Frobenius-Perron theory for projective schemes
J.M. Chen, Z.B. Gao, E. Wicks, J. J. Zhang, X-.H. Zhang, H. Zhu
aa r X i v : . [ m a t h . R A ] A p r FROBENIUS-PERRON THEORY FOR PROJECTIVE SCHEMES
J.M. CHEN, Z.B. GAO, E. WICKS, J. J. ZHANG, X-.H. ZHANG AND H. ZHU
Abstract.
The Frobenius-Perron theory of an endofunctor of a k -linear cat-egory (recently introduced in [CG]) provides new invariants for abelian andtriangulated categories. Here we study Frobenius-Perron type invariants forderived categories of commutative and noncommutative projective schemes.In particular, we calculate the Frobenius-Perron dimension for domestic andtubular weighted projective lines, define Frobenius-Perron generalizations ofCalabi-Yau and Kodaira dimensions, and provide examples. We apply thistheory to the derived categories associated to certain Artin-Schelter regularand finite-dimensional algebras. Introduction
The Frobenius-Perron dimension of an endofunctor of a category was introducedby the authors in [CG]. It can be viewed as a generalization of the Frobenius-Perrondimension of an object in a fusion category introduced by Etingof-Nikshych-Ostrik[ENO2005] in early 2000 (also see [EGNO2015, EGO2004, Nik2004]). It is shownin [CG] that the Frobenius-Perron dimension of either Ext or the suspension of atriangulated category is a useful invariant in several different topics such as embed-ding problem, Tame and wild dichotomy, complexity of categories. In particular,the Frobenius-Perron invariants have strong connections with the representationtype of a category [CG, ZZ].The definition of the Frobenius-Perron dimension of a category will be recalledin Section 2. In the present paper we continue to develop Frobenius-Perron theory,but we restrict our attention to the bounded derived category of coherent sheavesover a projective scheme. A projective scheme could be a classical (or commutative)one, or a noncommutative one in the sense of [AZ], or a weighted projective linein the sense of [GL]. We refer to Section 3 for some basics concerning weightedprojective lines.Our first goal is to understand the Frobenius-Perron dimension, denoted by fpd,of the bounded derived category of coherent sheaves over a weighted projectiveline, which is also helpful for understanding the Frobenius-Perron dimension of thepath algebra of an acyclic quiver of e A e D e E type via the derived equivalence given inLemma 2.1(2). Let X be a weighted projective line (respectively, a commutative ornoncommutative projective scheme). We use coh ( X ) to denote the abelian categoryof coherent sheaves over X and D b ( coh ( X )) to denote the bounded derived categoryof coh ( X ). Here is the main result in this topic. Mathematics Subject Classification.
Primary 16E35, 16E65, 16E10, Secondary 16B50.
Key words and phrases.
Frobenius-Perron dimension, derived category, projective scheme,weighted projective line, noncommutative projective scheme.
Theorem 0.1 (Theorem 2.13) . Let X be a weighted projective line that is eitherdomestic or tubular. Then fpd( D b ( coh ( X ))) = 1 . Note that Theorem 0.1 is a “weighted” version of [CG, Proposition 6.5(1,2)]. ByLemma 2.1(2), we obtain the Frobenius-Perron dimension of the bounded derivedcategory of finite dimensional representations of acyclic quivers of e A e D e E type.Our second goal is to introduce Frobenius-Perron (“fp” for short) version of someclassical invariants. We will focus on fp-analogues of two important invariants inprojective algebraic geometry, namely,Calabi-Yau dimension, andKodaira dimension.Let fp κ (respectively, fpcy) denote the fp version of the Kodaira dimension [Defini-tion 3.5(1)] (respectively, the Calabi-Yau dimension [Definition 3.1(3)]). Both aredefined for bounded derived categories of smooth projective schemes or more gener-ally triangulated categories with Serre functor. In algebraic geometry, a Calabi-Yauvariety has the trivial canonical bundle. In noncommutative algebraic geometry, a“skew Calabi-Yau” scheme may not have a trivial canonical bundle. Our fp versionof the Calabi-Yau dimension covers the case even when the canonical bundle is nottrivial. Below is one of the main results in this direction. Note that the definitionof fp κ is dependent on a chosen structure sheaf. Theorem 0.2 (Propositions 3.3 and 3.6) . Let X be a smooth projective schemeand T be the triangulated category D b ( coh ( X )) with structure sheaf O X . Then thefollowing hold. (1) fp κ ( T , O X ) = κ ( X ) . (2) fpcy( T ) = dim X . As a consequence, if X is Calabi-Yau, then fpcy T equalsthe Calabi-Yau dimension of X . In the noncommutative case we have
Theorem 0.3 (Theorem 4.5) . Let A be a noetherian connected graded Artin-Schelter Gorenstein algebra of injective dimension d ≥ and AS index ℓ that isgenerated in degree 1. Suppose that X := Proj A has finite homological dimensionand that the Hilbert series of A is rational. Let T be the bounded derived categoryof coh ( X ) and A be the image of A in Proj A . (1) fpcy( T ) = d − . (2) If ℓ > , then fp κ ( T , A ) = −∞ and fp κ − ( T , A ) = GKdim A − . (3) If ℓ < , then fp κ ( T , A ) = GKdim A − and fp κ − ( T , A ) = −∞ . (4) If ℓ = 0 , then fp κ ( T , A ) = fp κ − ( T , A ) = 0 . Similar results are proved for Piontkovski projective lines, see Section 4.Note that the definitions of fp Calabi Yau dimension and fp Kodaira dimensionmake sense for bounded derived category of left modules over a finite dimensionalgebra of finite global dimension. However, we don’t have a complete result forthat case. Some examples are given in Section 5.When we are working with finite dimensional algebras, it is well-known that theglobal dimension is not a derived invariant. By definition, fp Calabi-Yau dimensionis a derived invariant. This suggests that the fp Calabi-Yau dimension is nicerthan the global dimension in some aspects. So it is important to study this newinvariant. To start we ask the following questions.
ROBENIUS-PERRON THEORY 3
Question 0.4.
Let A be a finite dimensional algebra of finite global dimension andlet T be the derived category D b (Mod f.d − A ).(1) Is fpcy( T ) always finite?(2) What is the set of possible values of fpcy( T ) when A varies?(3) What is the set of possible values of fp κ ( T , A ) when A varies?More questions and some partial answers are given in Section 5. Some otherexamples are given in [Wi, ZZ].We have outlined several important applications of Frobenius-Perron invariantsin [CG]. Next we would like to mention one surprising application of the Frobenius-Perron curvature defined in [CG, Definition 2.3(4)]. Theorem 0.5. [ZZ, Corollary 0.6]
Suppose that the bounded derived categories ofrepresentations of two finite acyclic quivers are equivalent as tensor triangulatedcategories. Then the quivers are isomorphic.
This result is striking because it is well-known that, for two Dynkin quiverswith the same underlying Dynkin diagram, their derived categories are triangu-lated equivalent [BGP, Ha], even if the quivers are non-isomorphic. We hope thatdifferent Frobenius-Perron invariants will become effective tools in the study oftriangulated categories and monoidal (or tensor) triangulated categories.This paper is organized as follows. Some background material are provided inSection 1. In Section 2, we review some facts about weighted projective lines andprove Theorem 0.1. In Section 3 we introduce fp-version of Calabi-Yau dimensionand Kodaira dimension for Ext-finite triangulated categories with Serre functorand prove Theorem 0.2. In Section 4, fp Calabi-Yau dimension and fp Kodairadimension are studied for noncommutative projective schemes and Theorem 0.3 isproved there. In Section 5, some partial results, comments and examples are givenconcerning finite dimensional algebras. Sections 6 and 7 are appendices. The proofof Theorem 0.1 is dependent on some linear algebra computation given in Section6. This paper can be viewed as a sequel of [CG].1.
Preliminaries and definitions
Throughout let k be a base field that is algebraically closed. Let everything beover k .We are mainly interested in the derived category D b ( coh ( X )) where X is a smoothcommutative or noncommutative projective scheme, but most definitions work formore general pre-triangulated (or abelian) categories.Part of this section is copied from [CG].1.1. Spectral radius of a square matrix.
Let A be an n × n -matrix over complexnumbers C . The spectral radius of A is defined to be ρ ( A ) = max {| r | , | r | , · · · , | r n |} where { r , r , · · · , r n } is the complete set of eigenvalues of A . When each entry of A is a positive real number, ρ ( A ) is also called the Perron root or the
Perron-Frobeniuseigenvalue of A .In order to include the “infinite-dimensional” setting, we extend the definitionof the spectral radius in the following way. J.M. CHEN, Z.B. GAO, E. WICKS, J. J. ZHANG, X-.H. ZHANG AND H. ZHU
Let A = ( a ij ) n × n be an n × n -matrix with entries a ij in R ∞ := R ∪ {±∞} .Define A ′ = ( a ′ ij ) n × n where a ′ ij = a ij a ij = ±∞ ,x ij a ij = ∞ , − x ij a ij = −∞ . In other words, we are replacing ∞ in the ( i, j )-entry by a finite real number, called x ij , in the ( i, j )-entry. Or x ij are considered as function or a variable mapping R → R . Definition 1.1.
Let A be an n × n -matrix with entries in R + . The spectral radius of A is defined to be ρ ( A ) := lim inf all x ij →∞ ρ ( A ′ ) . See [CG, Remark 1.3 and Example 1.4].1.2.
Frobenius-Perron dimension of a quiver.Definition 1.2. [CG, Definition 1.6] Let Q be a quiver.(1) If Q has finitely many vertices, then the Frobenius-Perron dimension of Q is defined to be fpd Q := ρ ( A ( Q ))where A ( Q ) is the adjacency matrix of Q .(2) Let Q be any quiver. The Frobenius-Perron dimension of Q is defined to befpd Q := sup { fpd Q ′ } where Q ′ runs over all finite subquivers of Q .1.3. Frobenius-Perron dimension of an endofunctor.
Let C denote a k -linearcategory. For simplicity, we use dim( A, B ) for dim Hom C ( A, B ) for any two objects A and B in C . Here the second dim is dim k .The set of finite subsets of nonzero objects in C is denoted by Φ and the set ofsubsets of n nonzero objects in C is denoted by Φ n for each n ≥
1. It is clear thatΦ = S n ≥ Φ n . We do not consider the empty set as an element of Φ. Definition 1.3. [CG, Definition 2.1] Let φ := { X , X , · · · , X n } be a finite subsetof nonzero objects in C , namely, φ ∈ Φ n . Let σ be an endofunctor of C .(1) The adjacency matrix of ( φ, σ ) is defined to be A ( φ, σ ) := ( a ij ) n × n , where a ij := dim( X i , σ ( X j )) ∀ i, j. (2) An object M in C is called a brick [ASS, Definition 2.4, Ch. VII] ifHom C ( M, M ) = k . If C is a pre-triangulated category [Ne, Definition 1.1.2] with suspension Σ,an object M in C is called an atomic object if it is a brick and satisfiesHom C ( M, Σ − i ( M )) = 0 , ∀ i > . ROBENIUS-PERRON THEORY 5 (3) φ ∈ Φ is called a brick set (respectively, an atomic set ) if each X i is a brick(respectively, atomic) anddim( X i , X j ) = δ ij for all 1 ≤ i, j ≤ n . The set of brick (respectively, atomic) n -object subsetsis denoted by Φ n,b (respectively, Φ n,a ). We write Φ b = S n ≥ Φ n,b (respec-tively, Φ a = S n ≥ Φ n,a ). Definition 1.4. [CG, Definition 2.3] Retain the notation as in Definition 1.3, andwe use Φ b as the testing objects. When C is a pre-triangulated category, Φ b isautomatically replaced by Φ a unless otherwise stated.(1) The n th Frobenius-Perron dimension of σ is defined to befpd n ( σ ) := sup φ ∈ Φ n,b { ρ ( A ( φ, σ )) } . If Φ n,b is empty, then, by convention, fpd n ( σ ) = 0.(2) The Frobenius-Perron dimension of σ is defined to befpd( σ ) := sup n { fpd n ( σ ) } = sup φ ∈ Φ b { ρ ( A ( φ, σ )) } . (3) The Frobenius-Perron growth of σ is defined to befpg( σ ) := sup φ ∈ Φ b { lim sup n →∞ log n ( ρ ( A ( φ, σ n ))) } . By convention, log n −∞ .(4) The Frobenius-Perron curvature of σ is defined to befpv( σ ) := sup φ ∈ Φ b { lim sup n →∞ ( ρ ( A ( φ, σ n ))) /n } . In this paper, we only use Φ b and Φ a as the testing objects. But in principalone can use other testing objects, see Section 7. We continue to review definitionsfrom [CG]. Definition 1.5. [CG, Definition 2.7](1) Let A be an abelian category. The Frobenius-Perron dimension of A isdefined to be fpd A := fpd( E )where E := Ext A ( − , − ) is defined as in [CG, Example 2.6(1)]. The Frobenius-Perron theory of A is the collection { fpd m ( E n ) } m ≥ ,n ≥ where E n := Ext n A ( − , − ) is defined as in [CG, Example 2.6(1)].(2) Let T be a pre-triangulated category with suspension Σ. The Frobenius-Perron dimension of T is defined to befpd T := fpd(Σ) . (3) The fp-global dimension of T is defined to befpgldim T := sup { n | fpd(Σ n ) = 0 } . J.M. CHEN, Z.B. GAO, E. WICKS, J. J. ZHANG, X-.H. ZHANG AND H. ZHU
Fix an endofunctor σ of a category C . For a set of bricks B in C (or a set ofatomic objects when C is triangulated), we definefpd n | B ( σ ) = sup { ρ ( A ( φ, σ )) | φ := { X , · · · , X n } ∈ Φ n,b , and X i ∈ B ∀ i } . Let Λ := { λ } be a totally ordered set. We say a set of bricks B in C has a σ -decomposition { B λ } λ ∈ Λ (based on Λ) if the following holds.(1) B is a disjoint union S λ ∈ Λ B λ .(2) If X ∈ B λ and Y ∈ B δ with λ < δ , Hom C ( X, σ ( Y )) = 0.The following is [CG, Lemma 6.1]. Lemma 1.6. [CG, Lemma 6.1]
Let n be a positive integer. Suppose that B has a σ -decomposition { B λ } λ ∈ Λ . Then fpd n | B ( σ ) ≤ sup λ ∈ Λ ,m ≤ n { fpd m | B λ ( σ ) } . Frobenius-Perron theory of weighted projective lines
The main goal of this section is to recall some facts about weighted projectivelines and then to prove Theorem 0.1.2.1.
Weighted projective lines.
First we recall the definition and some basicsabout weighted projective lines. Details can be found in [GL, Section 1].For t ≥
1, let p := ( p , p , · · · , p t ) be a ( t + 1)-tuple of positive integers, calledthe weight sequence . Let D := ( λ , λ , · · · , λ t ) be a sequence of distinct points ofthe projective line P over k . We normalize D so that λ = ∞ , λ = 0 and λ = 1(if t ≥ S := k [ X , X , · · · , X t ] / ( X p i i − X p + λ i X p , i = 2 , · · · , t ) . The image of X i in S is denoted by x i for all i . Let L be the abelian group of rank1 generated by −→ x i for i = 0 , , · · · , t and subject to the relations p −→ x = · · · = p i −→ x i = · · · = p t −→ x t =: −→ c . The algebra S is L -graded by setting deg x i = −→ x i . The corresponding weightedprojective line , denoted by X ( p , D ) or simply X , is a noncommutative space whosecategory of coherent sheaves is given by the quotient category coh ( X ) := gr L − S gr L f.d. − S .
The weighted projective lines are classified into the following three classes: X is domestic if p is ( p, q ) , (2 , , n ) , (2 , , , (2 , , , (2 , , tubular if p is (2 , , , (3 , , , (2 , , , (2 , , , wild otherwise . In [Sc, Section 4.4], domestic (respectively, tubular, wild) weighted projective linesare called parabolic (respectively, elliptic, hyperbolic ). Let X be a weighted projec-tive line. A sheaf F ∈ coh ( X ) is called torsion if it is of finite length in coh ( X ). Let T or ( X ) denote the full subcategory of coh ( X ) consisting of all torsion objects. By ROBENIUS-PERRON THEORY 7 [Sc, Lemma 4.16], the category
T or ( X ) decomposes as a direct product of orthog-onal blocks(E2.0.1) T or ( X ) = Y x ∈ P \{ λ ,λ , ··· ,λ t } T or x × t Y i =0 T or λ i where T or x is equivalent to the category of nilpotent representations of the Jordanquiver (with one vertex and one arrow) over the residue field k x and where T or λ i is equivalent to the category of nilpotent representations over k of the cyclic quiverof length p i , see Example 2.3. A simple object in coh ( X ) is called ordinary simple (see [GL]) if it is the skyscraper sheaf O x of a closed point x ∈ P \ { λ , λ , · · · , λ t } .Let V ect ( X ) be the full subcategory of coh ( X ) consisting of all vector bundles.Similar to the elliptic curve case [BB, Section 4], one can define the concepts of degree , rank and slope of a vector bundle on a weighted projective line X ; detailsare given in [Sc, Section 4.7] and [LM, Section 2]. For each µ ∈ Q , let V ect µ ( X ) bethe full subcategory of V ect ( X ) consisting of all semistable vector bundles of slope µ . By convention, V ect ∞ ( X ) denotes T or ( X ). By [Sc, Comments after Corollary4.34], every indecomposable object in coh ( X ) is in [ µ ∈ Q ∪{∞} V ect µ ( X ) . Below we collect some nice properties of weighted projective lines. The definitionof a stable tube (or simply tube) was introduced in [Ri].
Lemma 2.1. [CG, Lemma 7.9]
Let X = X ( p , D ) be a weighted projective line. (1) coh ( X ) is noetherian and hereditary. (2) D b ( coh ( X )) ∼ = D b (Mod f.d. − k e A p,q ) if p = ( p, q ) ,D b (Mod f.d. − k e D n ) if p = (2 , , n ) ,D b (Mod f.d. − k e E ) if p = (2 , , ,D b (Mod f.d. − k e E ) if p = (2 , , ,D b (Mod f.d. − k e E ) if p = (2 , , . (3) Let S be an ordinary simple object in coh ( X ) . Then Ext X ( S , S ) = k . (4) fpd ( coh ( X )) ≥ . (5) If X is tubular or domestic, then Ext X ( X, Y ) = 0 for all X ∈ V ect µ ′ ( X ) and Y ∈ V ect µ ( X ) with µ ′ < µ . (6) If X is domestic, then Ext X ( X, Y ) = 0 for all X ∈ V ect µ ′ ( X ) and Y ∈ V ect µ ( X ) with µ ′ ≤ µ < ∞ . As a consequence, fpd(Σ | V ect µ ( X ) ) = 0 for all µ < ∞ . (7) Suppose X is tubular or domestic. Then every indecomposable vector bundle X is semi-stable. (8) Suppose X is tubular and let µ ∈ Q . Then each V ect µ ( X ) is a uniserial cate-gory. Accordingly indecomposables in V ect µ ( X ) decomposes into Auslander-Reiten components, which all are stable tubes of finite rank. In fact, forevery µ ∈ Q , V ect µ ( X ) ∼ = V ect ∞ ( X ) = T or ( X ) . Proof. (1) This is well-known, see [Le, Theorem 2.2](2) See [KLM, Proposition 5.1(i)] and [GL, 5.4.1].
J.M. CHEN, Z.B. GAO, E. WICKS, J. J. ZHANG, X-.H. ZHANG AND H. ZHU (3) Let S be an ordinary simple object which is of the form O x for some x ∈ P \ { λ , · · · , λ t } . Then S is a brick and Ext X ( S , S ) = Ext X ( O x , O x ) = k .(4) Follows from (3) by taking φ := {S} .(5) This is [Sc, Corollary 4.34(i)].(6) This is [Sc, Comments after Corollary 4.34]. The consequence is clear.(7) [GL, Theorem 5.6(i)].(8) See [Sc, Theorem 4.42] and [GL, Theorem 5.6(iii)]. (cid:3) Our main goal in this section is to prove Theorem 0.1. Eventually one shouldask the following question.
Question 2.2. [CG, Question 7.11] Let X be a weighted projective line. What isthe exact value of fpd n D b ( coh ( X )), for n ≥
1, in terms of other invariants of X ?2.2. Standard stable tubes [LS, Ri] . In this subsection we would like to under-stand the (standard) stable tubes in
T or λ i in the decomposition (E2.0.1), which isthe Auslander-Reiten quiver of the x -nilpotent (or x -torsion) representations of thealgebra in the following example. Example 2.3.
Let ξ be a primitive n th root of unity. Let T n be the algebra T n := k h g, x i ( g n − , gx − ξxg ) . This algebra can be expressed by using a group action. Let G be the group { g | g n = 1 } ∼ = Z / ( n )acting on the polynomial ring k [ x ] by g · x = ξx . Then T n is naturally isomorphicto the skew group ring k [ x ] ∗ G . Let −−−→ A n − denote the cycle quiver with n vertices,namely, the quiver with one oriented cycle connecting n vertices. It is also knownthat T n is isomorphic to the path algebra of the quiver −−−→ A n − .Let A be the category of finite dimensional left T n -modules that are x -torsion.In this subsection we will show that fpd( A ) = 1 [Corollary 2.12]. We start withsomewhat more general setting.Let A be any algebra and let Mod f.d. − A be the category of finite dimen-sional left A -modules. Let Γ(Mod f.d. − A ) denote the Auslander-Reiten quiver withAuslander-Reiten translation τ .Let C be a component of Γ(Mod f.d. − A ). We say C is a self-hereditary component of Γ(Mod f.d. − A ) if for each pair of indecomposable A -modules X and Y in C , wehave Ext A ( X, Y ) = 0.We now recall some results from the book [SS]. The definitions can be found in[SS]. Let φ := { E , · · · , E r } be a brick set in C . (In [SS], φ is called a finite familyof pairwise orthogonal bricks.) The extension category [SS, p.13] of φ , denoted by E , E A , or EX T A ( E , · · · , E r ), is defined to be the full subcategory of Mod f.d. − A whose nonzero objects are all the objects M such that there exists a chain ofsubmodules M = M ) M ) · · · ) M l = 0 , for some l ≥
1, with M i /M i +1 isomorphic to one of the bricks E , · · · , E r for all0 ≤ i < l . We say { E , · · · , E r } is a τ -cycle if τ ( E i ) = E i − for all i ∈ Z / ( r ).We will be using the notation introduced in [SS]. For example, E i [ j ] representssome uniserial object, which is nothing to do with the j th suspension of E i . ROBENIUS-PERRON THEORY 9
Theorem 2.4. [SS, Lemma 2.1 and Theorem 2.2 in Ch. X]
Let φ := { E , · · · , E r } ,with r ≥ , be a brick set on Mod f.d. − A . Suppose that φ is a τ -cycle and a self-hereditary family [SS, p.14] . Then the extension category E is an abelian categorywith the following properties. (1) For each pair ( i, j ) , with ≤ i ≤ r and j ≥ , there exist a uniserial object E i [ j ] of E -length l E ( E i [ j ]) = j in the category E , and homomorphisms u ij : E i [ j − −→ E i [ j ] , and p ij : E i [ j ] −→ E i +1 [ j − , for j ≥ , such that we have two short exact sequences in Mod f.d. − A −→ E i [ j − u ij −−−→ E i [ j ] p ′ ij −−−→ E i + j − [1] −→ , −→ E i [1] u ′ ij −−−→ E i [ j ] p ij −−−→ E i +1 [ j − −→ , where p ′ ij = p i + j − , ◦ · · · ◦ p ij and u ′ ij = u ij ◦ · · · ◦ u i . Moreover, for each j ≥ , there exists an almost split sequence → E i [ j − p i,j − u ij −−−−−−−−→ E i +1 [ j − ⊕ E i [ j ] ( u i +1 ,j − p ij ) −−−−−−−−−→ E i +1 [ j − → , in Mod f.d. − A , where we set E i [0] = 0 and E i + kr [ m ] = E i [ m ] , for m ≥ and all k ∈ Z . (2) The indecomposable uniserial objects E i [ j ] , with i ∈ { , · · · , r } and j ≥ ,of the category E , connected by the homomorphisms u ij : E i [ j − → E i [ j ] and p ij : E i [ j ] → E i +1 [ j − , form the infinite diagram presented below. (cid:0)(cid:0)✒ (cid:0)(cid:0)✒ (cid:0)(cid:0)✒❅❅❘ ❅❅❘ ❅❅❘(cid:0)(cid:0)✒ ❅❅❘(cid:0)(cid:0)✒ (cid:0)(cid:0)✒❅❅❘ ❅❅❘(cid:0)(cid:0)✒ ❅❅❘ · · · · · ·· · · · · · · · · · · ·· · · · · · · · · E i [1] E i [2] E i [ j − E i [ j − E i [ j ] E i +1 [1] E i +1 [ j − E i +1 [ j − E i +2 [ j − E i + j − [1] E i + j − [2] E i + j − [3] E i + j − [1] E i + j − [2] E i + j − [1] u i p i u i + j − , p i + j − , u i + j − , p i + j − , u i + j − , p i + j − , u i,j − p i,j − u i +1 ,j − p i +1 ,j − u i,j p i,j ❅❅❘ · · · ❅❅❘ · · · ❅❅❘ · · · ❅❅❘ · · · ❅❅❘ · · · (cid:0)(cid:0)✒ · · · (cid:0)(cid:0)✒ · · · (cid:0)(cid:0)✒ · · · (cid:0)(cid:0)✒ · · · (cid:0)(cid:0)✒ · · ·· · · (3) Ext A ( X, Y ) = 0 , for each pair of objects X and Y of E . Theorem 2.5. [SS, Theorem 2.6 in Ch. X]
Retain the hypothesis as in Theorem2.4. Then the abelian category E has the following properties. (1) Every indecomposable object M of the category E is uniserial and is of theform M ∼ = E i [ j ] , where i ∈ { , · · · , r } and j ≥ . (2) The collection of indecomposable objects forms a self-hereditary component,denoted by T E , of Γ(Mod f.d. − A ) . (3) The component T E is a standard stable tube of rank r [SS, Definition 1.1 inCh. X] . (4) The objects E , · · · , E r form the complete set of objects lying on the mouth [SS, Definition 1.2 in Ch. X] of the tube T E . Let T be the T E defined as in Theorem 2.5. Let D := Hom k ( − , k ) be the usual k -linear dual. Corollary 2.6. [SS, Corollary 2.7 in Ch. X]
Retain the hypothesis as in Theorem2.4. (1)
The only homomorphisms between two indecomposable modules in T are k -linear combinations of compositions of the homomorphisms u ij , p ij , and theidentity homomorphisms, and they are only subject to the relations arisingfrom the almost split sequences in Theorem 2.4(1). (2) Given i ∈ { , · · · , r } and j ≥ , we have (2a) End A ( E i [ j ]) ∼ = k [ t ] / ( t m ) , for some m ≥ , (2b) End A ( E i [ j ]) ∼ = k if and only if j ≤ r , and (2c) Ext A ( E i [ j ] , E i [ j ]) ∼ = D Hom A ( E i [ j ] , τ E i [ j ]) = 0 if and only if j ≤ r − . (3) If the tube T is homogeneous ( namely, r = 1) , then Ext A ( M, M ) = 0 , forany indecomposable M in C . Brick objects in T are determined by Corollary 2.6(2b). To work out all bricksets in T , we need to understand the Hom between brick objects. Part (2) of thefollowing theorem describes these Homs. Theorem 2.7.
Let T be a standard stable tube of rank r as used in Theorem 2.5and Corollary . Keep the notation as above and assume ≤ i, j, i ′ , j ′ ≤ r . Thenthe following hold. (1) End T ( E i [ j ]) ∼ = k . (2) Hom T ( E i [ j ] , E i ′ [ j ′ ]) = 0 if and only if ( i ′ , j ′ ) satisfies one of the followingconditions: (2a) i ≤ i ′ ≤ i + j − and i + j ≤ i ′ + j ′ , (2b) i ′ ≤ i + j − − r and i + j ≤ i ′ + j ′ + r . Here, if i + j − − r < , then { ≤ i ′ ≤ i + j − − r } = ∅ . Moreover, if
Hom T ( E i [ j ] , E i ′ [ j ′ ]) = 0 , then Hom T ( E i [ j ] , E i ′ [ j ′ ]) ∼ = k . Proof. (1) See Corollary 2.6(2).(2) Since T is a standard stable tube, it is a mesh category [SS, Definition2.4 in Ch. X]. Moreover, by Corollary 2.6(a), the only homomorphisms betweentwo indecomposable modules in T are k -linear combinations of compositions ofthe homomorphisms u ij , p ij , and the identity homomorphisms, which subject tothe relations arising from the almost split sequences in Theorem 2.4(a). By meshrelationship [SS, Definition 2.4 in Ch. X], we obtain the description (2a) and (2b)for all objects E i ′ [ j ′ ] that satisfy Hom T ( E i [ j ] , E i ′ [ j ′ ]) = 0.Moreover, if j ′ < j , Hom T ( E i [ j ] , E i ′ [ j ′ ]) is generated by composition morphisms u i ′ ,j ′ · · · u i ′ ,i + j − i ′ +1+ l · · · u i ′ ,i + j − i ′ +2 u i ′ ,i + j − i ′ +1 p i ′ − ,i + j − i ′ +1 · · · p i + k,j − k · · · p i +1 ,j − p ij , where 0 ≤ k ≤ i ′ − i, ≤ l ≤ ( i ′ + j ′ ) − ( i + j ) −
1. Here, if i + k > r , then theindex i + k means i + k − r .If j ′ ≥ j , Hom T ( E i [ j ] , E i ′ [ j ′ ]) is generated by p i ′ − ,j ′ +1 · · · p i + l,i ′ + j ′ − i − l · · · p i +1 ,i ′ + j ′ − i − p i,i ′ + j ′ − i u i,i ′ + j ′ − i · · · u i,j + k · · · u i,j +2 u i,j +1 , ROBENIUS-PERRON THEORY 11 where 0 ≤ k ≤ ( i ′ + j ′ ) − ( i + j ) , ≤ l ≤ i ′ − i − . Here, if i + l > r , then the index i + l means i + l − r . Therefore Hom T ( E i [ j ] , E i ′ [ j ′ ]) ∼ = k . (cid:3) Part (1) of Corollary 2.8 next is just a re-interpretation of Theorem 2.7(2).
Corollary 2.8.
Let T be a standard stable tube of rank r . Keep the notation asabove and put E i [ j ] , ≤ i, j ≤ r in order E [1] , E [1] , · · · , E r [1]; E [2] , E [2] , · · · , E r [2]; · · · ; E [ r ] , E [ r ] , · · · , E r [ r ] , and denote them by X , · · · , X n where n = r . (1) The n × n matrix (dim Hom T ( X j , X i )) n × n has the following form (E2.8.1) P P P P · · · P r − P P i =0 P i P i =1 P i P i =2 P i · · · r − P i = r − P i P P i =0 P i P i =0 P i P i =1 P i · · · r − P i = r − P i P P i =0 P i P i =0 P i P i =0 P i · · · r − P i = r − P i · · · · · · · · · · · · P P i =0 P i P i =0 P i P i =0 P i · · · r − P i =0 P i where P = · · · · · · · · · · · · r × r and where P is the identity matrix I r × r of order r . (2) The n × n -matrix (cid:0) dim Ext T ( X j , X i ) (cid:1) n × n has the following form (E2.8.2) P r − P r − P r − P r − · · · P r − P r − r − P i = r − P i r − P i = r − P i r − P i = r − P i · · · r − P i = r − P i P r − r − P i = r − P i r − P i = r − P i r − P i = r − P i · · · r − P i = r − P i P r − r − P i = r − P i r − P i = r − P i r − P i = r − P i · · · r − P i = r − P i · · · · · · · · · · · · P P i =0 P i P i =0 P i P i =0 P i · · · r − P i =0 P i . Proof. (1) This follows from Theorem 2.7(2).(2) The assertion follows from part (1) and the Serre dualityExt T ( E i [ j ] , E i ′ [ j ′ ]) ∼ = D Hom T ( E i ′ [ j ′ ] , τ E i [ j ]) = D Hom T ( E i ′ [ j ′ ] , E i − [ j ]) . Some detailed matching of entries is omitted. (cid:3)
We use the following example to illustrate the results in Corollary 2.8.
Example 2.9.
Let T be a standard stable tube of rank 3: (cid:0)(cid:0)✒ (cid:0)(cid:0)✒ (cid:0)(cid:0)✒❅❅❘ ❅❅❘ ❅❅❘❅❅❘ (cid:0)(cid:0)✒ ❅❅❘ (cid:0)(cid:0)✒ ❅❅❘ (cid:0)(cid:0)✒(cid:0)(cid:0)✒ (cid:0)(cid:0)✒ (cid:0)(cid:0)✒❅❅❘ ❅❅❘ ❅❅❘❅❅❘ (cid:0)(cid:0)✒ ❅❅❘ (cid:0)(cid:0)✒ ❅❅❘ (cid:0)(cid:0)✒ E [1] E [1] E [1] E [1] E [2] E [2] E [2] E [3] E [3] E [3] E [3] E [4] E [4] E [4]................................. ................................. · · · · · · · · · Put E i [ j ] , ≤ i, j ≤ E [1] , E [1] , E [1]; E [2] , E [2] , E [2]; E [3] , E [3] , E [3] . Denote this list as X , · · · , X . Then we have Table 1.
Hom T ( X j , X i ) Hom T ( X j , X i ) X j = E [1] E [1] E [1] E [2] E [2] E [2] E [3] E [3] E [3] X i = E [1] k k k E [1] 0 k k k E [1] 0 0 k k k E [2] k k k k k E [2] 0 k k k k k E [2] 0 0 k k k k k E [3] k k k k k k E [3] 0 k k k k k k E [3] 0 0 k k k k k k Table 2.
Ext T ( X j , X i ) Ext T ( X j , X i ) X j = E [1] E [1] E [1] E [2] E [2] E [2] E [3] E [3] E [3] X i = E [1] 0 k k k E [1] 0 0 k k k E [1] k k k E [2] 0 0 k k k k k E [2] k k k k k E [2] 0 k k k k k E [3] k k k k k k E [3] 0 k k k k k k E [3] 0 0 k k k k k k ROBENIUS-PERRON THEORY 13
The corresponding Hom-dimension and Ext -dimension matrices are(dim Hom T ( X j , X i )) × = P P P P P i =0 P i P i =1 P i P P i =0 P i P i =0 P i and (cid:0) dim Ext T ( X j , X i ) (cid:1) × = P P P P P i =1 P i P i =1 P i P P i =0 P i P i =0 P i where P = . The next lemma shows that if H ( i , · · · , i s ) is a principal submatrix of (E2.8.1)such that H ( i , · · · , i s ) = I s × s , then the corresponding principal submatrix of(E2.8.2) (with the same rows and columns) has spectral radius at most 1. Lemma 2.10 (Lemma 6.4) . Retain the above notation. Suppose the
Hom -matrixis given as in (E2.8.1) and
Ext -matrix is given as in (E2.8.2) , then ρ ( A ( φ )) ≤ for every brick set φ . The proof of Lemma 2.10 is given in Appendix A. Note that (E2.8.1)-(E2.8.2)are in fact the transpose of the usual Hom and Ext-matrices. By [CG, Lemma3.7] (by considering the opposite category), Lemma 2.10 holds for the transposematrices of (E2.8.1)-(E2.8.2) too.
Theorem 2.11.
Let E and T be as in Theorem 2.5. Then fpd E = 1 .Proof. By Corollary 2.6(2b), all brick objects in E are E i [ j ] for all i ∈ Z / ( r ) and1 ≤ j ≤ r . We can determine all brick sets by using the matrix in Corollary 2.8(1).For each brick set, its Ext -matrix M was determined by using (E2.8.2). By Lemma2.10, ρ ( M ) ≤
1. On the other hand, let φ = { E [1] , · · · , E r [1] } , then M = P r − and hence fpd E = ρ ( A ( φ, Ext )) = 1. Therefore the assertion follows. (cid:3) We have an immediate consequence. Let T r be the algebra in Example 2.3. Corollary 2.12.
Let A be the category of finite dimensional left x -nilpotent T r -modules. Then fpd A = 1 .Proof. It is well-known that A is equivalent to the category E of rank r . To see thiswe set deg x = 1 and deg g = 0. Then the degree zero component of T r is isomorphicto k ⊕ r with primitive idempotents { e , · · · , e r } . Under this setting, E i [ j ] is identifywith ( T r / ( x i )) e j for all i, j . Now the result follows from Theorem 2.11. (cid:3) Proof of Theorem 0.1.
Now we are ready to show Theorem 0.1.
Theorem 2.13.
Let X be a domestic or tubular weighted projective line. Then fpd D b ( coh ( X )) = 1 . Proof.
By Lemma 2.1(4), it suffices to show that fpd( D b ( coh ( X ))) ≤
1. By [CG,Theorem 3.5(4)], it is enough to show thatfpd( σ ) = fpd( coh ( X )) ≤ σ is Ext X ( − , − ).By [Sc, Corollary 4.34(iii)], every brick (or indecomposable) object is semistable.By Lemma 2.1(5), the class { V ect µ ( X ) } µ ∈ Q ∪{∞} is a σ -decomposition (see thedefinition before Lemma 1.6). By Lemma 1.6, it is enough to show the claim thatfpd | V ect µ ( X ) ( σ ) ≤ µ .Case 1: X is domestic. If µ is finite, then fpd | V ect µ ( X ) ( σ ) = 0 by Lemma2.1(6). If µ = ∞ , then, by (E2.0.1), V ect ∞ ( X ) := T or ( X ) has a decompositioninto Auslander-Reiten components, which all are tubes of finite rank. By Theorem2.11, fpd | T or ( X ) ( σ ) = 1. The claim follows.Case 2: X is tubular. By Lemma 2.1(8), V ect µ ( X ) ∼ = V ect ∞ ( X ) = T or ( X )for all µ . Then the proof of Case 1 applies. Therefore the claim follows. (cid:3) As usual we use O X for the structure sheaf of X [GL, Sect. 1.5]. Proposition 2.14.
Let X be a weighted projective line of wild type. Then fpd D b ( coh ( X )) ≥ dim Hom X ( O X , O X ( −→ ω )) where −→ ω is the dualizing element [GL, Sec. 1.2] .Proof. Let φ = {O X } which is a brick and atomic object. Then, by definition andSerre duality [GL, Theorem 2.2],fpd D b ( coh ( X )) ≥ ρ ( A ( φ, Ext )) = dim Ext X ( O X , O X )= dim Hom X ( O X , O X ( −→ ω )) . (cid:3) Dimension theory for classical projective schemes
The aim of this section is to introduce Frobenius-Perron versions of two im-portant and related invariants – Calabi-Yau dimension and Kodaira dimension of D b ( coh ( X )) where X is a smooth projective scheme.3.1. A result from [CG] . Let X be a smooth (irreducible) projective scheme over C of positive dimension. By [CG, Proposition 6.5 and 6.7],fpd( D b ( coh ( X ))) = ( X is P or an elliptic curve , ∞ otherwise.3.2. Calabi-Yau dimension.
Recall from [Ke2, Section 2.6] that if a Hom-finitecategory C has a Serre functor S , then there is a natural isomorphismHom C ( X, Y ) ∗ ∼ = Hom C ( Y, S ( X ))for all X, Y ∈ C . A (pre-)triangulated Hom-finite category C with Serre functor S is called n -Calabi-Yau if there is a natural isomorphism S ∼ = Σ n =: [ n ] ROBENIUS-PERRON THEORY 15 where Σ is the suspension of C . In this case n is called the Calabi-Yau dimensionof C . (In [Ke2, Section 2.6] it is called weakly n -Calabi-Yau .) More generally, C iscalled a fractional Calabi-Yau category if there is an m > S m ∼ = Σ n = [ n ]for some n , see [vR, p.2708] and [Ku, Definition 1.2]. In this case we say C hasCalabi-Yau dimension nm . Abelian hereditary fractionally Calabi-Yau categoriesare classified in [vR]. One key property of Calabi-Yau varieties is that the canon-ical bundle of these varieties are trivial. However, our definition of fp Calabi-Yaudimension (see Definition 3.1 below) applies to projective schemes that do not havethe trivial canonical bundle.Throughout the rest of this section, let T be a Hom-finite (pre-)triangulatedcategory with Serre functor S . We will define a version of (fractional) Calabi-Yaudimension for T which is not necessarily a (fractional) Calabi-Yau category.Recall from Definition 1.4(3) that the Frobenius-Perron growth of a functor σ isdefined to be fpg( σ ) := sup φ ∈ Φ a { lim sup n →∞ log n ( ρ ( A ( φ, σ n ))) } . By convention, log n −∞ . Similarly, we define a slightly modified version of fpgas follows, which is used to define the fp Calabi-Yau dimension. Definition 3.1.
Let σ be an endofunctor of T with Serre functor S .(1) The lower Frobenius-Perron growth of σ is defined to befpg( σ ) := sup φ ∈ Φ a n lim inf n →∞ log n ( ρ ( A ( φ, σ n ))) o . By convention, log n −∞ .(2) The spectrum of T is defined to be Sp ( T ) := (cid:8) ( m, n ) ∈ Z × | fpg( S m ◦ Σ − n ) > −∞ (cid:9) . (3) The fp Calabi-Yau dimension of T is defined to befpcy( T ) := lim M →∞ ( sup | m |≥ M n nm | ( m, n ) ∈ Sp ( T ) o) . Next we show that the fp Calabi-Yau dimension exists for various cases.
Lemma 3.2.
Suppose T is a pre-triangulated category satisfying the following con-ditions: (a) T is Ext-finite [BV, Definition 2.1] , namely, for all objects X, Y ∈ T , X s ∈ Z dim Hom T ( X, Σ s ( Y )) < ∞ . (b) T has a Serre functor S . (c) T is fractional Calabi-Yau of dimension d = a/b ∈ Q .Then the following holds. (1) Sp ( T ) ⊆ ( b, a ) Q . (2) If T contains at least one atomic object, there exists w ∈ N such that ( bwt, awt ) ∈ Sp ( T ) for all t ∈ Z . (3) Under the hypothesis of part (2), we have fpcy( T ) = d . Proof. (1) Let ( m, n ) be a pair of integers that is not in ( b, a ) Q , and let σ = S m ◦ Σ − n . Since T is fractional Calabi-Yau of dimension a/b , we can assume that S b ◦ Σ − a is the identity functor. Then, for each t , σ tb = Σ t ( ma − nb ) where ma − nb = 0. Byhypothesis (a), for any two objects X and Y in T , Hom T ( X, Σ t ( ma − nb ) Y ) = 0 for | t | ≫
0. Then, for every atomic set φ , it implies that A ( φ, σ tb ) = 0 for | t | ≫ tb ( ρ ( A ( φ, σ tb ))) = −∞ for all | t | ≫
0. This implies, by definition, thatfpg( σ ) = −∞ , or ( m, n ) Sp ( T ). The assertion follows.(2) Let φ be the set of a single atomic object X in T . Replacing ( b, a ) by( bw, aw ) for some positive integer w if necessary, we can assume that σ := S b ◦ Σ − a is the identity functor. Then A ( φ, σ n ) is the 1 × I for all n .Then log n ( ρ ( A ( φ, σ n ))) = 0 for all n . This implies that fpg( σ ) = 0 > −∞ or( b, a ) ∈ S ( T ). Similarly, one sees that ( bt, at ) ∈ S ( T ) for all integer t . Theassertion follows.(3) This follows from the definition and parts (1,2). (cid:3) The next proposition is part (2) of Theorem 0.2, which shows that fpcy is indeeda generalization of Calabi-Yau dimension.
Proposition 3.3.
Let X be a smooth irreducible projective variety of dimension d ∈ N and let T be D b ( coh ( X )) . Then fpcy( T ) = d .Proof. When d = 0, then T = D bf.d (Vect k ). It is easy to see that the Serre functor S is the identity. So the assertion is easily shown. Now we assume that d > Sp ( T ) = { ( t, td ) | t ∈ Z } .By [BO, (7)], the Serre functor S is equal to − ⊗ X ω X [ d ] where d is the dimensionof X and ω X is the canonical bundle of X . Let σ = S ◦ Σ − d . Then σ is the functor − ⊗ X ω X . Let O a be the skyscraper sheaf of a closed point a ∈ X . Then it is anatomic object and σ ( O a ) ∼ = O a . Let φ = {O a } . Then A ( φ, σ n ) is the 1 × I . This implies that log n ( ρ ( A ( φ, σ n ))) = 0 and that fpg( σ ) = 0 > −∞ .Therefore (1 , d ) ∈ S ( T ). Similarly, one sees that ( t, dt ) ∈ S ( T ) for all t ≥ m, n ) ∈ Z × \ { ( t, td ) | t ∈ Z } . Let σ = S m ◦ Σ − n .We need to show that that fpg( σ ) = −∞ . Note that σ = − ⊗ X ω ⊗ m X ◦ Σ − n + dm where − n + dm = 0. Since coh ( X ) has global dimension d , for all objects A and B in T , Hom T ( A, σ t ( B )) = Hom T ( A, ( B ⊗ X ω tm X )[ t ( − n + dm )]) = 0for all t ≫
0. This implies that fpg( σ ) = −∞ as required. (cid:3) Kodaira dimension.
First we review the classical definition of the Kodairadimension.
Definition 3.4. [La, Definition 2.1.3 and Example 2.1.5] Let X be a smooth pro-jective variety and let ω X be the canonical bundle of X .(1) The Kodaira dimension of X is defined to be κ ( X ) := lim n →∞ log n (cid:0) dim H ( X , ω ⊗ n X ) (cid:1) . (2) More generally, for a line bundle M , the Kodaira-Iitaka dimension of M isdefined to be κ ( X , M ) := lim n →∞ log n (cid:0) dim H ( X , M ⊗ n ) (cid:1) . ROBENIUS-PERRON THEORY 17 (3) The anti-Kodaira dimension of X is defined to be κ − ( X ) := κ ( X , ω − X ) . The anti-Kodaira dimension of a scheme was defined in [Sa]. It is classicaland well-known that κ ( X ) , κ ( X , M ) ∈ {−∞ , , , · · · , dim X } and that there are0 < c < c such that,(E3.4.1) c n κ ( X , M ) ≤ dim Hom X ( O X , M ⊗ n ) ≤ c n κ ( X , M ) ∀ n ≫ . See [La, Corollary 2.1.37]. By Proposition 3.3, dim X = fpcy( T ), which suggeststhe following definition.In the following we use the order t < t if t divides t . Since we are mainlyinterested in commutative and noncommutative projective schemes, T is equippedwith the structure sheaf, denoted by O . Definition 3.5.
Let T be a pre-triangulated category and T ∗ denote the pair ( T , O )where O is a given special object in T . Suppose that T is Hom-finite with Serrefunctor S and that fpcy( T ) = d = a/b for some integers a, b .(1) The fp Kodaira dimension of T ∗ is defined to befp κ ( T ∗ ) := lim t (cid:26) lim sup n →∞ dim Hom T ( O , ( S bnt ◦ Σ − ant )( O )) (cid:27) where the first limit ranges over all positive integers t with order < asdefined before the definition.(2) The fp anti-Kodaira dimension of T ∗ is defined to befp κ − ( T ∗ ) := lim t (cid:26) lim sup n →∞ dim Hom T ( O , ( S − bnt ◦ Σ ant )( O )) (cid:27) where the first limit ranges over all positive integers t with order < asdefined before the definition.The following proposition justifies the above definition. Proposition 3.6.
Let X be a smooth irreducible projective variety of dimension d ∈ N and let T be D b ( coh ( X )) with structure sheaf O := O X . Then fp κ ( T ∗ ) = κ ( X ) and fp κ − ( T ∗ ) = κ − ( X ) . Proof.
By Proposition 3.3, fpcy( T ) = d = dim X . So we take b = 1 and a = d inDefinition 3.5. By (E3.4.1), for each t ≥ n →∞ dim Hom T ( O , ( S btn ◦ Σ − ant )( O )) = κ ( X ) . The first assertion follows by the definition.The proof for anti-Kodaira dimension is similar. (cid:3)
Theorem 0.2 follows from Propositions 3.3 and 3.6.
Remark 3.7.
Assume the hypothesis of Lemma 3.2. Then one can check easilythat(E3.7.1) fp κ ( T ∗ ) = fp κ − ( T ∗ ) = 0 . So abstractly (E3.7.1) should be part of the definition of a fractional Calabi-Yauvariety (even in the noncommutative setting).
Proposition 3.8. If T is a triangulated category such that either fp κ ( T ∗ ) = ∞ or fp κ − ( T ∗ ) = ∞ or fpcy( T ) = ∞ or −∞ , then T ∗ is not triangulated equivalent tothe bounded derived category of a smooth projective scheme.Proof. By definition, fpcy is an invariant of a triangulated category, and fp κ is aninvariant of a triangulated category with O . The assertion follows from Propositions3.3 and 3.6. (cid:3) Invariants of noncommutative projective schemes
In this section we study fp Calabi-Yau dimension and fp Kodaira dimension ofnoncommutative projective schemes in the sense of [AZ]. An algebra A is said tobe connected graded over k if A = k ⊕ A ⊕ A ⊕ · · · with A i A j ⊆ A i + j for all i, j, ∈ N . Let A be a noetherian connected graded algebra. The noncommutativeprojective scheme associated to A is denoted by X := Proj A , see [AZ] for thedetailed definition of a noncommutative projective scheme. Let coh ( X ) be thecategory of noetherian objects in Proj A and let T be the triangulated category D b ( coh ( X )). Here is a restatement of a nice result of Bondal-Van den Bergh [BV,Theorem 4.2.13]. Theorem 4.1. [BV, Theorem 4.2.13]
Suppose A is noetherian and has a balanceddualizing complex and that Proj A has finite homological dimension. Then T has aSerre functor. One special class of connected graded algebras are the Artin-Schelter regularalgebras (see the next definition).
Definition 4.2. [AS] Let A be a connected graded algebra over the base field k . Wesay A is Artin-Schelter Gorenstein (or
AS Gorenstein ) if the following conditionshold:(a) A has finite injective dimension d on both sides,(b) Ext iA ( k , A ) = Ext iA op ( k , A ) = 0 for all i = d where k = A/A ≥ , and(c) Ext dA ( k , A ) ∼ = k ( ℓ ) and Ext dA op ( k , A ) ∼ = k ( ℓ ) for some integer ℓ . This integer ℓ is called the AS index of A .If moreover(d) A has finite global dimension,then A is called Artin-Schelter regular (or
AS regular ).We collect some well-known facts below. Let π : Gr A → Proj A be the canonicalquotient functor. By abuse of notation, we also apply π to some graded A -bimodulessuch as µ A in Lemma 4.3 below. Lemma 4.3.
Let A be a noetherian connected graded algebra, let X be Proj A , andlet T be D b ( coh ( X )) . (1) [Ye, Corollary 4.14] If A is Artin-Schelter Gorenstein, then A has a balanceddualizing complex. (2) [YZ, Corollary 4.3] Suppose A is Artin-Schelter Gorenstein such that X hasfinite homological dimension. Let d = injdim A , ℓ be the AS index of A and µ be the Nakayama automorphism of A . Then the Serre functor of T is − ⊗ O π ( µ A )( − ℓ )[ d − . ROBENIUS-PERRON THEORY 19
Let M be a locally finite Z -graded module or vector space. The Hilbert series of M is defined to be H M ( t ) := X n ∈ Z dim M n t n . Let A be a graded algebra and s be a positive integer. The s th Veronese subalgebra of A is defined to be A ( s ) := ⊕ n ∈ Z A sn . The following lemma is well-known.
Lemma 4.4.
Let A be a noetherian connected graded algebra generated in degree1. Let s be a positive integer. (1) A is a finitely generated module over A ( s ) on both sides. (2) GKdim A = GKdim A ( s ) . (3) If the Hilbert series of A is a rational function, then so is the Hilbert seriesof A ( s ) . (4) If the Hilbert series of A is a rational function, then lim sup n →∞ log n (dim A n ) = GKdim A − . (5) If A is an Artin-Schelter regular algebra, then the Hilbert series of A is arational function.Proof. (1) Connected graded noetherian algebras are finitely generated. So A isgenerated by ⊕ s − i =0 A i over A ( s ) on both sides.(2) It follows from part (1) and [MR, Proposition 8.2.9(i)].(3) This follows from the fact that H A ( t ) = 1 s s − X i =0 H A ( ξ i t )where ξ is an s th primitive root of unity.(4) When H A ( t ) is a rational function, dim A n is a multi-polynomial function of n in the sense of [Zh, p.399]. Say d is the degree of the multi-polynomial functiondim A n of n . Then GKdim A = d + 1 by using [Zh, (E7)]. This implies thatlim sup n →∞ log n (dim A n ) = d = GKdim A − . (5) This is [StZ, Proposition 3.1]. (cid:3) Theorem 4.5.
Let A be a noetherian connected graded Artin-Schelter Gorensteinalgebra of injective dimension d ≥ that is generated in degree 1. Suppose that X := Proj A has finite homological dimension. Let T be the bounded derived categoryof coh ( X ) . In parts (2,3,4,5,6), we further assume that the Hilbert series of A isrational. Let ℓ be the AS index of A . (1) fpcy( T ) = d − . (2) If ℓ > , then fp κ ( T ∗ ) = −∞ and fp κ − ( T ∗ ) = GKdim A − . (3) If ℓ < , then fp κ ( T ∗ ) = GKdim A − and fp κ − ( T ∗ ) = −∞ . (4) If ℓ = 0 , then fp κ ( T ∗ ) = fp κ − ( T ∗ ) = 0 . (5) For all objects C and D in T , lim sup n →∞ log n (dim Hom T ( C , ( S ◦ Σ − d ) n ( D ))) ≤ GKdim A − . (6) For all objects C and D in T , lim sup n →∞ log n (dim Hom T ( C , ( S ◦ Σ − d ) − n ( D ))) ≤ GKdim A − . Proof. (1) Let ℓ be the AS index of A . There are two different cases: ℓ ≤ ℓ ≥
0. The proofs are similar, so we only prove the assertion for the first case.First we claim that Sp ( T ) ⊆ (1 , d − Z . Suppose ( m, n ) (1 , d − Z . Let σ = S m ◦ Σ − n = − ⊗ M ⊗ m [( d − m − n ] where M = π ( µ A )( − ℓ ) and where( d − m − n = 0. Since Proj A has finite global dimension, we have that, for allobjects A and B in T ,Hom T ( A, σ t ( B )) = Hom T ( A, ( B ⊗ M ⊗ tm )[ t (( d − m − n )]) = 0for all t ≫
0. This implies that fpg( σ ) = −∞ . Hence ( m, n ) Sp ( T ) and hencewe have proven the claim.Second we claim that (1 , d − N ⊆ Sp ( T ). Let O = π ( A ). It is a brick objectas Hom T ( O , O ) = A = k . For every ( m, n ) = ( s, s ( d − ∈ (1 , d − N , let σ = S m ◦ Σ − n = − ⊗ M ⊗ s . Then σ ( O ) ∼ = O ( − ℓs )and Hom T ( O , σ ( O )) ∼ = A − ℓs = 0when ℓ ≤
0. This implies that ρ ( A ( φ, σ )) ≥ φ = {O} . Similarly, ρ ( A ( φ, σ n )) ≥ n . Consequently, fpg( σ ) ≥ > −∞ . Therefore ( m, n ) ∈ Sp ( T ) as desired. Now we have(1 , d − N ⊆ Sp ( T ) ⊆ (1 , d − Z which implies that fpcy( T ) = d − w = GKdim A which is at least 2. By Lemma 4.4(2,3,4), for every integer s ≥ n →∞ log n (dim A sn ) = GKdim A − w − . Now assume that ℓ is positive. By Lemma 4.3(2), σ := S ◦ Σ − ( d − is equivalentto − ⊗ O ( − ℓ ) when applied to O . Thus(E4.5.2) Hom T ( O , σ n ( O )) = A − ℓn = 0when n ≥ T ( O , σ − n ( O )) = A ℓn for n ≥
0. Now (E4.5.2) implies that κ ( T , O ) = −∞ , and (E4.5.3) together with(E4.5.1) implies that κ − ( T , O ) = w − A − ℓ = 0. Then σ := S ◦ Σ − ( d − is equivalent to − ⊗ O ( − ℓ ) whenapplied to O . Thus Hom T ( O , σ n ( O )) = A − ℓn = A = k for all n ∈ Z . This implies that fp κ ( T ∗ ) = fp κ − ( T ∗ ) = 0.(5,6) The proofs are similar. We only consider (5). Note that S ◦ Σ − ( d − = − ⊗ π ( µ A )( − ℓ ). By [BV, Lemmas 4.2.3 and 4.3.2], T is generated by {O ( n ) } n ∈ Z .Hence we can assume that C = O and D = O ( a )[ b ] for some a and b . Note that [AZ, ROBENIUS-PERRON THEORY 21
Theorem 8.1(3)] holds for Artin-Schelter Gorenstein algebras. Then [AZ, Theorem8.1(3)] implies thatdim Ext b X ( O , ( S ◦ Σ − ( d − ) ⊗ n ( O ( a ))) ≤ cn w − for some constant c only dependent on a, b . Therefore the assertion follows. (cid:3) The noncommutative projective scheme in the sense of [AZ] can be defined forconnected graded coherent algebras that are not necessarily noetherian. Here weconsider a family of noncommutative projective schemes of non-noetherian Artin-Schelter regular algebras of global dimension two.Let W n be the Artin-Schelter regular algebra k h x , · · · , x n i / ( P ni =1 x i ) of globaldimension 2. When n ≥
3, this algebra is non-noetherian [Zh, Theorem 0.2(1)], butcoherent [Pi, Theorem 1.2]. Let P n denote the noncommutative projective schemeassociated to W n defined in [Pi], which is also denoted by Proj W n . We call P n aPiontkovski projective line of rank n . See [Pi, SSm] for basic properties of P n . Themain result concerning P n is the following. See [CG] for the definition of fpcx andfpv. Theorem 4.6.
Let P n be a Piontkovski projective line of rank n ≥ . Let T n bethe derived category D b ( coh ( P n )) . Then (1) fpd( T n ) = 1 for all n ≥ . (2) fpgldim( T n ) = 1 for all n ≥ . (3) fpcx( T n ) = 0 for all n ≥ . (4) fpv( T n ) = 0 for all n ≥ . (5) fpcy( T n ) = 1 for all n ≥ . (6) fp κ − (( T n ) ∗ ) = ( n = 2 , ∞ n ≥ . (7) fp κ (( T n ) ∗ ) = −∞ for all n ≥ .Proof. Let O denote the object π ( W n ) in X := Proj W n . Then coh ( X ) is hereditary,and T n is Ext-finite with Serre functor S := − ⊗ O ( − T n ) = fpd( coh ( X )). Since coh ( X ) is hereditary,fpd( coh ( X )) ≤
1. It remains to show that fpd( coh ( X )) ≥
1. Let A be the simpleobject in coh ( X ) of the form π ( W/I )) where I is the right ideal of W generated by W ( x , · · · , x n ) + x . Then it is routine to verify that Ext coh ( X ) ( A, A ) = k . Thisimplies that fpd( coh ( X )) ≥ T n ) ≤ gldim coh ( X ) = 1. By part (1),fpgldim( T n ) ≥
1. The assertion follows.(3,4) These follow from the fact that coh ( X ) is hereditary.(5) Since the Serre functor S is of the form − ⊗ O ( − S , we can adapt the proofs ofProposition 3.6 and Theorem 4.5. (cid:3) Comments on fp-invariants of finite dimensional algebras
In this section we give some remarks, comments and examples concerning finitedimensional algebras. Let T ( A ) be the derived category D b (Mod f.d − A ). The fpCalabi-Yau dimension fpcy( T ) is defined as in Definition 3.1. Example 5.1.
Let Q be a finite acyclic quiver and A be the path algebra k Q . Let T be the bounded derived category D b (Mod f.d − A ).(1) If Q is of ADE type, then T is fractional Calabi-Yau and by Lemma 3.2(3)and [Ke1, Example 8.3(2)],fpcy( T ) = h − h where h is the Coxeter number of Q . In this case, fpcy( T ) is strictly between0 and 1.(2) If Q is of e A e D e E type, then, by using Lemma 2.1(2), T is equivalent to D b ( coh ( X )) for a weighted projective line X . Similar to the proof of Theorem4.5(1), one can show thatfpcy( D b ( coh ( X ))) = 1for every weighted projective line X . (The proof is slightly more complicatedand details are omitted). Thus we obtain thatfpcy( T ) = 1 . If Q is of wild representation type, in some cases, one can shows that(E5.1.1) fpcy( T ) = 1 . Such an example is the Kronecker quiver with two vertices and n arrows from thefirst vertex to the second, see Example 5.7. It is not clear to us if (E5.1.1) holdsfor all acyclic quivers of wild representation type. Lemma 5.2.
Let A and B be finite dimensional algebras of finite global dimension.Suppose that both T ( A ) and T ( B ) are fractional Calabi-Yau of dimension d and d respectively. Then T ( A ⊗ B ) is fractional Calabi-Yau of dimension d + d . The proof of Lemma 5.2 is straightforward and hence omitted. One immediatequestion is
Question 5.3.
Let A and B be finite dimensional algebras of finite global dimen-sion. Is then fpcy( T ( A ⊗ B )) = fpcy( T ( A )) + fpcy( T ( B ))?To define fp (anti-)Kodaira dimension of T as in Definition 3.5, we need to specifyan object O which plays the role of the structure sheaf in algebraic geometry. Onechoice for O is the left A -module A . So we let T ( A ) ∗ be ( T ( A ) , A ). Definition 5.4.
Let A be a finite dimensional algebra of finite global dimension.Suppose that T ( A ) has a fractional fp Calabi-Yau dimension ab ∈ Q .(1) The fp Kodaira dimension of A is defined to befp κ ( A ) := fp κ ( T ( A ) ∗ ) = lim t (cid:26) lim sup n →∞ dim Hom T ( A, ( S bnt ◦ Σ − ant )( A )) (cid:27) where the first limit ranges over all positive integers t with order < asdefined before Definition 3.5.(2) The fp anti-Kodaira dimension of A is defined to befp κ − ( A ) := fp κ − ( T ( A ) ∗ ) = lim t (cid:26) lim sup n →∞ dim Hom T ( A, ( S − bnt ◦ Σ ant )( A )) (cid:27) ROBENIUS-PERRON THEORY 23 where the first limit ranges over all positive integers t with order < asdefined before Definition 3.5. Remark 5.5.
Suppose T ( A ) is fractional Calabi-Yau. Then one can check easilythat(E5.5.1) fp κ ( A ) = fp κ − ( A ) = 0 . So if there is a notion of a fractional Calabi-Yau algebra, (E5.5.1) should be a partof the definition.Let A be a finite dimensional algebra of finite global dimension. Then theSerre functor is given by − ⊗ LA A ∗ . It is unknown if fpcy( T ( A )) always exists.If fpcy( T ( A )) exists and is a rational number, then we can define and calculate fpKodaira dimension (respectively, anti-Kodaira dimension) of A . Here is a list ofquestions that are related to the fp Kodaira dimension. Question 5.6.
Let A and B be two finite dimensional algebra of finite globaldimension. Suppose that fpcy( T ( A )) and fpcy( T ( B )) are rational numbers.(1) Are fp κ ( A ) and fp κ − ( A ) less than ∞ ?(2) If T ( A ) is triangulated equivalent to T ( B ), is then fp κ ( A ) = fp κ ( B ) andfp κ − ( A ) = fp κ − ( B )?(3) Suppose both fp κ ( A ) and fp κ ( B ) are finite, is thenfp κ ( A ⊗ B ) = fp κ ( A ) + fp κ ( B )?The first question has a negative answer. Example 5.7.
Let Q n be the Kronecker quiver with two vertices and n arrowsfrom the first vertex to the second. Let U n be the path algebra of Q n . By [Min,Theorem 0.1], if n ≥ D b ( M od f.d. − U n ) ∼ = D b ( coh ( P n )) =: T n where P n is given in Theorem 4.6. By Theorem 4.6(3), fpcy( D b ( M od f.d. − U n )) = 1for all n ≥
2. On the other hand, fpcy( D b ( M od f.d. − U )) = h − h = where h = 3is the Coxeter number of the quiver A (which is Q ), see [Ke1, Example 8.3(2)].We claim that fp κ ( U n ) = −∞ and that fp κ − ( U n ) = ∞ when n ≥
3. We onlyprove the second assertion. By the noncommutative Beilinson’s theorem given in[Min, Theorem 0.1] (also see (E5.9.1)), the equivalent (E5.7.1) sends U n to O⊕O (1)where O is the structure sheaf of P n .Note that the Serre functor is S := − ⊗ O ( − A = U n . Thenfp κ − ( A ) = lim t (cid:26) lim sup m →∞ dim Hom T ( A ) ( A, ( S − tm ◦ Σ tm )( A )) (cid:27) = lim t (cid:26) lim sup m →∞ dim Hom T n ( O ⊕ O (1) , ( S − tm ◦ Σ tm )( O ⊕ O (1))) (cid:27) ≥ lim t (cid:26) lim sup m →∞ dim Hom T n ( O , ( S − tm ◦ Σ tm )( O )) (cid:27) = fp κ − (( T n ) ∗ )= ∞ where the last equation is Theorem 4.6(6).We add a few more questions to Question 0.4. Question 5.8.
Let A be a finite dimensional algebra of finite global dimension andlet T be the derived category D b (Mod f.d − A ).(1) By Example 5.7, if A is the path algebra of the quiver Q , then fpcy( T ) = .Is the minimum value of fpcy( T ) equal to for an arbitrary A ?(2) Is there a value of fpcy( T ) outside of the set R := X h ≥ h − h N ∩ Q > ?By Lemma 5.2 and [Ke1, Example 8.3(2)], every number in R can be realizedas fpcy( T ) for some finite dimensional algebra. But we don’t have examplesof fpcy that are outside this range.Recall that Definition 5.9. [HP, p. 1230] Let X be a smooth projective scheme.(1) A coherent sheaf E on X is called exceptional if Hom X ( E , E ) ∼ = k andExt i X ( E , E ) = 0 for every i ≥ E , · · · , E n of exceptional sheaves is called an exceptional se-quence if Ext k X ( E i , E j ) = 0 for all k and for all i > j .(3) If an exceptional sequence generates D b ( coh ( X )), then it is called full .(4) If an exceptional sequence satisfiesExt k X ( E i , E j ) = 0for all k > i, j , then it is called a strongly exceptional sequence .The existence of a full exceptional sequence has been proved for many projectiveschemes. However, on Calabi-Yau varieties there are no exceptional sheaves. When X has a full exceptional sequence E , · · · , E n , then there is a triangulated equivalence(E5.9.1) D b ( coh ( X )) ∼ = D b (Mod f.d − A )where A is the finite dimensional algebra End X ( ⊕ ni =1 E i ). In this setting, the fpCalabi-Yau dimension of D b (Mod f.d − A ) is equal to dim X , which exists and isfinite. In many examples in algebraic geometry, a full exceptional sequence consistsof line bundles. Assume this is true. Via (E5.9.1), one sees easily that fp κ ± ( A ) ≥ fp κ ± ( X ). In fact, in many examples, we have fp κ ± ( A ) = fp κ ± ( X ).6. Appendix A: Proof of Lemma 2.10
As in Section 2, let r be a positive integer. Suppose that a hereditary abeliancategory T has r brick objects as given in Corollary 2.8, now labeled as1 , , , . . . , r − , r , where the matrices H = ( H ij ) r × r = (dim Hom( i, j )) r × r , see (E2.8.1), and E = ( E ij ) r × r = (dim Hom( i, Σ j )) r × r , see (E2.8.2), are given by the following block matrices: ROBENIUS-PERRON THEORY 25 (E6.0.1) H = (1) (2) (3) (4) · · · ( r ) (1) P P P P · · · P r − (2) P P i =0 P i P i =1 P i P i =2 P i · · · r − P i = r − P i (3) P P i =0 P i P i =0 P i P i =1 P i · · · r − P i = r − P i (4) P P i =0 P i P i =0 P i P i =0 P i · · · r − P i = r − P i ... ... · · · · · · · · · · · · ...( r ) P P i =0 P i P i =0 P i P i =0 P i · · · r − P i =0 P i ,E = (1) (2) (3) (4) · · · ( r ) (1) P r − P r − P r − P r − · · · P r − (2) P r − r − P i = r − P i r − P i = r − P i r − P i = r − P i · · · r − P i = r − P i (3) P r − r − P i = r − P i r − P i = r − P i r − P i = r − P i · · · r − P i = r − P i (4) P r − r − P i = r − P i r − P i = r − P i r − P i = r − P i · · · r − P i = r − P i ... ... · · · · · · · · · · · · ...( r ) P P i =0 P i P i =0 P i P i =0 P i · · · r − P i =0 P i = (1) (2) (3) (4) · · · ( r ) (1) P r − P r − P r − P r − · · · P r − (2) P r − P i =1 P r − i P i =1 P r − i P i =1 P r − i · · · P i =1 P r − i (3) P r − P i =2 P r − i P i =1 P r − i P i =1 P r − i · · · P i =1 P r − i (4) P r − P i =3 P r − i P i =2 P r − i P i =1 P r − i · · · P i =1 P r − i ... ... · · · · · · · · · · · · ...( r ) P r P i = r − P r − i r P i = r − P r − i r P i = r − P r − i · · · r P i =1 P r − i , where P is the r × r permutation matrix , and P denotes the r × r identity matrix. Note that P r = P . We will show that ρ ( A ( φ )) ≤ φ .The Hom and Ext matrices given in (E2.8.1)-(E2.8.2) are actually the transposeof the usual Hom and Ext matrices since they follow the convention of Corollary2.8. See the remark before Theorem 2.11.First, notice that H T = (1) (2) (3) (4) · · · ( r ) (1) P r P r P r P r · · · P r (2) P r − P i =0 P r − i P i =0 P r − i P i =0 P r − i · · · P i =0 P r − i (3) P r − P i =1 P r − i P i =0 P r − i P i =0 P r − i · · · P i =0 P r − i (4) P r − P i =2 P r − i P i =1 P r − i P i =0 P r − i · · · P i =0 P r − i ... ... · · · · · · · · · · · · ...( r ) P r − P i = r − P r − i r − P i = r − P r − i r − P i = r − P r − i · · · r − P i =0 P r − i . Define the following non-negative r × r matrices: F := (1) (2) (3) (4) · · · ( r ) (1) P r − P r − P r − P r − · · · P r − (2) P r − P r − P r − P r − · · · P r − (3) P r − P r − P r − P r − · · · P r − (4) P r − P r − P r − P r − · · · P r − ... ... · · · · · · · · · · · · ...( r ) P P P P · · · P ,G := (1) (2) (3) (4) · · · ( r ) (1) P r P r P r P r · · · P r (2) P r − P r P r P r · · · P r (3) P r − P r − P r P r · · · P r (4) P r − P r − P r − P r · · · P r ... ... · · · · · · · · · · · · ...( r ) P P P P · · · P r . We can see that
ROBENIUS-PERRON THEORY 27 H T + F =(E6.0.2) (1) (2) (3) (4) · · · ( r ) (1) P i =0 P r − i P i =0 P r − i P i =0 P r − i P i =0 P r − i · · · P i =0 P r − i (2) P i =1 P r − i P i =0 P r − i P i =0 P r − i P i =0 P r − i · · · P i =0 P r − i (3) P i =2 P r − i P i =1 P r − i P i =0 P r − i P i =0 P r − i · · · P i =0 P r − i (4) P i =3 P r − i P i =2 P r − i P i =1 P r − i P i =0 P r − i · · · P i =0 P r − i ... ... · · · · · · · · · · · · ...( r ) r P i = r − P r − i r P i = r − P r − i r P i = r − P r − i r P i = r − P r − i · · · r P i =0 P r − i = E + G. To finish the proof we need a few lemmas.
Lemma 6.1.
Let φ be a brick set which is a subset of { , , · · · , r } . Suppose that I, J ∈ φ and that the ( I, J ) -entry E IJ = 0 . Then F IJ = E IJ = 1 .Proof. Since E is a matrix whose entries consist of zeros and ones, E IJ = 1. Weconsider two different cases. Case 1: I = J . By (E6.0.2),1 + F IJ = ( H T ) IJ + F IJ = E IJ + G IJ = E IJ + 1which implies that F IJ = E IJ = 1. Case 2: I = J . Since
I, J are distinct elements in the same brick set, we have( H T ) IJ = 0. By (E6.0.2), F IJ = ( H T ) IJ + F IJ = E IJ + G IJ ≥ . Since F is a matrix whose entries are contained in { , } , F IJ = 1 . (cid:3) Lemma 6.2. If φ is a brick set, for each row I there exists at most one column J such that A ( φ ) IJ = 0 . If there exists a row J such that A ( φ ) IJ = 0 , then A ( φ ) IJ = 1 .Proof. Assume for contradiction that we have a brick set φ containing elements I, J, J ′ such that J = J ′ and E IJ = 0 = E IJ ′ . Notice that we have the followinggeneral formula for n, m ∈ { , . . . , r − } , i, j ∈ { , . . . , r } : F nr + i,mr + j = ( P r − − n ) ij . We can always write I = nr + i, J = mr + j, J ′ = m ′ r + j ′ , for some i, j, j ′ ∈ { , . . . , r } , n, m, m ′ ∈ { , . . . , r − } .Assume without loss of generality that m ≥ m ′ . We will show that H JJ ′ = H mr + j,m ′ r + j ′ = 0 . Therefore,
J, J ′ cannot be in the same brick set, a contradiction. By Lemma 6.1,1 = F IJ = F nr + i,mr + j = ( P r − − n ) ij = ( P n +1 − r ) ji , F IJ ′ = F nr + i,m ′ r + j ′ = ( P r − − n ) ij ′ . Then 1 = ( P n +1 − r ) ji ( P r − − n ) ij ′ = r X k =1 ( P n +1 − r ) jk ( P r − − n ) kj ′ = ( P n +1 − r P r − − n ) jj ′ = δ jj ′ which implies that j = j ′ .By examination of the H matrix (E6.0.1), we can see that for m ≥ m ′ , H JJ ′ = H mr + j,m ′ r + j ′ = ( P ) jj ′ + A jj ′ for some non-negative r × r matrix A . Therefore, H JJ ′ = H mr + j,m ′ r + j ′ = H mr + j,m ′ r + j = ( P ) jj + A jj = δ jj + A jj ≥ (cid:3) Lemma 6.3. If φ is a brick set, for each column J there exists at most one row I such that A ( φ ) IJ = 0 . If there exists a row I such that A ( φ ) IJ = 0 , then A ( φ ) IJ = 1 .Proof. The proof is similar to the proof of Lemma 6.2 and omitted. (cid:3)
Lemma 6.4. If φ is a brick set, ρ ( A ( φ )) ≤ .Proof. By Lemmas 6.2 and 6.3, we have shown that if φ is a brick set, A ( φ ) isa matrix with at most one non-zero entry in each row and at most one non-zeroentry in each column, such that if any entry is non-zero, it is 1. Then A ( φ ) isalmost a permutation matrix. In this case the quiver corresponding to A ( φ ) hascycle number at most 1. By [CG, Theorem 1.8], ρ ( A ( φ )) ≤ (cid:3) Appendix B: Some variants
Recall that the set of subsets of n nonzero objects in C is denoted by Φ n foreach n ≥
1. And let Φ = S n ≥ Φ n . In this paper, we use either Φ b or Φ a astesting objects in the definition of fp-invariants [Definition 1.4]. Depending on thesituation, we might want to choose a testing set different from Φ b or Φ a . Here is alist of possible alternative testing sets. Example 7.1. (1) Φ = S n ≥ Φ n .(2) If the category C is abelian, we can consider “simple sets” as follows. LetΦ n,s be the set of n -object subsets of C , say φ := { X , X , · · · , X n } , wherethe X i are non-isomorphic simple objects in C . Let Φ s = S n ≥ Φ n,s .(3) A subset φ = { X , X , · · · , X n } is called a triangular brick set if each X i isa brick object and, up to a permutation, Hom C ( X i , X j ) = 0 for all i < j .Let Φ n,tb be the set of all triangular brick n -sets, and let Φ tb = S n ≥ Φ n,tb .(4) Now assume that C is a triangulated category with suspension functor Σ.A subset φ = { X , X , · · · , X n } is called a triangular atomic set if each X i is an atomic object and, up to a permutation, Hom C ( X i , X j ) = 0 forall i < j . Let Φ n,ta be the set of all triangular atomic n -sets, and letΦ ta = S n ≥ Φ n,ta . ROBENIUS-PERRON THEORY 29
Basically, for any property P , we can define Φ n,P b (respectively, Φ n,P a ) andlet Φ P b (respectively, Φ
P a ) be S n ≥ Φ n,P b (respectively, S n ≥ Φ n,P a ). All of thedefinitions in this paper and in [CG] can be modified after we redefine ρ ( A ( φ, σ ))as follows. Definition 7.2.
Let C be a k -linear category and let φ be a set of n nonzero objects,say { X , · · · , X n } , in C . Let σ be a k -linear endofunctor of C . We define ρ ( A ( φ, σ )) := ρ ((dim Hom C ( X i , σ ( X j ))) n × n ) ρ ((dim Hom C ( X i , X j )) n × n ) . Note that ρ ( A ( φ, σ )) agrees with the original definition when φ is a brick set.One reason to introduce P -versions of fp-invariants is to extend these invariantseven if the category contains no brick objects. Acknowledgments.
The authors would like to thank Jarod Alper for many usefulconversations on the subject. J. Chen was partially supported by the NationalNatural Science Foundation of China (Grant No. 11971398) and the FundamentalResearch Funds for Central Universities of China (Grant No. 20720180002). Z.Gao was partially supported by the National Natural Science Foundation of China(Grant No. 61971365). E. Wicks and J.J. Zhang were partially supported bythe US National Science Foundation (grant Nos. DMS-1402863, DMS-1700825and DMS-2001015). X.-H. Zhang was partially supported by the National NaturalScience Foundation of China (Grant No. 11401328). H. Zhu was partially supportedby a grant from Jiangsu overseas Research and Training Program for universityprominent young and middle-aged Teachers and Presidents, China.
References [AS] M. Artin and W.F. Schelter,
Graded algebras of global dimension 3 . Adv. Math. (1987),no. 2, 171–216.[AZ] M. Artin, J.J. Zhang, Noncommutative projective schemes , Adv. Math. (1994), 228–287.[ASS] I. Assem, D. Simson and A. Skowro´nski,
Elements of the representation theory of associativealgebras , Vol. 1. Techniques of representation theory. London Mathematical Society StudentTexts, 65. Cambridge University Press, Cambridge, 2006.[BGP] I. N. Bernstein, I. M. Gelfand and V. A. Ponomarev,
Coxeter functors and Gabriel’stheorem , Uspehi Mat. Nauk :2(170) (1973), 19–33. In Russian; translated in Russ. Math.Surv. :2 (1973), 17–32.[BO] A. Bondal and D. Orlov, Reconstruction of a variety from the derived category and groupsof autoequivalences , Compositio Math. (2001), no. 3, 327–344.[BV] A. Bondal and M. van den Bergh,
Generators and representability of functors in commuta-tive and noncommutative geometry
Mosc. Math. J. (2003), no. 1, 1–36, 258.[BB] K. Br¨uning and I. Burban, Coherent sheaves on an elliptic curve , (English summary) Inter-actions between homotopy theory and algebra, 297–315, Contemp. Math., 436, Amer. Math.Soc., Providence, RI, 2007.[CG] J.M. Chen, Z.B. Gao, E. Wicks, J. J. Zhang, X-.H. Zhang and H. Zhu,
Frobenius-Perrontheory of endofunctors , Algebra Number Theory, (2019), no. 9, 2005–2055.[EGNO2015] P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, Tensor categories , MathematicalSurveys and Monographs, . American Mathematical Society, Providence, RI, 2015.[EGO2004] P. Etingof, S. Gelaki and V. Ostrik,
Classification of fusion categories of dimension pq , Int. Math. Res. Not., 2004, no. , 3041–3056.[ENO2005] P. Etingof, D. Nikshych and V. Ostrik, On fusion categories , Ann. of Math., (2) (2005), no. 2, 581–642.[GL] W. Geigle and H. Lenzing,
A class of weighted projective curves arising in representationtheory of finite-dimensional algebras , Singularities, representation of algebras, and vectorbundles (Lambrecht, 1985), 265–297, Lecture Notes in Math., 1273, Springer, Berlin, 1987. [Ha] D. Happel,
On the derived category of a finite-dimensional algebra , Comment. Math. Helv., (3), (1987), 339–389.[HP] L. Hille and M. Perling, Exceptional sequences of invertible sheaves on rational surfaces ,Compos. Math. (2011), no. 4, 1230–1280.[Ke1] B. Keller,
On triangulated orbit categories , Doc. Math. (2005), 551–581.[Ke2] B. Keller, Derived categories and tilting , Handbook of tilting theory, 49–104, London Math.Soc. Lecture Note Ser., , Cambridge Univ. Press, Cambridge, 2007.[KLM] D. Kussin, H. Lenzing and H. Meltzer,
Triangle singularities, ADE-chains, and weightedprojective lines , Adv. Math. (2013), 194–251.[Ku] A. Kuznetsov,
Calabi-Yau and fractional Calabi-Yau categories , J. Reine Angew. Math., (2019), 239–267.[La] R. Lazarsfeld,
Positivity in algebraic geometry. I. Classical setting: line bundles and linearseries.
A Series of Modern Surveys in Mathematics [Results in Mathematics and RelatedAreas. 3rd Series. A Series of Modern Surveys in Mathematics], 48. Springer-Verlag, Berlin,2004.[Le] H. Lenzing,
Weighted projective lines and applications , Representations of algebras and re-lated topics, 153–187, EMS Ser. Congr. Rep., Eur. Math. Soc., Z¨urich, 2011.[LM] H. Lenzing, H. Meltzer,
Sheaves on a weighted projective line of genus one, and representa-tions of a tubular algebra , Representations of algebras (Ottawa, Canada, 1992), CMS Conf.Proc. , 313–337, (1994).[LS] Z. Leszczy´nski and A. Skowro´nski, Tame generalized canonical algebras , J. Algebra 273(2004), no. 1, 412–433.[MR] J.C. McConnell and J.C . Robson,
Noncommutative Noetherian Rings , Wiley, Chichester,1987.[Min] H. Minamoto,
A noncommutative version of Beilinson’s theorem , J. Algebra (2008)238–252.[Ne] A. Neeman,
Triangulated categories , Annals of Mathematics Studies, 148. Princeton Univer-sity Press, Princeton, NJ, 2001.[Nik2004] D. Nikshych,
Semisimple weak Hopf algebras , J. Algebra, (2004), no. 2, 639–667.[Pi] D. Piontkovski,
Coherent algebras and noncommutative projective lines , J. Algebra (2008) 3280–3290.[Ri] C.M. Ringel,
Tame algebras and integral quadratic forms , Lecture Notes in Mathematics, , Springer-Verlag, Berlin, 1984.[Sa] F. Sakai,
Anti-Kodaira dimension of ruled surfaces , Sci. Rep. Saitama Univ. (1982), 1–7.[Sc] O. Schiffmann, Lectures on Hall algebras , Geometric methods in representation theory. II,1–141, S´emin. Congr., 24-II, Soc. Math. France, Paris, 2012.[SS] D. Simson and A. Skowro´nski,
Elements of the representation theory of associative algebras ,Volume 2, Tubes and Concealed Algebras of Euclidean type. London Mathematical SocietyStudent Texts 71. Cambridge University Press, Cambridge, 2007.[SSm] G. Sisodia and P.S. Smith,
The Grothendieck group of non-commutative non-noetheriananalogues of P and regular algebras of global dimension two , J. Algebra (2015), 188–210.[StZ] D.R. Stephenson, J.J. Zhang, Growth of graded Noetherian rings , Proc. Amer. Math. Soc. (1997), 1593–1605.[vR] A.-C. van Roosmalen,
Abelian hereditary fractionally Calabi-Yau categories , Int. Math. Res.Not. IMRN 2012, no. , 2708–2750.[Wi] E. Wicks, Frobenius-Perron theory of modified ADE bound quiver algebras , J. Pure Appl.Algebra (2019), no. 6, 2673–2708.[Ye] A. Yekutieli,
Dualizing complexes over noncommutative graded algebras , J. Algebra (1992), no. 1, 41–84.[YZ] A. Yekutieli and J.J. Zhang,
Serre duality for noncommutative projective schemes , Proc.Amer. Math. Soc. (1997), no. 3, 697–707.[Zh] J.J. Zhang,
Non-Noetherian regular rings of dimension 2 , Proc. Amer. Math. Soc. (1998)1645–1653.[ZZ] J.J. Zhang and J.-H. Zhou,
Frobenius-Perron theory of representations of quivers , preprint(2020), arXiv:2004.09111.
ROBENIUS-PERRON THEORY 31
Chen: School of Mathematical Science, Xiamen University, Xiamen, 361005, Fujian,China
E-mail address : [email protected] Gao: Department of Communication Engineering, Xiamen University, Xiamen, 361005,Fujian, China
E-mail address : [email protected] Wicks: Department of Mathematics, Box 354350, University of Washington, Seattle,Washington 98195, USA
E-mail address : [email protected] J.J. Zhang: Department of Mathematics, Box 354350, University of Washington,Seattle, Washington 98195, USA
E-mail address : [email protected] X.-H. Zhang: College of Sciences, Ningbo University of Technology, Ningb, 315211,Zhejiang, China
E-mail address : [email protected] Zhu: Department of Information Sciences, the School of Mathematics and Physics,Changzhou University, Changzhou, 213164, Jiangsu, China
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