈ T ( T ) = Gen( T ). (cid:3) OMINANT DIMENSION AND TILTING MODULES 15
Lemma 3.2.3.
Let Λ be an artin algebra, T be any tilting Λ -module, and B := End Λ ( T ) op . (1) Let M be a Λ -module where M ∈ T ( T ) . Then projdim B (Hom Λ ( T, M )) ≤ projdim Λ M . (2) Let U be a non-projective B -module and M be a Λ -module such that Ω U ∼ = Hom Λ ( T, M ) .Then projdim B U ≤ projdim Λ M + 1 . (3) gldim B ≤ gldim Λ + 1 . (4) gldim Λ ≤ gldim B + 1 . (5) | gldim B − gldim Λ | ≤ .Proof. (1) follows from [1, VI, Lemma 4.1].(2) projdim B U ≤ projdim B Ω U + 1 = projdim B (Hom Λ ( T, M )) + 1 ≤ projdim Λ M + 1.(3) This is a consequence of (2).(4) Since B T is a tilting B -module and Λ ∼ = End B ( T ) op , it follows from (3) that gldim Λ ≤ gldim B + 1. Finally, (5) is just a combination of (3) and (4). (cid:3) Now for our unique tilting module T C in C Γ , we have more precise results: 3.2.4 and 3.2.5. Lemma 3.2.4.
Let Γ be an artin algebra with domdim Γ ≥ , T C be the unique tilting modulein C Γ , and B C = End Γ ( T C ) op . If M ∈ Gen( T C ) and projdim Γ M ≥ , then projdim B C (Hom Γ ( T C , M )) = (projdim Γ M ) − . Proof.
We use induction on projdim Γ M . First, assume projdim Γ M = 1. Since M ∈ Gen( T C ) =Gen( e Q ), we know there is some Q ∈ add e Q such that Q → M is an epimorphism. Sinceprojdim Γ M = 1, M has a projective resolution:0 → P → Q → M → . Since P ∈ add Γ and domdim Γ ≥
2, the module M is a submodule of a projective-injectiveΓ-module Q . Therefore M is in C Γ . Additionally, as projdim Γ M = 1 by assumption, itfollows from Proposition 1.2.5 that M ∈ add T C . Hence projdim B C (Hom Γ ( T C , M )) = 0 . Now assume projdim Γ M = d >
1. In particular,
M / ∈ add T C . There is an exact sequence:0 → L → T f → M → , where f is a right add T C -approximation and L = 0. This induces an exact sequence of B C -modules: 0 → Hom Γ ( T C , L ) → Hom Γ ( T C , T ) → Hom Γ ( T C , M ) → . It follows that, Ext ( T C , L ) = 0, which implies L ∈ Gen( T C ) by (*).Since projdim Γ T ≤ Γ M = d >
1, it follows that projdim Γ L ≤ d − B C (Hom Γ ( T C , L )) = d −
2. Therefore,due to the above exact sequence, projdim B C (Hom Γ ( T C , M )) = d − (cid:3) Corollary 3.2.5.
Let Γ be an artin algebra with domdim Γ ≥ , T C be the unique tilting modulein C Γ , and B C = End Γ ( T C ) op . Then gldim B C ≤ gldim Γ .Proof. Let U be a B C -module. Then by Lemma 3.2.2(2) and Lemma 3.2.4, it follows thatprojdim B C U ≤ projdim B C (Hom Γ ( T C , M )) + 1 = (projdim Γ M −
1) + 1 ≤ gldim Γ . (cid:3) Remark 3.2.6.
Let Γ, T C , and B C be as in Corollary 3.2.5.(1) The sharp inequality gldim B C < gldim Γ does not always hold, see the Example 3 . . B C = gldim Γ or gldim B C = gldim Γ − Example 3.2.7.
This is an example of Γ such that gldim B C = gldim Γ. Let Γ be theNakayama algebra given by the following quiver and relations γβα = δγβ = αǫ = 0. We omit the modules when drawing the AR-quiver.1 5432 α βγδ ǫ with the AR-quiver • • • • •• • • •• The subcategory C Γ contains a tilting module T C = P ⊕ P ⊕ P ⊕ S ⊕ . One can check thatdomdim Γ = 2, gldim Γ = 4, and gldim End Γ ( T C ) = 4.In the next discussion, we are going to show exactly when it holds that gldim B C < gldim Γ.The proof relies on the following easy fact about homological dimensions. Lemma 3.2.8.
Let Λ be an artin algebra with gldim Λ = d . If N is a Λ -submodule of M with projdim N = d , then projdim M = d . Theorem 3.2.9.
Let Γ be an artin algebra with domdim Γ ≥ , T C be the unique tilting modulein C Γ , and B C = End Γ ( T C ) op . Then gldim B C < gldim Γ if and only if projdim Γ ( τ T C ) < gldim Γ . Proof.
Let gldim Γ = d . Then:“ ⇐ =:” Suppose projdim Γ ( τ T C ) < d . We prove by contradiction: assume gldim B C = d . So,there is a B C -module U with projdim B C U = d . We have an exact sequence of B C -modules:0 → Ω U → Hom Γ ( T C , T ) → U → . Then the first syzygy Ω U ∼ = Hom Γ ( T C , M ), for some M ∈ Gen( T C ) by Lemma 3.2.2. Thus,projdim B C (Hom Γ ( T C , M )) = d − Γ M = d due to Lemma 3 . . B C (Hom Γ ( T C , M ) , Hom Γ ( T C , T )) ∼ = Hom Γ ( M, T ), the embedding Hom Γ ( T C , M ) → Hom Γ ( T C , T ) is induced by a morphism f : M → T . Since T is in Sub( e Q ) and projdim Γ M = d , the map f is not a monomorphism. But Hom Γ ( T C , Ker( f )) = 0 since Hom Γ ( T C , M ) → Hom Γ ( T C , T ) is a monomorphism. Hence, Ker( f ) ∈ F ( T C ) = Sub( τ T C ). From Lemma 3 . . Γ ( τ T C ) < d , we know projdim Γ Ker( f ) < d .Consider the exact sequences:0 → Ker( f ) → M → Im( f ) → , and0 → Im( f ) → T → Coker( f ) → . Since projdim Γ Ker( f ) < d and projdim Γ M = d , we have projdim Γ Im( f ) = d . However, byLemma 3 . . Γ T = d ≥
2, whichis a contradiction. Therefore, we must have gldim B C < d .“= ⇒ :” By Proposition 2.4.7, it follows that Gdim Γ = d . Since T C is the direct sum ofthe first cosyzygy of the injective resolution of Γ and e Q , it follows that injdim T C = d − Γ τ T C < d . (cid:3) Example 3.2.10.
This example illustrates Theorem 3 . .
9. Let Q be the quiver1 2 o o o o o o o o and Γ = k Q/ rad ( k Q ). Then gldim Γ = 4 = domdim Γ, projdim Γ ( τ T C ) = 0 and gldim B C = 3.Finally, we give a digression on the condition projdim Γ ( τ T C ) < gldim Γ. OMINANT DIMENSION AND TILTING MODULES 17
Lemma 3.2.11.
Suppose Γ is an artin algebra with gldim Γ = d . Assume the tilting module T C exists in C Γ , then the following statements are equivalent: (1) projdim Γ ( τ T C ) < d , (2) Ext d Γ ( τ T C , M ) = 0 for all Γ -modules M , (3) Ext d Γ ( τ T C , S ) = 0 for all simple Γ -modules S such that injdim S = d , (4) τ − (Σ d − S ) ∈ Gen( e Q ) , for all simple Γ -modules S satisfying injdim S = d , where e Q is an additive generator of projective-injective Γ -modules and Σ d − S is the ( d − -thsyzygy of S in the minimal injective resolution.Proof. (1) ⇐⇒ (2), (2) = ⇒ (3) Trivial.(3) = ⇒ (2) We use induction on the length l ( M ). If l ( M ) = 1, M is simple. If injdim M < d ,it is clear that Ext d Γ ( τ T C , M ) = 0. Otherwise injdim M = d , the assertion follows directly. Nowassume l ( M ) >
1. There is a simple Γ-module S and an exact sequence:0 → M ′ → M → S → . Since l ( M ′ ) < l ( M ) and l ( S ) < l ( M ), then by induction hypothesis Ext d Γ ( τ T C , M ′ ) = 0 andExt d Γ ( τ T C , S ) = 0. Hence Ext d Γ ( τ T C , M ) = 0.(3) ⇐⇒ (4) Notice that Ext d Γ ( τ T C , S ) ≃ Ext ( τ T C , Σ d − S ). Because projdim T C = 1 andinjdim Σ d − S = 1, it follows by [1, IV,2.14] that Ext ( τ T C , Σ d − S ) ≃ Ext ( T C , τ − (Σ d − S )).Hence Ext d Γ ( τ T C , S ) = 0 if and only if Ext ( T C , τ − (Σ d − S )) = 0 if and only if τ − (Σ d − S ) ∈ Gen( T C ) = Gen( e Q ). (cid:3) Remark 3.2.12.
Notice that if a simple Γ-module S satisfies injdim S = gldim Γ < ∞ , thenits projective cover P ( S ) is not injective.Let 0 → S → I ( S ) → I ( S ) → · · · be the minimal injective resolution of a simple module S , ν = D Hom Γ ( − , Γ) be the
Nakayama functor and ν − = Hom Γ ( D Γ , − ) be its quasi-inverse. Proposition 3.2.13.
Suppose Γ is an artin algebra with gldim Γ = d . Assume the tiltingmodule T C exists in C Γ , then projdim Γ ( τ T C ) < d if and only if ν − I d ( S ) is injective, for anysimple Γ -module S with injdim Γ S = d .Proof. By Lemma 3.2.11, projdim( τ T C ) < d if and only if τ − (Σ d − S ) ∈ Gen( e Q ), for all simplemodules S such that injdim S = d .Applying ν − to the following minimal injective resolution0 → Σ d − S → I d − ( S ) → I d ( S ) → , we have an exact sequence:0 → ν − Σ d − S → ν − I d − ( S ) → ν − I d ( S ) → τ − (Σ d − S ) → . Hence the assertion follows from the fact that τ − (Σ d − S ) ∈ Gen( e Q ) if and only if its projectivecover ν − I d ( S ) is injective. (cid:3) Proposition 3.2.14.
Let Λ be an artin algebra and M ≃ L i M i be a generator-cogenerator,where the M i are non-isomorphic indecomposable Λ -modules. Denote Γ = End Λ ( M ) op . Then (1) The complete set of representatives of non-isomorphic indecomposable projective Γ -modules is given by { D Hom Λ ( M i , M ) } . (2) There are Γ -module isomorphisms ν − D Hom Λ ( M i , M ) ≃ Hom Λ ( M, M i ) . (3) The module ν − D Hom Λ ( M i , M ) is injective if and only if M i is an injective Λ -module. A relation to the Finitistic Dimension Conjecture.
In this section, we obtain anapplication of the results in Section 3.2 to the Finitistic Dimension Conjecture for a certainclass of artin algebras of representation dimension at most 4. First, let us recall:
Definition 3.3.1.
Let Λ be an artin algebra. Then the finitistic dimension of
Λ is:findim Λ := sup { projdim M | M ∈ mod Λ and projdim M < ∞} . The representation dimension of
Λ is:repdim Λ := inf { gldim End Λ ( X ) | X is a generator-cogenerator in mod Λ } . Iyama [16] proved that for any artin algebra Λ, its repdim Λ < ∞ always. On the otherhand, the long-standing Finitistic Dimension Conjecture says that for any artin algebra Λ, itsfindim Λ is finite. In [15], Igusa and Todorov proved a partial result of this conjecture. Inparticular, they proved that findim Λ < ∞ provided repdim Λ ≤
3. Their proof relied on thefollowing result using the IT function ψ defined as: Definition 3.3.2.
Let Λ be an artin algebra. Let K be the abelian group generated by [ X ],for all finitely generated Λ-modules X , modulo the relations:(a) [ C ] = [ A ] + [ B ] if C = A ⊕ B and (b) [ P ] = 0 for projective Λ-modules [ P ].Let L : K → K be the group homomorphism defined by L [ X ] = [Ω X ].For any Λ-module M , denote by h add M i the subgroup of K generated by [ M i ] where M i ’sare indecomposable summands of M . Then the IT-functions are: φ ( M ) := min { m | L m h add M i ∼ = L m +1 h add M i} ψ ( M ) := φ ( M ) + sup { projdim X | projdim X < ∞ , X is a summand of Ω φ ( M ) M } . Remark 3.3.3.
For any Λ-module M , the IT-functions φ ( M ) and ψ ( M ) are always finite.When projdim Λ M < ∞ , it is easy to see that φ ( M ) = ψ ( M ) = projdim Λ M . So IT-functionsare generalizations of projective dimension.The next lemma is also a generalization of the well-known result about projective dimen-sions: projdim C ≤ projdim( A ⊕ B ) + 1 when projdim A and projdim B are finite. Lemma 3.3.4.
Suppose that → A → B → C → is a short exact sequence of finitelygenerated Λ -modules and C has finite projective dimension. Then projdim Λ C ≤ ψ ( A ⊕ B ) + 1 . Corollary 3.3.5. [15]
Let Γ be an artin algebra with gldim Γ ≤ . Let Λ = End Γ ( P ) op ,where P is a projective Γ -module. Then findim Λ ≤ ψ (Hom Γ ( P, Γ)) + 3 , where
Hom Γ ( P, Γ) isconsidered as a Λ -module. Motivated by Igusa-Todorov’s result, Wei introduced in [26] the notion of m -IT algebra ,for any non-negative integer m : Definition 3.3.6. [26] Let Λ be an artin algebra. Then Λ is said to be m -IT if there exists amodule V such that for any Λ-module M there is an exact sequence0 → V → V → Ω m M ⊕ P → , where V , V ∈ add V and P is a projective Λ-module.Applying Lemma 3 . .
4, it is easy to see that the finitistic dimension of an m -IT algebra Λis bounded as findim Λ ≤ m + 1 + ψ ( V ). Consequently, the Finitistic Dimension Conjectureholds for m -IT algebras, [26, Theorem 1.1].It was shown that the class of 2-IT algebras is closed under taking endomorphism algebrasof projective modules, [26, Theorem 1.2]. Artin algebras with global dimension d are ( d − OMINANT DIMENSION AND TILTING MODULES 19
If repdim Λ ≤
3, then Λ is 2-IT. However, there exist algebras which are not IT algebras, forexample, the exterior algebra of a 3-dimensional vector space.In the following, we are going to see how the endomorphism algebra B C := End Γ ( T C ) op studied in Section 3.2 relates to 2-IT algebras.To fix the notation, let Λ be a basic artin algebra. Since repdim Λ < ∞ , there existsa (multiplicity-free) generator-cogenerator X ∈ mod Λ such that Γ = End Λ ( X ) op has finiteglobal dimension, say gldim Γ = d . Additionally, Γ has dominant dimension at least 2. Let T C be the unique tilting module in C Γ . Denote by e Q the sum of the representatives of isomor-phism classes of indecomposable projective-injective Γ-modules. We have Λ ∼ = End Γ ( e Q ) op byProposition 3.1.5.From the endomorphism algebra B C := End Γ ( T C ) op , we can also recover the algebra Λ asfollows: Let R := Hom Γ ( T C , e Q ) which is a projective B C -module. Then Lemma 3.3.7.
End B C ( R ) op ∼ = Λ .Proof. End B C ( R ) = Hom B C (Hom Γ ( T C , e Q ) , Hom Γ ( T C , e Q )) ∼ = Hom Γ ( e Q, e Q ) = End Γ ( e Q ); see [1,VI, § ∼ = End Γ ( e Q ) op ∼ = End B C ( R ) op . (cid:3) Combining this fact with Corollary 3 . . Corollary 3.3.8.
Suppose gldim B C ≤ . Then B C is -IT and hence Λ ∼ = End B C ( R ) op is also -IT, and findim Λ ≤ ψ (Hom B C ( R, B C )) + 3 . As a consequence of this result and Theorem 3.2.9, we conclude with the following specialcase of the Finitistic Dimension Conjecture: for an artin algebra Λ, let X be its Auslandergenerator-cogenerator. Let Γ := End Λ ( X ) op , then Γ is of dominant dimension at least 2 andsatisfies gldim Γ = repdim Λ. Let T C be the unique tilting module in C Γ . Then: Corollary 3.3.9. If repdim Λ ≤ and projdim Γ ( τ T C ) ≤ , then findim Λ < ∞ .Proof. With our setting and assumptions, gldim Γ = repdim Λ ≤ Γ ( τ T C ) ≤ B C = End Γ ( T C ) op . By Theorem 3 . .
9, we must have gldim B C ≤
3. It follows fromCorollary 3 . . ≤ ψ (Hom B C ( R, B C )) + 3, which is finite. (cid:3) Example 3.3.10.
We give an example of an algebra of representation dimension 4 whichsatisfies the hypothesis in Corollary 3 . . x / / x > > x (cid:29) (cid:29) x / / x > > x (cid:29) (cid:29) I = h x i x j − x j x i i for all 1 ≤ i, j ≤
3. It was studied and shown by Krause-Kussin, Iyama, and Oppermann that repdim Λ = 4 (see for example [22, Examples 7.3 andA.8]). To check that Λ satisfies the hypothesis in Corollary 3 . .
9, one needs to apply and checkPropositions 3.2.13 and 3.2.14(3).4.
Special class: Extensions of Auslander algebras by injective modules
We now describe a procedure to create algebras Λ which will have tilting modules that aregenerated and cogenerated by projective-injective modules, that is, those tilting modules arein C Λ . In particular, we describe a class of algebras, constructed from Auslander algebras by“extending” the Auslander algebras by certain injective modules. General triangular matrix construction.
We now recall the construction of an alge-bra Λ from algebras R and S and a bimodule S M R as investigated in [12]. We also recall somebasic properties of these algebras, together with the description of modules over such algebras. Definition 4.1.1.
Let R and S be finite dimensional algebras over the field k . Let S M R bean S - R -bimodule. Define an algebra Λ, which we will often denote by Λ = T ( R, S, S M R ):Λ = T ( R, S, S M R ) := (cid:20) R S M R S (cid:21) , where the multiplication is defined using the bimodule structure of S M R .A convenient way of viewing Λ-modules is using the category of triples T in [12]. We nowrecall this definition and some of the basic properties. Definition 4.1.2.
Let Λ = T ( R, S, S M R ). Define the category of triples T as follows:Objects of T are triples: ( R W, S V, f : S M R ⊗ R W → S V ), where R W and S V are left R and S -modules respectively, and f is an S -homomorphism. Morphisms in T are pairs:( α, β ) : ( R W, S V, f : S M R ⊗ R W → S V ) → ( R W ′ , S V ′ , f ′ : S M R ⊗ R W ′ → S V ′ ), where α : R W → R W ′ is an R -homomorphism and β : S V → S V ′ is an S -homomorphism makingappropriate diagrams commute. Remark 4.1.3.
Let Λ = T ( R, S, S M R ) and let T be the associated category of triples. Thenthe categories mod Λ and T are equivalent. Using this fact we refer to triples as Λ-modules. Proposition 4.1.4. [6, Proposition III.2.5] , [12] Let
Λ = T ( R, S, S M R ) and let T be theassociated category of triples. (1) Indecomposable projective objects in T are (0 , S P, and ( R Q, S M R ⊗ R Q, Id S M R ⊗ R Q ) ,where S P and R Q are indecomposable projective S -modules and R -modules respectively. (2) Indecomposable injective objects in T are ( R J, , and (Hom S ( M, I ) , S I, η : S M R ⊗ Hom S ( M, I ) ≃ −→ S I ) , where R J and S I are indecom-posable injective R -modules and S -modules respectively and η ( m ⊗ f ) := f ( m ) is an S -isomorphism. The above proposition has a description of all indecomposable projective and injectiveobjects in T and hence, using equivalence, projective and injective Λ-modules. In addition tothis, we will also be using the following functor which relates categories of S and Λ-modules. Proposition 4.1.5.
Let
Λ = T ( R, S, S M R ) . Then Ψ( S X ) := (0 , S X, defines a functor Ψ : mod S → mod Λ , which has the following properties: (1) Ψ is fully-faithful. (2) Ψ preserves kernels. (3) Ψ preserves projective resolutions. Triangular matrix construction from Auslander algebras.
In this section, we willlook at the triangular matrix where S = A is an Auslander algebra, A E is a special injective A -module and R = End A ( E ) op , then A E R is an A - R -bimodule. For the simplicity of notationwe will denote the algebra T ( R, A, A E R ) by A [ E ]. Definition 4.2.1.
Let A be an Auslander algebra and let e Q = L ti =1 Q i , where { Q , . . . , Q t } is a set of representatives of isomorphism classes of all indecomposable projective-injective A -modules. We choose an A -module E which satisfies the following conditions:(1) E = I ⊕ · · · ⊕ I r , where the I i are indecomposable injective A -modules for all i ,(2) End A ( I i ) = K i , where K i is a field, OMINANT DIMENSION AND TILTING MODULES 21 (3) Hom A ( I i , I j ) = 0 for all i = j ,(4) Hom A ( E, e Q ) = 0.Let A [ E ] := T ( R, A, A E R ) be the triangular matrix algebra as in Definition 4.1.1. A [ E ] := (cid:20) R A E R A (cid:21) . We now use the fact that the category of A [ E ]-modules is equivalent to the category oftriples ( R W, A V, f : A E R ⊗ R W → A V ), as described in Definition 4.1.2. Lemma 4.2.2.
Let E , I i , K i and A [ E ] be as above. The representatives of the isomorphismclasses of indecomposable projective-injective A [ E ] -modules correspond to the following triples: (1) There are t indecomposable projective-injective modules of type (0 , A Q i , , where Q i arethe indecomposable projective-injective A -modules, and (2) There are r indecomposable projective-injective modules of type ( K i , A I i , η i : A E R ⊗ R K i ∼ = −→ A I i ) , where I i are the indecomposable summand of the A -module E .Proof. (1) Using Proposition 4.1.4(1), it is clear that the A [ E ]-modules (0 , A Q i ,
0) are in-decomposable and projective. To see that they are injective: by Proposition 4.1.4(2), theindecomposable injective A [ E ]-modules are given as(Hom A ( E, Q i ) , A Q i , η i : A E R ⊗ Hom A ( E, Q i ) → A Q i ) , which are equal to (0 , A Q i , A ( E, e Q ) = 0 by condition (4) in Definition4.2.1. Therefore, (0 , A Q i ,
0) are indecomposable projective-injective A [ E ]-modules.(2) By Proposition 4.1.4(2) and conditions (2) and (3) in Definition 4.2.1, it is clear thatthe A [ E ]-modules ( K i , A I i , η i : A E R ⊗ R K i ∼ = −→ A I i ) are indecomposable and injective. Theyare also projective by Proposition 4.1.4(1) and the fact that K i are projective R -modules. (cid:3) By [10, Lemma 1.1] and Corollary 2.4.2, we know that for an Auslander algebra A , thetilting module T C exists in C A . We now relate this tilting module T C to a module in C A [ E ] andshow that there exists a tilting module in C A [ E ] . Lemma 4.2.3.
Let T C be a tilting module in C A . Then (0 , T C , is a module in C A [ E ] .Proof. The A [ E ]-module (0 , T C ,
0) is a submodule and a quotient module of the projective-injective A [ E ]-modules since T C is submodule and quotient module of the modules in add e Q . (cid:3) Lemma 4.2.4.
Let T C be a tilting module in C A . Then (1) projdim A [ E ] (0 , T C , ≤ , (2) Ext A [ E ] ((0 , T C , , (0 , T C , .Proof. Part (1) follows from Proposition 4 . . . . A [ E ] ((0 , T C , , (0 , T C , A ( T C , T C ) = 0. (cid:3) Corollary 4.2.5.
Let T C be a tilting module in C A . Then (0 , T C , is a partial tilting modulein C A [ E ] , with n A summands, where n A is the number of non-isomorphic simple A -modules. Theorem 4.2.6.
Let A be an Auslander algebra and A [ E ] be the algebra described in Definition . . . Then there is a tilting module in C A [ E ] .Proof. Let T C be a tilting module in C A . Then T C A [ E ] := (0 , T C , ⊕ ( L ri =1 Y i ) is a tiltingmodule in C A [ E ] , where Y i = ( K i , A I i , η i : A E R ⊗ R K i ∼ = −→ A I i ). The number of indecomposablesummands of T C A [ E ] equals n A + r which is the number of non-isomorphic simple A [ E ]-modules. (cid:3) Special class: Tilting modules in C Λ for Nakayama algebras Nakayama algebras.
In this section, let Λ be any Nakayama algebra. We will showcriteria for the subcategory C Λ to contain a tilting module T C . Due to Theorem 2 . .
4, it isequivalent to finding Nakayama algebras with dominant dimension at least 2. Notice that sucha class of algebras has been classified by Fuller in [11, Lemma 4.3] in a module theoretic way.However, using Auslander-Reiten theory, our descriptions in Corollary 5.2.5 and Theorem 5.3.1can be regarded as a combinatorial approach.First, we recall some well known facts about Nakayama algebras. A module M over anartin algebra is called a uniserial module if the set of its submodules is totally ordered byinclusion, or equivalently, there is a unique composition series of M . An artin algebra Λ is saidto be Nakayama algebra if both the indecomposable projective and indecomposable injectivemodules are uniserial. One can show that all indecomposable modules over a Nakayama algebraare uniserial [6, VI, Theorem 2.1].Moreover, we have the following classification of Nakayama algebras [1, Theorem V.3.2]: Abasic connected artin algebra Λ is a Nakayama algebra if and only if its ordinary quiver Q Λ iseither a quiver of type A n with straight orientation or a complete oriented cycle. Accordingto [19], Nakayama algebras whose ordinary quiver is A n with straight orientation are called Linear-Nakayama algebras and Nakayama algebras whose ordinary quiver is a completeoriented cycle are called the
Cyclic-Nakayama algebras . In this section, we always assumeΛ to be basic and connected.For any Λ-module M , denote by l ( M ) the length of M . For a Nakayama algebra Λ, thereexists an ordering { P , P , . . . , P n } of non-isomorphic indecomposable projective Λ-modulessuch that:(a) P i +1 / rad P i +1 ∼ = τ − ( P i / rad P i ), for 1 ≤ i ≤ n −
1; andif l ( P ) = 1, then P / rad P ∼ = ( P n / rad P n ),(b) l ( P i ) ≥
2, for 2 ≤ i ≤ n ,(c) l ( P i +1 ) ≤ l ( P i ) + 1, for 1 ≤ i ≤ n − l ( P ) ≤ l ( P n ) + 1.Such an ordering is called a Kupisch series for Λ and ( l ( P ) , l ( P ) , . . . , l ( P n )) is called thecorresponding admissible sequence for Λ. Remark 5.1.1. (1) The Kupisch series (and hence the admissible sequence) for a Nakayama algebra isalways unique up to a cyclic permutation (or simply unique if l ( P ) = 1).(2) Let ( c , c , . . . , c n ) be a sequence of integers such that c j ≥ j ≥
2, and c j +1 ≤ c j for j ≤ n −
1, and c ≤ c n + 1. There is a Nakayama algebra Λ such that( c , c , . . . , c n ) is the admissible sequence for Λ.5.2. Cyclic-Nakayama algebras with dominant dimension at least . Suppose Λ is aCyclic-Nakayama algebra with n simple modules. We always label the vertices of its ordinaryquiver in such a way that arrows are ( i + 1 → i ) for 1 ≤ i ≤ n − → n ). It is easyto check that { P , P , . . . , P n } is a Kupisch series, where P i := P ( S i ) is the projective cover ofthe simple module S i .Let Λ be a Cyclic-Nakayama algebra with Kupisch series ( P , P , . . . , P n ). Then we canview the corresponding admissible sequence ( c , c , . . . , c n ) as a function c : Z n → Z sending i c i and satisfying c i +1 ≤ c i + 1 and c i ≥
2. On the other hand, each such function givesrise to a Cyclic-Nakayama algebra.According to our labeling, it is easy to see:
Lemma 5.2.1.
Let P i := P ( S i ) be the projective cover of the simple module S i . Then OMINANT DIMENSION AND TILTING MODULES 23 (1) S i ∼ = τ S i +1 , (2) soc P i ∼ = S i − c i +1 , where the index i − c i + 1 is regarded as an element in Z n . Lemma 5.2.2.
Suppose P i := P ( S i ) is projective non-injective. The injective envelope I ( P i ) is a projective-injective module. Then soc I ( P i ) /P i ∼ = S i +1 , where the index i + 1 is regardedas an element in Z n .Proof. The exact sequence: 0 → P i → I ( P i ) → I ( P i ) /P i → I ( P i ) /P i ∼ = τ − top P i = τ − S i . Then the assertion follows from Lemma 5 . . (cid:3) Definition 5.2.3.
Define Q c := { i ∈ Z n | c i +1 ≤ c i } and P c := { i ∈ Z n | c i +1 = c i + 1 } .By definition, Q c ∪ P c = Z n . An indecomposable projective module P i is also injective ifand only if i ∈ Q c . Theorem 5.2.4.
Let P i be an indecomposable projective module with the index i ∈ P c . Then domdim P i ≥ if and only if i ∈ { j − c j ∈ Z n | j ∈ Q c } .Proof. The dominant dimension domdim P i ≥ I ( P i ) /P i is a submodule of P j for some j ∈ Q c , which is equivalent tosoc I ( P i ) /P i ∼ = soc P j . By Lemmas 5 . . . .
2, it is equivalent to say i + 1 = j − c j + 1. Therefore, domdim P i ≥ i ∈ { j − c j ∈ Z n | j ∈ Q c } . (cid:3) Corollary 5.2.5.
Suppose Λ is a Cyclic-Nakayama algebra with n simple modules. Then domdim Λ ≥ if and only if P c ⊆ { j − c j ∈ Z n | j ∈ Q c } . Corollary 5.2.6. If domdim Λ ≥ then |Q c | ≥ n . At last, we point out that this provides us with a method to find all Cyclic-Nakayamaalgebras with dominant dimension at least 2.Suppose c and c ′ are admissible sequences of Cyclic-Nakayama algebras Λ and Λ ′ . We saythat c and c ′ are in the same difference class (see [19]) if c ′ i = c i + n for all i . From Corollary5 . .
5, it is easy to see that if c and c ′ are in the same difference class, then domdim Λ ≥ ′ ≥
2. In fact, the dominant dimension of Λ only depends on the differenceclass of the admissible sequence [19, Theorem 1.1.4].Therefore, to find all the Cyclic-Nakayama algebras with dominant dimension at least 2, itis enough to find those with “minimal” admissible sequences.
Definition 5.2.7.
Suppose c is an admissible sequence of Cyclic-Nakayama algebras Λ. We saythat c is elementary if min ≤ i ≤ n { c i } ≤ n + 1, and c is absolutely elementary if min ≤ i ≤ n { c i } = 2. Example 5.2.8.
When n = 3, the following are all the (absolutely) elementary admissiblesequences for Cyclic-Nakayama algebras with dominant dimension at least 2 (up to cyclicpermutations):Absolutely elementary: (2 , , , , , , , , , , , , , , Linear-Nakayama algebras with dominant dimension at least . Most of theresults for Cyclic-Nakayama algebras also works for Linear-Nakayama algebras. For complete-ness, we will state the criteria for Linear-Nakayama algebras Λ having domdim Λ ≥ A n with n simplemodules. We always label the vertices of its quiver in such a way that arrows are ( i + 1 → i ) for 1 ≤ i ≤ n −
1. It is easy to check that { P , P , . . . , P n } is a Kupisch series, where P i := P ( S i )is the projective cover of simple module S i .The corresponding admissible sequence ( c , c , . . . , c n ) satisfies c = 1, c i +1 ≤ c i + 1, and c i ≥ ≤ i ≤ n . On the other hand, each such sequence gives rise to a Linear-Nakayamaalgebra. Define Q c := { i | c i +1 ≤ c i } and P c := { i | c i +1 = c i + 1 } as in Definition 5.2.3. Noticethat for Linear-Nakayama algebras c = 2, c i ≤ i and 1 ∈ P c . Theorem 5.3.1.
Suppose Λ is a Linear-Nakayama algebra with n simple modules. Then domdim Λ ≥ if and only if P c ⊆ { j − c j | j ∈ Q c } . The tilting module T C for Nakayama algebras. Lastly, for a Nakayama algebra Λ,we give a description of the tilting module T C in the subcategory C Λ if it exists.Let Λ be a Nakayama algebra with Kupisch series ( P , P , . . . , P n ) and admissible sequence( c , c , . . . , c n ). Let Q c and P c be the sets as defined in the previous sections. Then for each i ∈ P c , define δ ( i ) := min { k ∈ N | i + k ∈ Q c } .According to Corollary 2.2.9, we have the following description of the tilting module T C : Theorem 5.4.1.
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E-mail address : [email protected] Institutt for matematiske fag, Norges Teknisk-Naturvitenskapelige Universitet, N-7491 Trond-heim, Norway
E-mail address : [email protected] Department of Mathematics, Northeastern University, Boston, MA 02115, USA
E-mail address : [email protected] Department of Mathematics, Northeastern University, Boston, MA 02115, USA
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