Auslander Correspondence for Triangulated Categories
aa r X i v : . [ m a t h . R T ] J u l AUSLANDER CORRESPONDENCE FOR TRIANGULATED CATEGORIES
NORIHIRO HANIHARA
Abstract.
We give analogues of the Auslander correspondence for two classes of triangulated cate-gories satisfying certain finiteness conditions. The first class is triangulated categories with additivegenerators and we consider their endomorphism algebras as the Auslander algebras. For the second one,we introduce the notion of [1]-additive generators and consider their graded endormorphism algebrasas the Auslander algebras. We give a homological characterization of the Auslander algebras for eachclass. Along the way, we also show that the algebraic triangle structures on the homotopy categoriesare unique up to equivalence. Introduction
The main concern in representation theory of algebras is to understand the module categories. Amongsuch categories, those with finitely many indecomposable objects, or equivalently the representation-finite algebras, are most fundamental. Let us recall the following famous theorem due to Auslander:
Theorem 1.1 ([Au], Auslander correspondence) . There exists a bijection between the set of Moritaequivalence classes of finite dimensional algebras Λ of finite representation type and the set of Moritaequivalence classes of finite dimensional algebras Γ such that gl . dim Γ ≤ and dom . dim Γ ≥ . This theorem states that a categorical property (=representation-finiteness) of mod
Λ can be charac-terized by homological invariants (=gl . dim and dom . dim) of Γ, called the Auslander algebra of mod Λ.Such relationships between categorical properties of those appearing naturally in representation theory,and homological properties of their ‘Auslander algebras’ has been established in [I1, I2, E].The aim of this paper is to find an analogue of these results for triangulated categories [N]. Let k be an arbitrary field and T be a k -linear, Hom-finite, idempotent-complete triangulated category. Weconsider two kinds of finiteness conditions on triangulated categories.The first one is a direct analogue of representation-finiteness: T is finite , that is, T has finitely manyindecomposable objects up to isomorphism. In this case, T has an additive generator M . We call End T ( M ) the Auslander algebra of T , which is uniquely determined by T up to Morita equivalence. Thefirst main result of this paper is the following homological characterization of the Auslander algebrasof triangulated categories. We say that a finite dimensional algebra A is twisted n -periodic if it is self-injective and there exists an automorphism α of A such that Ω n ≃ ( − ) α as functors on mod A . We referto Corollary 2.2 for equivalent characterizations. Theorem 1.2.
Let k be a perfect field. The following are equivalent for a basic finite dimensional k -algebra A : (1) A is the Auslander algebra of a k -linear, Hom -finite, idempotent-complete triangulated categorywhich is finite. (2) A is twisted -periodic. This result shows a close connection between periodic algebras [ES] and triangulated categories. Ourproof depends on Amiot’s result (Proposition 3.2). This is a complement of Heller’s classical observation[He, 16.4] which gives a parametrization of pre-triangle structures on a pre-triangulated category T interms of isomorphisms Ω ≃ [ −
1] on mod T . Later practice of this property of the third syzygy inrepresentation theory can be seen in [AR, Yo2, Am, IO]. Mathematics Subject Classification.
Key words and phrases. triangulated category; Auslander correspondence; periodic algebra; Cohen-Macaulay module.
Moreover, with some additional assumptions on T , we give a bijection between finite triangulatedcategories and certain algebras, which is a more precise form of the above theorem; see Theorem 3.3.The second finiteness condition is the following:(S1) There is an object M ∈ T such that T = add { M [ n ] | n ∈ Z } .(S2) For any X, Y ∈ T , Hom T ( X, Y [ n ]) = 0 holds for almost all n .If these conditions are satisfied, we say T is [1] -finite and call M as in (S1) a [1] -additive generator . Forexample, the bounded derived categories of representation-finite hereditary algebras are [1]-finite, andadditive generators for module categories are [1]-additive generators for the derived categories. Thereare various studies on [1]-finite triangulated categories, for example [Ro, XZ, Am]. Note that [1]-finitetriangulated categories have infinitely many indecomposable objects unless T = 0.For a [1]-finite triangulated category T with a [1]-additive generator M , we call C = M n ∈ Z Hom T ( M, M [ n ])the [1] -Auslander algebra of T , which is naturally a Z -graded algebra and is uniquely determined by T up to graded Morita equivalence. Thanks to our condition (S2), C is finite dimensional. To study it, weprepare some results on ‘graded projectivization’ in Section 4 (see Proposition 4.2). Such constructionsof graded algebras appear naturally in various contexts [AZ, As].Our second main result is the Auslander correspondence for [1]-finite triangulated categories. To stateit, we have to restrict to a nice class of triangulated categories called algebraic . Recall that they are thestable categories of Frobenius categories [Ha, I.2.6]. Algebraic triangulated categories are enhanced bydifferential graded categories [Ke3], and play a central role in tilting theory [AHK].Now we can formulate the following second main result of this paper in terms of algebraic triangulatedcategories and graded algebras. We say that a finite dimensional Z -graded algebra A is ( a ) -twisted n -periodic if it is self-injective and there exists a graded automoprhism α of A such that P α ≃ P for all P ∈ proj Z A and Ω n ≃ ( − ) α ( a ) as functors on mod Z A . We refer to Corollary 2.4 for equivalent conditions. Theorem 1.3.
Let k be an algebraically closed field. There exists a bijection between the following. (1) The set of triangle equivalence classes of k -linear, Hom -finite, idempotent-complete, algebraictriangulated categories T which are [1] -finite. (2) The graded Morita equivalence classes ( see Definition of finite dimensional graded k -algebra C which are ( − -twisted -periodic. (3) A disjoint union of Dynkin diagrams of type A, D, and E.The correspondences are given as follows: • From (1) to (2) : Taking the [1] -Auslander algebra of T . • From (1) to (3) : Taking the tree type of the AR-quiver of T . • From (2) to (1) : C proj Z C . • From (3) to (1) : Q k ( Z Q ) , where k ( Z Q ) is the mesh category associated with Z Q . Moreover, we have the following explicit descriptions of (1) and (2) in the above theorem.
Theorem 1.4 (Theorem 5.3, Proposition 6.1) . The classes (1) and (2) in Theorem 1.3 are the same as (1 ′ ) and (2 ′ ) , respectively. (1 ′ ) The set of triangle equivalence classes of the bounded derived categories D b ( mod kQ ) of the pathalgebra kQ for a disjoint union Q of Dynkin quivers of type A, D, and E. (2 ′ ) The orbit algebras k ( Z Q ) / [1] for a disjoint union Q of Dynkin quivers of type A, D, and E. Comparing with Theorem 1.2, Theorem 1.3 is more strict in the point that the Auslander algebras C corresponds bijectively to the triangulated categories. This can be done by the classification of [1]-finitetriangulated categories as is stated in (1 ′ ). These results suggest that [1]-finite triangulated categoriesare easier than finite ones in controlling their triangle structures as well as their additive structures.Our classification is deduced from the following uniqueness of the triangle structures on the homotopycategories. USLANDER CORRESPONDENCE FOR TRIANGULATED CATEGORIES 3
Theorem 1.5 (Theorem 5.1) . Let Λ be a ring such that K b ( proj Λ) is a Krull-Schmidt category and Λ does not have a semisimple ring summand, and let C be an algebraic triangulated category. If C and K b ( proj Λ) are equivalent as additive categories, then they are equivalent as triangulated categories. For example, K b ( proj Λ) is Krull-Schmidt if Λ is a module-finite algebra over a complete Noetherianlocal ring. We actually see that the possible triangle structure on a given Krull-Schmidt additive categoryis unique in the sense that the suspensions and the mapping cones are uniquely determined as objects,see Proposition 5.5 for details.As an application of our classification Theorem 1.4 of [1]-finite triangulated categories, we recover themain result of [CYZ] stating that any finite dimensional algebra over an algebraically closed field withderived dimension 0 is piecewise hereditary of Dynkin type.We also apply Theorem 1.4 to Cohen-Macaulay representation theory. A rich source of [1]-finitetriangulated categories is given by CM-finite Iwanaga-Gorenstein algebras [CR1, CR2, LW, S, Yo1], forexample, simple singularities and trivial extension algebras of representation-finite hereditary algebras.We consequently obtain the following result, which states that CM Z Λ is triangle equivalent to the derivedcategory of a Dynkin quiver under some mild assumptions.
Corollary 1.6 (Theorem 7.3) . Let k be an algebraically closed field and Λ = L n ≥ Λ n be a positivelygraded CM-finite Iwanaga-Gorenstein algebra such that each Λ n is finite dimensional over k and Λ hasfinite global dimension. Then, the stable category CM Z Λ is [1] -finite and therefore, it is triangle equivalentto D b ( mod kQ ) for a disjoint union Q of some Dynkin quivers of type A, D, and E. This partially recovers [KST], [BIY, 2.1] in a quite different way. Note that our result is more general,but less explicit in the sense that Corollary 1.6 does not give the type of Q from given Λ.As this application suggests, our classification shows that the ‘easiest’ triangulated categories are verylikely to be the derived category of Dynkin quivers, and provides a completely different method (from adirect construction of tilting objects) of giving a triangle equivalence for such categories. Acknowledgement.
The author is deeply grateful to his supervisor Osamu Iyama for many helpfulsuggestions and careful instructions.
Notations and conventions.
We denote by k a field. For a category C , we denote by Hom C ( − , − )or simply C ( − , − ) the Hom-spaces between the objects and by J C ( − , − ) the Jacobson radical of C . A C -module is a contravariant functor from C to the category of abelian groups. A C -module M is finitelypresented if there is an exact sequence C ( − , X ) → C ( − , Y ) → M → X, Y ∈ C . We denote by mod C the category of finitely presented C -modules. If C is graded by a group G , the category of finitelypresented graded functor is denoted by mod G C , and its projectives by proj G C . The morphism space in mod G C is denoted by Hom C ( − , − ) or C ( − , − ) . The category mod G C is endowed with the grade shiftfunctor ( g ) for each g ∈ G , defined by M ( g ) = M as an ungraded module and ( M ( g )( X )) h = ( M X ) gh for each X ∈ C .Similarly, for a k -algebra A , the Jacobson radical of A is denoted by J A . A module over A meansa finitely generated right module. We denote by mod A (resp. proj A ) the category of (projective) A -modules. If A is graded, the category of graded (projective) A -modules is denoted by mod G A (resp. proj G A ). 2. Periodicity of syzygies
Let A be a k -algebra. We denote by A e the enveloping algebra A op ⊗ k A and by Ω A (resp. Ω A e ) thesyzygy, that is, the kernel of the projective cover in mod A (resp. mod A e ). In this section, we generalizefor our purpose the result of Green-Snashall-Solberg [GSS] which relates the periodicity of syzygy ofsimple A -modules and that of A considered as a bimodule over itself. The following theorem and itsproof is a graded and twisted version of [GSS, 1.4]. Theorem 2.1.
Let G be an abelian group and A be a finite dimensional, ring-indecomposable, non-semisimple G -graded k -algebra. Assume that J A = J A ⊕ ( L i =0 A i ) and that A/J A is separable over k .Then, the following are equivalent for a ∈ G and n > . (1) Ω nA ( A/J A ) ≃ A/J A ( a ) in mod G A . NORIHIRO HANIHARA (2) A is self-injective and there exists a graded algebra automorphism α of A such that Ω n ≃ ( − ) α ( a ) as functors on mod G A . (3) There exists a graded algebra automorphism α of A such that Ω nA e ( A ) ≃ A α ( a ) in mod G A e .Proof. By the original case, we have that A is self-injective under the assumption (3). Then the implica-tion (3) ⇒ (2) follows. Also, (2) ⇒ (1) is clear.It remains to prove (1) implies (3). Note that by our assumption on J A , it is graded and any simpleobject in mod G A is simple in mod A . Assume (1) holds and set B = Ω nA e ( A ). This is a projective A -module on each side. Step 1: S ⊗ A B is simple for all graded simple (right) A -modules S . Let S be a graded simple A -module. Then, applying S ⊗ A − to the minimal projective resolution P : · · · → P i d i −→ P i − → · · · → P of A in mod G A e yields the minimal projective resolution of S in mod G A . Indeed, since A/J A is separable over k , we have J A e = J A ⊗ k A + A ⊗ k J A . Then, Im d i ⊂ P i − J A e = J A P i − + P i − J A by the minimality of P and therefore, Im ( S ⊗ d i ) ⊂ S ⊗ A P i − J A by S ⊗ A J A P i − = 0. This shows S ⊗ A P is minimal. Therefore we have S ⊗ A B ≃ Ω nA ( S ), which is simpleby assumption (1).It follows by induction that the exact functor − ⊗ A B preserves length. Step 2: B ≃ A ( a ) in mod G A . Consider the exact sequence 0 → J A → A → A/J A → mod G A . Applying − ⊗ A B yields B → A/J A ( a ) →
0. This shows that the module B contains A/J A ( a ) in its top. But since B is aprojective (right) A -module having the same length as A by the remark following Step 1, we see that B ≃ A ( a ) in mod G A . Step 3: There exists a graded algebra automorphism α of A such that Ω nA e ( A ) ≃ A α ( a ) . By Step 2, there exists a graded algebra endomorphism α of A such that B ≃ α A ( a ) in mod G A e .Indeed, fix an isomorphism ϕ : A ( a ) → B in mod G A , put x = ϕ (1), and set α ( u ) = ϕ − ( ux ) for u ∈ A .Then, α is of degree 0, since x and ϕ are, and it is easily checked that α is an algebra endomorphismand that ϕ : α A ( a ) → B is an isomorphism in mod G A e . Now we show that α is an isomorphism. Let I be the kernel of α . Since B ≃ α A is a projective left A -module, the inclusion I ⊂ A in mod G A staysinjective by applying − ⊗ A α A . But since that map I ⊗ A α A → α A is zero, we have I ⊗ A α A = 0, andwe conclude that I = 0 by the remark following Step 1.This finishes the proof of (1) ⇒ (3). (cid:3) We need the following two particular cases. The first one, which we will use in Section 3 is the followingresult for G = { } , where the special case ‘Ω n ( S ) ≃ S for all simples’ is [GSS, 1.4]. Corollary 2.2.
Let A be a ring-indecomposable, non-semisimple finite dimensional k -algebra such that A/J A is separable over k . Then, the following are equivalent for n > . (1) Ω nA ( A/J A ) ≃ A/J A . (2) A is self-injective and there exists an automorphism α of A such that Ω n ≃ ( − ) α as functors on mod A . (3) There exists an automorphism α of A such that Ω nA e ( A ) ≃ A α in mod A e . We call such algebras as follows.
Definition 2.3.
A finite dimensional algebra is twisted n -periodic if it is a direct product of simplealgebras or algebras satisfying the equivalent conditions in Corollary 2.2.The second one is the following for G = Z and the permutation of simples is the identity, which willbe used in Section 6. Corollary 2.4.
Let A be a finite dimensional, ring-indecomposable, non-semisimple Z -graded k -algebrasuch that A/J A is separable over k . Then, the following are equivalent for a ∈ Z and n > . (1) Ω nA ( S ) ≃ S ( a ) in mod Z A for any simple objects in mod Z A . (2) A is self-injective and there exists a graded algebra automorphism α of A such that Ω n ≃ ( − ) α ( a ) as functors on mod Z A and P α ≃ P in mod Z A for all P ∈ proj Z A .. USLANDER CORRESPONDENCE FOR TRIANGULATED CATEGORIES 5 (3)
There exists a graded algebra automorphism α of A such that Ω nA e ( A ) ≃ A α ( a ) in mod Z A e and P α ≃ P in mod Z A for all P ∈ proj Z A . Similarly, we name these algebras as follows.
Definition 2.5.
A finite dimensional graded algebra is ( a ) -twisted n -periodic if it is a direct product ofsimple algebras or algebras satisfying the equivalent conditions in Corollary 2.4.3. Auslander correspondence
We now prove the first main result Theorem 1.2 of this paper, which gives a homological characteri-zation of the Auslander algebras of finite triangulated categories.First, we give the properties of the endomorphism algebra of a basic additive generator for a finitetriangulated category, proving Theorem 1.2 (1) ⇒ (2). Proposition 3.1.
Let T be a k -linear, Hom -finite idempotent-complete triangulated category. Assume T has an additive generator M . Take M to be basic and set C = End T ( M ) . Let σ be the automorphismof C induced by [1] ; precisely, fix an isomorphism a : M → M [1] and define α by α ( f ) = a − ◦ f [1] ◦ a for f ∈ End T ( M ) . Then, C is a finite dimensional algebra which is twisted -periodic.Proof. Since mod
T ≃ mod C and mod T is a Frobenuis category (see [Kr, 4.2]), C is self-injective. Also,since the triangles in T yield projective resolutions of C -modules, the third syzygy is induced by theautomorphism α , that is, we have Ω ≃ ( − ) α on mod C . Then C is twisted 3-periodic by Corollary2.2. (cid:3) For the converse implication, we need the following result due to Amiot, which allows one to introducea triangle structure on the category of projectives in a Frobenius category.
Proposition 3.2. [Am, 8.1]
Let P be an idempotent complete k -linear category such that the functorcategory mod P is naturally a Frobenius category. Let S be an autoequivalence of P and extend this to mod P → mod P . Assume there exists an exact sequence of exact functors from mod P to mod P / / / / X / / X / / X / / S / / , where X i take values in P = proj P . Then, P has a structure of a triangulated category with suspension S . The triangles are ones isomorphic to X M → X M → X M → SX M for M ∈ mod P . Combining this with Corollary 2.2, we can prove Theorem 1.2 (2) ⇒ (1). Let us summarize the proofbelow. Proof of Theorem 1.2. (1) ⇒ (2) is Proposition 3.1.(2) ⇒ (1) Since A is self-injective, Ω permutes the simples, so by Corollary 2.2, there exists an exactsequence 0 / / A / / P / / P / / P / / A α / / A, A )-bimodules, with P i ’s projective and α is an automorphism of A . Then, we can apply Proposition3.2 for P = proj A , S = − ⊗ A A α , and X i = − ⊗ A P i . (cid:3) Applying a recent result of Keller [Ke4], we can formulate Theorem 1.2 in terms of bijection betweentriangulated categories and algebras under the standardness T . Theorem 3.3.
Let k be an algebraically closed field. Then, there exists a bijection between the following. (1) The set of triangle equivalence classes of k -linear, Hom -finite, idempotent-complete triangulatedcategories which are finite, algebraic, and standard. (2)
The set of isomorphism classes of finite dimensional mesh algebras over k .The correspondence from (1) to (2) is given by taking the basic Auslander algebra, and from (2) to (1)by taking the category of projective modules. NORIHIRO HANIHARA
Proof.
We first check that each map is well-defined.Let T be a triangulated category as in (1). Then, the standardness of T implies that its basic Auslanderalgebra is a mesh algebra.Suppose next that A is a finite dimensional mesh algebra. We want to show that proj A has theunique structure of an algebraic triangulated category up to equivalence. Since the third syzygy ofsimple A -modules are simple, T = proj A has a structure of a triangulated category by Theorem 1.2.Also, this is standard since A is a mesh algebra. We claim that proj A admits a triangle structure whichis algebraic. Since T is a finite, standard triangulated category, there exists a Dynkin quiver Q , a k -linear automorphism F of D b ( mod kQ ), and a k -linear equivalence D b ( mod kQ ) /F ≃ proj A [Ri]. Asin the proof of [Ke4], F is isomorphic to − ⊗ LkQ X for some ( kQ, kQ )-bimodule complex X . Then by[Ke2], D b ( mod kQ ) /F admits an algebraic triangle structure as a triangulated orbit category, hence sodoes proj A . This finishes the proof of the claim. Now, this algebraic triangle structure is unique up toequivalence by the main result of [Ke4]. This shows the well-definedness.It is clear that these maps are mutually inverse. (cid:3) Graded projectivization
In this section, we formulate the method of realizing certain additive categories, which we call G -finite additive categories on which a group G acts with some finiteness conditions, as the category of gradedprojective modules over a G -graded algebra. This generalizes the classical ‘projectivization’ [ARS, II.2],which realizes a finite additive category as the category of projectives over an algebra.Let A be an additive category with an action of a group G . Precisely, an automorphism F g of A is given for each g ∈ G so that F gh = F h ◦ F g for all g, h ∈ G . Then the action of G extends to anautomorphism of mod A by F g M = M ◦ F − g . For example, the action on the representable functors is F g A ( − , X ) = A ( − , F g X ).Recall that the orbit category A /G has the same objects as A and the morphism space( A /G )( X, Y ) = M g ∈ G A ( X, F g Y )and the composition b ◦ a of a ∈ T ( X, F g Y ) and b ∈ T ( Y, F h Z ) is given by b ◦ a = F g ( b ) a , where the righthand side is the composition in A . Then, A /G is naturally a G -graded category whose degree g part is A ( X, F g Y ). Proposition 4.1.
Let A be an additive category with an action of a group G . Consider the orbit category C = A /G . Then, the following assertions hold: (1) The Yoneda embedding
A → proj G C is fully faithful. It is an equivalence if A is idempotent-complete. (2) There exists an equivalence mod
A ≃ mod G C such that the action of F g on mod A corresponds tothe grade shift ( g ) on mod G C , that is, we have the following commutative diagram of functors: mod A ≃ / / F g (cid:15) (cid:15) mod G C ( g ) (cid:15) (cid:15) mod A ≃ / / mod G C . Proof. (1) We have the Yoneda lemma for graded functors:
Hom mod G C ( C ( − , X ) , M ) = ( M X ) . It followsthat the Yoneda embedding A → proj G C is fully faithful. Also, if A is idempotent-complete, the projec-tives in mod G C are representable, and therefore the Yoneda embedding is dense.(2) It is clear that the functor in (1) induces an equivalence mod A ≃ mod G C . Also, the degree h partof the functor C ( − , F g X ) is A ( − , F h F g X ) = A ( − , F gh X ), which is equal to the same degree part of C ( − , X )( g ). Thus we have the commutative diagram. (cid:3) Now we impose the following finiteness conditions on the G -action:(G1) There is M ∈ A such that A = add { F g M | g ∈ G } .(G2) For any X, Y ∈ A , Hom A ( X, F g Y ) = 0 for almost all g ∈ G . USLANDER CORRESPONDENCE FOR TRIANGULATED CATEGORIES 7
If these conditions are satisfied, we say that an additive category A with an action of G is G -finite . If A is a G -finite additive category, we say M ∈ A as in (G1) is a G -additive generator . If G is generatedby a single element F , we use the term F -finite for G -finiteness, and F -additive generator for G -additivegenerator. Note that if G is the trivial group, G -finiteness is nothing but finiteness, and a G -additivegenerator is an additive generator.Let us reformulate Proposition 4.1 in terms of the graded endomorphism algebra below. Note thatthis generalizes the classical ‘projectivization’ for finite additive categories, which is the case G is trivial,to ‘graded projectivization’ for G -finite categories. Although this is rather formal, it will be useful in thesequel. Proposition 4.2.
Let A be a k -linear, Hom -finite, idempotent-complete category with an action of G ,which is G -finite. Let M ∈ A be a G -additive generator and set C = End A /G ( M ) . Then, the followingassertions hold: (1) C is a finite dimensional G -graded algebra. (2) The functor
A → proj G C , X L g ∈ G Hom A ( M, F g X ) is an equivalence. (3) There exists an equivalence mod
A ≃ mod G C such that the action of g on mod A corresponds tothe grade shift ( g ) on mod G C .Proof. (1) C is finite dimensional by (G2).(2) Since we have an equivalence proj G A /G → proj G C by substituting M , the assertion follows fromProposition 4.1 (1).(3) This is the same as Proposition 4.1 (2). (cid:3) Definition 4.3. G -Graded rings A and B are graded Morita equivalent if there is an equivalence mod G A ≃ mod G B which commutes with grade shift functors ( g ) for all g ∈ G .Let us note the following remark. Proposition 4.4.
Assume (G1) is satisfied and set C = End A /G ( M ) . (1) The ungraded algebra C does not depend on the choice of M up to Morita equivalence. (2) The graded algebra C does not depend on the choice of M up to graded Morita equivalence.Proof. (1) Since C is the endomorphism algebra of an additive generator of the category A /G , theassertion follows.(2) This follows from Proposition 4.2 (3). (cid:3) As a direct application of this graded projectivization, we present as an example the following gradedversion of the Auslander correspondence. For simplicity, we consider Z -graded algebras. A graded algebraΛ is representation-finite if mod Z Λ has finitely many indecomposables up to grade shift. This is equivalentto the representation-finiteness of the ungraded algebra Λ [GG].
Proposition 4.5.
There exists a bijection between the following. (1)
The set of graded Morita equivalence classes of finite dimensional Z -graded algebras Λ of finiterepresentation type. (2) The set of graded Morita equivalence classes of finite dimensional Z -graded algebras Γ with gl . dim Γ ≤ ≤ dom . dim Γ .The correspondence is given as follows: • From (1) to (2) : Γ =
End Λ ( M ) = L n ∈ Z Hom Λ ( M, M ( n )) for a (1) -additive generator M for mod Z Λ . • From (2) to (1) : Λ =
End Γ ( Q ) = L n ∈ Z Hom Γ ( Q, Q ( n )) for a (1) -additive generator Q for thecategory of graded projective-injective Γ -modules.Proof. Note that Γ (resp. Λ) does not depend on the choice of M (resp. Q ) by Proposition 4.4 (2). Therest of the proof follows by the same argument as in Theorem 1.1; see [ARS, VI.5]. (cid:3) Notice that this correspondence Λ ↔ Γ is the same as the ungraded case, thus it is a refinement ofTheorem 1.1 on how much grading Λ or Γ have up to graded Morita equivalence.
NORIHIRO HANIHARA Uniqueness of triangle structures
The aim of this section is to prove some results which state the uniqueness of triangle structures oncertain additive categories. We say that an additive category C has a unique algebraic triangle structureup to equivalence if C = ( C , [1] , △ ) and C = ( C , [1] ′ , △ ′ ) are algebraic triangle structures on C , then thereexists a triangle equivalence F : C ≃ −→ C such that F ( X ) ≃ X in C for all X ∈ C .The following is the main result of this section. Theorem 5.1.
Let Λ be a ring with no simple ring summands such that K b ( proj Λ) is Krull-Schmidt.Then, the additive category K b ( proj Λ) has a unique algebraic triangle structure up to equivalence. We give applications of Theorem 5.1. For a quiver Q , let Z Q be the associated infinite translationquiver [ASS, Ha], and let k ( Z Q ) be its mesh category [Ha]. Corollary 5.2.
Let Q be a disjoint union of Dynkin quivers which does not contain A . Then, the meshcategory k ( Z Q ) has a unique algebraic triangle structure up to equivalence. As a consequence, we have the classification of [1]-finite algebraic triangulated categories.
Theorem 5.3.
Let k be an algebraically closed field. Any [1] -finite algebraic triangulated category over k is triangle equivalent to the bounded derived category D b ( mod kQ ) of the path algebra kQ for a disjointunion Q of Dynkin quivers of type A, D, and E. Now we start the preparations for the proofs of the above results. Recall that an additive category is
Krull-Schmidt if any object is a finite direct sum of objects whose endomorphism rings are local. Thisis the case if the category is idempotent-complete and Hom-finite over a complete Noetherian local ring.A Krull-Schmidt category C is purely non-semisimple if for each X ∈ C , J C ( − , X ) = 0 or J C ( X, − ) = 0holds. Note that these conditions are equivalent if C is triangulated.First we observe that the suspension and the terms appearing in triangles in a triangulated categoryare determined by its additive structure under some Krull-Schmidt assumptions. Lemma 5.4.
Let C be a Krull-Schmidt additive category. Assume C has a structure of a triangulatedcategory. Let f : X → Y be a right minimal morphism in J C . (1) The mapping cone of f is the minimal weak cokernel of f . (2) X [1] is the minimal weak cokernel of the minimal weak cokernel of f .Proof. Complete f to a triangle X f −→ Y g −→ Z h −→ X [1].(1) We have to show that g is the minimal weak cokernel of f . We only have to show the (left) minimality.If this is not the case, then h has a summand W W −−→ W for a common non-zero summand W of Z and X [1]. This contradicts the right minimality of f .(2) We want to show that h is the minimal weak cokernel of g . Again, we only have to show the minimality.If this is not the case, then f [1] has a summand V V −−→ V for a common non-zero summand V of X [1]and Y [1]. This contradicts f ∈ J C ( X, Y ). (cid:3) We deduce that the possible triangle structures on a given purely non-semisimple Krull-Schmidt ad-ditive category is roughly unique in the following sense. We denote by cone △ ( f ) the mapping cone of f in a triangle structure △ . Proposition 5.5.
Let C be a purely non-semisimple Krull-Schmidt additive category. If ( C , [1] , △ ) and ( C , [1] ′ , △ ′ ) are triangle structures on C , then we have the following. (1) X [1] ≃ X [1] ′ for all objects X ∈ C . (2) cone △ ( f ) ≃ cone △ ′ ( f ) in C for all morphisms f in C .Proof. (1) Let X ∈ C be an indecomposable object. Since C is purely non-semisimple, there exists anon-zero morphism f : X → Y in J C . Then, f is a right minimal radical map, and hence the assertionfollows from Lemma 5.4 (2).(2) Let f : X → Y be an arbitrary morphism in C . By removing the summands isomorphic to W −→ W , which does not affect the mapping cone, we may assume f ∈ J C . Then, f has a decomposition USLANDER CORRESPONDENCE FOR TRIANGULATED CATEGORIES 9 X ⊕ X f , −−−−→ Y with right minimal f ∈ J C and the mapping cone of f is the direct sum of thatof f and X [1]. Now the mapping cone of f is determined by Lemma 5.4 (1) and since C is purelynon-semisimple, [1] is determined by the additive structure by (1). This proves the assertion. (cid:3) For Theorem 5.1, we need the following result of Keller on algebraic triangulated categories.
Proposition 5.6. [Ke1, 4.3]
Let T be an algebraic triangulated category and T ∈ T be a tilting object.Then, there exists a triangle equivalence T ≃ K b ( proj End T ( T )) . Note that we have the following observation, which will be crucial for the proof.
Lemma 5.7.
Let C be a purely non-semisimple Krull-Schmidt additive category. Assume C = ( C , [1] , △ ) and C = ( C , [1] ′ , △ ′ ) are triangle structures on C . Then, an object T ∈ C is a tilting object in C if andonly if it is a tilting object in C .Proof. Indeed, we have C ( T, T [ n ]) = C ( T, T [ n ] ′ ) by Proposition 5.5 (1), which shows that the vanishingof extensions does not depend on the triangle structure. Also, by Proposition 5.5 (2), T generates C ifand only if T generates C . This shows the assertion. (cid:3) Now we are ready to prove our results.
Proof of Theorem 5.1.
Assume C is triangulated. We show that C is triangle equivalent to K = K b ( proj Λ)by finding a tilting object whose endomorphism algebra is Λ. Fix an additive equivalence
C ≃ K . Then, K is purely non-semisimple and Krull-Schmidt by our assumption on Λ. Let T ∈ C be the objectcorresponding to Λ ∈ K . Then, T is a tilting object by Lemma 5.7 and clearly End C ( T ) = Λ. By ourassumption that C is algebraic, we deduce that C is triangle equivalent to K by Proposition 5.6. (cid:3) For the proof of Corollary 5.2, let us recall the following standardness theorem of Riedtmann.
Proposition 5.8. [Ri]
Let k be a field and T be a k -linear, Hom -finite idempotent-complete triangulatedcategory whose AR-quiver is Z Q for some acyclic quiver Q . Assume the endomorphism algebra of anindecomposable object of T is k . Then, T is k -linearly equivalent to the mesh category k ( Z Q ) . A well known application of this result is an equivalence K b ( proj kQ ) ≃ k ( Z Q ) for a Dynkin quiver Q [Ha, I. 5.6]. Proof of Corollary 5.2.
Since k ( Z Q ) ≃ K b ( proj kQ ) as additive categories, Theorem 5.1 gives the result. (cid:3) A k -linear triangulated category T is locally finite [XZ] if for each indecomposable X ∈ T , we have P Y :indec. dim k Hom T ( X, Y ) < ∞ . This condition is equivalent to its dual [XZ]. Clearly, our [1]-finitetriangulated categories are locally finite. The classification of [1]-finite triangulated category depends onthe following result. Proposition 5.9. [XZ, 2.3.5]
Let k be an algebraically closed field and T be a locally finite triangulatedcategory which does not contain a non-zero finite triangulated subcategory. Then, the AR-quiver of T is Z Q for a disjoint union Q of Dynkin quivers of type A, D, and E.Proof of Theorem 5.3. The AR-quiver of a [1]-finite triangulated category is Z Q for some Dynkin quiver Q by Proposition 5.9. Moreover, it is equivalent to k ( Z Q ) by Proposition 5.8. Thus Corollary 5.2applies. (cid:3) We end this section by noting the following lemma, which we use later. This lemma states in particular,that for mesh categories, the suspension is unique up to isomorphism of functors
Lemma 5.10.
Let Q be a Dynkin quiver and α be an automorphism of the mesh category k ( Z Q ) suchthat αX ≃ X for all X ∈ k ( Z Q ) . Then, α is isomorphic as functors to the identity functor.Proof. Since Q is Dynkin, we can inductively construct a natural isomorphism between α and id. (cid:3)
6. [1] -Auslander correspondence
In this section, we prove the second main result Theorem 1.3 of this paper. In the first subsection,we give the correspondence from triangulated categories to algebras, and the converse one in the secondsubsection. We will prove the main theorem in the final subsection.6.1.
From triangulated categories to algebras.
We apply the graded projectivization prepared inSection 4 to triangulated categories. Let T be a k -linear, Hom-finite, idempotent-complete triangulatedcategory. Consider the action on T of G = Z , generated by the suspension [1]. Then, the G -finiteness inthis case are(S1) There is M ∈ T such that T = add { M [ n ] | n ∈ Z } .(S2) For any X, Y ∈ T , Hom T ( X, Y [ n ]) = 0 for almost all n .According to the terminology in Section 4, we say T is [1] -finite , and call M as in (S1) a [1] -additivegenerator .The following theorem gives the correspondence from triangulated categories to algebras. Proposition 6.1.
Let T be a k -linear, Hom -finite, idempotent-complete, triangulated category whichis [1] -finite. Let M ∈ T be a [1] -additive generator and set C = End T / [1] ( M ) . Then, C is a finite-dimensional graded self-injective algebra such that Ω L ≃ L ( − for any graded C -module L .Proof. C is finite dimensional by (S2). Also, since mod T ≃ mod Z C by Proposition 4.2 (2) and mod T is Frobenius, C is self-injective. It remains to show the statement on the third syzygy. Let L be agraded C -module and let Q → R → L → L in mod Z C . Take themap X → Y in T corresponding to Q → R and complete it to a trianle W → X → Y → W [1]. Put P Z = L n ∈ Z Hom T ( M, Z [ n ]) for each Z ∈ T . This is the graded projective C -module corresponding to Z . Note that P Z [1] = P Z (1), where (1) is the grade shift functor on mod Z C . The triangle above yieldsan exact sequence P X ( − → P Y ( − → P W → P X → P Y → P W (1). Since P X = Q and P Y = R , wesee that Ω L ≃ L ( − (cid:3) Example 6.2.
Let Q be a Dynkin quiver and T = D b ( mod kQ ). Let M be an additive generator for mod kQ . Then, M is a [1]-additive generator for T and we have C = End kQ ( M ) ⊕ Ext kQ ( M, M ). Thedegree 0 part of C is the Auslander algebra of mod kQ .Let Q ′ be another Dynkin quiver with the same underlying graph ∆ as Q . Since kQ and kQ ′ arederived equivalent, we have D b ( mod kQ ′ ) = T . Similarly as above, an additive generator M ′ for mod kQ ′ is a [1]-additive generator for T . The corresponding graded algebra C ′ is End kQ ′ ( M ′ ) ⊕ Ext kQ ′ ( M ′ , M ′ ),with the Auslander algebra of mod kQ ′ in the degree 0 part.By Lemma 4.4, C and C ′ are isomorphic as ungraded algebras (but not as graded algebras). In thisway, C ≃ C ′ contains the Auslander algebras of module categories over ∆ for any orientation of ∆.Let us give a more specific example. Example 6.3.
Let Q be the following Dynkin quiver of type A , and T be its derived category D b ( mod kQ ). a b o o c o o Then, the AR-quiver of T is as follows: · · · ◦ $ $ ■■■■ $ $ ■■■■ $ $ ■■■ $ $ ■■■ $ $ ■■■ · · ·· · · ◦ $ $ ■■■■ : : ✉✉✉✉ ◦ $ $ ■■■■ : : ✉✉✉✉ $ $ ■■■■ : : ✉✉✉ $ $ ■■■ : : ✉✉✉✉ $ $ : : ◦ $ $ ■■■ : : ✉✉✉ · · · ◦ : : ✉✉✉✉ ◦ : : ✉✉✉✉ : : ✉✉✉✉ : : : : ✉✉✉ · · · , where 1 , . . . , mod kQ . Take M = L i =1 M i , where M i is the indecomposable kQ -module corresponding to the vertex i . Then, C = End kQ ( M ) ⊕ Ext kQ ( M, M ). It is easily verified
USLANDER CORRESPONDENCE FOR TRIANGULATED CATEGORIES 11 that C is presented by the quiver Z A / [1] and the mesh relations. The quiver of C looks as follows:4 (cid:0) (cid:0) ✁✁✁ ~ ~ ⑥⑥⑥ ~ ~ ⑥⑥⑥ ~ ~ ⑥⑥⑥ ~ ~ ⑥⑥⑥ ` ` ❆❆❆ ~ ~ ⑥⑥⑥ ` ` ❆❆❆ ~ ~ ⑥⑥⑥ ` ` ❆❆❆ ~ ~ ⑥⑥⑥ ` ` ❆❆❆ ~ ~ ⑥⑥⑥ ^ ^ ❂❂❂ (cid:0) (cid:0) ✁✁✁ ^ ^ ❂❂❂ ` ` ❆❆❆ ` ` ❆❆❆ ` ` ❆❆❆ ` ` ❆❆❆ where the vertices with the same number are identified, with mesh relations along the dotted lines. Thearrows 1 → → Q ′ be the quiver obtained by reflecting Q at vertex a ; a / / b c o o Fix an equivalence D b ( mod kQ ′ ) ≃ D b ( mod kQ ) so that M ′ = M ⊕ · · · ⊕ M ⊕ M [1] is an additivegenerator for mod kQ ′ . Then, C ′ = End kQ ′ ( M ′ ) ⊕ Ext kQ ′ ( M ′ , M ′ ) is presented by the same quiver withrelations as C , with arrows 2 → → C ≃ C ′ as ungraded algebras but not as graded algebras.Nevertheless, C and C ′ are graded Morita equivalent. Here we give a direct equivalence mod Z C → mod Z C ′ . Let e i be the idempotent of C corresponding to M i (1 ≤ i ≤
6) and set P = e C ⊕ · · · ⊕ e C ⊕ e C (1). Then, we have End C ( P ) ≃ C ′ as graded algebras and Hom C ( P, − ) gives a desired equivalence.6.2. From algebras to triangulated categories.
We can give the converse correspondence as inSection 3. Setting a = − Proposition 6.4.
Let A be a finite dimensional graded algebra such that A/J A is separable over k and Ω S ≃ S ( a ) for any graded simple module S . Then, proj Z A has a structure of a triangulated category. If k is algebraically closed and a = 0 , then the suspension is isomorphic to ( − a ) and the algebraic trianglestructure on proj Z A is unique up to equivalence.Proof. By Corollary 2.4, A is self-injective and there exists an exact sequence0 / / A / / P / / P / / P / / A α ( − a ) / / mod Z A e , where P i , i = 0 , , α is a graded algebra automorphism of A such that P α ≃ P for all P ∈ proj Z A . Then, we can apply Proposition 3.2 for P = proj Z A , X i = − ⊗ A P i and S = ( − ) α ( − a ) to see that proj Z A is triangulated with suspension ( − ) α ( − a ). Now assume k is algebraicallyclosed and a = 0. Since we have Hom proj Z A ( X, Y ( − na )) = 0 for almost all n ∈ Z for each X, Y ∈ proj Z A ,the triangulated category proj Z A is [1]-finite, and therefore, it is equivalent to the mesh category k ( Z Q )for some Dynkin diagram Q by Propositions 5.9 and 5.8. Then, by changing the triangle structure ifnecessary, proj Z A has a structure of an algebraic triangulated category, which is unique up to equivalenceby Corollary 5.2. Also, ( − ) α ( − a ) and ( − a ) are isomorphic as functors by Lemma 5.10. (cid:3) Proof of Theorem 1.3.
Combining the previous results, we can now prove the second main resultof this paper.
Proof of Theorem 1.3.
For M as in (1), C is as stated in (2) by Proposition 6.1. Also, the gradedMorita equivalence class of C does not depend on the choice of M by Proposition 4.4. This shows thewell-definedness of (1) to (2).For the map from (2) to (1), it is well-defined since proj Z C has the unique structure of an algebraictriangulated category up to equivalence by Proposition 6.4.It is easily checked that these maps are mutually inverse.The bijection between (1) and (3) is Proposition 5.9 and Corollary 5.3. (cid:3) Remark . The algebra C in Theorem 1.3 satisfies [3] ≃ (1) as functors on mod Z C by Proposition 6.1. Applications to Cohen-Macaulay modules
Applying our classification in Theorem 5.3 of [1]-finite triangulated categories, we show that the stablecategories CM Z Λ of some CM-finite Iwanaga-Gorenstein algebras, in particular, of (commutative) gradedsimple singularities are triangle equivalent to the derived categories of Dynkin quivers.A Noetherian algebra Λ is
Iwanaga-Gorenstein if id Λ Λ = id Λ op Λ < ∞ . A typical example of Iwanaga-Gorenstein algebra is given by commutative Gorenstein rings of finite Krull dimension. For an Iwanaga-Gorenstein algebra Λ, we have the category CM Λ = { X ∈ mod Λ | Ext i Λ ( X, Λ) = 0 for all i > } of Cohen-Macaulay
Λ-modules. It is naturally a Frobenius category and we have a triangulated category CM Λ.Now consider the case Λ is graded: let Λ = L n ≥ Λ n is a positively graded Noetherian algebra suchthat each Λ n is finite dimensional over a field k . If Λ is a graded Iwanaga-Gorenstein algebra, we similarlyhave the category CM Z Λ = { X ∈ mod Z Λ | Ext i Λ ( X, Λ) = 0 for all i > } of graded Cohen-Macaulay modules. It is again Frobenius and hence the stable category CM Z Λ is trian-gulated. A graded Iwanaga-Gorenstein algebra is
CM-finite if CM Z Λ has finitely many indecomposableobjects up to grade shift.We now show that CM-finite Iwanaga-Gorenstein algebras give a large class of examples of [1]-finitetriangulated categories.
Proposition 7.1.
Let Λ be a positively graded CM-finite Iwanaga-Gorenstein algebra with gl . dim Λ < ∞ . Then, the triangulated category CM Z Λ is [1] -finite. To prove this, we need an observation for general Noetherian algebras, which is motivated by [Ya, 3.5].Let us fix some notations. We denote by
Ext i Λ ( − , − ) the Ext groups on mod Z Λ. Note that for
M, N ∈ mod Z Λ, the Ext groups on mod
Λ are graded k -vector spaces: Ext i Λ ( M, N ) = L n ∈ Z Ext i Λ ( M, N ( n )) ,( i ≥ M ∈ mod Z Λ and n ∈ Z , we denote by M ≥ n the Λ-submodule of M consisting ofcomponents of degree ≥ n . Lemma 7.2.
Let Λ be a positively graded Noetherian algebra with gl . dim Λ < ∞ . Then, for any X, Y ∈ mod Z Λ , we have Hom Λ ( X, Ω n Y ) = 0 for sufficiently large n .Proof. Take a minimal graded projective resolution of Y : · · · → P → P → P → Y →
0. We will showthat for each i ∈ Z , P n = ( P n ) ≥ i holds for n ≫
0. For this, it suffices to show that P n = ( P n ) ≥ for Y = Y ≥ . Note that the degree 0 part of the minimal projective resolution of Y yields a Λ -projectiveresolution of Y . By our assumption that gl . dim Λ < ∞ , we have ( P n ) = 0, hence ( P n ) ≥ = P n forsufficiently large n . Now, we have Hom Λ ( X, Λ( − n )) = Hom Λ ( X, Λ) − n = 0 for n ≫
0. Indeed, thisis certainly true if X is projective. For general X , take a surjection P ։ X from a projective module P . Then we have an injection Hom Λ ( X, Λ) ֒ → Hom Λ ( P, Λ) and our assertion follows from the case X isprojective. Therefore, we conclude that Hom Λ ( X, P n ) = 0, thus Hom Λ ( X, Ω n +1 Y ) = 0 for sufficienlylarge n . (cid:3) Proof of Proposition 7.1.
We verify the conditions (S1) and (S2).First we show (S2):
Hom Λ ( X, Ω n Y ) = 0 for almost all n for each X, Y ∈ CM Z (Λ). The case n ≫ n ≪
0. Since Λ is CM-finite, CM Z Λ has the ARduality, and we have D Hom ( X, Ω n Y ) ≃ Hom ( Y, Ω − n − τ X ) , hence the assertion follows from the caseof n ≫ CM Z Λ has only finitely many indecomposables up to suspension. Since Λ is offinite CM type, there exists 0 = n ∈ Z such that Ω n X ≃ X up to grade shift for any indecomposable X ∈ CM Z Λ. By (S2), Ω n X and X are not actually isomorphic in CM Z Λ. Therefore, CM Z Λ has onlyfintely many indecomposables up to Ω n , in particular up to Ω − .These assertions show that CM Z Λ is [1]-finite. (cid:3)
As an application of Theorem 5.3, we immediately obtain the following result.
USLANDER CORRESPONDENCE FOR TRIANGULATED CATEGORIES 13
Theorem 7.3.
Let k be algebraically closed and let Λ = L n ≥ Λ n is a positively graded Iwanaga-Gorenstein algebra such that each Λ n is finite dimensional over k . Suppose Λ is CM-finite and gl . dim Λ < ∞ . Then, the AR-quiver of CM Z Λ is Z ∆ for a disjoint union ∆ of some Dynkin diagrams of type A, Dand E. Moreover, CM Z Λ is triangle equivalent to D b ( mod kQ ) for any orientation Q of ∆ .Proof. The statement for the AR-quiver follows from Proposition 7.1 and Proposition 5.9. The triangleequivalence follows from Proposition 7.1 and Theorem 5.3. (cid:3)
A well-known class of commutative Gorenstein rings of finite representation type is given by simplesingularities. Here we assume that k is algebraically closed of characteristic 0. Then, they are classifiedup to isomorphism by the Dynkin diagrams for each d = dim Λ and have the form k [ x, y, z , . . . , z d ] / ( f )with( A n ) f = x + y n +1 + z + · · · + z d , ( n ≥ D n ) f = x y + y n − + z + · · · + z d , ( n ≥ E ) f = x + y + z + · · · + z d ,( E ) f = x + xy + z + · · · + z d ,( E ) f = x + y + z + · · · + z d ,see [LW, Chapter 9]. We admit any gradings on Λ so that each variable and f are homogeneous ofpositive degrees. Then, Λ is CM-finite (in the graded sense) since its completion b Λ at the maximal idealΛ > is CM-finite, that is, CM b Λ has only finitely many indecomposable objects [Yo1, Chapter 15].
Corollary 7.4.
Let k be an algebraically closed field of characteristic zero and Λ = k [ x, y, z , . . . , z d ] / ( f ) with f one of above. Give a grading on Λ so that each variable and f are homogeneous of positivedegrees. Then, the stable category CM Z Λ is triangle equivalent to the derived category D b ( mod kQ ) of thepath algebra kQ of a disjoint union Q of Dynkin quivers. We give several more examples. First we consider the case Λ is finite dimensional.
Example 7.5.
Let Λ = Λ n = k [ x ] / ( x n )with deg x = 1. Then, Λ is a finite dimensional self-injective algebra. In this case we have CM Z Λ = mod Z Λ.It is of finite representation type with indecomposable Λ-modules Λ i (1 ≤ i ≤ n ), and Λ = k has finiteglobal dimension. We can easily compute its AR-quiver (for n = 4) to be · · · ◦ (cid:31) (cid:31) ❄❄❄❄ Λ ( − (cid:31) (cid:31) ❄❄❄❄ Λ (cid:31) (cid:31) ❄❄❄❄ Λ (1) (cid:31) (cid:31) ❄❄❄❄ ◦ (cid:31) (cid:31) ❄❄❄❄❄ · · ·· · · ◦ (cid:31) (cid:31) ❄❄❄❄❄ ? ? ⑧⑧⑧⑧⑧ Λ ( − (cid:31) (cid:31) ❄❄❄❄ ? ? ⑧⑧⑧⑧ Λ ( − (cid:31) (cid:31) ❄❄❄❄ ? ? ⑧⑧⑧⑧ Λ (cid:31) (cid:31) ❄❄❄❄ ? ? ⑧⑧⑧⑧ Λ (1) (cid:31) (cid:31) ❄❄❄❄ ? ? ⑧⑧⑧⑧ ◦ (cid:31) (cid:31) ❄❄❄❄❄ ? ? ⑧⑧⑧⑧⑧ · · · ◦ ? ? ⑧⑧⑧⑧ Λ ( − ? ? ⑧⑧⑧⑧ Λ ( − ? ? ⑧⑧⑧⑧ Λ ? ? ⑧⑧⑧⑧ Λ (1) ? ? ⑧⑧⑧⑧ · · · , where the top of Λ i is in degree 0. We see that the AR-quiver of mod Z Λ is Z A n − . Consequently, wehave a triangle equivalence mod Z Λ ≃ D b ( mod kQ ) for a quiver Q of type A n − .The next one is a finite dimensional Iwanaga-Gorenstein algebra. Example 7.6.
Let Λ be the algebra presented by the following quiver with relations:1 x (cid:30) (cid:30) ❃❃❃❃❃❃❃❃❃❃ a o o y (cid:30) (cid:30) ❃❃❃❃❃❃❃❃❃❃ b o o c O O f o o d O O g o o e O O da = f c, eb = gd,ax = yg,cx = 0 , xd = 0 , xf = 0 , dy = 0 , by = 0 , ye = 0 , with deg x = deg y = 1 and all other arrows having degree 0. Then, it is an Iwanaga-Gorenstein algebraof dimension 1. (In fact, this is the 3-preprojective algebra [IO] of its degree 0 part.) We can compute the AR-quiver of mod Z Λ to be the following. ?>=<89:; (cid:31) (cid:31) ❄❄❄❄ (cid:31) (cid:31) ❄❄❄❄❄ WVUTPQRS
65 34 21 (cid:31) (cid:31)
GFED@ABC
15 (1) (cid:31) (cid:31) ❄❄❄ (cid:31) (cid:31) ❄❄❄❄❄ ?>=<89:; (cid:31) (cid:31) ❄❄❄❄ ? ? ⑧⑧⑧⑧⑧ ? ? ⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ (cid:31) (cid:31) ❄❄❄ WVUTPQRS ? ? (cid:31) (cid:31) ❄❄❄
65 34 2 (cid:31) (cid:31) ❄❄ ?>=<89:; ? ? ⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄ (cid:31) (cid:31) ❄❄❄ ? ? ⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄ ?>=<89:; · · · / / ? ? ⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄ GFED@ABC
21 65 / /
21 6 ? ? ⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄ GFED@ABC / / ? ? ⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ GFED@ABC
54 21 / / / / ? ? ⑧⑧ (cid:31) (cid:31) ❄❄❄ / / / / ? ? ⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ / /
65 5 34 2 / / ? ? ⑧⑧⑧ (cid:31) (cid:31) ❄❄
65 32 / /
65 3 ? ? ⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄ WVUTPQRS / / ? ? ⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄ WVUTPQRS
21 65 (1) / / · · · − ? ? ⑧⑧⑧⑧ ?>=<89:; ? ? ⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄ GFED@ABC
34 21 ? ? ⑧⑧ (cid:31) (cid:31) ❄❄❄
54 2 ? ? ⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄ ? ? ⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄
65 34 ? ? ⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ ?>=<89:; ? ? ⑧⑧⑧⑧ GFED@ABC ? ? ⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄ · · · ? ? ⑧⑧⑧⑧⑧ ?>=<89:; ? ? ⑧⑧⑧ ? ? ⑧⑧⑧⑧ ? ? ⑧⑧⑧⑧ ? ? ⑧⑧⑧ ? ? ⑧⑧⑧⑧ · · · Here, each module is graded so that its top is concentrated in degree 0, or equivalently, its lowest degree isat 0. We then compute the category CM Z Λ to be the circled modules and it is verified that the AR-quiverof CM Z Λ is · · · (cid:31) (cid:31) ❄❄❄❄ (cid:31) (cid:31) ❄❄❄❄❄ (cid:31) (cid:31) ❄❄❄ (cid:31) (cid:31) ❄❄❄ (cid:31) (cid:31) ❄❄❄❄ (cid:31) (cid:31) ❄❄❄❄❄ ? ? ⑧⑧⑧⑧ ? ? ⑧⑧⑧⑧
34 21 ? ? ⑧⑧ ? ? ⑧⑧⑧⑧⑧ ? ? ⑧⑧⑧⑧ · · · We see that this is Z A and consequently CM Z Λ ≃ D b ( mod kQ ) for a quiver Q of type A .We consider as a final example a Gorenstein order : let R = k [ x , . . . , x d ] be a polynomial ring. ANoetherian R -algebra Λ is an R -order if it is projective as an R -module. An R -order Λ is Gorenstein if Hom R (Λ , R ) is projective as a Λ-module. In this case, Cohen-Macaulay Λ-modules are Λ-modules whichare projective as R -modules. Example 7.7.
Let R = k [ x ] be a graded polynomial ring with deg x = 1 and letΛ = (cid:18) R R ( x n ) R (cid:19) . This is a Gorenstein R -order of dimension 1. Its indecomposable CM modules up to grade shift are givenby the row vectors M i = (cid:0) ( x i ) R (cid:1) for 0 ≤ i ≤ n , and M and M n are the projectives. We define thegradings on M i ’s so that their top (cid:0) k (cid:1) is in degree 0. Then, the AR-quiver of CM Z Λ (for n = 4) iscomputed to be · · · (cid:31) (cid:31) ❄❄❄❄❄ ◦ (cid:31) (cid:31) ❄❄❄❄ M ( − (cid:31) (cid:31) ❄❄❄❄ M (cid:31) (cid:31) ❄❄❄❄ · · ·◦ (cid:31) (cid:31) ❄❄❄❄ ? ? ⑧⑧⑧⑧⑧ M ( − (cid:31) (cid:31) ❄❄❄❄ ? ? ⑧⑧⑧⑧ M (cid:31) (cid:31) ❄❄❄❄ ? ? ⑧⑧⑧⑧ M (1) (cid:31) (cid:31) ❄❄❄❄ ? ? ⑧⑧⑧⑧ · · · (cid:31) (cid:31) ❄❄❄❄ ? ? ⑧⑧⑧⑧⑧ M ( − (cid:31) (cid:31) ❄❄❄❄ ? ? ⑧⑧⑧⑧ M (cid:31) (cid:31) ❄❄❄❄ ? ? ⑧⑧⑧⑧ M (1) (cid:31) (cid:31) ❄❄❄❄ ? ? ⑧⑧⑧⑧ · · · M ( − (cid:31) (cid:31) ❄❄❄❄ ? ? ⑧⑧⑧⑧ M (cid:31) (cid:31) ❄❄❄❄ ? ? ⑧⑧⑧⑧ M (1) (cid:31) (cid:31) ❄❄❄❄ ? ? ⑧⑧⑧⑧ ◦ (cid:31) (cid:31) ❄❄❄❄❄ ? ? ⑧⑧⑧⑧⑧ · · · ? ? ⑧⑧⑧⑧ M ? ? ⑧⑧⑧⑧ M (1) ? ? ⑧⑧⑧⑧ ◦ ? ? ⑧⑧⑧⑧⑧ · · · , where the upgoing arrows are natural inclusions, the downgoing arrows are the multiplications by x ,and the dotted lines indicate the AR-translations. By deleting the projective vertices, we see that theAR-quiver of CM Z Λ is Z A n − , and consequently CM Z Λ ≃ D b ( mod kQ ) for a quiver Q of type A n − . References [Am] C. Amiot,
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