An Upper Bound for the Dimension of Bounded Derived Categories
aa r X i v : . [ m a t h . R A ] A p r An Upper Bound for the Dimension of BoundedDerived Categories ∗† Junling Zheng a,b , Zhaoyong Huang b, ‡ a Department of Mathematics, China Jiliang University, Hangzhou 310018, Zhejiang Province, P.R. China b Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, P.R. China
Abstract
Let Λ be an artin algebra. We give an upper bound for the dimension of the boundedderived category of the category mod Λ of finitely generated right Λ-modules in terms ofthe projective and injective dimensions of certain class of simple right Λ-modules as well asthe radical layer length of Λ. In addition, we give an upper bound for the dimension of thesingularity category of mod Λ in terms of the radical layer length of Λ.
Given a triangulated category T , Rouquier introduced in [19] the dimension dim T of T underthe idea of Bondal and van den Bergh in [6]. This dimension and the infimum of the Orlovspectrum of T coincide, see [3, 16]. Roughly speaking, it is an invariant that measures howquickly the category can be built from one object. Many authors have studied the upper boundof dim T , see [3, 5, 7, 9, 13, 15, 18, 19] and so on. There are a lot of triangulated categories havinginfinite dimension, for instance, Oppermann and ˇSt’ov´ıˇcek proved in [15] that all proper thicksubcategories of the bounded derived category of finitely generated modules over a Noetherianalgebra containing perfect complexes have infinite dimension.Let Λ be an artin algebra. Let mod Λ be the category of finitely generated right Λ-modulesand let D b (mod Λ) and D bsg (mod Λ) be the bounded derived category and singularity categoryof mod Λ respectively. The upper bounds for the dimensions of these two categories can be givenin terms of the Loewy length LL(Λ) and the global dimension gl . dim Λ of Λ. Theorem 1.1.
Let Λ be an artin algebra. Then we have(1) ([19, Proposition 7.37]) dim D b (mod Λ) LL(Λ) − (2) ([19, Proposition 7.4] and [13, Proposition 2.6]) dim D b (mod Λ) gl . dim Λ; (3) ([5, Lemma 4.5]) dim D bsg (mod Λ) LL(Λ) − . ∗ † Keywords: Dimensions, Bounded derived categories, Upper bounds, Projective dimension, Injective dimen-sion, Radical layer length. ‡ Email: [email protected], [email protected] D b (mod Λ) and dim D bsg (mod Λ) are finite;however, Theorem 1.1(2) does not provide any information when gl . dim Λ is infinite.For a length-category C , generalizing the Loewy length, Huard, Lanzilotta and Hern´andezintroduced in [10, 11] the (radical) layer length associated with a torsion pair, which is a newmeasure for objects of C . Let Λ be an artin algebra and V a set of some simple modules in mod Λ.Let t V be the torsion radical of a torsion pair associated with V (see Section 3 for details). Weuse ℓℓ t V (Λ) to denote the t V -radical layer length of Λ. For a module M in mod Λ, we use pd M and id M to denote the projective and injective dimensions of M respectively; in particular, setpd M = − M if M = 0. For a subclass B of mod Λ, the projective dimension pd B of B is defined as pd B = ( sup { pd M | M ∈ B} , if B 6 = ∅ ; − , if B = ∅ . Dually, the injective dimension id B of B is defined. Note that V is a finite set. So, if eachsimple module in V has finite projective (resp. injective) dimension, then pd V (resp. id V )attains its (finite) maximum.The aim of this paper is to prove the following Theorem 1.2. (Theorems 3.12 and 3.14)
Let Λ be an artin algebra and V a set of some simplemodules in mod Λ with ℓℓ t V (Λ) = n . Then we have(1) if d = min { pd V , id V} , then dim D b (mod Λ) ( d + 2)( n + 1) − (2) dim D bsg (mod Λ) max { , n − } . In Section 3, we give the proof of Theorem 1.2. In fact, Theorem 1.1 is some special cases ofTheorem 1.2 (see Remark 3.16). Moreover, by choosing some suitable V and applying Theorem1.2, we may obtain more precise upper bounds for dim D b (mod Λ) and dim D bsg (mod Λ) thanthat in Theorem 1.1. We give in Section 4 two examples to illustrate this and that the differencebetween the upper bounds obtained from Theorems 1.1 and 1.2 may be arbitrarily large. We recall some notions from [14, 18, 19]. Let T be a triangulated category and I ⊆ Ob T . Let hIi be the full subcategory consisting of T of all direct summands of finite direct sums of shiftsof objects in I . Given two subclasses I , I ⊆ Ob T , we denote I ∗ I by the full subcategoryof all extensions between them, that is, I ∗ I = { X | X −→ X −→ X −→ X [1] with X ∈ I and X ∈ I } . Write I ⋄ I := hI ∗ I i . Then ( I ⋄ I ) ⋄ I = I ⋄ ( I ⋄ I ) for any subclasses I , I and I of T by the octahedral axiom. Write hIi := 0 , hIi := hIi and hIi n +1 := hIi n ⋄ hIi for any n > . n upper bound for the dimension of derived category Definition 2.1. ([19, Definiton 3.2])
The dimension dim T of a triangulated category T is theminimal d such that there exists an object M ∈ T with T = h M i d +1 . If no such M exists forany d , then we set dim T = ∞ . Lemma 2.2. ([17, Lemma 7.3])
Let T be a triangulated category and let X, Y be two objects of T . Then h X i m ⋄ h Y i n ⊆ h X ⊕ Y i m + n for any m, n > . Lemma 2.3. ([1, Proposition 3.2])
Let A be an abelian category admitting enough projectiveobjects. Let X = ( X i , d i ) be a bounded complex in A such that the homology H i ( X ) has projectivedimension at most n for all i . Then X ∈ hPi n +1 ⊆ D b ( A ) for the subcategory P ⊆ A of projectiveobjects.
Dually, we have
Lemma 2.4.
Let A be an abelian category admitting enough injective objects. Let X = ( X i , d i ) be a bounded complex in A such that the homology H i ( X ) has injective dimension at most n forall i .Then X ∈ hEi n +1 ⊆ D b ( A ) for the subcategory E ⊆ A of injective objects.
We recall some notions from [11]. Let C be a length-category , that is, C is an abelian, skeletallysmall category and every object of C has a finite composition series. We use End Z ( C ) to denotethe category of all additive functors from C to C , and use rad to denote the Jacobson radicallying in End Z ( C ). For any α ∈ End Z ( C ), set the α -radical functor F α := rad ◦ α . Definition 2.5. ([11, Definition 3.1]) For any α, β ∈ End Z ( C ), we define the ( α, β ) -layer length ℓℓ βα : C −→ N ∪ {∞} via ℓℓ βα ( M ) = inf { i > | α ◦ β i ( M ) = 0 } ; and the α -radical layer length ℓℓ α := ℓℓ F α α . Lemma 2.6.
Let α, β ∈ End Z ( C ) . For any M ∈ C , if ℓℓ βα ( M ) = n , then ℓℓ βα ( M ) = ℓℓ βα ( β j ( M ))+ j for any j n ; in particular, if ℓℓ α ( M ) = n , then ℓℓ α ( F nα ( M )) = 0 .Proof. If ℓℓ βα ( M ) = n , then n = inf { i > | αβ i ( M ) = 0 } . By Definition 2.5, for any 0 j n ,we have ℓℓ βα ( β j ( M )) = inf { i > | αβ i + j ( M ) = 0 } = n − j, that is, ℓℓ βα ( M ) = ℓℓ βα ( β j ( M )) + j . In particular, if ℓℓ α ( M ) = n , then putting β = F α we have ℓℓ α ( F nα ( M )) = ℓℓ α ( M ) − n = n − n = 0.Recall that a torsion pair (or torsion theory ) for C is a pair of classes ( T , F ) of objectsin C satisfying the following conditions.(1) Hom C ( M, N ) = 0 for any M ∈ T and N ∈ F ;(2) an object X ∈ C is in T if Hom C ( X, − ) | F = 0; (3) an object Y ∈ C is in F if Hom C ( − , Y ) | T = 0.For a subfunctor α of the identity functor 1 C , we write q α := 1 C /α . Let ( T , F ) be a torsionpair for C . Recall that the torsion radical t is a functor in End Z ( C ) such that0 −→ t ( M ) −→ M −→ q t ( M ) −→ q t ( M ) = M/t ( M ) ∈ F . In this section, Λ is an artin algebra. Then mod Λ is a length-category. For a module M inmod Λ, we use rad M , soc M and top M to denote the radical, socle and top of M respectively.For a subclass W of mod Λ, we use add W to denote the subcategory of mod Λ consisting ofdirect summands of finite direct sums of modules in W , and if W = { M } for some M ∈ mod Λ,we write add M := add W .Let S be the set of all simple modules in mod Λ, and let V be a subset of S and V ′ the setof all the others simple modules in mod Λ, that is, V ′ = S\V . We write F ( V ) := { M ∈ mod Λ | there exists a finite chain 0 = M ⊆ M ⊆ · · · ⊆ M m = M of submodules of M such that each quotient M i /M i − is isomorphic to some module in V} . By[11, Lemma 5.7 and Proposition 5.9], we have that ( T V , F ( V )) is a torsion pair, where T V = { M ∈ mod Λ | top M ∈ add V ′ } . We use t V to denote the torsion radical of the torsion pair ( T V , F ( V )). Then t V ( M ) ∈ T V and q t V ( M ) ∈ F ( V ) for any M ∈ mod Λ. By [11, Propositions 5.3 and 5.9(a)], we have Proposition 3.1. (1) F ( V ) = { M ∈ mod Λ | t V ( M ) = 0 } ;(2) T V = { M ∈ mod Λ | t V ( M ) = M } ;(3) top M ∈ add V ′ if and only if t V ( M ) = M . As a consequence, we get the following
Proposition 3.2. If V = ∅ , then ℓℓ t V ( M ) = LL( M ) for any M ∈ mod Λ .Proof. If V = ∅ , then the torsion pair ( T V , F ( V )) = (mod Λ , M ∈ mod Λ we have t V ( M ) = M and ℓℓ t V ( M ) = LL( M ). Lemma 3.3. (1) F ( V ) is closed under extensions, submodules and quotient modules.(2) The functor t V preserves monomorphisms and epimorphisms. n upper bound for the dimension of derived category Proof. (1) It is [11, Lemma 5.7].(2) By [11, Lemma 3.6(a)], we have that t V preserves monomorphisms. Since F ( V ) is closedunder quotient modules by (1), we have that t V preserves epimorphisms by [4, Proposition1.3].We use D to denote the usual duality between mod Λ and mod Λ op . Proposition 3.4.
Let G be a generator and E a cogenerator for mod Λ . Then ℓℓ t V ( G ) = ℓℓ t V ( E ) . In particular, for any M ∈ mod Λ , we have ℓℓ t V ( M ) ℓℓ t V (Λ) = ℓℓ t V ( D (Λ)) . Proof.
By Lemma 3.3(2) and [11, Lemma 3.4(b)(c)].The following lemma is essentially contained in [14, Lemma 2.2.4]. A similar result also holdstrue for objects in the bounded derived category of a hereditary abelian category (see [12, 1.6]for details).
Lemma 3.5.
Let X : · · · d i − / / X i − d i − / / X i d i / / X i +1 d i +1 / / · · · be a bounded complex in mod Λ with all X i seimisimple. Then X ∼ = ⊕ i H i ( X )[ i ] and X ∈h Λ / rad Λ i in D b (mod Λ) .Proof. By assumption, there exist two integers r and t such that X i ∈ add(Λ / rad Λ), where X i = 0 for any i / ∈ [ r, t ], where [ r, t ] is the integer interval with endpoints r and t . By [2,Theorem 9.6], the exact sequence0 −→ Ker d t − −→ X t − −→ Im d t − −→ / / X r d r / / X r +1 d r +1 / / X r +2 d r +2 / / · · · d t − / / X t − d t − / / X t / / / / X r d r / / X r +1 d r +1 / / X r +2 d r +2 / / · · · d t − / / Ker d t − / / / / / / / / / / / / · · · / / Im d t − / / X t / / . ( ∗ )Note that the complex ( ∗ ) is isomorphic to the stalk complex H t ( X )[ t ] in D b (mod Λ). Byinduction, we have X ∼ = ⊕ ti = r H i ( X )[ i ] in D b (mod Λ). dim D b (mod Λ) We use S < ∞ to denote the set of the simple modules in mod Λ with finite projective dimension,and use S ∞ to denote the set of the simple modules in mod Λ with infinite projective dimension.Thus S < ∞ ∪ S ∞ = S . For a subset V of S , it is easy to see that pd F ( V ) pd V and id F ( V ) id V . We will use this observation in the sequel freely. Lemma 3.6.
Let V be a subset of S < ∞ and pd V = a . Then the following complex X : · · · d i − / / X i − d i − / / X i d i / / X i +1 d i +1 / / · · · with all X i in mod Λ induces a complex q t V ( X ) : · · · q t V ( d i − ) / / q t V ( X i − ) q t V ( d i − ) / / q t V ( X i ) q t V ( d i ) / / q t V ( X i +1 ) q t V ( d i +1 ) / / · · · such that pd H i ( q t V ( X )) a for all i .Proof. Since q t V is a covariant functor, we can obtain the complex q t V ( X ). For any i , since q t V ( X i ) ∈ F ( V ), it follows from Lemma 3.3(1) that all Ker q t V ( d i ), Im q t V ( d i − ) and H i ( q t V ( X ))are in F ( V ). Thus we have pd H i ( q t V ( X )) a . Lemma 3.7.
Let V be a subset of S < ∞ and pd V = a . For a bounded complex X = ( X i , d i ) in mod Λ , if ℓℓ t V (Λ) = n , then F nt V ( X ) ∈ h Λ i a +1 .Proof. By Proposition 3.4, we have ℓℓ t V ( X i ) ℓℓ t V (Λ) = n for all i . Then by Lemma 2.6 andProposition 3.1(1), we have ℓℓ t V ( F nt V ( X i )) = 0 and F nt V ( X i ) ∈ F ( V ), which implies H i ( F nt V ( X )) ∈ F ( V ) by Lemma 3.3(1), and hence pd H i ( F nt V ( X )) a for all i . It follows from Lemma 2.3 that F nt V ( X ) ∈ h Λ i a +1 .We now are in a position to prove the following Theorem 3.8.
Let V be a subset of S < ∞ and pd V = a . If ℓℓ t V (Λ) = n , then dim D b (mod Λ) ( a + 2)( n + 1) − . Proof. If V = ∅ , then ℓℓ t V (Λ) = LL(Λ) by Proposition 3.2. Now the assertion follows fromTheorem 1.1(1).If n = 0, that is, t V (Λ) = 0, then Λ ∈ F ( V ) by Proposition 3.1(1). Since V contains everysimple module by the definition of F ( V ) and since the composition series of Λ does, we have V = S and gl . dim Λ = a . It follows from Theorem 1.1(2) that dim D b (mod Λ) a .Let X ∈ D b (mod Λ) and n >
1. Since both q t V and t V are covariant functors, we have that0 −→ t V ( X ) −→ X −→ q t V ( X ) −→ Y ∈ D b (mod Λ), we have the following shortexact sequence of complexes 0 −→ rad Y −→ Y −→ top Y −→ . n upper bound for the dimension of derived category Y = t V ( X ), we have h X i ⊆ h t V ( X ) i ⋄ h q t V ( X ) i⊆ h t V ( X ) i ⋄ h Λ i a +1 (by Lemmas 3.6 and 2.3) ⊆ h rad t V ( X ) i ⋄ h top t V ( X ) i ⋄ h Λ i a +1 = h F t V ( X ) i ⋄ h top t V ( X ) i ⋄ h Λ i a +1 ⊆ h F t V ( X ) i ⋄ h Λ / rad Λ i ⋄ h Λ i a +1 (by Lemma 3.5) ⊆ h F t V ( X ) i ⋄ h Λ ⊕ (Λ / rad Λ) i a +2 . (by Lemma 2.2)By replacing X with F it V ( X ) for any 1 i n −
1, we get h X i ⊆ h F nt V ( X ) i ⋄ h Λ ⊕ (Λ / rad Λ) i n ( a +2) . By Lemma 3.7, we have F nt V ( X ) ∈ h Λ i a +1 . Thus h X i ⊆ h Λ ⊕ (Λ / rad Λ) i ( n +1)( a +2) − . It follows that D b (mod Λ) = h Λ ⊕ (Λ / rad Λ) i ( a +2)( n +1) − anddim D b (mod Λ) ( a + 2)( n + 1) − . We use S < ∞ inj to denote the set of the simple modules in mod Λ with finite injective dimension.The following two lemmas are dual to Lemmas 3.6 and 3.7 respectively, we omit their proofs. Lemma 3.9.
Let V be a subset of S < ∞ inj and id V = c . Then the following complex X : · · · d i − / / X i − d i − / / X i d i / / X i +1 d i +1 / / · · · with all X i in mod Λ induces a complex q t V ( X ) : · · · q t V ( d i − ) / / q t V ( X i − ) q t V ( d i − ) / / q t V ( X i ) q t V ( d i ) / / q t V ( X i +1 ) q t V ( d i +1 ) / / · · · such that id H i ( q t V ( X )) c for all i . Lemma 3.10.
Let V be a subset of S < ∞ inj and id V = c . For a bounded complex X = ( X i , d i ) in mod Λ , if ℓℓ t V ( D (Λ)) = n , then F nt V ( X ) ∈ h D (Λ) i c +1 . The following result is dual to Theorem 3.8.
Theorem 3.11.
Let V be a subset of S < ∞ inj and id V = c . If ℓℓ t V ( D (Λ)) = n , then dim D b (mod Λ) ( c + 2)( n + 1) − . Proof.
Though the proof is similar to that of Theorem 3.8, we still give it here for the readers’convenience.If V = ∅ , then ℓℓ t V ( D (Λ)) = LL( D (Λ)) = LL(Λ) by Proposition 3.2. Now the assertionfollows from Theorem 1.1(1). If n = 0, that is, t V ( D (Λ)) = 0, then D (Λ) ∈ F ( V ) by Proposition 3.1(1). Since V containsevery simple module by the definition of F ( V ) and since the composition series of D (Λ) does, wehave V = S and gl . dim Λ = c . It follows from Theorem 1.1(2) that dim D b (mod Λ) c .Let X, Y ∈ D b (mod Λ) and n >
1. Just like the argument in Theorem 3.8, we have thefollowing two short exact sequence of complexes0 −→ t V ( X ) −→ X −→ q t V ( X ) −→ , −→ rad Y −→ Y −→ top Y −→ . Now by letting Y = t V ( X ), we have h X i ⊆ h t V ( X ) i ⋄ h q t V ( X ) i⊆ h t V ( X ) i ⋄ h D (Λ) i c +1 (by Lemmas 3.9 and 2.4) ⊆ h rad t V ( X ) i ⋄ h top t V ( X ) i ⋄ h D (Λ) i c +1 = h F t V ( X ) i ⋄ h top t V ( X ) i ⋄ h D (Λ) i c +1 ⊆ h F t V ( X ) i ⋄ h Λ / rad Λ i ⋄ h D (Λ) i c +1 (by Lemma 3.5) ⊆ h F t V ( X ) i ⋄ h D (Λ) ⊕ (Λ / rad Λ) i c +2 . (by Lemma 2.2)By replacing X with F it V ( X ) for any 1 i n −
1, we get h X i ⊆ h F nt V ( X ) i ⋄ h D (Λ) ⊕ (Λ / rad Λ) i n ( c +2) . By Lemma 3.10, we have F nt V ( X ) ∈ h D (Λ) i c +1 . Thus h X i ⊆ h D (Λ) ⊕ (Λ / rad Λ) i ( n +1)( c +2) − . It follows that D b (mod Λ) = h D (Λ) ⊕ (Λ / rad Λ) i ( c +2)( n +1) − anddim D b (mod Λ) ( c + 2)( n + 1) − . Combining Theorems 3.8 and 3.11, we get the following
Theorem 3.12.
Let V be a subset of S and min { pd V , id V} = d . If ℓℓ t V (Λ) = n , then dim D b (mod Λ) ( d + 2)( n + 1) − . Proof.
The case for d = ∞ is trivial. Since ℓℓ t V (Λ) = ℓℓ t V ( D (Λ)) by Proposition 3.4, the casefor d < ∞ follows from Theorems 3.8 and 3.11. dim D bsg (mod Λ) Recall that the singularity category D bsg (mod Λ) of mod Λ is defined as D b (mod Λ) /K b (proj Λ),where K b (proj Λ) is the bounded homotopy category of the subcategory proj Λ of mod Λ con-sisting of projective modules. For any M ∈ mod Λ and m >
1, we use Ω m ( M ) to denote the m -th syzygy of M ; in particular, Ω ( M ) = M . n upper bound for the dimension of derived category Lemma 3.13. (1) ℓℓ t S < ∞ (Λ) = 0 if and only if gl . dim Λ < ∞ ;(2) ℓℓ t S < ∞ (Λ) = 1 .Proof. (1) If ℓℓ t S < ∞ (Λ) = 0, then t S < ∞ (Λ) = 0. So Λ ∈ F ( S < ∞ ) by Proposition 3.1(1), whichimplies S < ∞ = S . Thus gl . dim Λ = pd S = pd S < ∞ < ∞ . Conversely, if gl . dim Λ < ∞ , then S < ∞ = S and the torsion pair ( T S < ∞ , F ( S < ∞ )) = ( T S , F ( S )) = (0 , mod Λ). By Proposition3.1(2), for any M ∈ mod Λ we have t S < ∞ ( M ) = 0 and ℓℓ t S < ∞ (Λ) = 0.(2) Suppose ℓℓ t S < ∞ (Λ) = 1. Then by (1), we have gl . dim Λ = ∞ and there exists a simplemodule S in mod Λ such that pd S = ∞ . Consider the following exact sequence0 −→ Ω ( S ) −→ P −→ S −→ , in mod Λ with P the projective cover of S . Because top S = S ∈ add S ∞ , we have t S < ∞ ( S ) = S by Proposition 3.1(3). It follows from [11, Lemma 6.3] that ℓℓ t S < ∞ (Ω ( S )) = ℓℓ t S < ∞ (Ω ( t S < ∞ ( S ))) ℓℓ t S < ∞ (Λ) − , that is, ℓℓ t S < ∞ (Ω ( S )) = 0, and Ω ( S ) ∈ F ( S < ∞ ), which induces pd Ω ( S ) < ∞ , a contradiction.In the following result, we give an upper bound for dim D bsg (mod Λ). Theorem 3.14.
Let V be a subset of S < ∞ with ℓℓ t V (Λ) = n . Then we have dim D bsg (mod Λ) max { , n − } . Proof. If V = ∅ , then ℓℓ t V (Λ) = LL(Λ) by Proposition 3.2. Now the assertion follows fromTheorem 1.1(3).Now suppose V 6 = ∅ . If n
1, then ℓℓ t S < ∞ (Λ) ℓℓ t S < ∞ (Λ) =0 and gl . dim Λ < ∞ by Lemma 3.13, which implies dim D bsg (mod Λ) = 0.Let n > a := pd V . From [8, Lemma 2.4(2)(a)], we know that every object in D bsg (mod Λ) is isomorphic to a stalk complex for some module. Let X ∈ mod Λ. If ℓℓ t V ( X ) = 0,then pd X < ∞ and X = 0 in D bsg (mod Λ). If ℓℓ t V ( X ) >
0, then by [11, Lemma 6.3], we have ℓℓ t V (Ω ( t V ( X ))) ℓℓ t V (Λ) − n −
1. By Lemma 2.6, we have ℓℓ t V ( F n − t V (Ω ( t V ( X )))) = 0.By Proposition 3.1(1), we have F n − t V (Ω ( t V ( X ))) ∈ F ( V ) and pd F n − t V (Ω ( t V ( X ))) a .For any Y ∈ mod Λ, we have the following two exact sequences0 −→ t V ( Y ) −→ Y −→ q t V ( Y ) −→ , −→ F t V ( Y ) −→ t V ( Y ) −→ top t V ( Y ) −→ . Since q t V ( Y ) ∈ F ( V ), we have pd q t V ( Y ) a . By the horseshoe lemma, we haveΩ a +1 ( Y ) ∼ = Ω a +1 ( t V ( Y )) , → Ω a +1 ( F t V ( Y )) → Ω a +1 ( t V ( Y )) ⊕ P → Ω a +1 (top t V ( Y )) → , P is projective in mod Λ. Thus we have h Ω a +1 ( Y ) i = h Ω a +1 ( t V ( Y )) i ⊆ h Ω a +1 ( F t V ( Y )) i ⋄ h Ω a +1 (top t V ( Y )) i⊆ h Ω a +1 ( F t V ( Y )) i ⋄ h Ω a +1 (Λ / rad Λ) i . By replacing Y with F it V ( Y ) for any 1 i n −
2, we get h Ω a +1 ( Y ) i ⊆ h Ω a +1 ( F n − t V ( Y )) i ⋄ h Ω a +1 (Λ / rad Λ) i n − . Let Y = Ω ( t V ( X )). Since pd F n − t V (Ω ( t V ( X ))) a , we haveΩ a +1 ( F n − t V (Ω ( t V ( X )))) = 0 , and so h Ω a +2 ( t V ( X )) i ⊆ h Ω a +1 (Λ / rad Λ) i n − . By [8, Lemma 2.4(2)(b)], we have X ∼ = Ω a +2 ( X )[ a + 2] in D bsg (mod Λ). Thus X ∼ = Ω a +2 ( X )[ a + 2] ∼ = Ω a +2 ( t V ( X ))[ a + 2] ∈ h Ω a +1 (Λ / rad Λ) i n − . It follows that D bsg (mod Λ) = h Ω a +1 (Λ / rad Λ) i n − and dim D bsg (mod Λ) n − ℓℓ t S < ∞ (Λ) LL(Λ), so this corollary improves Theorem 1.1(3).
Corollary 3.15. If ℓℓ t S < ∞ (Λ) = n , then we have dim D bsg (mod Λ) max { , n − } . Now we explain why Theorem 1.1 is a special case of our results.
Remark 3.16. (1) If V = ∅ , then ℓℓ t V (Λ) = LL(Λ) by Proposition 3.2. Since c = min { pd V , id V} = −
1, byTheorem 3.12 we havedim D b (mod Λ) ( c + 2)( n + 1) − − − − . This is Theorem 1.1(1).By Theorem 3.14, we havedim D bsg (mod Λ) max { , LL(Λ) − } . This is Theorem 1.1(3).(2) If V = S < ∞ = S , then the torsion pair ( T V , F ( V )) = (0 , mod Λ). By Proposition 3.1(2),for any M ∈ mod Λ we have t V ( M ) = 0 and ℓℓ t V (Λ) = 0. Because c = min { pd V , id V} =gl . dim Λ < ∞ , by Theorem 3.12 we havedim D b (mod Λ) ( c + 2)( ℓℓ t V (Λ) + 1) − . dim Λ + 2)(0 + 1) − . dim Λ . This is Theorem 1.1(2). In addition, since gl . dim Λ < ∞ , we have dim D bsg (mod Λ) = 0. n upper bound for the dimension of derived category By choosing some suitable V and applying Theorems 3.12 and 3.14, we may obtain more preciseupper bounds for dim D b (mod Λ) and dim D bsg (mod Λ) than that in Theorem 1.1. We give twoexamples to illustrate this. The global dimension of the algebra in the first example is infiniteand that in the second one is finite. Example 4.1.
Consider the bound quiver algebra Λ = kQ/I , where k is an algebraically closedfield and Q is given by 1 α (cid:18) (cid:18) α / / α m +1 | | ②②②②②②②②② α m +2 " " ❊❊❊❊❊❊❊❊❊ α / / α / / α / / · · · α m / / mm + 1 m + 2and I is generated by { α , α α m +1 , α α m +2 , α α , α α · · · α m } with m ≥
10. Then the inde-composable projective Λ-modules are1 ♠♠♠♠♠♠♠♠♠♠♠✇✇✇✇✇ ■■■■■■■ m + 1 m + 2 2 3 3 P (1) = 3 P (2) = 4 P (3) = 4 P ( m + 1) = m + 1 , P ( m + 2) = m + 2... ... ... m − , m, m, and P ( i + 1) = rad P ( i ) for any 2 i m −
1; and the indecomposable injective Λ-modules are2 13 2 1 1 1 I ( m ) = ... I ( m −
1) = ... I (1) = 1 , I ( m + 1) = m + 1 , I ( m + 2) = m + 2 m, , and I ( i ) = I ( i + 1) / soc I ( i + 1) for any 2 i m − S ( i ) = ∞ , if i = 1;1 , if 2 i m − , if m i m + 2 . So S ∞ = { S (1) } and S < ∞ = { S ( i ) | i m + 2 } . We also haveid S ( i ) = ( ∞ , if i = 1 , , m, m + 1 , m + 2;1 , if 3 i m − . V := { S ( i ) | i m − } ⊆ S < ∞ . Then a := pd S = 1 , c := id S = 1 and d := min { a, c } = 1 . Let V ′ be all the others simple modules in mod Λ, that is, V ′ = { S (1) , S (2) , S ( m ) , S ( m +1) , S ( m +2) } . By [11, Lemma 3.4(a)] and Λ = ⊕ m +2 i =1 P ( i ), we have ℓℓ t V (Λ) = max { ℓℓ t V ( P ( i )) | i m + 2 } . In order to compute ℓℓ t V ( P (1)), we need to find the least non-negative integer i such that t V F it V ( P (1)) = 0. Since top P (1) = S (1) ∈ add V ′ , we have t V ( P (1)) = P (1) by Proposition3.1(3). Thus F t V ( P (1)) = rad t V ( P (1)) = rad( P (1)) = S (1) ⊕ S ( m + 1) ⊕ S ( m + 2) ⊕ T, T = 3... m − . Since top S (1) = S (1) ∈ add V ′ , we have t V ( S (1)) = S (1) by Proposition 3.1(3). Similarly, t V ( S ( m + 1)) = S ( m + 1), t V ( S ( m + 2)) = S ( m + 2) and t V ( T ) = T . So t V F t V ( P (1)) = t V ( S (1) ⊕ S ( m + 1) ⊕ S ( m + 2) ⊕ T ) = S (1) ⊕ S ( m + 1) ⊕ S ( m + 2) ⊕ T, and hence F t V ( P (1)) = rad t V F t V ( P (1)) = rad( S (1) ⊕ S ( m + 1) ⊕ S ( m + 2) ⊕ T ) = rad T. It is easy to see that rad T ∈ F ( V ), so t V (rad T ) = 0 by Proposition 3.1(1). Moreover, t V F t V ( P (1)) = 0. It follows that ℓℓ t V ( P (1)) = 2. Similarly, we have ℓℓ t V ( P ( i )) = ( , if i = 2;1 , if 3 i m + 2 . Thus n := ℓℓ t V (Λ) = max { ℓℓ t V ( P ( i )) | i m + 2 } = 2.(1) Because LL(Λ) = m −
1, we havedim D b (mod Λ) LL(Λ) − m − D b (mod Λ). By Theorem 1.1(3), we havedim D bsg (mod Λ) LL(Λ) − m − . (2) By Theorem 3.12, we havedim D b (mod Λ) ( d + 2)( n + 1) − . By Theorem 3.14, we have dim D bsg (mod Λ) = 0 . n upper bound for the dimension of derived category Example 4.2.
Consider the bound quiver algebra Λ = kQ/I , where k is an algebraically closedfield and Q is given by 1 α / / α m +1 (cid:15) (cid:15) α / / α / / · · · α m − / / mm + 1 α m +2 / / m + 2 α m +3 / / m + 3 α m +4 / / · · · α m − / / m − I is generated by { α i α i +1 | m + 1 i m − } with m >
9. Then the indecomposableprojective Λ-modules are1 ⑥⑥⑥⑥ m + 1 2 3 3 jP (1) = 3 P (2) = 4 P (3) = 4 P ( j ) = j + 1 , P (2 m −
1) = 2 m − , ... ... ... m, m, m, where m +1 j m − P ( i +1) = rad P ( i ) for any 2 i m −
1; and the indecomposableinjective Λ-modules are1 12 2 1 j − I ( m ) = ... I ( m −
1) = ... I ( m + 1) = m + 1 , I ( j ) = j,m, m − , where m + 2 j m − I ( i ) = I ( i + 1) / soc I ( i + 1) for any 1 i m − S ( i ) = m − , if i = 1;1 , if 2 i m − , if i = m ;2 m − − i, if m + 1 i m − , and S < ∞ = S . We also haveid S ( i ) = , if i = 1;1 , if 2 i m ; i − m, if m + 1 i m − . Let V := { S ( i ) | i m } ⊆ S < ∞ . Then a := pd V = 1 , c := id V = 1 and d := min { a, c } = 1 . V ′ be all the others simple modules in mod Λ, that is, V ′ = { S ( i ) | i = 1 or m + 1 i m − } . Similar to the computation in Example 4.1, we have n := ℓℓ t V (Λ) = 2.(1) Because LL(Λ) = m , we havedim D b (mod Λ) LL(Λ) − m − . dim Λ = m −
1, we also havedim D b (mod Λ) gl . dim Λ = m − D bsg (mod Λ) LL(Λ) − m − D b (mod Λ) ( d + 2)( n + 1) − . By Theorem 3.14, we have dim D bsg (mod Λ) = 0 . In the above two examples, the upper bounds in (2) are smaller than that in (1) and thedifference between them may be arbitrarily large.
Acknowledgements.
This research was partially supported by National Natural ScienceFoundation of China (Grant Nos. 11971225, 11571164) and a Project Funded by the PriorityAcademic Program Development of Jiangsu Higher Education Institutions. The authors wouldlike to thank Dong Yang for his helpful discussions, and thank the referees for very useful anddetailed suggestions.
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