On standard derived equivalences of orbit categories
aa r X i v : . [ m a t h . R T ] J a n On standard derived equivalences of orbit categories
Yury Volkov ∗ and Alexandra Zvonareva † Abstract
Let k be a commutative ring, A and B – two k -linear categories with an action ofa group G . We introduce the notion of a standard G -equivalence from K b p B to K b p A .We construct a map from the set of standard G -equivalences to the set of standardequivalences from K b p B to K b p A and a map from the set of standard G -equivalencesfrom K b p B to K b p A to the set of standard equivalences from K b p ( B /G ) to K b p ( A /G ).We investigate the properties of these maps and apply our results to the case where A = B = R is a Frobenius k -algebra and G is the cyclic group generated by itsNakayama automorphism ν . We apply this technique to obtain the generating set ofthe derived Picard group of a Frobenius Nakayama algebra over an algebraically closedfield. Let A and B be two derived equivalent categories. The notion of a standard equivalence from DB to DA was introduced in [1]. This notion generalizes the notion of a standard equiva-lence for algebras [3]. We define such standard equivalences in terms of tilting subcategoriesinstead of tilting complexes of bimodules. We denote by TrPic( A , B ) the set of standardequivalences from K b p B to K b p A , standard equivalences from DB to DA correspond bijectivelyto standard equivalences from K b p B to K b p A . In [1] it is proved that the composition of stan-dard equivalences and the inverse equivalence of a standard equivalence are again standard.In particular, composition defines a group structure on TrPic( A ) = TrPic( A , A ). We call thisgroup the derived Picard group of A . In the case where a group G acts on A and B , we intro-duce the notion of a standard G -equivalence from K b p B to K b p A . We denote by TrPic G ( A , B )the set of such equivalences. It appears that the composition of standard G -equivalences isdefined and it determines a group structure on TrPic G ( A ) = TrPic G ( A , A ). We constructthe maps Φ A , B : TrPic G ( A , B ) → TrPic( A , B ) and Ψ A , B : TrPic G ( A , B ) → TrPic( A /G, B /G )which respect the composition. Here A /G is the orbit category defined in [2]. We investigatethe properties of these maps. We prove that Φ A , B sends a standard G -equivalence to a Moritaequivalence iff Ψ A , B sends this standard G -equivalence to a Morita equivalence. We provea theorem which gives a necessary and sufficient condition for an element of TrPic( A , B ) to ∗ The author was partially supported by RFFI 13-01-00902. † The author was partially supported by RFFI 13-01-00902 and by the Chebyshev Laboratory (De-partment of Mathematics and Mechanics, St. Petersburg State University) under RF Government grant11.G34.31.0026. A , B . In the case of a finite group G we provide necessary and sufficientcondition for an element of TrPic( A /G, B /G ) to lie in the image of Ψ A , B .It was proved in [3] that the Nakayama functor commutes with any standard derivedequivalence. Suppose that R is a Frobenius algebra with a Nakayama automorphism ν .Suppose that ord ν = n < ∞ . Then the cyclic group G = h ν i ∼ = C n acts on R and wecan define an algebra R/G . We prove that the homomorphism Φ R = Φ R,R : TrPic G ( R ) → TrPic( R ) is an epimorphism if for any a ∈ Z ( R ) ∗ there is an element b ∈ Z ( R ) ∗ such that a = b n . Moreover, we prove that Cok(Φ R ) is generated by the classes of elements from thePicard group of R if for any a ∈ Z ( R ) ∗ there is an element b ∈ R ∗ and an automorphism σ of R such that a = bν ( b ) . . . ν n − ( b ) and σνσ − ( x ) = ν ( bxb − ) for all x ∈ R . We apply thesefacts to find a generating set of the derived Picard group of a Frobenius Nakayama algebrausing the generating set of the derived Picard group of a symmetric Nakayama algebra from[4]. Throughout this paper k is a fixed commutative ring. We assume everywhere that A and B are small, k -linear and k -flat categories ( A is called k -flat if A ( x, y ) is a flat k -module forany x, y ∈ A ). We simply write ⊗ for ⊗ k . In this section we recall some basic definitionsand results on standard equivalences of derived categories. Definition 1.
Contravariant functors from A to Mod k are called A -modules . We denote byMod A the category of A -modules. An A -module is called projective if it is a direct summandof a direct sum of representable functors (a representable functor is a functor isomorphic to A ( − , x ) for some x ∈ A ). An A -module is called finitely generated if it is an epimorphicimage of a finite direct sum of representable functors.A morphism of A -modules d : V → V is called differential if d = 0. A Z -graded A -module is a module V with a decomposition V = ⊕ n ∈ Z V n . We say that a morphism f : V → V ′ is of degree m if f = P n ∈ Z f n for some morphisms f n : V n → V ′ n + m . We denoteby V [ m ] the module V with the following grading: V [ m ] n = V n + m . Definition 2. An A -complex is a Z -graded module V with a differential d V : V → V of degree 1. A morphism of A -complexes is a morphism of A -modules of degree 0 whichcommutes with differentials. We denote by CA the category of A -complexes. A morphism f : V → V ′ is called null homotopic if f = hd V + d V ′ h for some morphism of modules h : V → V ′ of degree −
1. We denote by H ( V, V ′ ) the space of all null homotopic morphismsfrom V to V ′ . The homotopy category of Mod A is the category KA with the same objects as CA and morphism spaces KA ( V, V ′ ) = CA ( V, V ′ ) /H ( V, V ′ ). The derived category of Mod A denoted by DA is the localization of KA at the set of quasi-isomorphisms (a morphism from V to V ′ is called a quasi-isomorphism if it induces an isomorphism from Ker d V / Im d V toKer d V ′ / Im d V ′ ).We denote by K p A the full subcategory of KA consisting of all projective complexes. Thecanonical functor from KA to DA induces an equivalence between categories K p A and DA .We denote by K b p A the full subcategory of KA consisting of all finitely generated projectivecomplexes V such that V n = 0 for large enough and small enough n ∈ Z .2 efinition 3. A full subcategory X of K b p A is called a tilting subcategory for A if • K b p A ( U, V [ i ]) = 0 for all U, V ∈ X , i = 0, • A ( − , x ) ( x ∈ A ) lies in the smallest full triangulated subcategory of K b p A containing X and closed under isomorphisms and direct summands (we denote this subcategoryof K b p A by thick X ). Definition 4.
The tensor product
A ⊗ B of A and B is a k -linear category defined in thefollowing way. Its objects are pairs ( x, y ) where x ∈ A , y ∈ B . Its morphism spaces are( A ⊗ B )(( x , y ) , ( x , y )) = A ( x , x ) ⊗ B ( y , y ) . The composition in
A ⊗ B is given by the formula( f ⊗ g )( f ⊗ g ) = f f ⊗ g g , where f ∈ A ( x , x ), f ∈ A ( x , x ), g ∈ B ( y , y ), g ∈ B ( y , y ) and x , x , x ∈ A , y , y , y ∈ B .Let X be an A ⊗ B op -complex. It defines a functor T X : CB → CA in the following way.If M ∈ CB , then ( T X M )( x ) ( x ∈ A ) is the cokernel of the map ρ X,M ( x ) : M y,z ∈B ( M ( z ) ⊗ B ( y, z ) ⊗ X ( x, y )) → M y ∈B ( M ( y ) ⊗ X ( x, y ))defined by the equality ρ X,M ( x )( u ⊗ g ⊗ v ) = M ( g )( u ) ⊗ v − u ⊗ X (Id x ⊗ g )( v )for u ∈ M ( z ), v ∈ X ( x, y ), g ∈ B ( y, z ). If f ∈ A ( x , x ) ( x , x ∈ A ), then ( T X M )( f ) is theunique map such that the diagram L y ∈B ( M ( y ) ⊗ X ( x , y )) / / ⊕ y ∈B ( Id M ( y ) ⊗ X ( f ⊗ Id y ) ) (cid:15) (cid:15) ( T X M )( x ) ( T X M )( f ) (cid:15) (cid:15) L y ∈B ( M ( y ) ⊗ X ( x , y )) / / ( T X M )( x )commutes. Finally, for α ∈ CB ( M, N ) we obtain T X α from the commutativity of diagrams L y ∈B ( M ( y ) ⊗ X ( x, y )) / / ⊕ y ∈B ( α y ⊗ Id X ( x,y ) ) (cid:15) (cid:15) ( T X M )( x ) ( T X α ) x (cid:15) (cid:15) L y ∈B ( N ( y ) ⊗ X ( x, y )) / / ( T X N )( x )We denote by L T X the left derived functor of T X , i.e. the composition DB p → K p B T X −→ KA π → DA , where p sends an object of DB to its projective resolution (see [1, Theorem 3.1] for details)and π is the canonical functor from KA to DA . If X is an A ⊗ B op -complex, then for any y ∈ B we can define an A -complex X y as follows: X y ( x ) = X ( x, y ) and X y ( f ) = X ( f ⊗ Id y )for objects and morphisms of A respectively. Then L T X is an equivalence iff the followingconditions hold: 3 X y is isomorphic to some object of K b p A in KA for any y ∈ B , • the full subcategory of K b p A consisting of objects isomorphic to some X y ( y ∈ B ) in KA is a tilting subcategory for A , • the map B ( y, z ) → KA ( X y , X z ) is an isomorphism for all y, z ∈ B .Moreover, L T X ∼ = L T Y iff X ∼ = Y in D ( A ⊗ B op ). See [1, 6.1] for details. If L T X is anequivalence, then X is called a tilting A ⊗ B op -complex .If F : A → B is a functor, then we denote by F ( A ) the full subcategory of B formed byobjects isomorphic to some F ( U ) ( U ∈ A ). Definition 5.
We call θ : B → K b p A a tilting functor if θ ( B ) is a tilting subcategory for A and θ induces an equivalence from B to θ ( B ).Categories A and B are said to be derived equivalent if the derived categories DA and DB are equivalent as triangulated categories. The following theorem is well known (see [1,9.2, Corollary] and [2, Theorem 4.6]). Theorem 1.
The following conditions are equivalent (1) A and B are derived equivalent; (2) there is a tilting A ⊗ B op -complex X ; (3) there is a tilting functor θ : B → K b p A . Remark 1.
Note that if X is a tilting A⊗B op -complex, then the objects of K b p A isomorphic to X y ( y ∈ B ) form a tilting subcategory which is equivalent to B . We denote the correspondingtilting functor (which is defined modulo natural isomorphism) by θ X . Conversely, if we havea tilting functor θ : B → K b p A , then we can construct a tilting A ⊗ B op -complex X such that θ X ∼ = θ (see [1, Section 9]). Definition 6.
We call an equivalence F : DB → DA standard if there is some
A ⊗ B op -complex X such that F ∼ = L T X . We denote by TrPic( A , B ) the set of all standard equiva-lences from DB to DA modulo natural isomorphisms.Let L T X be a standard equivalence. Define a B ⊗ A op -complex X T as follows: X T ( y, x ) = Mod A ( X y , A ( − , x )) , X T ( g ⊗ f ) = Mod A ( X (Id − ⊗ g ) , A ( − , f ))for x ∈ A , y ∈ B , a morphism f in A and a morphism g in B . By [1, 6.2, Lemma] the functor L T X T is quasi-inverse to L T X . Moreover, if L T X and L T Y are standard equivalences (whichcan be composed), then by [1, 6.3, Lemma] we have L T X L T Y ∼ = L T Z , where Z = T p X Y and p X is the projective resolution of X over A ⊗ B op . Definition 7.
The derived Picard group of A is the setTrPic( A ) := TrPic( A , A )with the operation of composition. It follows from the arguments above that it is actually agroup.Note that the functor L T X : DB → DA is an equivalence iff it’s restriction to K b p B induces an equivalence to K b p A . Moreover, L T X ∼ = L T Y iff the corresponding equivalencesfrom K b p B to K b p A are isomorphic. So we denote by L T X the corresponding equivalence from K b p B to K b p A too. From here on we consider only standard equivalences and identify theset TrPic( A , B ) with the set of standard equivalences from K b p B to K b p A modulo naturalisomorphisms. 4 Standard equivalences and tilting subcategories
For a subcategory E of K b p A we denote by add E the full subcategory of K b p A consisting ofdirect summands of finite direct sums of copies of objects of E . Let us define a category C b add E . Objects of C b add E are objects of add E with a decomposition V = ⊕ n ∈ Z V n anda differential d V = P n ∈ Z d V,n where V n ∈ add E , d V,n ∈ KA ( V n , V n +1 ) and V n = 0 for largeenough and small enough n . If V, V ′ ∈ C b add E , then the set C b add E ( V, V ′ ) is formed bymaps f = P n ∈ Z f n such that f n ∈ KA ( V n , V ′ n ) and f d V − d V ′ f equals 0 in KA . A morphism f ∈ C b add E ( V, V ′ ) is called null homotopic if f = hd V + d V ′ h for some h = P n ∈ Z h n where h n ∈ KA ( V n , V ′ n − ). We denote the set of null homotopic morphisms from V to V ′ by B ( V, V ′ ) again. Then K b add E is a category whose objects are the same as the objects of C b add E and whose morphism spaces are K b add E ( V, V ′ ) = C b add E ( V, V ′ ) /B ( V, V ′ ).We denote by Y A : A → K b p A the Yoneda embedding, i.e. Y A ( x ) = A ( − , x ) and Y A ( f ) = A ( − , f ) for an object x and a morphism f in A . Let θ : B → K b p A be a tiltingfunctor. Our aim is to define an equivalence F θ : K b p B → K b p A in such a way that F θ Y B = θ .Denote by X the category θ ( B ). Let P B be the category of finitely generated projective B -modules. Let us define an equivalence S : P B → add X . Define S on direct sums ofrepresentable functors as follows: S (cid:0) ⊕ mi =1 B ( − , x i ) (cid:1) = ⊕ mi =1 θ ( x i ) . If f = (cid:0) B ( − , f i,j ) (cid:1) i m, j l : ⊕ mi =1 B ( − , x i ) → ⊕ lj =1 B ( − , y j )is a morphism in P B , then S ( f ) = (cid:0) θ ( f i,j ) (cid:1) i m, j l : ⊕ mi =1 θ ( x i ) → ⊕ lj =1 θ ( y j ) . Let us consider an arbitrary object U ∈ P B . There is some direct sum of representablefunctors W U such that U is a direct summand of W U . Let ι U : U → W U and π U : W U → U be the corresponding direct inclusion and projection (for convenience we assume that W U = U and ι U = π U = Id U if U is a direct sum of representable functors). It followsfrom [5] that idempotents split in K b p A . In particular, they split in add X . Since S ( ι U π U ) : S ( W U ) → S ( W U ) is an idempotent in add X , there is some object X U ∈ add X and morphisms ι ′ U : X U → S ( W U ) and π ′ U : S ( W U ) → X U such that π ′ U ι ′ U = Id X U and ι ′ U π ′ U = S ( ι U π U ). Wedefine S ( U ) = X U . If f : U → V is a morphism in P B , then we define S ( f ) by the formula S ( f ) = π ′ V S ( ι V f π U ) ι ′ U . It is clear that S is an equivalence. Then S induces an equivalence ¯ S : K b p B → K b add X .Note also that S ( ι U ) = ι ′ U and S ( π U ) = π ′ U .Let us now translate some results of [6] from the case of algebras to the case of categories.Since the arguments for the case of categories are analogous to the case of algebras, we omitmost of the proofs and give only references to the corresponding results of [6].Let V be an object of C b add X . Then we can consider V as a bigraded module V = ⊕ i,j ∈ Z V i,j (where V i,j = ( V j ) i ) with a differential τ = P j ∈ Z d V j : V → V of degree (1 , τ : V → V of degree (0 ,
1) such that τ | V i,j = ( − i + j d V | V i,j . Such objectssatisfy the conditions 5 V i,j = 0 if i or j is large enough or small enough; • τ = 0; • τ τ + τ τ = 0; • τ = 0 in KA if we consider V as an object of add X .We write ( V, τ , τ ) for such object. Note that for two such objects KA ( V, V ′ [ i ])) = 0, i = 0 if we consider them as objects of add X . A morphism from ( V, τ , τ ) to ( V ′ , τ ′ , τ ′ )in C b add X is a morphism α : V → V ′ of degree (0 ,
0) such that ατ = τ ′ α and ατ − τ ′ α is null homotopic if we consider V and V ′ as objects of add X . Moreover, a morphism α : ( V, τ , τ ) → ( V ′ , τ ′ , τ ′ ) is equal to 0 in C b add X if it is null homotopic as a morphism inadd X . If ( V, τ , τ ) ∈ C b add X , then we can define morphisms τ i : V → V of degree (1 − i, i )in such a way that l X i =0 τ i τ l − i = 0for any l > α : ( V, τ , τ ) → ( V ′ , τ ′ , τ ′ ), then there is a sequenceof maps α i : V → V ′ of degree ( − i, i ) such that α = α and l X i =0 α i τ l − i = l X i =0 τ ′ i α l − i for any l > V, τ , τ ) is an object of C b add X , then we defineTot( V, τ , τ ) ∈ CA by the formulasTot( V, τ , τ ) n = M i + j = n V i,j , d Tot(
V,τ ,τ ) = X i > τ i . If α : ( V, τ , τ ) → ( V ′ , τ ′ , τ ′ ), then we define Tot α : Tot( V, τ , τ ) → Tot( V ′ , τ ′ , τ ′ ) by theformula Tot α = X i > α i . Thus we define a functor Tot : C b add X → K b p A (see [6, Proposition 2.10]). By [6, Proposition2.11] the functor Tot factors through some functor ¯ Q : K b add X → K b p A . We define F θ asthe composition K b p B ¯ S → K b add X ¯ Q → K b p A . (3.1)Note that F θ is defined modulo isomorphism. We fix some representative of this equivalencefor each tilting functor. Proposition 1.
Let θ : B → K b p A be a tilting functor and X be a tilting A ⊗ B op -complex.If θ X ∼ = θ , then L T X ∼ = F θ . Proof
Let F θ be defined by the composition (3.1). Then it is enough to prove that T X | K b p B ∼ = ι ¯ Q ¯ S , where ι is the canonical embedding of K b p A to KA . We know that there is an isomor-phism ξ : T X Y B ∼ = ιθ . Let define an isomorphism ζ : T X | K b p B ∼ = ι ¯ Q ¯ S . Let U = ⊕ mi =1 B ( − , x i )be a direct sum of representable functors. Then we define ζ U : T X ( U ) = ⊕ mi =1 T X Y B ( x i ) → ⊕ mi =1 ιθ ( x i ) = ι ¯ Q ¯ S ( U )6y the equality ζ U = ⊕ mi =1 ξ x i . For U ∈ P B we define ζ U = S ( π U ) ζ W U T X ( ι U ) : T X ( U ) → S ( U ) = ι ¯ Q ¯ S ( U ) . It is easy to see that ζ defines an isomorphism from T X | P B to ι ¯ Q ¯ S | P B .Let now U = ⊕ n ∈ Z U n be an arbitrary object of K b p B . Then T X ( U ) is a totalization ofa bigraded module V = ⊕ i,j ∈ Z T X ( U j ) i with differential τ = P j ∈ Z d T X ( U j ) of degree (1 , τ of degree (0 ,
1) defined by the equality τ | V i,j = ( − i + j T X ( d U ) | T X ( U j ) i .At the same time ι ¯ Q ¯ S ( U ) is a totalization of a bigraded module V ′ = ⊕ i,j ∈ Z S ( U j ) i withsome differentials τ ′ i ( i >
0) of degree (1 − i, i ) such that τ ′ = P j ∈ Z d S ( U j ) and τ ′ | V i,j =( − i + j S ( d U ) | S ( U j ) i . Here we write S ( d U ) for some representative of homotopy class of it.If we choose a representative of homotopy class of ζ U j for all j ∈ Z , then we obtain adifferential ζ : V → V ′ of degree (0 ,
0) such that τ ′ ζ = ζ τ and τ ′ ζ is homotopic to ζ τ if we consider V and V ′ as objects of K b p A (i.e. if we forget the grading on U ). Analogouslyto [6, Proposition 2.7] we can construct ζ i for i > − i, i ) such that ζ l τ + ζ l − τ = l X i =0 τ ′ i ζ l − i for any l >
1. We define ζ U = P i > ζ i : Tot( V, τ , τ ) → Tot( V ′ , τ ′ , τ ′ ). If U, U ′ ∈ K b p B and f ∈ C b B ( U, U ′ ), then it is clear that ζ U ′ T X ( f ) − ¯ Q ¯ S ( f ) ζ U equals P i > υ i , where υ i isof degree ( − i, i ) and υ is null homotopic. Then it follows from arguments above that ζ : T X | K b p B → ι ¯ Q ¯ S is a morphism of functors. It is clear that it is actually an isomorphism. (cid:3) Note that by Proposition 1 and Remark 1 an equivalence F : K b p B → K b p A is standardiff F ∼ = F θ for some tilting functor θ : B → K b p A . Moreover, F θ ∼ = F θ ′ iff θ ∼ = θ ′ . G -functors and orbit categories We say that G acts on the category A if there is a homomorphism of groups ∆ : G → Aut( A ).In this case we simply write g instead of ∆( g ) for g ∈ G . Throughout this section we assumethat A and B are categories with G -action. Now we recall some definitions from [2]. Definition 8.
A family η = ( η g ) g ∈ G of natural isomorphisms η g : F ◦ g → g ◦ F is called a G -equivariance adjuster for the functor F : A → B if the diagram( F ◦ gh ) x η g,hx / / η gh,x ' ' ❖❖❖❖❖❖❖❖❖❖❖ ( g ◦ F ◦ h ) x g ( η h,x ) (cid:15) (cid:15) ( gh ◦ F ) x commutes for all g, h ∈ G and x ∈ A . We say that F is a G -equivariant functor if there is a G -equivariance adjuster for F . The functor F is called strictly G -equivariant if F ◦ g equals g ◦ F , i.e. if the G -equivariance adjuster can be set to be identity.7 efinition 9. A G -functor from A ′ to A is a pair ( F, η ), where F is a functor from A ′ to A and η is a G -equivariance adjuster for F . A morphism from ( F, η ) to ( F ′ , η ′ ) is a morphismof functors α : F → F ′ such that the diagram F g αg / / η g (cid:15) (cid:15) F ′ g η ′ g (cid:15) (cid:15) gF gα / / gF ′ commutes for any g ∈ G . It is clear that a morphism of G -functors is an isomorphismiff it is an isomorphism of functors. A G -functor ( F, η ) is called a G -equivalence if F isan equivalence. If F is strictly G -equivariant, we simply write F for the corresponding G -functor.If ( F, η ) : A ′ → A and ( F ′ , η ′ ) : A ′′ → A ′ are G -functors, then we define their compositionby the formula ( F, η ) ◦ ( F ′ , η ′ ) = ( F F ′ , ηF ( η ′ )) : A ′′ → A , where (cid:0) ηF ( η ′ ) (cid:1) g,x = η g,F ′ x ◦ F ( η ′ g,x ) : F F ′ gx → gF F ′ x. It is easy to see that the composition defined above is associative (see [7, Lemma 2.8]).Moreover, it respects isomorphisms of G -functors. If ( F, η ) : A ′ → A is a G -equivalence and¯ F is an equivalence quasi-inverse to F , then there is ¯ η = (¯ η g ) g ∈ G (¯ η g : ¯ F g → g ¯ F ) such that( ¯ F , ¯ η ) : A → A ′ is a G -equivalence and( F, η ) ◦ ( ¯ F , ¯ η ) ∼ = Id A and ( ¯ F , ¯ η ) ◦ ( F, η ) ∼ = Id A ′ . This follows from the proof of [7, Theorem 9.1]. We call this G -equivalence ( ¯ F , ¯ η ) (which isdefined by ( F, η ) modulo isomorphism of G -equivalences) the quasi-inverse G -equivalence to( F, η ). Remark 2.
Let
F, F ′ : A → A ′ be functors, η – a G -equivariance adjuster for F and ξ : F → F ′ – an isomorphism. Then η ′ = ( η ′ g ) g ∈ G , where η ′ g,x = g ( ξ x ) ◦ η g,x ◦ ξ − gx , is a G -equivariance adjuster for F ′ . Moreover, ( F ′ , η ′ ) ∼ = ( F, η ) . Definition 10.
The orbit category A /G is defined as follows. • The class of objects of A /G is equal to that of A . • Let x, y ∈ A /G . The set A /G ( x, y ) consists of f = ( f h,g ) g,h ∈ G such that – f h,g ∈ A ( gx, hy ); – the sets { g ∈ G | f g,h = 0 } and { g ∈ G | f h,g = 0 } are finite for any h ∈ G ; – f lh,lg = l ( f h,g ) for all g, h, l ∈ G . • The composition in A /G is defined by the fomula( f ′ h,g ) g,h ∈ G ( f h,g ) g,h ∈ G = X l ∈ G f ′ h,l f l,g ! g,h ∈ G .
8e can define the action of G on the category Mod A by the formula g X := X ◦ g − for X ∈ Mod A and in the obvious way for morphisms. Note that g A ( − , x ) ∼ = A ( − , g ( x )). Thisaction of G induces an action of G on the category K b p A . Let θ : B → K b p A be a tiltingfunctor. If θ is G -equivariant, then the categories A /G and B /G are derived equivalent by[2, Theorem 4.11]. Remark 3.
Let X be a tilting A ⊗ B op -complex. The category A ⊗ B op can be equipped withthe diagonal action of G , i.e. for g ∈ G we put g ( x, y ) = ( gx, gy ) and g ( f ⊗ f ′ ) = gf ⊗ gf ′ for x ∈ A , y ∈ B and morphisms f in A , f ′ in B . Then it is easy to see that a G -equivarianceadjuster for θ X : B → K b p A is the same thing as a family of maps φ g ∈ C ( A ⊗ B op )( X, g X ) such that the diagram X y ( φ g ) y / / ( φ gh ) y ❋❋❋❋❋❋❋❋❋ ( g X ) y ( g φ h ) y (cid:15) (cid:15) ( gh X ) y commutes in KA for all g, h ∈ G and y ∈ B . G -equivalences From here on we equip the category A /G with the trivial action of G (i.e. ∆( g ) = Id A /G forall g ∈ G ) for any category A with a G -action. We equip the category K b p ( A /G ) with thetrivial action of G as well. Definition 11.
The canonical functor P : A → A /G is defined by P ( x ) = x and P ( f ) = (cid:0) δ g,h g ( f ) (cid:1) g,h ∈ G for x, y ∈ A and f ∈ A ( x, y ). Let s = ( s g ) g ∈ G be the collection of maps s g : P g → P , where s g,x = ( δ hg,h ′ Id h ′ x ) h,h ′ ∈ G : P gx → P x . Then s is a G -equivarianceadjuster for P . We call ( P, s ) the canonical G -functor .By [2, Proposition 2.6] every morphism in f ∈ A /G ( x, y ) can be uniquely presented inthe form f = X g ∈ G s g,y ◦ P f g (5.1)for some f g ∈ A ( x, gy ). Definition 12.
We define the pullup functor P • : Mod A /G → Mod A by the formula P • ( X ) = X ◦ P for all X ∈ Mod A /G . The pushdown functor P • : Mod A →
Mod A /G is thefunctor left adjoint to P • . It also induces a functor P • : K b p A → K b p ( A /G ).We will use the explicit description of P • obtained in [2, Theorem 4.3]. In particular,we have ( P • X )( x ) = ⊕ g ∈ G X ( gx ). The same theorem says that the map s • defined by thecommutative diagram ( P • g X )( x ) s • ,g,X,x / / ( P • X )( x ) L h ∈ G X ( g − hx ) ( δ g − h,h ′ Id X ( h ′ x ) ) h,h ′∈ G / / L h ′ ∈ G X ( h ′ x )9s a G -equivariance adjuster for P • . Moreover, by [2, Theorem 4.4] every morphism f ∈K b p ( A /G )( P • X, P • Y ) can be uniquely presented in the form f = X g ∈ G s • ,g,Y ◦ P • f g (5.2)for some f g ∈ K b p A ( X, g Y ). In addition, we have an isomorphism γ y : B /G ( − , P y ) → P • B ( − , y ) ( y ∈ B ) defined by the formula γ y ( s g,y P f ) = g − ( f ) ∈ B ( g − x, y ) for f ∈ B ( x, gy ).Note also that the Yoneda embedding Y A : A → K b p A admits a G -equivariance adjuster φ = ( φ g ) g ∈ G defined by the formula φ g,x ( f ) = g − ( f ) for f ∈ A ( y, gx ). Lemma 1.
For all g ∈ G , x, y ∈ A and f ∈ A ( x, gy ) the following equality holds γ − y s • ,g, A ( − ,y ) P • φ g,y P • Y A ( f ) γ x = Y A /G ( s g,y P f ) . (5.3) Proof
It is enough to prove that the left and the right parts of the equality (5.3) sendthe element s h,x P f ′ ∈ A /G ( P z, P x ) to the same element of A /G ( P z, P y ) for all h ∈ G , z ∈ A and f ′ ∈ A ( z, hx ). Direct calculations show that both parts of (5.3) send s h,x P f ′ to s hg,y P ( h ( f ) f ′ ). (cid:3) Definition 13. A G -equivalence ( F, η ) : K b p B → K b p A is called standard G -equivalence if F is a standard equivalence and there is a standard equivalence F ′ : K b p ( B /G ) → K b p ( A /G )such that there is an isomorphism of G -functors( P • , s • ) ◦ ( F, η ) ∼ = F ′ ( P • , s • ) . (5.4)We denote by TrPic G ( A , B ) the set of isomorphism classes of standard G -equivalences from K b p B to K b p A .It is clear that the composition of standard G -equivalences and the quasi-inverse G -equivalence to a standard G -equivalence are standard. Definition 14.
The derived Picard G -group of A is the setTrPic G ( A ) := TrPic G ( A , A )with the operation of composition. It follows from the arguments above that it is actually agroup.Let θ : B → K b p A be a tilting functor and ψ be a G -equivariance adjuster for θ . Wedenote θ ( B ) by X , note that by agreement θ ( B ) is closed under isomorphism. Then thecategory P • X is a tilting subcategory for A /G (see the proof of [2, Theorem 4.7]). We willconstruct a tilting functor µ θ,ψ : B /G → K b p ( A /G ) which induces an equivalence from B /G to P • X . We define it on objects by the formula µ θ,ψ ( P y ) = P • θ ( y ) for y ∈ B . Let us considera morphism f = P g ∈ G s g,y P f g ∈ B /G ( P x, P y ). Define µ θ,ψ ( f ) = X g ∈ G s • ,g,θ ( y ) ◦ P • ψ g,y ◦ P • θ ( f g ) . It is easy to check that µ θ,ψ is a functor. Since θ induces an equivalence to X , µ θ,ψ induces anequivalence to P • X by the arguments above. So an equivalence F µ θ,ψ : K b p ( B /G ) → K b p ( A /G )is defined. Note that if ( θ, ψ ) ∼ = ( θ ′ , ψ ′ ), then µ θ,ψ ∼ = µ θ ′ ,ψ ′ .10 roposition 2. There is a G -equivariance adjuster η for F θ such that F µ θ,ψ ◦ ( P • , s • ) ∼ = ( P • , s • ) ◦ ( F θ , η ) . Proof
Note that g U lies in X for all g ∈ G and U ∈ X . Indeed, since θ induces anequivalence to X , there is some y ∈ B such that θ ( y ) ∼ = U . Then g U ∼ = g θ ( y ) ∼ = θ ( gy ) and g U lies in X because X is closed under isomorphisms. Then it is easy to see that the action of G on K b p A induces an action on K b add X and the G -functor ( P • , s • ) : K b p A → K b p ( A /G ) inducesa G -functor ( P • , s • ) : K b add X → K b add P • X (here we equip the category K b add P • X withthe trivial action of G ). Let us consider the diagram K b p B ( ¯ S,η S ) / / ( P • ,s • ) (cid:15) (cid:15) K b add X ( ¯ Q,η Q ) / / ( P • ,s • ) (cid:15) (cid:15) K b p A ( P • ,s • ) (cid:15) (cid:15) K b p ( B /G ) ¯ S G / / K b add P • X ¯ Q G / / K b p ( A /G ) (5.5)where the rows are the compositions corresponding to (3.1) from the construction of F θ and F µ θ,ψ (if we omit the G -equivariance adjusters in the upper row). It is enough to show that η S and η Q can be constructed in such a way that the diagram (5.5) becomes commutativemodulo isomorphism as a diagram of G -functors. Here we consider the functors in the lowerrow as strict G -functors.It is clear that K b p B = K b add Y B ( B ). Let us define a G -equivariance adjuster η forthe functor S : add Y B ( B ) = P B → add X (see section 3) in the following way. If U = ⊕ mi =1 B ( − , x i ) is a direct sum of representable functors, then η g,U = ( ⊕ mi =1 ψ g,x i ) S ( ⊕ mi =1 φ − g,x i ) : S ( g U ) ∼ = ⊕ mi =1 θ ( gx i ) → ⊕ mi =1 g θ ( x i ) = g S ( U ) . Let us now consider an arbitrary U ∈ P B . Then we define η g,U by the formula η g,U = g S ( π U ) η g,W U S ( g ι U )(see the construction of the functor S for notation). Direct calculations involving formula(5.3) show that ( S, η ) is a G -functor. Then η induces a G -equivariance adjuster η S for ¯ S in the obvious way. To prove the commutativity of the first square in (5.5) it is enough toprove that the diagram P B ( S,η ) / / ( P • ,s • ) (cid:15) (cid:15) add X ( P • ,s • ) (cid:15) (cid:15) P B /G S G / / add P • X commutes modulo isomorphism of G -functors. Let us construct an isomorphism χ : P • S → S G P • . If U = ⊕ mi =1 B ( − , x i ) is a direct sum of representable functors, then P • S ( U ) = ⊕ mi =1 P • θ ( x i ) = S G ( ⊕ mi =1 B /G ( − , P x i )) . We set χ U = S G ( ⊕ mi =1 γ x i ). For an arbitrary U ∈ P B we define χ U by the formula χ U = S G ( P • π U ) χ W U P • S ( ι U ) . χ is the required isomorphism of G -functors.It remains to check the commutativity of the second square in (5.5). Let us take an objectof K b add X represented by a triple ( U, τ , τ ) (see section 3). Then P • U can be represented bythe triple ( P • U, P • τ , P • τ ). Suppose that ¯ Q sends ( U, τ , τ ) to the totalization ( U, P i > τ i )and ¯ Q G sends ( P • U, P • τ , P • τ ) to the totalization ( P • U, P i > υ i ). It is clear that υ = P • τ and that υ is homotopic to P • τ if we consider P • U as an object of K b p ( A /G ). By the resultsof section 3 there is a sequence of A /G -module morphisms α i : P • U → P • U ( i >
0) suchthat • α i is of degree ( − i, i ), • α = Id P • U , • l P i =0 α i P • τ l − i = l P i =0 υ i α l − i .In this case define the isomorphism ζ U from the totalization ( P • U, P i > P • τ i ) to the total-ization ( P • U, P i > υ i ) by the formula ζ U = P i > α i . Let U, V ∈ K b add X , f : U → V be amorphism in C b add X . It is clear that the map ζ V P • ¯ Q ( f ) − ¯ Q G P • ( f ) ζ U is a totalization of amap from P • U to P • V which have nonzero components only in degrees ( − i, i ) for i >
0. Itfollows from the results of section 3 that the totalization of such a map is null homotopic.So ζ gives an isomorphism from P • ¯ Q to ¯ Q G P • . It remains to construct a G -equivarianceadjuster η Q for ¯ Q such that the diagram P • ¯ Q g U ζ gU / / s • ,g, ¯ QU ◦ P • η Q,g,U (cid:15) (cid:15) ¯ Q G P • g U ¯ Q G ( s • ,g,U ) (cid:15) (cid:15) P • ¯ QU ζ U / / ¯ Q G P • U commutes for all U ∈ K b add X and g ∈ G . The construction of such isomorphisms η Q,g,U :¯ Q g U → g ¯ QU is analogous to the construction of ζ U and so it is left to the reader. (cid:3) Corollary 1.
Let θ : B → K b p A be a tilting functor. Then the following statements areequivalent: there is a G -equivariance adjuster for θ ; there is a G -equivariance adjuster for F θ ; there is a G -equivariance adjuster η for F θ such that ( F θ , η ) is a standard G -equivalence. Proof
The implication ”1) ⇒ ⇒ ⇒ (cid:3) Let ( F θ , η ) : K b p B → K b p A be a G -equivalence. We define ψ g,y : θ ( gy ) → g θ ( y ) by theformula ψ g,y = η g,B ( − ,y ) F θ ( φ g,y ) . It is clear that ψ = ( ψ g ) g ∈ G is a G -equivariance adjuster for θ .12 heorem 2. Let ( F θ , η ) : K b p B → K b p A be a standard G -equivalence. Then F µ θ,ψ is deter-mined by the condition (5.4) uniquely modulo isomorphism. Proof
Since ( F θ , η ) is a standard G -equivalence, there is some standard equivalence F ′ : K b p ( B /G ) → K b p ( A /G ) satisfying the condition (5.4).It is enough to prove that F ′ Y B /G ∼ = µ θ,ψ . From (5.4) we have a natural isomorphism ξ : F ′ P • → P • F θ such that the diagram F ′ P • g U ξ gU / / F ′ ( s • ,g,U ) (cid:15) (cid:15) P • F θg U s • ,g,FθU ◦ P • ( η g,U ) (cid:15) (cid:15) F ′ P • U ξ U / / P • F θ U (5.6)commutes for any U ∈ K b p B . Let us define ζ P y : F ′ Y B /G ( P y ) → µ θ,ψ ( P y ) by the formula ζ P y = ξ B ( − ,y ) F ′ γ y . Then using (5.3), (5.6) and the fact that ξ is a morphism of functors we get ζ P y F ′ Y B /G ( s g,y P f ) = ζ P y F ′ (cid:0) γ − y s • ,g,B ( − ,y ) P • ( φ g,y ) γ gy B /G ( − , P f ) (cid:1) = s • ,g,θ ( y ) ◦ P • η g, B ( − ,y ) ◦ ξ g B ( − ,y ) F ′ P • ( φ g,y B ( − , f )) F ′ ( γ x ) = µ θ,ψ ( s g,y P f ) ζ P x . for all x, y ∈ B , g ∈ G , f ∈ B ( x, gy ). Since any morphism in B /G is of the form (5.1), ζ isthe required isomorphism from F ′ Y B /G to µ θ,ψ . (cid:3) Φ and Ψ In this section we define two maps:Ψ A , B : TrPic G ( A , B ) → TrPic( A /G, B /G ) and Φ A , B : TrPic G ( A , B ) → TrPic( A , B ) . Then we investigate some of their properties.Let (
F, η ) : K b p B → K b p A be a standard G -equivalence. Then we define Φ A , B as follows:Φ A , B ( F, η ) = F and define Ψ A , B ( F, η ) to be the unique standard equivalence F ′ satisfying the condition (5.4).The correctness of the definition of Ψ A , B follows from Theorem 2. It is clear thatΦ A , B ′ (cid:16) ( F, η ) ◦ ( F ′ , η ′ ) (cid:17) = Φ A , B ( F, η ) ◦ Φ B , B ′ ( F ′ , η ′ )and Ψ A , B ′ (cid:16) ( F, η ) ◦ ( F ′ , η ′ ) (cid:17) = Ψ A , B ( F, η ) ◦ Ψ B , B ′ ( F ′ , η ′ ) . In particular, Φ A := Φ A , A and Ψ A := Ψ A , A are homomorphisms of groups.13 efinition 15. A standard equivalence F : K b p B → K b p A is called a Morita equivalence if F ∼ = F θ where θ ( y ) is isomorphic to some object U concentrated in degree 0 ( U n = 0for n = 0) for any y ∈ B . We denote by Pic( A , B ) the set of Morita equivalences from B to A modulo isomorphisms. It is clear that the composition of Morita equivalences andthe inverse to a Morita equivalence are again Morita equivalences. In particular, the setPic( A ) := Pic( A , A ) is a subgroup of TrPic( A ). This group is called the Picard group of A . Theorem 3.
Let ( F, η ) : K b p B → K b p A be a standard G -equivalence. Then Φ A , B ( F, η ) ∈ Pic( A , B ) ⇔ Ψ A , B ( F, η ) ∈ Pic( A /G, B /G ) . In particular, Φ − A (cid:0) Pic( A ) (cid:1) = Ψ − A (cid:0) Pic( A /G ) (cid:1) . Proof
By Remark 2 we can assume that F = F θ for some tilting functor θ . Denote θ ( B ) by X . By Theorem 2 we have Ψ A , B ( F, η ) ∼ = F µ for some equivalence µ : B /G → K b p ( A /G ) suchthat µ ( P y ) = P • θ ( y ) for any y ∈ B .Suppose that F ∈ Pic( A , B ). Let us consider y ∈ B . There is some object U ∈ K b p A concentrated in degree 0 such that θ ( y ) ∼ = U . Then µ ( P y ) = P • θ ( y ) ∼ = P • U . It is clear that P • U is concentrated in degree 0. Consequently, Ψ A , B ( F, η ) ∈ Pic( A /G, B /G ).Suppose now that Ψ A , B ( F, η ) ∈ Pic( A /G, B /G ). It is enough to prove that any objectof X is isomorphic in K b p A to some object concentrated in degree 0. Any object of P • X isisomorphic in K b p ( A /G ) to some object concentrated in degree 0 by our assumption. Considersome U ∈ X . We know that P • U is isomorphic to an object concentrated in degree 0. Then P • P • U is isomorphic to an object concentrated in degree 0 in K p A . Since U is a directsummand of P • P • U (see the proofs of [2, Theorems 4.3 and 4.4]), U is isomorphic to someobject concentrated in degree 0. (cid:3) Definition 16.
The center of a category A is the set of natural transformations from Id A to itself. We denote the center of a category A by Z ( A ). By Z ( A ) ∗ we denote the subset of Z ( A ) formed by natural isomorphisms. If θ : A → B is a functor, then α ∈ Z ( A ) determinesa natural transformation θ ( α ) : θ → θ by the formula θ ( α ) x = θ ( α x ). It is clear that if θ isan equivalence, then any natural isomorphism from θ to θ is of the form θ ( α ) ( α ∈ Z ( A ) ∗ ).Now let F θ be an element of TrPic( A , B ). We want to determine when F θ lies in the imageof Φ A , B . Let the group G be given by generators and relations G = < { a } a ∈ A |{ b } b ∈ B > . Weknow from Proposition 2 that F θ ∈ Im (Φ A , B ) iff there is a G -equivariance adjuster for θ . Inparticular, if F θ ∈ Im (Φ A , B ), then there is some natural isomorphism ϕ a : θa → aθ for any a ∈ A .Define ϕ a − : θa − → a − θ by the formula ϕ a − ,y = a − ( ϕ − a,a − y ) . Denote ˜ A := A ∪ { a − | a ∈ A } . Let us define natural isomorphisms ϕ a ,...,a n : θa . . . a n → a . . . a n θ for all families a , . . . , a n ∈ ˜ A . We have done this for the case n = 1. Let ϕ a ,...,a n − : θa . . . a n − → a . . . a n − θ be defined. Then we define ϕ a ,...,a n by the formula ϕ a ,...,a n ,x = a ...a n − ϕ a n ,x ϕ a ,...,a n − ,a n x . a , . . . , a n ∈ ˜ A be such elements that a . . . a n ∈ B . Then ϕ a ,...,a n is a natural isomor-phism from θ to itself. So there is a family α = ( α b ) b ∈ B of elements of Z ( B ) ∗ such that ϕ a ,...,a n = θ ( α b ) for b = a . . . a n ∈ B . As it was mentioned above any σ ∈ Aut B inducesan automorphism σ ∈ Aut( K b p B ). It is clear that σ is a standard derived equivalence lyingin Pic( B ). Let ǫ a : σa → aσ ( a ∈ A ) be a family of natural isomorphisms. We define ǫ a ,...,a n : σa . . . a n → a . . . a n σ for a , . . . , a n ∈ ˜ A analogously to the definition of ϕ a ,...,a n .For b = a . . . a n ∈ B we define ǫ b by the formula ǫ b = ǫ a ,...,a n . Definition 17.
In the above notation the family of isomorphisms ϕ = ( ϕ a ) a ∈ A is called an approximate equivariance adjuster for θ . The family α is called an equivariance error for ϕ .The family ǫ = ( ǫ a ) a ∈ A is called an equivariance σ -correction for α if α b,σy ǫ b,y = Id σy for any b ∈ B and y ∈ B . Theorem 4.
Suppose that G is given by generators and relations. Let θ : B → K b p A be a tilting functor. Suppose that ϕ is an approximate equivariance adjuster for θ withequivariance error α . Let σ be some automorphism of B . Then F θ σ lies in the image of Φ A , B iff there exists an equivariance σ -correction for α . Proof
By Proposition 2 F θ σ ∈ Im Φ A , B iff there is a G -equivariance adjuster for θσ . Supposethat ψ is a G -equivariance adjuster for θσ . Then direct calculations show that ǫ defined bythe equalities ψ a,x = ϕ a,σx ◦ θǫ a,x is an equivariance σ -correction for α .Assume now that ǫ is an equivariance σ -correction for α . For a , . . . , a n ∈ ˜ A we define ψ a ...a n : θσa . . . a n → a . . . a n θσ by the formula ψ a ...a n ,x = ϕ a ,...,a n ,σx ◦ θǫ a ,...,a n ,x . The correctness of this definition follows from the fact that ǫ is an equivariance σ -correctionfor α . It can be verified by direct calculations that ψ is a G -equivariance adjuster for θσ . (cid:3) G -grading and the image of Ψ In this section we give a description of the image of Ψ A , B in the case where G is a finitegroup. We need the finiteness of G to prove the following lemma. Lemma 2.
Let G be a finite group and X be a subcategory of K b p A such that for any U ∈ X and any g ∈ G there is V ∈ X such that g U ∼ = V . Then X is a tilting subcategory for A iff P • X is a tilting subcategory for A /G . Proof
For the proof of the fact that P • X is a tilting subcategory for A /G if X is a tiltingsubcategory for A see the proof of [2, Theorem 4.7].Now let P • X be a tilting subcategory for A /G . Let us prove that K b p A ( U, V [ i ]) = 0for U, V ∈ X , i = 0. Suppose that it is not true. Then there are U, V ∈ X and i = 0such that K b p A ( U, V [ i ]) = 0. Let f be a nonzero element of K b p A ( U, V [ i ]). Then P • f is anonzero element of K b p A /G ( P • U, P • V [ i ]) and so P • X is not a tilting subcategory for A /G .It remains to prove that any representable functor lies in the subcategory thick X . Notethat if | G | < ∞ , then P • sends finitely generated modules to finitely generated modules15nd so it induces a functor P • : K b p ( A /G ) → K b p A . Let us consider a functor A ( − , x )( x ∈ A ). By our assumption A ( − , P x ) ∼ = P • (cid:0) A ( − , x ) (cid:1) lies in thick P • X . Let us consider thesubcategory thick P • P • X of K b p A . It contains the subcategory P • thick P • X and so contains P • P • (cid:0) A ( − , x ) (cid:1) ∼ = ⊕ g ∈ G A ( − , gx ). Since thick P • P • X is closed under direct summands itcontains A ( − , x ). It remains to prove that thick X contains P • P • U for any U ∈ X . Butthick X contains g U for any U ∈ X and any g ∈ G . So it contains P • P • U ∼ = ⊕ g ∈ Gg U for any U ∈ X . (cid:3) Definition 18. A G -graded category is a category A having a family of direct sum de-compositions A ( x, y ) = ⊕ g ∈ G A ( x, y ) ( g ) ( x, y ∈ A ) of k -modules such that the compositionof morphisms gives the inclusions A ( y, z ) ( g ) A ( x, y ) ( h ) ⊂ A ( x, z ) ( gh ) for all x, y, z ∈ A and g, h ∈ G . If f ∈ A ( x, y ) ( g ) , then we say that f is of G -degree g . A functor F : A → A ′ between G -graded categories is called degree-preserving if F ( A ( x, y ) ( g ) ) ⊂ A ′ ( F x, F y ) ( g ) forall x, y ∈ A and g ∈ G .By [2, Lemma 5.4] there is a G -grading on B /G such that P f is of degree 1 G and s g,x isof degree g − . Definition 19.
Let A be a G -graded category. A G -graded A -complex is an A -complex U with a family of direct sum decompositions U ( x ) = ⊕ g ∈ G U ( x ) ( g ) ( x ∈ A ) such that d U,x ( U ( x ) ( g ) ) ⊂ U ( x ) ( g ) and U ( f )( U ( x ) ( g ) ) ⊂ U ( y ) ( gh ) for all x, y ∈ A , g, h ∈ G , f ∈A ( y, x ) ( h ) . If U, V are G -graded complexes, then we say that f : U → V is of degree h if f x ( U ( x ) ( g ) ) ⊂ V ( x ) ( hg ) . Let us now define a category K b p,G A . It’s objects are G -graded A -complexes U which lie in K b p A considered as complexes without grading. If U, V ∈ K b p,G A ,then K b p,G A ( U, V ) = K b p A ( U, V ).Let A be a G -graded category and U, V ∈ K b p,G A . Then we denote by K b p,G A ( U, V ) ( g ) the set of morphisms in K b p A ( U, V ) which can be presented by a morphism of degree g in CA ( U, V ). It is not hard to check that K b p,G A ( x, y ) = ⊕ g ∈ G K b p,G A ( U, V ) ( g ) and thisdecomposition turns K b p,G A into a G -graded category. For U ∈ K b p,G A we denote by ¯ U thecorresponding object of K b p A . Note that any equivalence θ : B → K b p,G A determines anequivalence ¯ θ : B → K b p A in an obvious way. Theorem 5.
Suppose that G is a finite group. Let F : K b p ( B /G ) → K b p ( A /G ) be a standardequivalence. Then F lies in the image of Ψ A , B iff there is a degree-preserving functor µ : B /G → K b p,G ( A /G ) such that ¯ µ is a tilting functor and F ∼ = F ¯ µ . Proof If F lies in the image of Ψ A , B , then it is isomorphic to F µ θ,ψ for some tilting functor θ : B → K b p A and a G -equivariance adjuster ψ for θ . Let us define a G -grading on µ θ,ψ ( P y ) = P • θ ( y ) ( y ∈ B ) as follows: ( P • θ ( y ))( P x ) ( g ) = θ ( y )( gx ). Then it can be easily verified that µ θ,ψ defines a degree-preserving functor from B /G to K b p,G ( A /G ). Note that this part of theprove does not require the finiteness of G .Now let µ : B /G → K b p,G ( A /G ) be a degree-preserving functor such that ¯ µ is a tiltingfunctor. Let us prove that F ¯ µ lies in the image of Ψ A , B . By Proposition 2 it is enough to finda tilting functor θ : B → K b p A and a G -equivariance adjuster ψ for θ such that F ¯ µ ∼ = F µ θ,ψ .16or U ∈ K b p,G ( A /G ) we define e U ∈ K b p A in the following way. It is defined on objectsby the formula e U ( x ) = U ( P x ) (1) and on morphisms by the formula e U ( f ) = U ( P f ) | U ( x ) (1) ( f ∈ A ( x, y )). The differential d e U is defined by the formula d e U,x = d U,P x | U ( P x ) (1) . Thecorrectness of this definition follows from the definition of a G -graded complex. Also wedefine a morphism ξ U : ¯ U → P • e U as follows: ξ U,x = M g ∈ G U ( s g,x ) | U ( P x ) ( g ) : M g ∈ G U ( P x ) ( g ) → M g ∈ G U ( P gx ) (1) . Let us now define θ : B → K b p A . We define it on objects by θ ( y ) = ^ µ ( P y ). For f ∈ B ( y, z )we define the natural transformation θ ( f ) by the formula θ ( f ) x = µ ( P f ) P x | µ ( P y )( P x ) (1) : θ ( y )( x ) → θ ( z )( x )for all x ∈ A . Let ψ g,y,x : θ ( gy )( x ) → g θ ( y )( x ) ( x ∈ A , y ∈ B , g ∈ G ) be the composition θ ( gy )( x ) = µ ( P gy )( P x ) (1) µ ( s g,y ) x −→ µ ( P y )( P x ) ( g − ) µ ( y )( s g − ,x ) −→ µ ( P y )( P g − x ) (1) = g θ ( y )( x ) . It is not hard to prove that the following conditions hold:1) ξ U is an isomorphism in K b p ( A /G );2) θ induces an equivalence from B to θ ( B );3) ψ is a G -equivariance adjuster for θ ;4) The family of morphisms ζ P y = ξ µ ( P y ) : µ ( P y ) → P • θ ( y ) = µ θ,ψ ( P y ) defines an isomor-phism from ¯ µ to µ θ,ψ .It follows from 1)–3) and Lemma 2 that θ ( B ) is a tilting subcategory for A , hence thetheorem is proved. (cid:3) From here on we assume that k is a field. Let R be an associative finite dimensional k -algebra.It can be considered as a category with one object and so the results of the previous sectionscan be applied to R . We denote the unique object of R by e . We apply these results in thecase where R is a Frobenius algebra and G is a finite cyclic group which acts on R by powersof a Nakayama automorphism. First, let us recall the definition of a Frobenius algebra. Definition 20.
An algebra R is called Frobenius if there is a linear map ǫ : R → k such thatthe bilinear form h a, b i = ǫ ( ab ) is nondegenerate. The Nakayama automorphism ν : R → R is the automorphism which satisfies the equation h a, b i = h b, ν ( a ) i for all a, b ∈ R . If thebilinear form on R can be chosen in such a way that h a, b i = h b, a i for all a, b ∈ R , then thealgebra R is called symmetric .From here on we fix some Frobenius algebra R , and some Nakayama automorphism ν of R . Moreover, we assume that there is an integer n > ν n = Id R . Then the17yclic group G = < g | g n > acts on R by the following rule: ∆( g ) = ν . Note that R/G isa symmetric algebra. Indeed, if h , i is the bilinear form on R , then h , i G defined by theformula < s g k P a, s g l P b > G = δ g k + l +1 , G < ν l a, b > is the desired bilinear form on R/G . So the maps Φ R and Ψ R allow us to transfer someinformation from the derived Picard group of a symmetric algebra to the derived Picardgroup of a Frobenius algebra.We have the following application of Theorem 4. Proposition 3. If for any a ∈ Z ( R ) ∗ there exists an element b ∈ Z ( R ) ∗ such that ab n = 1 ,then Φ R is surjective. If for any a ∈ Z ( R ) ∗ there exists an element b ∈ R ∗ and an automorphism σ ∈ Aut R suchthat bν ( b ) . . . ν n − ( b ) = a and σνσ − ( c ) = bν ( c ) b − for any c ∈ R , then Cok Φ R is generatedby the images of elements from Pic( R ) . Proof
Note that the functor ν ( − ) : K b p R → K b p R is isomorphic to the Nakayama functor.Then by [3, Proposition 5.2] we have an isomorphism η F : F ◦ ν ∼ = ν ◦ F for any standardequivalence F : K b p R → K b p R . If F is given by a tilting functor θ : R → K b p R , then θ ∼ = F Y R . So the isomorphism η F, Y ( e ) ◦ F ( φ g,e ) : F Y R ν ( e ) → νF Y R ( e ) gives an isomorphism ϕ g,e : θν ( e ) → νθ ( e ). Then ϕ is an approximate equivariance adjuster for θ .1) Note that ν ( b ) = b for any b ∈ Z ( R ). Then it follows from the assumption that thereis an equivariance Id R -correction for any equivariance error. So Φ R is surjective by Theorem4. 2) It follows from the assumption that for any equivariance error a there is some σ ∈ Aut R such that there exists an equivariance σ -correction for a . So by Theorem 4 for any F ∈ TrPic( R ) there is some σ ∈ Aut R such that F σ lies in the image of Φ R . Then theimage of F in Cok Φ R equals the image of σ − . So the assertion follows from the fact that σ − ∈ Pic( R ). (cid:3) Corollary 2.
If the field k is algebraically closed and it’s characteristic does not divide n ,then Φ R is surjective. Proof
We may assume that R is an indecomposable algebra. Then any element a ∈ Z ( R ) ∗ is of the form a = κ (1 + Q ) for some κ ∈ k and nilpotent Q ∈ Z ( R ). Since k is algebraicallyclosed, there is some ¯ κ ∈ k such that κ = ¯ κ n . Then a = b n for b = ¯ κ X i > i − Q j =0 (cid:0) n − j (cid:1) i ! Q i . (cid:3) Application: generators of the derived Picard groupof a self-injective Nakayama algebra
From here on we assume that k is an algebraically closed field. In this section we apply themethods of the previous sections to obtain generators of the derived Picard group of algebras N ( nm, tm ) defined in the following way. Let m, n, t > n and t are coprime. Let Q ( nm ) be a cyclic quiver with nm vertices, i.e. the quiver whosevertex set is Z nm and whose arrows are β i : i → i + 1 ( i ∈ Z nm ). Let I ( nm, tm ) be anideal in the path algebra of Q ( nm ) generated by all paths of length tm + 1. We denote N ( nm, tm ) := k Q ( nm ) / I ( nm, tm ). For i ∈ Z nm we denote by e i the primitive idempotentcorresponding to the vertex i and by P i the projective module e i N ( nm, tm ). For a path w from the vertex i to the vertex j we denote by w the unique homomorphism from P i to P j which sends e i to w as well. Also we introduce the following auxiliary notation: β i,k = β i + k − . . . β i . It is well-known that N ( nm, tm ) is a Frobenius algebra with a Nakayama automorphism ν defined as follows: ν ( e i ) = e i − tm and ν ( β i ) = β i − tm . If U is a module, then we also denoteby U the corresponding complex concentrated in degree 0. For i ∈ Z nm , 1 k m − X i := P i − tm β i − tm → P i − tm +1 β i − tm +1 ,tm → P i +1 concentrated in degrees -2, -1 and 0 and the complexes Y i,k := P i β i,k → P i + k concentrated in degrees 0 and 1.If m >
1, then for 0 l m − N ( nm, tm )-complex H nml = (cid:16) M i ∈ Z nm ,m ∤ i − l P i (cid:17) ⊕ (cid:16) M i ∈ Z nm ,m | i − l X i (cid:17) . In this case we can define an algebra isomorphism θ nml : N ( nm, tm ) → K b p ( N ( nm, tm ))( H nml , H nml ) . We define it on idempotents by the formula θ nml ( e i ) = Id P i if m ∤ i − l and m ∤ i − − l ,Id P i +1 if m | i − l ,Id X i if m | i − − l .We define θ nml ( β i ) ( i ∈ Z nm ) in the obvious way (it equals 0 in all degrees except for the zerodegree and equals β i , β i β i +1 or Id P i +1 depending on i in the zero degree).If m > t = 1, then for 0 l m − N ( nm, m )-complex Q nml = M i ∈ Z nm ,m | i − l (cid:16) P i ⊕ m − M k =1 Y i,k (cid:17) .
19n this case we can define an algebra isomorphism ε nml : N ( nm, m ) → K b p ( N ( nm, m ))( Q nml , Q nml ) . We define it on idempotents by the fomula ε nml ( e i ) = ( Id P i if m | i − l ,Id Y i + k,m − k if m | i + k − l for some k , 1 k m − ε nml ( β i ) ( i ∈ Z nm ) in the following way. It equals 0 in all degrees except 0 and 1.In degree 0 it equals Id P i + k if m | i + k − l for 1 k m − β i,m if m | i − l . Indegree 1 it equals β i + m if m ∤ i − l and m ∤ i + 1 − l and equals 0 if m | i − l or m | i + 1 − l .In this section we prove the following theorem. Theorem 6. If m = 1 , then TrPic( N ( n, t )) is generated by the shift and Pic( N ( n, t )) ; If m > , t > , then TrPic( N ( nm, tm )) is generated by the shift, Pic( N ( nm, tm )) and F θ nml ( l ∈ Z m ); If m > , t = 1 , then TrPic( N ( nm, m )) is generated by the shift, Pic( N ( nm, m )) , F θ nml and F ε nml ( l ∈ Z m ). It is clear that N ( nm, tm ) is symmetric iff n = 1. If in addition m = 1, then N (1 , t ) is alocal algebra and so TrPic( N (1 , t )) is generated by the shift and Pic( N (1 , t )) by the resultsof [8], [9]. So the first assertion of Theorem 6 holds for n = 1. The assertions 2) and 3)of the theorem for n = 1 follow from the results of [4], where the set of generators of thederived Picard group was described in the case n = 1, m >
1. Moreover, it was proved therethat any element of TrPic( N ( m, tm )) is of the form U V , where V ∈ Pic( N ( m, tm )) and U is a product of elements listed in the points 2)–3) except for Pic( N ( m, tm )).Now let us consider n >
1. Let G = h g | g n i be a cyclic group which acts on N ( nm, tm )by the rule ∆( g ) = ν . It is well-known that N ( nm, tm ) /G is Morita equivalent to N ( m, tm ).We need the explicit formula for this equivalence to obtain the isomorphism of the derivedPicard groups defined by it. Let W = ⊕ j ∈ Z n W j where W j is isomorphic to N ( m, tm ) asa right N ( m, tm )-module. As it was mentioned above every path w in Q ( m ) defines ahomomorphism w : W j → W j . Thus, there is a left N ( m, tm )-module structure on W j . Letus define a left N ( nm, tm ) /G -module structure on W . Let s i,j : W i → W j ( i, j ∈ Z n ) bethe isomorphism arising from Id N ( m,tm ) . Let i ∈ Z nm be represented by an integer number0 ¯ i nm −
1. Present ¯ i in the form ¯ i = ¯ qm + ¯ r , where 0 ¯ r < m . Let q ∈ Z n and r ∈ Z m be elements represented by ¯ q and ¯ r respectively. Consider an element x ∈ W . Suppose that x ∈ W j for some j ∈ Z n . Then we define( P e i ) x = ( δ q,j e r ) x, ( P β i ) x = ( δ q,j β r ) x and s g l x = s j,j + ltm ( x ) . It is clear that in such a way W becomes a N ( nm, tm ) /G − N ( m, tm )-bimodule whichinduces a Morita equivalence T W = − ⊗ N ( nm,tm ) /G W : K b p ( N ( nm, tm ) /G ) → K b p ( N ( m, tm )) . We define L : TrPic( N ( m, tm )) → TrPic( N ( nm, tm ) /G ) by the formula L ( F ) = ¯ T W ◦ F ◦ T W , T W is a quasi-inverse equivalence for T W . It is clear that L sends the shift to the shiftand Pic( N ( m, tm )) to Pic( N ( nm, tm ) /G ).There are G -equivariance adjusters ψ nml for θ nml and ϕ nml for ε nml . The maps ψ nml,g p and ϕ nml,g p can be constructed in the obvious way as the sums of the isomorphisms of the form P i − ptm ∼ = ν p P i , X i − ptm ∼ = ν p X i and Y i − ptm,k ∼ = ν p Y i,k . Then we have the following lemma.
Lemma 3. If m > , then for l m − we have L ( F θ ml ) ∼ = F µ θnml ,ψnml . If m > and t = 1 , then for l m − we have L ( F ε ml ) ∼ = F µ εnml ,ϕnml . Proof
1) We will prove that F θ ml ◦ T W ∼ = T W ◦ F µ θnml ,ψnml . (9.1)Let us describe the left part of this equality. Let H = L j ∈ Z n H j , where H j ∼ = H ml as a right N ( m, tm )-complex. Denote by s ′ i,j : H i → H j ( i, j ∈ Z n ) the isomorphism arising from Id H ml .In addition, for u ∈ K b p ( N ( m, tm ))( H ml , H ml ) we denote by u the corresponding morphismfrom H j to H j . Let us define θ : N ( nm, tm ) /G → K b p ( N ( m, tm ))( H, H ). Consider i ∈ Z nm .Let q ∈ Z n and r ∈ Z m be as above. Then for x ∈ H j ( j ∈ Z n ) we define θ ( e i )( x ) = δ j,q θ ml ( e r )( x ) , θ ( β i )( x ) = δ j,q θ ml ( β r )( x ) and θ ( s g l )( x ) = s ′ j,j + ltm ( x ) . Then the left part of the equality (9.1) gives F θ . It is not hard to construct an isomorphism ξ : H → ( P • H nml ) ⊗ N ( nm,tm ) /G W such that ξθ ( c ) = ( µ θ nml ,ψ nml ( c ) ⊗ N ( nm,tm ) /G Id W ) ξ for any c ∈ N ( nm, tm ) /G . The existenceof such ξ gives the isomorphism (9.1).2) The proof is similar and so it is left to the reader. (cid:3) Let us now apply the results of the previous sections to the algebra N ( nm, tm ). Lemma 4. If char k ∤ n , then Φ N ( nm,tm ) is surjective. If char k | n , then Cok Φ N ( nm,tm ) is generated by images of elements from Pic( N ( nm, tm )) . Proof
1) Follows directly from Corollary 2.2) Note, that in this case n >
1. It is enough to prove that the condition of the secondpart of Proposition 3 is satisfied. Consider a ∈ Z ( N ( nm, tm )) ∗ . Let us introduce thenotation u := P i ∈ Z nm β i,nm , ¯ u := P mi =1 β i,nm . It can be easily proved that a = P k tn c k u k for some c k ∈ k , c = 0. Since k is algebraically closed we may assume that c = 1. Then a = bν ( b ) . . . ν n − ( b ) for b = 1 + P k tn c k ¯ u k . Let us denote by γ the automorphism of N ( nm, tm ) defined by the formula γ ( x ) = bν ( x ) b − for x ∈ N ( nm, tm ). It remains to findsuch σ ∈ Aut N ( nm, tm ) that σ − νσ = γ . 21he automorphism γ equals ν on the idempotents and is defined on the arrows by theformula γ ( β i ) = β i − tm , if i = tm and i = ( t + 1) m,aβ , if i = tm,a − β m , if i = ( t + 1) m. Let 0 < p < n be such a number that n | pt −
1. We define σ in the following way. It isidentical on the idempotents and is defined on arrows in such a way that σ − ( β i ) = ( a − β i , if i = (1 + k ) tm for some 0 k < p,β i , otherwise . It is easy to verify that σ − νσ = γ . (cid:3) Proof of Theorem 6
It was mentioned above that the theorem is true for n = 1. Consider n >
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