Stable functors of derived equivalences and Gorenstein projective modules
aa r X i v : . [ m a t h . R T ] D ec Stable functors of derived equivalences and Gorenstein projectivemodules
WEI HU and
SHENGYONG PAN
Abstract
From certain triangle functors, called non-negative functors, between the bounded derived categories of abeliancategories with enough projective objects, we introduce their stable functors which are certain additive functors be-tween the stable categories of the abelian categories. The construction generalizes a previous work by Hu and Xi. Weshow that the stable functors of non-negative functors have nice exactness property and are compatible with compo-sition of functors. This allows us to compare conveniently the homological properties of objects linked by the stablefunctors. Particularly, we prove that the stable functor of a derived equivalence between two arbitrary rings providesan explicit triangle equivalence between the stable categories of Gorenstein projective modules. This generalizes aresult of Y. Kato. Our results can also be applied to provide shorter proofs of some known results on homologicalconjectures.
Derived equivalences were introduced by Grothendieck and Verdier in 1960s, and play an important role nowadays inmany branches of mathematics and physics, especially in representation theory and in algebraic geometry. A derivedequivalence is a triangle equivalence between the derived categories of complexes over certain abelian categories suchas the module category of a ring or the category of coherent sheaves over some variety. For derived equivalent abeliancategories, it is very hard to directly compare the objects in the given abelian categories, since a derived equivalencetypically takes objects in one abelian category to complexes over the other.For an arbitrary derived equivalence F between two Artin algebras, a functor ¯ F between the stable module cate-gories were introduced in [HX10], called the stable functor of F . This functor allows us to compare the modules overone algebra with the modules over the other. Another nice property of this functor is that ¯ F is a stable equivalence ofMorita type in case that F is an almost ν -stable standard derived equivalence, This generalizes a classic result [Ric91] ofRickard which says that a derived equivalence between two selfinjective algebras always induces a stable equivalenceof Morita type. However, in [HX10], many basic questions on the stable functor remain. For instance, we even don’tknow whether the stable functor is uniquely determined by the given derived equivalence, and whether the definitionof the stable functor is compatible with composition of derived equivalences.In this paper, we shall look for a more general and systematical definition of stable functors, and generalize thenotion of stable functors in two directions. One direction is that, instead of module categories of Artin algebras, weconsider arbitrary abelian categories with enough projective objects. The other direction is that, instead of derivedequivalences, we consider certain triangle functors, called non-negative functors, between the derived categories. Notethat this condition is not restrictive: all derived equivalences between rings are non-negative up to shifts. We shallprove, in this general framework, that the stable functor is uniquely determined by the given non-negative functor(Theorem 4.8) and is compatible with the composition of non-negative functors (Theorem 4.9).Our theory of stable functors can be applied to study stable categories of Gorenstein projective modules of derivedequivalent rings, namely, the stable functor of a derived equivalence between two arbitrary rings provides an explicittriangle equivalence between their stable categories of Gorenstein projective modules (Corollary 5.4). Gorensteinprojective modules go back to a work of Auslander and Bridger [AB69]. Since then they have attracted more attentionand have also nice applications in commutative algebra, algebraic geometry, singularity theory and relative homologicalalgebra. In general, the size and homological complexity of the stable category of Gorenstein projective modulesmeasure how far the ring is from being Gorenstein. A nice feature of the stable category of Gorenstein projectivemodules is that it is a triangulated category, and admits a full triangulated embedding into the singularity category inOrlov’s sense, which is an equivalence if and only if the ring is Goresntein.1his paper is organized as follows. In Section 2 we recall some basic definitions and facts required in proofs.Section 3 is devoted to studying for which complexes the localization functor from the homotopy category to thederived category preserves homomorphism spaces. The theory of stable functors will be given in Section 4, and willbe applied to study stable category of Gorenstein projective modules in Section 5. An example is given in Section 6to illustrate how we can compute the Gorenstein projective modules over an algebra via the stable functor. Finally, westress in Section 7 that our results can be used to give shorter proofs of some known results on homological conjectures. In this section, we recall some basic definitions and collect some basic facts for later use.Throughout this paper, unless specified otherwise, all categories are additive categories, and all functors are additivefunctors. The composite of two morphisms f : X → Y and g : Y → Z in a category C will be denoted by f g . If f : X → Y is a map between two sets, then the image of an element x ∈ X will be denoted by ( x ) f . However, we willdeal with functors in a di ff erent manner. The composite of two functors F : C → D and G : D → E will be denoted by GF . For each object X in C , we write F ( X ) for the corresponding object in D , and for each morphism f : X → Y in C we write F ( f ) for the corresponding morphism in D from F ( X ) to F ( Y ). For an object M in an additive category C , weuse add( M ) to denote the full subcategory of C consisting of direct summands of finite direct sums of copies of M .Let A be an additive category. A complex X • over A is a sequences d iX between objects X i in A : · · · −→ X i − d i − X −→ X i d iX −→ X i + d i + X −→ · · · such that d iX d i + X = i ∈ Z . The category of complexes over A , in which morphismsare chain maps, is denoted by C ( A ), and the corresponding homotopy category is denoted by K ( A ). When A is anabelian category, we write D ( A ) for the derived category of A . We also write K b ( A ), K − ( A ) and K + ( A ) for thefull subcategories of K ( A ) consisting of complexes isomorphic to bounded complexes, complexes bounded above,and complexes bounded below, respectively. Similarly, for ∗ ∈ { b , − , + } , we have D ∗ ( A ). Moreover, for integers m ≤ n and for a collection of objects X , we write D [ m , n ] ( X ) for the full subcategory of D ( A ) consisting of complexes X • isomorphic in D ( A ) to complexes with terms in X of the form0 −→ X m −→ · · · −→ X n −→ . For each complex X • over A , its i th cohomology is denoted by H i ( X • ).The homotopy category of an additive category, and the derived category of an abelian category are both triangulatedcategories. For basic facts on triangulated categories, we refer to Neeman’s book [Nee01]. However, the shift functorof a triangulated category will be denoted by [1] in this paper. In the homotopy category, or the derived category of anabelian category, the shift functor acts on a complex by moving the complex to the left by one degree, and changingthe sign of the di ff erentials.Suppose that A is an abelian category. There is a full embedding A ֒ → D ( A ) by viewing an object in A as acomplex in D ( A ) concentrated in degree zero. Let X be a collection of objects in D ( A ) and let n be an integer. Wedefine a full subcategory of D ( A ): ⊥ > n X : = { Z • ∈ D ( A ) | Hom D ( A ) ( Z • , X • [ i ]) = i > n and for all X • ∈ X } , For simplicity, we write ⊥ X for ⊥ > X .Suppose that A is an abelian category with enough projective objects. Let P A be the full subcategory of A consistingof all projective objects. The stable category of A , denoted by A , is defined to be the additive quotient A / P A , where theobjects are the same as those in A and the morphism space Hom A ( X , Y ) is the quotient space of Hom A ( X , Y ) moduloall morphisms factorizing through projective objects. Two objects X and Y are isomorphic in A if and only if there areprojective objects P and Q such that X ⊕ Q ≃ Y ⊕ P in A . This seems not so obvious. Indeed, first of all, it is easy tocheck that the injection X → X ⊕ Q is an isomorphism in A . So, if X ⊕ Q ≃ Y ⊕ P in A with P , Q projective, then X and Y are isomorphic in A . Conversely, suppose that f : X → Y is a morphism in A such that its image f : X → Y in Hom A ( X , Y ) is an isomorphism. Then there is a morphism g : Y → X such that 1 X − f g factorizes through someprojective object P . Namely, there exist morphisms α : X → P and β : P → X such that 1 X = f g + αβ . Then we canform a split exact sequence 0 −→ X [ f ,α ] −→ Y ⊕ P (cid:2) uv (cid:3) −→ Q −→ .
2t follows that f u = − α v factorizes through the projective object P . This implies that f u =
0. However, the morphism f is an isomorphism. Hence u =
0, and therefore u factorizes through a projective object P ′ , say, u = ab for somemorphisms a : Y → P ′ and b : P ′ → Q . Thus (cid:2) uv (cid:3) factorizes through the morphism P ′ ⊕ P h bv i −→ Q . The above split exactsequence indicates that 1 Q factorizes through (cid:2) uv (cid:3) , and consequently factorizes through (cid:2) bv (cid:3) . Hence Q is isomorphic toa direct summand of P ′ ⊕ P and has to be projective. This establishes that X ⊕ Q ≃ Y ⊕ P with P , Q projective.Let A be an arbitrary ring with identity. The category A -Mod of unitary left A -modules is an abelian category withenough projective objects. We use A -mod to denote the full subcategory of A -Mod consisting of finitely presented A -modules, that is, A -modules X admitting a projective presentation P → P −→ X → P i finitely generatedprojective for i = ,
1. The category A -mod is abelian when A is left coherent. The full subcategory of A -Mod consistingof all projective modules is denoted by A -Proj, and the category of finitely generated A -modules is written as A -proj.Note that A -proj are precisely those projective modules in A -mod. The stable category of A -Mod is denoted by A -Mod,in which morphism space is denoted by Hom A ( X , Y ) for each pair of A -modules X and Y . For a full subcategory X of A -Mod, we denote by X the full subcategory of A -Mod consisting of all modules in X . However, the full subcategoryof A -Mod consisting of finitely presented modules is denoted by A -modTwo rings A and B are said to be derived equivalent if the following equivalent conditions are satisfied.(1). D ( A -Mod) and D ( B -Mod) are equivalent as triangulated categories.(2). D b ( A -Mod) and D b ( B -Mod) are equivalent as triangulated categories.(3). K b (cid:0) A -Proj (cid:1) and K b (cid:0) B -Proj (cid:1) are equivalent as triangulated categories.(4). K b (cid:0) A -proj (cid:1) and K b (cid:0) B -proj (cid:1) are equivalent as triangulated categories.(5). There is a complex T • in K b (cid:0) A -proj (cid:1) satisfying the conditions:(a). Hom K b ( A - proj )( T • , T • [ n ]) = n , T • ) generates K b (cid:0) A -proj (cid:1) as a triangulated category,such that the endomorphism algebra of T • in K b (cid:0) A -proj (cid:1) is isomorphic to B .For the proof that the above conditions are indeed equivalent, we refer to [Ric89, Kel94]. If the algebras A and B areleft coherent, then the above equivalent conditions are further equivalent to the following condition.(6). D b ( A -mod) and D b ( B -mod) are equivalent as triangulated categories.A complex T • satisfying the conditions (a) and (b) above is called a tilting complex . A triangle equivalence functor F : D b ( A -Mod) → D b ( B -Mod) is called a derived equivalence . In this case, the image F ( A ) is isomorphic in D b ( B -Mod) to a tilting complex, and there is a tilting complex T • over A such that F ( T • ) is isomorphic to B in D b ( B -Mod). The complex T • is called an associated tilting complex of F . The following is an easy lemma for theassociated tilting complexes. For the convenience of the reader, we provide a proof. Lemma 2.1.
Let A and B be two rings, and let F : D b ( A- Mod) −→ D b ( B- Mod) be a derived equivalence. Then F ( A ) is isomorphic in D b ( B- mod) to a complex ¯ T • ∈ K b (cid:0) B- proj (cid:1) of the form −→ ¯ T −→ ¯ T −→ · · · −→ ¯ T n −→ for some n ≥ if and only if F − ( B ) is isomorphic in D b ( A- Mod) to a complex T • ∈ K b (cid:0) A- proj (cid:1) of the form −→ T − n −→ · · · −→ T − −→ T −→ . Proof.
We prove the necessity, the proof of the su ffi ciency is similar. Suppose that F ( A ) is isomorphic to a complex ¯ T • in K b (cid:0) B -proj (cid:1) of the form 0 −→ ¯ T −→ ¯ T −→ · · · −→ ¯ T n −→ , and T • is a complex in K b (cid:0) A -proj (cid:1) such that F ( T • ) ≃ B . ThenHom D b ( A - Mod) ( A , T • [ i ]) ≃ Hom D b ( B - Mod) ( ¯ T • , B [ i ]) = i >
0. Hence T • has zero homology in all positive degrees. Since all the terms of T • are projective, the complex T • is split in all positive degrees, and is isomorphic in K b (cid:0) A -proj (cid:1) to a complex with zero terms in all positive degrees.3hus, we can assume that T i = i >
0. To prove that T • is isomorphic to a complex in K b (cid:0) A -proj (cid:1) with zeroterms in all degrees < − n , it su ffi ces to show that Hom D b ( A - Mod) ( T • , P [ i ]) = i > n and for all finitely generatedprojective A -module P . Actually, since F ( P ) is in add( ¯ T • ), we can deduce thatHom D b ( A - Mod) ( T • , P [ i ]) ≃ Hom D b ( B - Mod) ( B , F ( P )[ i ]) = i > n . (cid:3) K ( A ) to D ( A ) Let A be an abelian category, let q : K ( A ) −→ D ( A ) be the localization functor. The morphisms in the derived cate-gory are “complicated”, while the morphisms in the homotopy category are relatively “simple”: they can be presentedby chain maps. It is very natural to ask the following question: For which complexes X • and Y • , the induced mapq ( X • , Y • ) : Hom K ( A ) ( X • , Y • ) −→ Hom D ( A ) ( X • , Y • ) is an isomorphism? It is known that this is true in case that X • is an above-bounded complex of projective objects, or Y • is a below-boundedcomplex of injective objects. In this section, we shall prove the following very useful proposition, which allows us toget morphisms between objects from morphisms between complexes in the derived category. It seems that this has notappeared elsewhere in the literature. Proposition 3.1.
Let A be an abelian category, and let X • and Y • be above-bounded and below-bounded complexesof objects in A , respectively. Suppose that X i ∈ ⊥ Y j for all integers j < i. Then the induced mapq ( X • , Y • [ n ]) : Hom K ( A ) ( X • , Y • [ n ]) −→ Hom D ( A ) ( X • , Y • [ n ]) is an isomorphism for all n ≤ , and is a monomorphism for n = . This proposition generalizes [HX10, Lemma 2.2], and its proof will be given after several lemmas.Let F : T −→ S be a triangle functor between two triangulated categories, and let M ∈ T be an object. We define U FM to be the full subcategory of T consisting of objects X satisfying the following two conditions.(1) F ( X , M [ i ]) : Hom T ( X , M [ i ]) −→ Hom S ( F ( X ) , F ( M )[ i ]) is an isomorphism for all i ≤ F ( X , M [1]) : Hom T ( X , M [1]) −→ Hom S ( F ( X ) , F ( M )[1]) is monic.Let T be a triangulated category, and let X and Y be full subcategories of T . We define X ∗ Y : = { Z ∈ T | There is a triangle X → Z → Y → X [1] with X ∈ X and Y ∈ Y } It is well known that “ ∗ ” is associative, that is, ( X ∗ Y ) ∗ Z = X ∗ ( Y ∗ Z ) for any full subcategories X , Y and Z of T . So, for full subcategories X , · · · , X n of T , we can simply write X ∗ · · · ∗ X n . Lemma 3.2.
Let F : T −→ S be a triangle functor between triangulated categories T and S . Then we have thefollowing. (1) . Suppose that M ∈ T , and X i ⊆ U FM for i = , · · · , n. Then X ∗ · · · ∗ X n ⊆ U FM . (2) . Suppose that M i ∈ T , and X ∈ U FM i for i = , · · · , n. Then X ∈ U FM for all M ∈ { M } ∗ · · · ∗ { M n } .Proof. (1). Clearly, we only need to prove the case that n =
2. Let X be an object in X ∗ X . There is a triangle X → X → X → X [1] in T with X i ∈ X i for i = ,
2. For simplicity, we write T ( − , − ) for Hom T ( − , − ). Then, foreach integer i , we can form a commutative diagram with exact rows. T ( X , M [ i − / / F ( X , M [ i − (cid:15) (cid:15) T ( X , M [ i ]) / / F ( X , M [ i ]) (cid:15) (cid:15) T ( X , M [ i ]) / / F ( X , M [ i ]) (cid:15) (cid:15) T ( X , M [ i ]) / / F ( X , M [ i ]) (cid:15) (cid:15) T ( X , M [ i + F ( X , M [ i + (cid:15) (cid:15) S (cid:0) FX , F M [ i − (cid:1) / / S ( FX , F M [ i ]) / / S ( FX , F M [ i ]) / / S ( FX , F M [ i ]) / / S ( FX , F M [ i + i ≤
0, then, by assumption, the maps F ( X , M [ i − , F ( X , M [ i ]) , F ( X , M [ i ]) are isomorphisms and F ( X , M [ n + is monic. ByFive Lemma, the map F ( X , M [ i ]) is an isomorphism in this case. Our assumption also indicates that F ( X , M [1]) and F ( X , M [1]) are monic, and F ( X , M ) is an isomorphism. By Five Lemma again, the map F ( X , M [1]) is monic. Hence X ∈ U FM . Theproof of (2) is similar to that of (1). We leave it to the reader. (cid:3) X and Y be two objects in an abelian category A , and let q : K ( A ) → D ( A ) be the localization functor.Then it is straightforward to check that X [ i ] ∈ U qY [ j ] for all i ≥ j . If Y ∈ X ⊥ , then X [ i ] ∈ U qY [ j ] for all integers i and j , since q ( X , Y [ m ]) : Hom K ( A ) ( X , Y [ m ]) → Hom D ( A ) ( X , Y [ m ]) is an isomorphism for all integers m in this case. If Y • is a complex with Y i = i < n , then X [ i ] ∈ U qY • for all i ≥ − n +
2. In this case Hom K ( A ) ( X [ i ] , Y • [ m ]) = Hom D ( A ) ( X [ i ] , Y • [ m ]) = m ≤
1. Keeping these basic facts in mind helps us to prove the following lemma.
Lemma 3.3.
Let A be an abelian category, X be an object in A , and let Y • be a below-bounded complex over A .Suppose that m ∈ Z and that Y i ∈ X ⊥ for all i < m. Then X [ i ] ∈ U qY • for all i ≥ − m.Proof. For i ≥ m , we have − m ≥ − i , and X [ − m ] ∈ U qY i [ − i ] . For each i < m , since Y i ∈ X ⊥ , we have X [ − m ] ∈ U qY i [ − i ] . Itfollows that X [ − m ] ∈ U qY i [ − i ] for all i ∈ Z . Note that there is some integer n < m such that Y i = i < n , since Y • isbounded below. Then σ ≤ m + Y • is in { Y m + [ − m − } ∗ · · · ∗ { Y n [ − n ] } . By Lemma 3.2 (2), we get that X [ − m ] ∈ U q σ ≤ m + Y • .Now it is clear that Hom K ( A ) (cid:0) X [ − m ] , ( σ > m + Y • )[ i ] (cid:1) = = Hom D ( A ) (cid:0) X [ − m ] , ( σ > m + Y • )[ i ] (cid:1) for all i ≤
1. Hence q ( X [ − m ] , ( σ > m + Y • )[ i ]) is an isomorphism for all i ≤
1. This establishes X [ − m ] ∈ U q σ > m + Y • . Since Y • is in { σ > m + Y • } ∗ { σ ≤ m + Y • } , we deduce that X [ − m ] ∈ U qY • by Lemma 3.2 (2). Finally, by definition, we have U qY • [1] ⊆ U qY • .Hence X [ i ] ∈ U qY • for all i ≥ − m . (cid:3) With the above lemmas, we can give a proof of Proposition 3.1.
Proof of Proposition 3.1.
What we need to prove is exactly X • ∈ U qY • . By Lemma 3.3, we have X i [ − i ] ∈ U qY • for all i ∈ Z . Note that there is an integer n such that X i = i > n , since X • is above-bounded. Thus for each integer m < n , the complex σ ≥ m X • belongs to { X n [ − n ] }∗· · ·∗{ X m [ − m ] } , and is consequently in U qY • by Lemma 3.2 (1). Taking m to be su ffi ciently small such that Y j = j < m +
1. Then for each integer i ≤
1, both Hom K ( A ) ( σ < m X • , Y • [ i ]) andHom D ( A ) ( σ < m X • , Y • [ i ]) vanish. Hence q ( σ < m X • , Y • [ i ]) is an isomorphism for all i ≤
1, and consequently σ < m X • ∈ U qY • .Note that X • ∈ { σ ≥ m X • } ∗ { σ < m X • } . It follows, by Lemma 3.2 (1) again, that X • ∈ U qY • . (cid:3) Proposition 3.1 has the following useful corollary.
Corollary 3.4.
Let A be an abelian category, and let f : X → Y be a homomorphism in A . Suppose that Z • is abounded complex over A such that Z i ∈ X ⊥ for all i < and that Z i ∈ ⊥ Y for all i > . If f factorizes through Z • in D b ( A ) , then f factorizes through Z in A .Proof. Suppose that f = gh for g ∈ Hom D b ( A ) ( X , Z • ) and h ∈ Hom D b ( A ) ( Z • , Y ). By Proposition 3.1, both g and h canbe presented by a chain map. Namely, g = g • and h = h • in D b ( A ) for some chain maps g • : X → Z • and h • : Z • → Y .Hence f = g • h • = g h in D b ( A ), and consequently f = g h since A ֒ → D b ( A ) is a fully faithful embedding. (cid:3) The stable functor of a derived equivalence between Artin algebras was introduced in [HX10]. In this section, wegreatly generalize this notion. Namely, we consider “ non-negative functors ” between derived categories of abeliancategories with enough projective objects, and develop a theory of their stable functors.Throughout this section, we assume that A and B are abelian categories with enough projective objects. The fullsubcategories of projective objects are denoted by P A and P B , respectively. The corresponding stable categories aredenoted by A and B , respectively. Definition 4.1.
A triangle functor F : D b ( A ) −→ D b ( B ) is called uniformly bounded if there are integers r < s suchthat F ( X ) ∈ D [ r , s ] ( B ) for all X ∈ A , and is called non-negative if F satisfies the following conditions:(1) F ( X ) is isomorphic to a complex with zero homology in all negative degrees for all X ∈ A .(2) F ( P ) is isomorphic to a complex in K b ( P B ) with zero terms in all negative degrees for all P ∈ P A . emark. The condition (1) is equivalent to saying that F sends objects in the part D ≥ ( A ) of the canonical t -structure( D ≤ ( A ) , D ≥ ( A )) of D b ( A ) to objects in the part D ≥ ( B ) of the canonical t -structure ( D ≤ ( B ) , D ≥ ( B )) of D b ( B ).The condition (2) indicates that F sends complexes in K b ( P A ) to complexes in K b ( P B ).For derived equivalences between module categories of rings, we have the following lemma. Lemma 4.2.
Let F : D b ( A- Mod) −→ D b ( B- Mod) be a derived equivalence between two rings A and B. Then (1)
F is uniformly bounded. (2)
F is non-negative if and only if the tilting complex associated to F is isomorphic in K b (cid:0) B- proj (cid:1) to a complexwith zero terms in all positive degrees. In particular, F [ i ] is non-negative for su ffi ciently small i.Proof. Let T • be a tilting complex associated to F , that is, F ( T • ) ≃ B . Since T • is a bounded complex, there areintegers r < s such that T i = i < r and for all i > s . Let X be an A -module. There is an isomorphism H i ( F ( X )) = Hom D b ( B - Mod) ( B , F ( X )[ i ]) ≃ Hom D b ( A - Mod) ( T • , X [ i ])for each integer i . It follows that H i ( F ( X )) = i < r and for all i > s , that is, F ( X ) ∈ D [ r , s ] ( B -Mod). This provesthat F is uniformly bounded.By [Ric89, Proposition 6.2], the derived equivalence F induces a triangle equivalence functor between K b (cid:0) A -Proj (cid:1) and K b (cid:0) B -Proj (cid:1) . Suppose that the tilting complex T • associated to F has T i = i >
0. By Lemma 2.1, theimage F ( A ) is isomorphic to a complex ¯ T • ∈ K [0 , n ] (cid:0) B -proj (cid:1) for some non-negative integer n . As an equivalence, thefunctor F preserves coproducts. Hence F ( ` A ) ∈ K [0 , n ] (cid:0) B -Proj (cid:1) , and consequently F ( A -Proj) ⊆ K [0 , n ] (cid:0) B -Proj (cid:1) .Finally, for each A -module X , we have Hom D b ( B - Mod) ( B , F ( X )[ i ]) ≃ Hom D b ( A - Mod) ( T • , X [ i ]) = i <
0. Thisimplies that H i ( F ( X )) = i < F ( X ) ∈ D ≥ ( B -Mod). Hence F is a non-negative functor.Conversely, suppose that F is a non-negative derived equivalence. Then F ( A ) is isomorphic to a bounded complex Q • in K ≥ (cid:0) B -Proj (cid:1) . Let T • be a tilting complex associated to F , that is, F ( T • ) ≃ B . ThenHom D b ( A - Mod) ( A , T • [ i ]) ≃ Hom D b ( B - Mod) ( F ( A ) , B [ i ]) = i . Hence T • has zero homology in all positive degrees. This shows that T • is split in all positive degreesand thus isomorphic to a complex in K b (cid:0) A -proj (cid:1) with zero terms in all positive degrees. (cid:3) In general, both statements in Lemma 4.2 may fail for a triangle functor F : D b ( A ) → D b ( B ) between thederived categories of abelian categories A and B , even if F is a derived equivalence. For instance, let A and B be thecategories of finitely generated graded modules over the polynomial algebra k [ x , x , · · · , x n ] and the exterior algebra V k ( e , e , · · · , e n ), respectively. Then there is a triangle equivalence F : D b ( A ) −→ D b ( B ), known as Koszul duality,such that F ( X h i i ) ≃ F ( X ) h− i i [ i ] for all X ∈ D b ( A ) and for all i ∈ Z , where h i i is the degree shifting functor of gradedmodules. The functor F is not uniformly bounded and F [ i ] cannot be non-negative for any i ∈ Z . Also the two notionsin Definition 4.1 are independent. Clearly, a uniformly bounded triangle functor F needs not to be non-negative. Thefollowing example gives a non-negative functor which is not uniformly bounded. Example . Let k be a field, and let Q be the infinite quiver • • α o o • α o o • α o o · · · o o A representation of Q over k is a collection of vector spaces V i for each vertex i together with linear maps f α i : V i → V i − for all i . Let A be the category of all finite dimensional representations ( V i , f α i + ) i ≥ of Q satisfying f α i f α i − = i >
0. Let P be the representation k ←− ←− ←− · · · , and, for each i >
0, let P i be the representation0 ←− · · · ←− k ←− k ←− ←− · · · , where the two k ’s correspond to the vertices i − , i . Then A is an abelian categorywith enough projective objects and P i , i ≥ A . Consider thefollowing complexes over A : T • i : 0 −→ P −→ · · · −→ P i − −→ P i −→ , i ≥ . It is easy to check that { T • i | i ≥ } is a tilting subcategory of D b ( A ), that is, the following two conditions are satisfied.a) Hom D b ( A ) ( T • i , T • j [ l ]) = i , j ∈ N and l , { T • i | i ≥ } = D b ( A ). 6he tilting subcategory { T • i | i ≥ } is equivalent as a category to the quiver Q T : • β / / • β / / • β / / • / / · · · For each i ≥
0, let P ∗ i be the representation 0 −→ · · · −→ −→ k −→ k −→ k −→ · · · , where the first k correspondsto the vertex i . Let B be the category of finitely generated representations of Q T over k . Then B is an abelian categorywith enough projective objects, and the indecomposable projective objects are P ∗ i , i ∈ N . Note that gl . dim B = D b ( B ) = K b ( P B ). By [Kel06, Theorem 3.6], there is a triangle equivalence F : D b ( B ) −→ D b ( A ) sending P ∗ i to T • i for all i ∈ N . This functor is non-negative, but not uniformly bounded. Lemma 4.3.
Let A and B be abelian categories with enough projective objects, and let F : D b ( A ) −→ D b ( B ) be auniformly bounded, non-negative triangle functor. Suppose that n > is such that F ( A ) ⊆ D [0 , n ] ( B ) . Then (1) If F admits a right adjoint G, then G is uniformly bounded and G ( B ) ⊆ D [ − n , ( A ) . (2) If F admits a left adjoint E, then E ( P B ) ⊆ K [ − n , ( P A ) . (3) If G is both a left adjoint and a right adjoint of F, then G [ − n ] is uniformly bounded and non-negative.Proof. (1) Let X be an object in B and P be a projective object in A . Then Hom D b ( A ) ( P , G ( X )[ i ]) ≃ Hom D b ( B ) ( F ( P ) , X [ i ])vanishes for all i < [ − n , F ( P ) is isomorphic to a complex in K [0 , n ] ( P B ). It fol-lows that G ( X ) ∈ D [ − n , ( A ) for all X ∈ B .(2) Let Q ∈ P B and let X be an object in A . Then Hom D b ( A ) ( E ( Q ) , X [ i ]) ≃ Hom D b ( B ) ( P , F ( X )[ i ]) vanishes for all i < [0 , n ]. This implies that E ( Q ) ∈ K [ − n , ( P A ).(3) This follows from (1) and (2) immediately. (cid:3) For the rest of this section, we assume that F : D b ( A ) −→ D b ( B )is a non-negative triangle functor. The following lemma describes the images of objects in A under F . Lemma 4.4.
For each X ∈ A , there is a triangleU • X i X −→ F ( X ) π X −→ M X µ X −→ U • X [1] in D b ( B ) with M X ∈ B and U • X ∈ D [1 , n X ] ( P B ) for some n X > .Proof. By definition, F ( X ) has no homology in negative degrees. Take a projective resolution of F ( X ) and then dogood truncation at degree zero. The lemma follows. (cid:3) Lemma 4.5.
Suppose that U • i α i −→ X • i β i −→ M i γ i −→ U • i [1] , i = , are triangles in D b ( B ) such that M , M are objectsin B and U • , U • ∈ D [1 , n ] ( P B ) . Then, for each morphism f : X • −→ X • in D b ( B ) , there is morphism b : M −→ M in B and a morphism a : U • −→ U • in D b ( B ) such that the diagramU • α / / a (cid:15) (cid:15) X • β / / f (cid:15) (cid:15) M γ / / b (cid:15) (cid:15) U • [1] a [1] (cid:15) (cid:15) U • α / / X • β / / M γ / / U • [1] is commutative. Moreover, if f is an isomorphism in D b ( B ) , then b is an isomorphism in B .Proof. The morphisms a and b exist because α f β must be zero, sinceHom D b ( B ) ( U • , M ) ≃ Hom K b ( B ) ( U • , M ) = . Now assume that f is an isomorphism in D b ( B ). Namely, there is a morphism g : X • −→ X • in D b ( B ) such that f g = X • and g f = X • . By the above discussion, there a morphism c : M −→ M such that β c = g β . Then β − β bc = β − f β c = β − f g β = , and 1 M − bc factorizes through U • [1]. It follows that 1 M − bc factorizes through the projective object U by Corollary3.4. Hence bc = M is the identity map of M in B . Similarly we have cb = M , and therefore b : M −→ M is anisomorphism in B . (cid:3) .2 The definition of the stable functor Keeping the notations above, we can define a functor ¯ F : A −→ B as follows. For each X ∈ A , we fix a triangle ξ X : U • X i X −→ F ( X ) π X −→ M X µ X −→ U • X [1]in D b ( B ) with M X ∈ B , and U • X a complex in D [1 , n X ] ( P B ) for some n X >
0. The existence is guaranteed by Lemma 4.4.For each morphism f : X → Y in A , by Lemma 4.5, we can form a commutative diagram in D b ( B ): U • X i X / / a f (cid:15) (cid:15) F ( X ) π X / / F ( f ) (cid:15) (cid:15) M X µ X / / b f (cid:15) (cid:15) U • X [1] a f [1] (cid:15) (cid:15) U Y • i Y / / F ( Y ) π Y / / M Y µ Y / / U Y • [1]If b ′ f is another morphism such that π X b ′ f = F ( f ) π Y , then π X ( b f − b ′ f ) =
0, and b f − b ′ f factorizes through U • X [1]. ByCorollary 3.4, the map b f − b ′ f factorizes through U X which is projective. Hence the morphism b f ∈ B ( M X , M Y ) isuniquely determined by f . Moreover, suppose that f factorizes through a projective object P in A , say f = gh for g : X → P and h : P → Y . Then π X ( b f − b g b h ) = F ( f ) π Y − F ( g ) π P b h = F ( f ) π Y − F ( g ) F ( h ) π Y =
0. Hence b f − b g b h factorizes through U • X [1], and factorizes through U X by Corollary 3.4. Thus b f factorizes through P ⊕ U X which isprojective. Hence b f =
0. Thus, we get a well-defined map φ : Hom A ( X , Y ) −→ Hom B ( M X , M Y ) , f b f . It is easy to say that φ is functorial in X and Y . Defining ¯ F ( X ) : = M X for each X ∈ A and ¯ F ( f ) : = φ ( f ) for eachmorphism f in A , we get a functor ¯ F : A −→ B which is called the stable functor of F . Example. (a). If k is a field, and if F = ∆ • L ⊗ A − is a standard derived equivalence given by a two-sided tilting complex ∆ • of B - A -bimodules. Assume that ∆ • has no homology in negative degrees. Take a projective resolution of ∆ • and dogood truncation at degree zero. Then ∆ • is isomorphic in D b ( B ⊗ k A op ) to a complex of the form0 −→ M −→ P −→ · · · −→ P n −→ P i projective for all i >
0. By [Ric91, Proposition 3.1], this complex is a one-sided tilting complex on both sides.It follows that B M A is projective as one-sided modules, and F ( X ) is isomorphic to 0 −→ M ⊗ A X −→ P ⊗ A X −→· · · −→ P n ⊗ A X −→ P i ⊗ A X projective for all i >
0. In this case, the stable functor ¯ F of F is induced by theexact functor B M ⊗ A − : A -Mod −→ B -Mod.(b). Let A be an abelian category with enough projective objects, and let n be a non-negative integer. The n thsyzygy functor Ω n A : A −→ A is a stable functor of the derived equivalence [ − n ] : D b ( A ) −→ D b ( A ). Proposition 4.6.
The following diagram is commutative up to isomorphism. A ¯ F (cid:15) (cid:15) Σ / / D b ( A ) / K b ( P A ) F (cid:15) (cid:15) B Σ / / D b ( B ) / K b ( P B ) , where Σ is the canonical functor induced by the embedding A ֒ → D b ( A ) .Proof. For each X ∈ A , the morphism π X : F ( X ) → ¯ F ( X ) in D b ( B ) can be viewed as a morphism π X : F Σ ( X ) −→ Σ ¯ F ( X ) in D b ( B ) / K b ( P B ). We claim that this gives a natural isomorphism from F ◦ Σ to Σ ◦ ¯ F . Since U • X in the triangle8 X is a complex in K b ( P B ), the morphism π X is an isomorphism in D b ( B ) / K b ( P B ). Moreover, for each morphism f : X → Y in A , one can check from the definition of ¯ F that there is a commutative diagram F ◦ Σ ( X ) π X / / F ◦ Σ ( f ) (cid:15) (cid:15) Σ ◦ ¯ F ( X ) Σ ◦ ¯ F ( f ) (cid:15) (cid:15) F ◦ Σ ( Y ) π X / / Σ ◦ ¯ F ( Y ) . This finishes the proof. (cid:3)
From the definition of the stable functor, it is unclear that whether the stable functor is independent of the choices ofthe triangles ξ X . In this subsection, we shall solve this problem. Actually, we will show that isomorphic non-negativefunctors have isomorphic stable functors.We keep the notations in the previous subsection. For each object X ∈ A , suppose that we choose and fix anothertriangle ξ ′ X : U ′ X • i ′ X −→ F ( X ) π ′ X → M ′ X µ ′ X −→ U ′ X • [1]in D b ( B ) with M ′ X ∈ B and U ′ X • a complex in D [1 , n ′ X ] ( P B ) for some n ′ X >
0. Let ¯ F ′ : A −→ B be the functor defined byusing the triangles ξ ′ X ’s. That is, ¯ F ′ ( X ) = M ′ X for each X ∈ A , and ¯ F ′ ( f ) = b ′ f for momorphism f : X → Y in A , where b ′ f : M ′ X → M ′ Y is a morphism in B such that π ′ X b ′ f = F ( f ) π ′ Y . Proposition 4.7.
The functors ¯ F and ¯ F ′ are isomorphic.Proof. For each X ∈ A , by Lemma 4.5, we can form a commutative diagram U • X i X / / α X (cid:15) (cid:15) F ( X ) π X / / M X µ X / / η X (cid:15) (cid:15) U • X [1] α X [1] (cid:15) (cid:15) U ′• X i ′ X / / F ( X ) π ′ X / / M ′ X µ ′ X / / U ′• X [1] , in D b ( B ) such that η X is an isomorphism in B . Now, for each morphism f : X → Y in A , we have π X b f η Y = F ( f ) π Y η Y = F ( f ) π ′ Y = π ′ X b ′ f = π X η X b ′ f . Hence π X ( η X b ′ f − b f η Y ) =
0, and η X b ′ f − b f η Y factorizes through U • X [1]. It follows that η X b ′ f − b f η Y factorizes throughthe projective object U X by Corollarly 3.4. This shows that η X b ′ f − b f η Y =
0, that is, η X ¯ F ′ ( f ) = ¯ F ( f ) η Y . Thus, we get a natural transformation η : ¯ F → ¯ F ′ with η X : = η X for all X ∈ A . Since we have shown that η X is anisomorphism for all X ∈ A , it follows that η : ¯ F −→ ¯ F ′ is an isomorphism of functors. (cid:3) The above proposition shows that, up to isomorphism, the stable functor ¯ F is independent of the choices of thetriangles ξ X ’s, and is uniquely determined by F . Actually, we can further prove the following theorem. Theorem 4.8.
Let F , F : D b ( A ) −→ D b ( B ) be two isomorphic non-negative triangle functors. Then their stablefunctors ¯ F and ¯ F are isomorphic. roof. Suppose that η : F → F is an isomorphism of triangle functors. For each X ∈ A , by definition, we have twotriangles U • X i X −→ F ( X ) π X −→ ¯ F ( X ) µ X −→ U • X [1] and V • X j X −→ F ( X ) p X −→ ¯ F ( X ) ρ X −→ V • X [1] with U • X and V • X in D [1 , n X ] ( P B )for some positive integer n X . By Lemma 4.5, we can form a commutative diagram U • X i X / / α X (cid:15) (cid:15) F ( X ) π X / / η X (cid:15) (cid:15) ¯ F ( X ) µ X / / δ X (cid:15) (cid:15) U • X [1] α X [1] (cid:15) (cid:15) V • X j X / / F ( X ) p X / / ¯ F ( X ) µ ′ X / / V • X [1]such that δ X is an isomorphism in B , since η X is an isomorphism in D b ( B ). Now for each morphism f : X −→ Y in A ,there are morphisms b f : ¯ F ( X ) −→ ¯ F ( Y ) and b ′ f : ¯ F ( X ) −→ ¯ F ( Y ) with π X b f = F ( f ) π Y , p X b ′ f = F ( f ) p Y such that ¯ F ( f ) = b f and ¯ F ( f ) = b ′ f . Now we have π X ( b f δ Y − δ X b ′ f ) = F ( f ) π Y δ Y − η X p X b ′ f = F ( f ) η Y p Y − η X F ( f ) p Y = (cid:0) F ( f ) η Y − η X F ( f ) (cid:1) p Y = . Hence b f δ Y − δ X b ′ f factorizes through U • X [1], and consequently factorizes through the projective object U X by Corollary3.4. Hence 0 = b f δ Y − δ X b ′ f = ¯ F ( f ) δ Y − δ X ¯ F ( f ). This gives rise to a natural transformation δ : ¯ F −→ ¯ F with δ X : = δ X for each X ∈ A . Recall that δ X is an isomorphism for all X ∈ A . This proves that δ : ¯ F −→ ¯ F is anisomorphism. (cid:3) Suppose that A , B and C are abelian categories with enough projective objects. Let F : D b ( A ) −→ D b ( B ) and G : D b ( B ) −→ D b ( C ) be non-negative triangle functors. It is easy to see that GF is also non-negative. The relationshipamong the stable functors of F , G and GF is the following theorem. Theorem 4.9.
The functors ¯ G ◦ ¯ F and GF are isomorphic.Proof.
For each X ∈ A , there are two triangles U • X i X −→ F ( X ) π X −→ ¯ F ( X ) µ X −→ U • X [1] , V • X j X −→ G ( ¯ F ( X )) p X −→ ¯ G ¯ F ( X ) ω X −→ V • X [1]with U • X ∈ D [1 , n X ] ( P B ) and V • X ∈ D [1 , m X ] ( P C ). By the octahedral axiom, we can form a commutative diagram V • Xj X (cid:15) (cid:15) V • X ǫ X (cid:15) (cid:15) G ( U • X ) u X (cid:15) (cid:15) G ( i X ) / / GF ( X ) G ( π X ) / / G ( ¯ F ( X )) G ( µ X ) / / p X (cid:15) (cid:15) G ( U • X )[1] u X [1] (cid:15) (cid:15) W • X α X / / GF ( X ) β X / / ¯ G ¯ F ( X ) γ X / / ω X (cid:15) (cid:15) W • X [1] v X (cid:15) (cid:15) V • X [1] V • X [1]with all rows and columns being triangles in D b ( C ). Since U • X is a complex in D [1 , n X ] ( P B ), and G ( U iX ) is isomorphic toa complex D [0 , t i ] ( P C ) for all i = , · · · , n X , it follows, by [Hu12, Lemma 2.1] for example, that G ( U • X ) is isomorphic in10 b ( C ) to a complex in D [1 , a X ] ( P C ) for some a X >
0. Recall that V • X is a complex in D [1 , m X ] ( P C ). As a result, the complex W • X [1], which is the mapping cone of ǫ X , is isomorphic to a complex in D [0 , m X + a X − ( P C ). Hence we can assume that W • X is a complex in D [1 , m X + a X ] ( P C ). Thus the stable functor of GF can be defined by fixing, for each X ∈ A , the triangle W • X α X −→ GF ( X ) β X −→ ¯ G ¯ F ( X ) γ X −→ W • X [1] . Therefore, for each X ∈ A , we have GF ( X ) = ¯ G ¯ F ( X ).Let f : X −→ Y be a morphism in A . By the construction of stable functor, there is a morphism b f : ¯ F ( X ) −→ ¯ F ( Y )in B such that π X b f = F ( f ) π Y , and ¯ F ( f ) = b f , and there is a morphism c f : ¯ G ¯ F ( X ) −→ ¯ G ¯ F ( Y )in C such that β X c f = GF ( f ) β Y and GF ( f ) = c f . Also there is a morphism c ′ f : ¯ G ¯ F ( X ) −→ ¯ G ¯ F ( Y )in C such that p X c ′ f = G ( b f ) p Y and ¯ G ( b f ) = c ′ f . Now we have the following β X ( c f − c ′ f ) = GF ( f ) β Y − G ( π X ) p X c ′ f = GF ( f ) β Y − G ( π X ) G ( b f ) p Y = GF ( f ) β Y − G ( π X b f ) p Y = GF ( f ) β Y − G ( F ( f ) π Y ) p Y = GF ( f ) β Y − GF ( f ) G ( π Y ) p Y = GF ( f ) β Y − GF ( f ) β Y = . Hence c f − c ′ f factorizes through W • X [1], and consequently c f − c ′ f factorizes through the projective object W X byCorollary 3.4. Therefore we have c f = c ′ f , and GF ( f ) = c f = c ′ f = ¯ G ( b f ) = ¯ G ¯ F ( f ) . This shows that, by choosing the triangles carefully, we get ¯ G ◦ ¯ F = GF . Since the stable functor is unique up toisomorphism, we are done. (cid:3) An immediate consequence is the following.
Corollary 4.10.
Keep the notations above. The functors ¯ F ◦ Ω A ≃ Ω B ◦ ¯ F.Proof.
Since F ◦ [ − ≃ [ − ◦ F , the corollary follows from Theorem 4.8 and Theorem 4.9. (cid:3) Although it is hard to say whether the stable functor ¯ F is an exact functor or not, the following proposition shows thatthe stable functor does have certain “exactness” property. Proposition 4.11.
Keep the notations above. Suppose that → X f → Y g → Z → is an exact sequence in A . Thenthere is an exact sequence −→ ¯ F ( X ) [ a , b ] −→ ¯ F ( Y ) ⊕ P (cid:2) u vs t (cid:3) −→ ¯ F ( Z ) ⊕ Q −→ in B for some projective objects P and Q such that ¯ F ( f ) = a and ¯ F ( g ) = u.Proof. For each X ∈ A , since F is a non-negative functor, we may assume that F ( X ) is a complex P • X ∈ D [0 , n X ] ( B ) with P X = ¯ F ( X ) and P iX ∈ P B for all i > → X f → Y g → Z → A , we get a triangle in D b ( A ): X f → Y g → Z h → X [1] . F results in a triangle in D b ( B ): P • X F ( f ) → P • Y F ( g ) → P • Z F ( h ) → P • X [1] . By Proposition 3.1, the morphisms F ( f ) and F ( g ) are induced by chain maps p • and q • , respectively. That is, F ( f ) = p • and F ( g ) = q • . There is a commutative diagram in D b ( B ): P • Z [ − / / P • X p • / / r (cid:15) (cid:15) P • Y p • / / P • Z P • Z [ − / / con( q • )[ − π • / / P • Y q • / / P • Z , for some isomorphism r , where π • = ( π i ) with π i : P iY ⊕ P i − Z → P iY the canonical projection for each integer i . ByProposition 3.1, the morphism r is induced by a chain map r • . Then cone( r • ) is of the form0 / / P X [ − d , r ] / / P X ⊕ P Y (cid:20) − d x y − d q (cid:21) / / P X ⊕ P Y ⊕ P Z / / · · · , where r = [ x , y ] : P X → P Y ⊕ P Z and P iX , P iY and P iZ are projective for i ≥
1. Since r = r • is an isomorphism in D b ( A ), the mapping cone con( r • ) is an acyclic complex. Thus dropping the split direct summands of con( r • ), we getan exact sequence 0 / / P X [ − d , r ] / / P X ⊕ P Y (cid:20) α βγ q (cid:21) / / Q ⊕ P Z h δη i / / V / / , where Q = P X ⊕ P Y and V is a projective object in B . Let [ ǫ, χ ] : V −→ Q ⊕ P Z be such that [ ǫ, χ ] (cid:2) δη (cid:3) = V . We claimthat the sequence 0 / / P X [0 , − d , r ] / / V ⊕ P X ⊕ P Y " ǫ χα βγ q / / Q ⊕ P Z / / , is exact. It su ffi ces to prove that the sequence is exact at the middle term. Clearly [0 , − d , r ] (cid:20) ǫ χα βγ q (cid:21) =
0. If [ x , x , x ]is a morphism from an object U to V ⊕ P X ⊕ P Y such that [ x , x , x ] (cid:20) ǫ χα βγ q (cid:21) =
0, then x = [ x , x , x ] (cid:20) ǫ χα βγ q (cid:21) (cid:2) δη (cid:3) = x , x ] h α βγ q i =
0. Thus [ x , x ] factorizes uniquely through [ − d , r ] by exactness, and [ x , x , x ] = [0 , x , x ] factorizes through [0 , − d , r ]. Setting P = V ⊕ P X , a = r , b = [0 , − d ] , u = q , v = γ, s = (cid:2) χβ (cid:3) and t = (cid:2) ǫα (cid:3) ,we get the desired exact seuqence. (cid:3) Let A be an abelian category with enough projective objects, and let P A be the full subcategory of A consisting of allprojective objects. An object X ∈ A is called Gorenstein projective if there is an exact sequence P • : · · · −→ P − d − −→ P d −→ P d −→ · · · in C ( P A ) such that Hom • A ( P • , Q ) is exact for all Q ∈ P A and X ≃ Im d . We denote by A - GP the full subcategoryconsisting of all Gorenstein projective objects. Then A - GP is a Frobenius category with projective ( = injective) objectsbeing the projective objects in A . The stable category A - GP is a triangulated category with shifting functor Ω − A . Thefollowing lemma is an alternative description of Gorenstein projective objects.12 emma 5.1. Let A be an abelian category with enough projective objects. Then an object X ∈ A is Gorensteinprojective if and only if there are short exact sequences −→ X i −→ P i + −→ X i + −→ in A with P i projective and X i ∈ ⊥ P A for i ∈ Z such that X = X. The following proposition shows that the stable functor of certain non-negative functor preserves Gorenstein pro-jective modules.
Proposition 5.2.
Let A and B be two abelian categories with enough projective objects. Suppose that F : D b ( A ) −→ D b ( B ) is a non-negative triangle functor admitting a right adjoint G with G ( Q ) ∈ K b ( P A ) for all Q ∈ P B . Let m be anon-negative integer. Then we have the following. (1) . If X ∈ ⊥ > m P A , then ¯ F ( X ) ∈ ⊥ > m P B . (2) . If X ∈ A - GP , then ¯ F ( X ) ∈ B - GP .Proof. For each Q ∈ P B , by assumption G ( Q ) ∈ K b ( P A ). We claim that G ( Q ) is isomorphic to a complex in K b ( P A )with zero terms in all positive degrees. This is equivalent to saying that Hom D b ( A ) ( P , G ( Q )[ i ]) = P ∈ P A andall i >
0. However, this follows from the isomorphismHom D b ( A ) ( P , G ( Q )[ i ]) ≃ Hom D b ( B ) ( F ( P ) , Q [ i ])and the assumption that F is non-negative.(1). Suppose that X ∈ ⊥ > m P A . Then P A ⊆ X ⊥ > m . It is clear that X ⊥ > m is closed under the shift functor [1] andextensions. It follows that each bounded complex in K b ( P A ), which has zero terms in all positive degrees, are in X ⊥ > m .In particular, G ( Q ) ∈ X ⊥ > m for all Q ∈ P B . By the definition of the stable functor ¯ F , there is a triangle U • X i X −→ F ( X ) π X −→ ¯ F ( X ) µ X −→ U • X [1]in D b ( B ) with U • X ∈ D [1 , n ] ( P B ) for some n >
0. Let Q ∈ P B , and let i be a positive integer. We haveHom D b ( B ) ( U • X [1] , Q [ i ]) = = Hom D b ( B ) ( U • X , Q [ i ]) . Applying Hom D b ( B ) ( − , Q [ i ]) to the above triangle results in an isomorphismHom D b ( B ) ( ¯ F ( X ) , Q [ i ]) ≃ Hom D b ( B ) ( F ( X ) , Q [ i ]) . The latter is further isomorphic to Hom D b ( A ) ( X , G ( Q )[ i ]), which vanishes for i > m . Hence ¯ F ( X ) ∈ ⊥ > m P B .(2). Suppose that X is Gorenstein projective. By Lemma 5.1, there are short exact sequences0 −→ X i − −→ P i −→ X i −→ , i ∈ Z with P i projective and X i ∈ ⊥ P A for all i such that X = X . It follows from Lemma 4.11 that there exist short exactsequences 0 −→ ¯ F ( X i − ) −→ Q i −→ ¯ F ( X i ) −→ , i ∈ Z in B with Q i projective for all i . Moreover, the objects ¯ F ( X i ) , i ∈ Z are all in ⊥ P B by (1). Hence, by Lemma 5.1, theobject ¯ F ( X ) is Gorenstein projective. (cid:3) It is well-known that, for an abelian category A with enough projective objects, there is a triangle embedding A - GP ֒ → D b ( A ) / K b ( P A ) induced by the canonical embedding A ֒ → D b ( A ). One may ask whether the stablefunctor is compatible with this embedding. The following theorem provides an a ffi rmative answer. Theorem 5.3.
Let A and B be abelian categories with enough projective objects, and let F : D b ( A ) −→ D b ( B ) be atriangle functor. Then we have the following. (1) . If F is non-negative and admits a right adjoint G with G ( Q ) ∈ K b ( P A ) for all Q ∈ P B , then there is acommutative diagram (up to natural isomorphism) of triangle functors. A - GP ¯ F (cid:15) (cid:15) (cid:31) (cid:127) / / D b ( A ) / K b ( P A ) F (cid:15) (cid:15) B - GP (cid:31) (cid:127) / / D b ( B ) / K b ( P B ) , If F is a uniformly bounded non-negative equivalence, then the functor ¯ F : A - GP −→ B - GP in the abovediagram is a triangle equivalence.Proof. The commutative diagram follows from Proposition 4.6 and Proposition 5.2. It follows from Corollary 4.10 andProposition 4.11 that ¯ F : A - GP −→ B - GP is a triangle functor.(2) Let G be a quasi-inverse of F . Then G is both a left adjoint and a right adjoint of F . By Lemma 4.3, there existssome integer n > G [ − n ] is non-negative. Note that F [ n ] is a right adjoint of G [ − n ], and sends projectiveobjects in A to complexes in K b ( P B ). By (1), the stable functor G [ − n ] of G [ − n ] induces a triangle functor from B - GP to A - GP . Thus, by Theorem 4.8 and Theorem 4.9, we have isomorphisms of functors¯ F ◦ G [ − n ] ≃ F ◦ G [ − n ] ≃ [ − n ] ≃ Ω n B and G [ − n ] ◦ ¯ F ≃ G [ − n ] ◦ F ≃ [ − n ] ≃ Ω n A . Note that Ω A and Ω B induce auto-equivalences of A - GP and B - GP , respectively. It follows that ¯ F : A - GP −→ B - GP is an equivalence. (cid:3) Let F : D b ( A -Mod) → D b ( B -Mod) be a derived equivalence between two rings such that the tilting complexassociated to F has zero terms in all positive degrees. By Lemma 4.2, the functor F satisfies the assumption ofTheorem 5.3 (2). Thus, we have the following corollary. Corollary 5.4.
Let A and B be rings, and let F : D b ( A- Mod) −→ D b ( B- Mod) be a non-negative derived equivalence.Then ¯ F : A- GP −→ B- GP is a triangle equivalence. For a given derived equivalence functor F between two rings, by Lemma 4.2, F [ m ] is non-negative when m issu ffi ciently small. The following corollary is then clear. Corollary 5.5.
Let A and B be derived equivalent rings. Then A- GP and B- GP are triangle equivalent. Recall that a ring A is called left coherent provided that the category A -mod of left finitely presented A -modules isan abelian category. In this case, the finitely generated Gorenstein projective A -modules coincide with those Gorensteinprojective modules in A -mod. By A - f GP we denote the category of finitely generated Gorenstein projective A -modules,and by A - f GP we denote its stable category.By Rickard’s result in [Ric89]. For left coherent rings A and B , D b ( A -mod) and D b ( B -mod) are triangle equivalentif and only if D b ( A -Mod) and D b ( B -Mod) are triangle equivalent, and every triangle equivalence between D b ( A -Mod)and D b ( B -Mod) restricts to a triangle equivalence between D b ( A -mod) and D b ( B -mod). Thus, we obtain the followingcorollary. Corollary 5.6.
Let A and B be left coherent rings, and let F : D b ( A- mod) → D b ( B- mod) be a non-negative derivedequivalence. Then ¯ F : A-f GP → B-f GP are triangle equivalence. Particularly, the stable categories of finitelygenerated Gorenstein projective modules of two derived equivalent coherent rings are triangle equivalent.Remark. This generalizes a result of Kato [Kat02], where it was proved that standard derived equivalences betweentwo left and right coherent rings induce triangle equivalences between stable categories of finitely generated Gorensteinprojective modules.
For a finite dimensional algebra Λ , in general, it is very hard to find all the indecomposable Gorenstein projectivemodules in Λ - f GP . However, if Λ is derived equivalent to another algebra Γ for which the Gorenstein projectivemodules are known, then the stable functor will be helpful to describe the Gorenstein projective modules in Λ - f GP .Let k be a field, and let k [ ǫ ] be the algebra of dual numbers, that is, the quotient algebra of the polynomial algebra k [ x ] modulo the ideal generated by x . Let A be the k -algebra given by the quiver • • • •• • • •· · · n − n +
10 2 2 n − n α (cid:7) (cid:7) ✎✎✎✎✎✎✎ β / / α (cid:7) (cid:7) ✎✎✎✎✎✎✎ β / / α (cid:7) (cid:7) ✎✎✎✎✎✎✎ α (cid:7) (cid:7) ✎✎✎✎✎✎✎ βα =
0. Then A is a tilted algebra, and is derived equivalent to the path algebra, denoted by B , of the following quiver. • • • •• • • •· · · n − n +
10 2 2 n − n G G ✎✎✎✎✎✎✎ (cid:23) (cid:23) ✴✴✴✴✴✴✴ G G ✎✎✎✎✎✎✎ G G ✎✎✎✎✎✎✎ (cid:23) (cid:23) ✴✴✴✴✴✴✴ G G ✎✎✎✎✎✎✎ Let Λ : = k [ ǫ ] ⊗ k A and let Γ : = k [ ǫ ] ⊗ k B . Then Λ and Γ are also derived equivalent. The Gorenstein projectivemodules over Γ have been described by Ringel and Zhang in [RZ11]. They also proved that Γ - f GP is equivalent tothe orbit category D b ( B ) / [1]. This means that the indecomposable non-projective Gorenstein projective Γ -modulesare one-to-one correspondent to the indecomposable B -modules. Then correspondence reads as follows. Let S be theunique simple k [ ǫ ]-module, and let Q i be the indecomposable projective B -module corresponding to the vertex i for all i . For each i ∈ { , , · · · , n + } , and 1 ≤ l ≤ n + − i , we denote by X ( i , l ) the indecomposable B -module with topvertex i and length l . We write M ( i , l ) for the corresponding Gorenstein projective Γ -module. If X ( i , l ) is projective, thatis, i + l = n +
2, then M ( i , l ) = S ⊗ Q i . If X ( i , l ) is not projective, then there is an short exact sequence0 −→ S ⊗ Q i + l −→ M ( i , l ) −→ S ⊗ Q i −→ . ( ∗ )Here we shall use the stable functor of the derived equivalence between Γ and Λ to get all the indecomposable Goren-stein projective modules over Λ .For each i ∈ { , , · · · , n + } , we denote by P i the indecomposable projective A -module corresponding to thevertex i . The derived equivalence between A and B is given by the tilting module n M i = (cid:0) P i + ⊕ τ − S i (cid:1) , where S i is the simple A -module corresponding to the vertex 2 i . Note that τ − S i has a projective resolution0 −→ P i −→ P i + −→ τ − S i −→ . Thus, we get a derived equivalence F : D b ( B ) −→ D b ( A ) such that F ( Q i + ) ≃ P i + [ −
1] and F ( Q i ) is0 −→ P i −→ P i + −→ P i in degree zero for all 0 ≤ i ≤ n . By [Ric91], there is a derived equivalence F ′ : D b ( Γ ) −→ D b ( Λ ), whichsends k [ ǫ ] ⊗ Q i + to k [ ǫ ] ⊗ P i + [ − k [ ǫ ] ⊗ Q i to the complex0 −→ k [ ǫ ] ⊗ P i −→ k [ ǫ ] ⊗ P i + −→ ≤ i ≤ n .For each i ∈ { , , · · · , n + } , and for each 1 ≤ l ≤ n + − i , let N ( i , l ) be the image of M ( i , l ) under the stablefunctor of F ′ . Then it is easy to see that the image N (2 i + , n − i +
1) of M (2 i + , n − i + = S ⊗ Q i + ) is Ω ( S ⊗ P i + ), which is isomorphic to S ⊗ P i + . The module N (2 i , n − i +
2) fits into the following pullback diagram N (2 i , n − i + / / (cid:15) (cid:15) k [ ǫ ] ⊗ P i + (cid:15) (cid:15) (cid:15) (cid:15) S ⊗ P i / / S ⊗ P i + , and can be diagrammatically presented as follows • • • •• • · · · i + n + i i (cid:7) (cid:7) ✎✎✎✎✎ / / / / _ _ ❄❄❄ Each vertex of the above diagram corresponds to a basis vector of the module, and the arrow from 2 i to 2 i correspondsto the action of ǫ . The other arrow corresponds to the action of the corresponding arrow in the quiver of A . Since thestable functor is a triangle equivalence, we can use the short exact sequence ( ∗ ) to get N ( i , l ) for 1 ≤ l < n + − i . Theresult can be listed as the following table. 15 , l N ( i , l ) i and l are even • • • i + l + • • •• • · · · · · ·• • • •• • · · · i + n + i i i + l + i + l i + l (cid:7) (cid:7) ✎✎✎✎✎ / / / / / / _ _ ❄❄❄ / / / / (cid:7) (cid:7) ✎✎✎✎✎ _ _ ❄❄❄ _ _ ❄❄❄ _ _ ❄❄❄ _ _ ❄❄❄ _ _ ❄❄❄ i is even, l > • • • i + l • • •• • · · · · · ·• • • •• · · · i + n + i i i + li + l − (cid:7) (cid:7) ✎✎✎✎✎ / / / / / / _ _ ❄❄❄ / / / / (cid:7) (cid:7) ✎✎✎✎✎ _ _ ❄❄❄ _ _ ❄❄❄ _ _ ❄❄❄ _ _ ❄❄❄ i is even, l = • i i is odd, l is even • • • i + l • • •• · · · · · ·• • • •• · · · i n + i − i + li + l − (cid:7) (cid:7) ✎✎✎✎✎ / / / / / / / / / / (cid:7) (cid:7) ✎✎✎✎✎ _ _ ❄❄❄ _ _ ❄❄❄ _ _ ❄❄❄ _ _ ❄❄❄ i is odd, l is odd • • • i + l + • • •• · · · · · ·• • • •• • · · · i n + i − i + l + i + l i + l (cid:7) (cid:7) ✎✎✎✎✎ / / / / / / / / / / (cid:7) (cid:7) ✎✎✎✎✎ _ _ ❄❄❄ _ _ ❄❄❄ _ _ ❄❄❄ _ _ ❄❄❄ _ _ ❄❄❄ In case that n =
1, the algebra A is given by the quiver • •• • (cid:7) (cid:7) ✎✎✎✎✎✎✎ / / (cid:7) (cid:7) ✎✎✎✎✎✎✎ The Auslander-Reiten quiver of Λ - f GP can be drawn as follows. ◦ •◦ • (cid:7) (cid:7) ✎✎✎ ◦ •◦ • • (cid:7) (cid:7) ✎✎✎ _ _ • •• ◦ (cid:7) (cid:7) ✎✎✎ / / • •• ◦• (cid:7) (cid:7) ✎✎✎ / / _ _ ◦ ◦◦ • • •• ◦ • • (cid:7) (cid:7) ✎✎✎ / / _ _ (cid:7) (cid:7) ✎✎✎ • •• ◦• • • (cid:7) (cid:7) ✎✎✎ / / _ _ _ _ (cid:7) (cid:7) ✎✎✎ ◦ •◦ • (cid:7) (cid:7) ✎✎✎ • •• ◦ • •• (cid:7) (cid:7) ✎✎✎ / / _ _ _ _ (cid:7) (cid:7) ✎✎✎ • •• ◦• • •• (cid:7) (cid:7) ✎✎✎ / / _ _ _ _ _ _ (cid:7) (cid:7) ✎✎✎ ◦ •◦ • • (cid:7) (cid:7) ✎✎✎ _ _ ◦ ◦• ◦ • •• ◦ (cid:7) (cid:7) ✎✎✎ / / • •• ◦• (cid:7) (cid:7) ✎✎✎ / / _ _ ? ? ⑧⑧⑧⑧⑧⑧⑧ ? ? ⑧⑧⑧⑧⑧⑧⑧ ? ? ⑧⑧⑧⑧⑧⑧⑧ ? ? ⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄❄ (cid:31) (cid:31) ❄❄❄❄❄❄❄ ? ? ⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ ? ? ⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄ ? ? ⑧⑧⑧⑧ ? ? ⑧⑧⑧⑧⑧ ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄ (cid:31) (cid:31) ❄❄❄❄❄❄ (cid:31) (cid:31) ❄❄❄❄❄❄❄ (cid:31) (cid:31) ❄❄❄❄❄❄❄ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ ❄ ❄ ❄ ❄ ❄ ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄❄⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧⑧⑧⑧⑧⑧❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ The modules in the two dashed frames are identified correspondingly.
Our results can be applied abelian categories with enough injective objects (e.g. Grothendieck categories). One justneed to consider their opposite categories, which are abelian categories with enough projective objects. Our results canalso be used to give shorter proofs of some known results on homological conjectures.16n the following, we assume that A and B are derived equivalent left coherent rings, and F : D b ( A -mod) → D b ( B -mod) is a derived equivalence. Without loss of generality, we can assume that F is non-negative and the tiltingcomplex associated to F has terms only in degrees 0 , · · · , n . Let G be a quasi-inverse of F . Then G [ − n ] is alsonon-negative. Finitistic dimension.
The finitistic dimension of a left coherent ring is the supremum of projective dimensions offinitely presented modules with finite projective dimensions. The finiteness of finitistic dimension is proved to bepreserved under derived equivalences in [PX09]. With the stable functor, the proof will be very easy. We claim that | fin . dim( A ) − fin . dim( B ) | ≤ n , where fin . dim stands for the finitistic dimension. To prove this, it is su ffi cient to provethat, for each A -module X , there are inequalities between the projective dimensions of X and ¯ F ( X ):proj . dim B ¯ F ( X ) ≤ proj . dim A X ≤ proj . dim B ¯ F ( X ) + n . We first prove the first inequality. Suppose that proj . dim A X = m . Then Ω mA ( X ) ≃ A -mod, and consequently Ω mB ◦ ¯ F ( X ) ≃ ¯ F ◦ Ω mA ( X ) ≃ B -mod, where the first isomorphism follows from Corollary 4.10. Henceproj . dim B ¯ F ( X ) ≤ m = proj . dim A X . The proof of the second inequality goes as follows. Suppose that proj . dim B ¯ F ( X ) = m . Then we have the followingisomorphisms in A -mod: Ω m + nA ( X ) ≃ [ − n − m ]( X ) ≃ G [ − n ] ◦ [ − m ] ◦ F ( X ) ≃ G [ − n ] ◦ Ω mB ◦ ¯ F ( X ) ≃ , where the third isomorphism follows from Theorem 4.9. This implies thatproj . dim A X ≤ m + n = proj . dim B ¯ F ( X ) + n . Syzygy finiteness.
A left coherent ring Λ is called Ω m -finite provided that add( Ω m Λ ( Λ -mod)) contains only finitely manyisomorphism classes of indecomposable Λ -modules, and is called syzygy-finite if A is Ω m -finite for some m . Clearly, asyzygy-finite algebra always has finite finitistic dimension. With the help of the stable functor, we can prove that: If A is Ω m -finite, then B is Ω m + n -finite. In particular A is syzygy-finite if and only if so is B. The proof of the above statement is almost trivial. Let X be a B -module. By assumption, there is an A -module M such that Ω mA ¯ F ( X ) ∈ add( M ). Applying the stable functor of G [ − n ], we see that Ω m + nB ( X ), which is isomorphic to G ◦ [ − n ] ◦ Ω mA ◦ ¯ F ( X ) in B -mod, is in add( B ⊕ G [ − n ]( M )), showing that B is Ω m + n -finite. Generalized Auslander-Reiten conjecture.
This conjecture says that a module X over an Artin algebra Λ satisfyingExt i Λ ( X , X ⊕ Λ ) = i > m ≥ ≤ m . Via the stable functor, the second author provedin [Pan13] that A satisfies the generalized Auslander-Reiten conjecture if and only so does B . This was also proved byWei [Wei12] and by Diveris and Purin [DP12] independently. Acknowledgements.
The research work of both authors are partially supported by NSFC and the Fundamental Re-search Funds for the Central Universities. W. Hu thanks BNSF(1132005, KZ201410028033) for partial support, andS.Y. Pan is also partially supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry.
References [AB69] M. A uslander and M. B ridger , Stable module theory , no. 94, American Mathematical Soc., 1969.[DP12] K. D iveris and M. P urin , The generalized auslandercreiten condition for the bounded derived category,
Arch.Math. (Basel) (2012), 507–511.[Hu12] W. H u , On iterated almost ν -stable derived equivalences, Comm. Algebra (2012), 3920–3932.[HX10] W. H u and C. C. X i , Derived equivalences and stable equivalences of Morita type, I, Nagoya Math. J. (2010), 107–152.[Kat02] Y. K ato , On derived equivalent coherent rings,
Comm. Algebra (2002), 4437–4454.17Kel94] B. K eller , Deriving dg categories, Ann. Sci. ´Ecole Norm. Sup. (1994), 63–102.[Kel06] B. K eller , On di ff erential graded categories, in International Congress of Mathematicians. Vol. II , Eur. Math.Soc., Z¨urich, 2006, pp. 151–190.[Nee01] A. N eeman , Triangulated categories , no. 148, Princeton University Press, 2001.[Pan13] S. P an , Generalized Auslander-Reiten conjecture and derived equivalences, Comm. Algebra (2013), 3695–3704.[PX09] S. P an and C. X i , Finiteness of finitistic dimension is invariant under derived equivalences, J. Algebra (2009), 21–24.[Ric89] J. R ickard , Morita theory for derived categories,
J. London Math. Soc. (1989), 436–456.[Ric91] J. R ickard , Derived equivalences as derived functors, J. London Math. Soc. (1991), 37–48.[RZ11] C. M. R ingel and P. Z hang , Representations of quivers over the algebras of dual numbers, (2011).[Wei12] J. W ei , Tilting complexes and Auslander-Reiten conjecture, Math. Z. (2012), 431–441.Wei HuSchool of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, MOE, Beijing Normal Uni-versity, 100875 Beijing, ChinaBeijing Center for Mathematics and Information Interdisciplinary Sciences, 100048 Beijing, China
Email: [email protected]
Shengyong PanDepartment of Mathematics, Beijing Jiaotong University, 100044 Beijing, ChinaBeijing Center for Mathematics and Information Interdisciplinary Sciences, 100048 Beijing, China