On equivariant derived categories

We study the equivariant category associated to a finite group action on the derived category of coherent sheaves of a smooth projective variety. In particular, we discuss decompositions of the equivariant category, prove the existence of a Serre functor, and give a criterion for the equivariant category to be Calabi–Yau. We describe an obstruction for a subgroup of the group of auto-equivalences to act on the derived category. As application we show that the equivariant category of any Calabi–Yau action on the derived category of an elliptic curve is equivalent to the derived category of an elliptic curve.


Introduction
Equivariant categories appear naturally when expressing sheaves on a quotient space X /G in terms of sheaves on the space X on which a given group G acts. For example, given a finite group G acting freely on a complex quasi-projective variety X the category of coherent sheaves on the quotient variety X /G is the G-equivariant category of coherent sheaves on X : Coh(X ) G = Coh(X /G).
More generally, if we start with a category of sheaves on a space and an (abstract) group action on the category, the equivariant category may be viewed as a form of 'non-commutative quotient'. Since the action does not have to come from an action on the underlying space, this quotient may exist only on a categorical level, i.e. it is not clear whether it is again the category of sheaves on some space. If we nevertheless hope for some geometric meaning of this category, it is natural to explore what geometric structures it possesses. The focus of this paper is to discuss several of these structures (decompositions, duality, cohomology) for finite group actions on the derived category of coherent sheaves on a smooth projective variety.
We give a short overview of the content of the paper: In Sect. 2 we define group actions on C-linear additive categories. We give a natural obstruction class in H 3 (G, C * ) that governs whether a subgroup of the group of auto-equivalences can define an action on the category. We also discuss criteria for the vanishing of this class, and give an example that the obstruction is effective.
In Sect. 3 we define equivariant categories, recall and prove their 2-categorical universal properties, show (following work of Romagny [20]) that the equivariant categories can be taken successively, and discuss the action of the dual group.
In Sect. 4 we study the decomposition of the equivariant category into decomposable components. Under suitable assumptions we show that each such component is the equivariant category of a related group action on the original category.
In Sect. 5 we prove that given a triangulated category with a Serre functor the equivariant category also carries a Serre functor.
In Sect. 6 we discuss properties of the Hochschild cohomology of equivariant categories. By relying on work of Perry [18] we give a criterion for the equivariant category of a Calabi-Yau category to be again Calabi-Yau. We also describe the induced action by the group of characters on Hochschild cohomology.
In Sect. 7 we illustrate our methods by determining the equivariant categories of an elliptic curve with respect to Calabi-Yau group actions.
We attempted to make the presentation of this paper as self-contained as possible, so that it can serve also as an introduction to the subject. Many of the topics we discuss here are scattered around in the literature, and we expect many others, where we have not found any references, to be folklore and known to the experts. Useful sources for the equivariant category of derived categories are the works of Elagin [8], Shinder [21], Perry [18], Kuznetsov and Perry [12] and others. We refer to these papers for related discussions and applications.
Studying equivariant categories of derived categories of smooth projective varieties becomes a rich subject only if interesting group actions are known beyond geometric automorphism. The motivation for us arose from symplectic group actions on the derived category of K3 and abelian surfaces. In this case string theory and holomorphic symplectic geometry alike provide a wide range of interesting examples. In many of these cases, the equivariant categories are in a non-trivial way equivalent to the derived category of another symplectic surface. We refer to [1] for more details and applications to fixed loci of holomorphic symplectic varieties.

Conventions
We always work over C. All categories are assumed to be small. A variety is a separated integral scheme of finite type over C.
such that for all triples g, h, k ∈ G we have the commutative diagram ρ g ρ h ρ k ρ g ρ hk ρ gh ρ k ρ ghk . ρ g θ h,k θ g,h ρ k θ g,hk θ gh,k (2.1) We will often write g for ρ g .
Recall the 2-category Cats of categories, where the objects are categories, the morphisms are functors between categories and the 2-morphisms are natural transformations. Similarly we have the 2-category G-Cats of categories with a G-action. A morphism or G-functor ( f , σ ): (D, ρ, θ) → (D , ρ , θ ) between categories with G-actions is a pair of a functor f : D → D together with 2-isomorphisms σ g : f • ρ g → ρ g • f such that ( f , σ ) intertwines the associativity relations on both sides, i.e. such that the following diagram commutes: is a 2-morphism t : f → f that inter-twines the σ g , i.e. such that the following diagram commutes: An action of G on D is strict if θ g,h = id for all g, h ∈ G. In particular, this implies that ρ 1 = id. By ( [21], Theorem 5.4) every G-action on D is equivalent to a strict G-action on some equivalent category D . Here we say that a G-action (ρ, θ ) on D is equivalent to a G-action (ρ , θ ) on D if we have an equivalence in G-Cats, i.e. (D, ρ, θ) ∼ = (D , ρ , θ ). Because of this, by passing to an equivalent category one can (and we often will) assume that the action is strict.

Obstruction to actions
Let Aut D be the group of isomorphism classes of auto-equivalences of D. Hence two equivalences f 1  − → f 2 . Every group action on D yields a subgroup of Aut D. For C-linear categories the converse however does not always hold and is obstructed by a class in the group cohomology of G as explained in the following theorem.
To state it we will need a different notion of equivalence for group actions. We say that G-actions (ρ, θ ) and (ρ , θ ) on D are isomorphic if there exists a G-functor of the form (id D , σ ): (D, ρ, θ) → (D, ρ , θ ).
(a) There exists a class in H 3 (G, C * ) canonically associated to the inclusion G ⊂ Aut D which vanishes if and only if there exists an action of G on D whose image in Aut D is G. Moreover, the set of isomorphism classes of such actions is a torsor under H 2 (G, C * ). (b) There exits a finite group G and a surjection G G such that G acts on D and the induced map G → Aut D factors over the quotient map to G.
In the group cohomology H i (G, C * ) above the group G acts trivially on the coefficient group C * . For cyclic groups one has Hence the obstruction in part (a) of Theorem 2.1 can be non-trivial even for cyclic groups. We refer to Sect. 3.6 for a class of such examples in a geometrically wellbehaved situation.
Proof (a) For every g ∈ G choose a functor ρ g : D → D with image g in Aut D and for every pair g, h ∈ G choose isomorphisms θ g,h : ρ g ρ h → ρ gh . Then for every triple g, h, k ∈ G consider the composition given by applying the maps in (2.1) once counterclockwise around the square, where we have written (g) for ρ g . The assumption Hom(id D , id D ) = C yields that this composition is a scalar multiple of the identity, which we call c(g, h, k). The first step of the proof is to prove that the assignment c : G 3 → C * is a cocycle. Since the G-action on C * is taken to be trivial here, this boilds down to showing that for all quadruples g, h, k, l ∈ G we have We start with the left-hand side. The constant c(g, h, kl) times the identity is the composition where the maps are all given by the corresponding θ 's as in (2.2). 1 Similarly to above, c(gh, k, l) id is the composition and precomsing it with its inverse yields (2.6) Composing (2.5) by (gh)(k)(l) → (gh)(kl) and precomposing by its inverse yields (2.7) By considering the composition (2.6) • (2.7) and noting that the last map in (2.7) is precisely the inverse of the first map in (2.6), we hence find that c(g, h, kl) c(gh, k, l) times the identity is given by the following composition: (2.8) 1 It is very suggestive to write this composition vertically: We invite the reader to re-write the other maps below in a similar form, in order to make the various compositions more clear. For brevity we will stick to the horizontal notation.
We now turn to the right-hand side of (2.3). Arguing in a similar manner shows that c(h, k, l) c(g, hk, l) c(g, h, k) times the identity is equal to the composition We see that except for the two respective outer arrows, the compositions (2.8) and (2.9) agree. Hence to prove the desired Eq. (2.3) it remains to prove that the compositions agree, or equivalently, that we have a commutative diagram This follows from the following.

Lemma 2.2
Let f , f , g, g : D → D be functors and let α : f → f and β : g → g be natural transformations. Then we have a commutative diagram commutes. This follows when we apply the condition that α is a natural transformation to the morphism β A : g A → g A. Therefore, the above defines a cocycle c : G 3 → C * . It depends on the choice of representative ρ g and the choice of θ g,h . By the assumption Hom(id D , id D ) = C for any second choice of isomorphism θ g,h : (g)(h) → (gh) we have θ g,h = λ(g, h)θ g,h for some λ(g, h) ∈ C * . The new cocycle c one obtains in this way is and hence c differs from c by a coboundary. Similarly, any other choice of representative ρ g changes c by at most a coboundary. Hence we obtain a well-defined class depending only on G ⊂ Aut D. By construction, it vanishes if and only if there is a choice of θ g,h which satisfies (2.1), hence if and only if there is an action (ρ, θ ) of G on the category D. Moreover, once an action (ρ, θ ) has been found, we obtain any other action θ by multiplying θ with an arbitrary λ : G 2 → C * which is a 2-cocycle. One checks that such a λ is a coboundary if and only if the actions θ and θ are isomorphic. This proves part (a).
(b) We will show that given α ∈ H 3 (G, C * ) there exists a surjection G G from a finite group G such that the restriction of α to H 3 (G , C * ) vanishes. Since H 3 (G, C * ) is finite, α is the image of some β ∈ H 3 (G, Z n ) for some n under the map induced by the inclusion Z n → C * . Recall that we have the commutative diagram We find that it is enough to construct a surjection G G such that the image of β in H 3 (G , Z n ) vanishes.
It is known that β corresponds to a crossed module where the action of G on Z n is trivial ( [4], IV.5). The pullback of β along E → G corresponds to the crossed module Since this has a section s : E → E × G E and G acts trivially on Z n , it is equivalent to the trivial crossed module. Concretely, there is a morphism of crossed modules, which by definition is a map of long exact sequences compatible with the actions of the groups in the crossed modules. The upper crossed module corresponds to the trivial crossed module. Therefore the pullback of the class β to E vanishes. Moreover, by ([9], p. 502) we can choose E to be finite, so setting G = E yields the claim.

Action via Fourier-Mukai transforms
Let X be a smooth complex projective variety and let be the bounded derived category of coherent sheaves on X . Given an object where p, q : X × X → X are the projections and all functors are derived. For a multitude of applications it is very useful to know whether a given endofunctor of D b (X ) is given by a Fourier-Mukai transform. For fully faithful exact functors this was answered affirmatively by Orlov [15].
such that for all g, h, k the diagram (2.1) commutes with ρ g replaced by E g .
By associating to a kernel its Fourier-Mukai transform we see that any Fourier-Mukai action on D b (X ) induces a group action on D b (X ) in the sense of Sect. 2.1. We have the following converse.

Lemma 2.4 Let X be smooth complex projective variety and let G be a finite group. Then any G-action on D b (X ) is induced by a unique Fourier-Mukai action.
Proof Given (ρ, θ ), by Orlov's theorem [15] for every g ∈ G there exists a kernel E g ∈ D b (X × X ) with FM E g ∼ = ρ g . By uniqueness of the kernels there also exists an isomorphism θ g,h : E g • E h ∼ = E gh . Since 3 Hom(E g , E g ) = C, arguing as in the proof of Theorem 2.1 yields a class α in H 3 (G, C * ) which vanishes if and only if after replacing θ by a boundary the pair (E g , θ ) defines a Fourier-Mukai action. By passing to Fourier-Mukai transforms one sees that α is the same class as the obstruction class defined by G ⊂ Aut D b (X ) and hence has to vanish (since G acts on D b (X )). This shows that there is a Fourier-Mukai action of A similar argument shows further that the possible isomorphism classes of such Fourier-Mukai G-actions are a torsor under H 2 (G, C * ). Moreover the map that associated to a Fourier-Mukai action the induced action on D b (X ) is equivariant with respect to the action of H 2 (G, C * ). This implies that we can also match (in a unique way, up to isomorphism) the 2-isomorphisms θ for the action on D b (X ) and for the Fourier-Mukai action.

Equivariant categories
Let (ρ, θ ) be an action of a finite group G on an additive C-linear category D.

Definition
The equivariant category D G is defined as follows: • Objects of D G are pairs (E, φ) where E is an object in D and φ = (φ g : E → ρ g E) g∈G is a family of isomorphisms such that The definition of morphism can be reformulated as follows. For any objects (E, φ) and (E , φ ) in D G consider the action of G on Hom D (E, E ) via Then we have

Induction and restriction functor
Given a subgroup H ⊂ G, we have a restriction functor defined by restricting the linearization of an equivariant object to the subgroup H .
In the opposite direction we have an induction functor which is constructed as follows: Let g i be representatives of the cosets G/H , where one of the g i equals 1 ∈ G (representing the unit coset). Then we set where for every g ∈ G the restriction of φ G g to the summand ρ g j E is defined by In the case of the trivial subgroup, H = 1, the restriction and induction functors specialize to the forgetful functor which forgets the linearization, and the linearization functor (3.2)

Universal property
Equivariant categories can be viewed as limits in the category Cats. To explain this let us view a finite group G as the 2-category G with one object, 1-morphisms given by the elements of G, and only identities as 2-morphisms. Giving a G-action on a category D is then equivalent to giving a 2-functor 4 G → Cats which takes the unique object in G to the category D. The equivariant category is the 2-limit of this morphism: This yields a universal property of the equivariant category that we state in explicit terms below. The proof follows by general theory, but for concreteness we will sketch a direct argument. We also refer to ( be the category whose objects are functors A → B and whose morphisms are natural transformations of such functors, and similarly for categories with G-action.

Proposition 3.1 Let G be a finite group which acts on a C-linear category D. Then for every category A we have a bifunctorial equivalence of categories
The proposition implies that every G-functor ι(A) → D can be factored via the equivariant category: Conversely, the forgetful functor p : D G → D carries a natural structure of a Gfunctor. Hence composing any functor A → D G with p yields a G-functor ι(A) → D.
Proof of Proposition 3. 1 We can assume that the G-action on D is strict.
Let A be a category and let ( f , σ ): ι(A) → D be a G-functor. By definition of a G-functor, the 2-isomorphisms σ g : f → ρ g f fit into the commutative diagram Thus for any E ∈ A the collection One further checks that any natural transformation of G-functors ( f , σ ) → ( f , σ ) yields a natural transformation F → F of the corresponding functors. These assignments define a functor Conversely, the 2-isomorphisms Similarly, for any natural transformation t : This yields an inverse to (3.4).
If D is an abelian category we also have a universal property with respect to the functor q. It says that D G is the 2-colimit in the 2-category of abelian categories where the morphisms are left-exact functors. We state it here for completeness, but it will not be essential later on. For any two abelian categories let us denote by

Proposition 3.2 Let G be a finite group which acts on an abelian category D. Then for any abelian category A we have the equivalence of categories
Similar results hold for dg-categories (or certain stable ∞-categories) and will play a role in determining the Hochschild cohomology of the equivariant category, see Sect. 6 below and [18].
Proof We assume that the action is strict. Since q(gE) = q E for all objects E in D, the linearization functor q : by pre-composing with q, i.e. by sending a functor F : D G → A to (Fq, F id), and a natural transformation t of such functors to tq.
Conversely, let ( f , σ ): D → ι(A) be a G-functor. One checks immediately (or see e.g. [21], Lemma 3.5) that ( f , σ ) naturally lifts to a functor f : where E is an object in A endowed with a G-action given by the linearization φ g : E → E. Since A is abelian and hence has finite limits, we have a functor Similarly, any natural transformation t : f → f lifts to a natural transformation

This yields a functor
We need to show that (3.5) and (3.8) are quasi-inverse to each other when restricted to the subcategory of left-exact functors. Consider a G-functor ( f , σ ) ∈ Hom G-Cats,l.e. (D, ι(A)) and let F be defined as in (3.7). Then F is a left-exact additive functor and we have the composition of G-functors Conversely, given F ∈ Hom Cats,l.e. (D G , A), define the G-functor which is left-exact and additive, and consider its lift f : This defines a G-action on the object q B, i.e. a homomorphism G → Hom is G-invariant with respect to the linearizations φ and φ can , and defines an isomorphism Hence it is precisely F applied to the morphism (3.9). Since F is left-exact and hence commutes with finite limits, we find The final step (which is left to the reader) is to show that (3.5) and (3.8) are inverse to each other on natural transformations.

Taking equivariant categories successively
The following result shows that when determining equivariant categories it is sufficient to consider simple groups G. We also refer to ([20], Remark 2.4) for the parallel statement for stacks.

Proof
We assume that the action is strict. For every g ∈ G define a functorρ g : D H → D H by letting it act on morphisms by ρ g and on objects bȳ One checks that the assignment g →ρ g defines a strict G-action on D H for which For the second part we define a new action ρ g . Choose representatives g 1 , . . . , g n ∈ G for the elements in G/H , where we take the identity element for the coset of the identity. Given any element g ∈ g i H we set For any two elements g ∈ g i H and g ∈ g j H write g i g j = g k h (here g k and h only depend on g i , g j ). Then define The adjunction follows by a direct check. Using Proposition 3.1 and the 2-categorical Yoneda lemma yields the isomorphism D G ∼ = (D H ) G/H .

Action of the dual group
The group of characters of G, acts on the equivariant category D G via the identity on morphisms and by The cocycle condition for χφ is satisfied because for any g, h ∈ G one has We discuss a typical example arising in geometry.

Example 3.4
Let G be a finite group acting on an complex quasi-projective algebraic variety X . The G-action induces an action on the category of Coh(X ) of coherent sheaves on X by sending a sheaf E to its pushforward g * E under the automorphism g : X → X . If G acts freely, then we have the following well-known equivalence of the equivariant category with the category of coherent sheaves on the quotient variety X /G 5 : The equivalence is given by pullback of sheaves along the quotient map π : X → X /G. Conversely the linearization φ of some (E, φ) is the descent datum of the sheaf E with respect to π . Under the equivalence the structure sheaf of O X /G corresponds to the equivariant sheaf (O X , 1), where we write 1 for the canonical linearization given by pullback of functions along g. For every character χ ∈ G ∨ consider the line bundle L χ ∈ Pic(X /G) which corresponds to the twisted linearization (O X , χ). Then tensoring with L χ on Coh(X /G) corresponds under the equivalence (3.10) to the dual action of χ ∈ G ∨ on the equivariant category. In particular, the line bundles L χ are all torsion.
If the group G is abelian, then G ∨ is called the dual group and is non-canonically isomorphic to G. In this case we have the following: The proposition can be applied for example to the categories Coh(X ) and D b (X ), since they are both idempotent complete.

Example 3.6
If X is an algebraic variety and H ⊂ Pic(X ) a finite subgroup which acts on Coh(X ) by tensoring, then the equivariant category Coh(X ) H is equivalent to Coh( X ) where X is the cover In particular, if G is taken to be abelian in Example 3.4, and we take the subgroup of line bundles L χ for all χ ∈ G ∨ , then one recovers Hence this is a basic example of the reversion principle of Proposition 3.5. 5 If the G-action is not free, then parallel statements apply to the stack quotient [X /G].
A G-linearization of an object E ∈ D is an object E ∈ D G such that p E ∼ = E. We say that an object E is G-linearizable if it admits a G-linearization. Equivalently, it lies in the essential image of the functor p.
By work of Ploog, the action of the dual group yields the following useful description of the set of G-linearizations of a simple object.

A subgroup of the group of auto-equivalences which does not act
Let D be a triangulated category and let τ : D → D be an auto-equivalence of order 4 (so τ k id unless 4 | k) which defines a strict Z 4 -action on D. Let τ 2 ∼ = Z 2 denote the subgroup generated by τ 2 and let D = D τ 2 be the equivariant category. As in Proposition 3.3, τ induces an auto-equivalencē τ : D → D together with a natural isomorphism t : id D ∼ =τ 2 . Concretely, for an equivariant object (A, φ) we define We also have an equivalence χ : D → D of order 2 obtained by twisting with the non-trivial character χ of Z 2 . The automorphismsτ and χ commute canonically and hence the composition χ •τ is of order 2 in Aut D . Suppose also that 6 Hom(id D , id D ) = Cid.

Claim 1
The subgroup Z 2 ⊂ Aut D generated by g = χ •τ does not act on D .
Proof of Claim 1 Since Hom(id D , id D ) = C, any isomorphism θ g,g : g 2 ∼ = − → id is a scalar multiple of t −1 . Hence it is enough to show that gt = tg. For any (A, φ) the map (gt) (A,φ) is obtained by applying χτ to φ τ 2 : (A, φ) →τ 2 (A, φ). Hence it is equal to τ φ τ 2 (twisting by χ acts by the identity on morphisms). On the other hand we have To translate the above into a simple concrete case, let τ : Coh(E) → Coh(E) be the translation by a 4-torsion point on an elliptic curve E. The equivariant category D τ 2 is then equivalent to Coh(E ) with E = E/ τ 2 being the quotient. The induced morphismτ is equivalent to translation t a by a 2-torsion point and χ is equivalent to tensoring with a 2-torsion line bundle L b = O E (b −0 E ) corresponding to a 2-torsion point b which is distinct from a. We find that the involution L b ⊗ t * a (−) does not define an action of Z 2 on Coh(E ), but only a Z 4 -action.

Equivariant triangulated categories and decompositions
We consider C-linear triangulated categories D with group actions. By work of Elagin (Sect. 4.2), if D has a dg-enhancement, then D G is also triangulated, in which case we can consider the decomposition of D G into components. The main result of this section is (under suitable conditions) a description of these components as equivariant categories for related group actions on D. As a consequence one can often restrict oneself to consider group actions for which the equivariant category is indecomposable.

Orthogonal decompositions
A triangulated category D is the orthogonal direct sum of n full subcategories D i if every object E ∈ D is isomorphic to a direct sum i E i with E i ∈ D i and there are no non-trivial morphisms between objects which lie in different subcategories. In this case we write D = i D i . The category D is indecomposable if in any such decomposition all except one summand is trivial. Given a finite decomposition where all D i are non-trivial and indecomposable, the summands D i are unique up to permutation and called the components of D.
Write Hom( f , g) for the vector space of a natural transformations f → g for functors f , g : D → D. We will use the following lemma. Proof If D admits an orthogonal decomposition n i=1 D i with D i non-trivial then the projections to the factors define n linearily independent elements in Hom D (id D , id D ).
The following example shows that the number of components of a triangulated category can be strictly smaller than dim Hom D (id D , id D ). 7

Example 4.3 Let
where for every A ∈ D, the morphism t A : A → A is multiplication by .

Equivariant triangulated categories
Let G be a finite group acting on a C-linear triangulated category D. We define a shift functor [1] : and we say a triangle in D G is distinguished if and only it it is distinguished after applying the forgetful functor p. By a result of Elagin ([8], Theorem 6.10), if D admits a dg-enhancement, 8 then these definitions make D G a triangulated category. The existence of a dg-enhancement is a technical condition [6], which is known for the cases we are most interested in. In particular, by a result of Lunts and Orlov [13] the bounded derived category D b (Coh(X )) of coherent sheaves on a smooth projective variety X has up to equivalence a unique dg-enhancement.
We can also ask for the stronger condition that the G-action on D lifts to a Gaction on a dg-enhancement of D. This is satisfied for example if G preserves a full abelian subcategory A ⊂ D such that D b (A) ∼ = D, see [6] and ([1], Section 2.1). This condition is useful since it will allow us to use methods from Hochschild cohomology.
From now on we will always assume that D G is triangulated. We state a useful lemma that shows that we can always assume that D is indecomposable when computing equivariant categories: The composition (D 0 ) K F − → D G p − → D is given by (E, φ) → g∈G/K gE. 7 We thank the referee for pointing out this example. 8 We refer to [6] for references on dg-enhancements.

Moreover, consider the action ρ K ∨ of K ∨ on D G induced by the dual group action of K ∨ on (D 0 ) K under the isomorphism (4.1). Then the natural map G ∨ → Aut D G associated to the dual group action of G ∨ factors as G
Proof Since D has a dg-enhancement, all equivariant categories are triangulated. The composition of Res G K : D G → D K followed by the restriction to (D 0 ) K is the adjoint (both left and right) to F. This and a straightforward calculation implies that F is fully faithful and hence yields a semi-orthogonal decomposition D G = (D 0 ) K , E . Moreover, for any object A ∈ E we have that the projection of p A to D 0 vanishes and since p A is G-invariant, one obtains that p A = 0, so A = 0. This proves the first claim. The second claim follows since any element of D G is of the form Ind G K (A) for some A ∈ (D 0 ) K . In particular, any element in the kernel of G ∨ → K ∨ acts as the identity.

An example: -trivial action
Let D be a triangulated category satisfying Hom(id D , id D ) = C. Let (ρ, θ ) be the action of a finite group G on D such that ρ g = id for all g ∈ G, but with θ arbitrary. By Theorem 2.1 there is an associated cocycle where the trivial action corresponds to the trivial class.
Let f i : G → GL(V i ) for i ∈ I be the (isomorphism classes of) projective irreducible representations of G of class α, see [7] for an introduction to the theory of these representations. By ( [7], Section 2) the index set I has order equal to the number of conjugacy classes of G which consists of α-elements. 9

Lemma 4.5 If D G is also triangulated, then we have the orthogonal decomposition
Proof The case of the trivial action (α = 0) can be found in ( [12], Proposition 3.3). The argument for the general case is completely parallel (note that for projective representations we also have that the regular (projective) representation C[G] decomposes into V ∨ i ⊗ V i , see [7], Corollary 3.11).

Faithful actions
We make the following new definition. Consider a G-action on an indecomposable derived category D such that D G is triangulated.

Definition 4.6
The G-action on D is called faithful if the associated equivariant category is indecomposable.  To apply this criterion in practice, we need to better understand Hom(id D G , id D G ):

Proposition 4.9 Let (ρ, θ ) be a G-action on a triangulated category D which lifts to an action on a dg-enhancement of D. Then there exists an isomorphism
where the action on the right is given by conjugation.
The G-action on the right is given as follows. An element g ∈ G acts by g : Hom(id D , ρ h ) → Hom(id D , ρ ghg −1 ) by sending t : id D → ρ h to g • t which is defined by the commutative diagram

Proof of Proposition 4.9
The left-hand side of (4.2) can be identified with the degree 0 Hochschild cohomology of D G and the result hence follows from the description of the Hochschild cohomology of D G by Perry, see [18] and Theorem 6.1 below.
However, to provide an idea of the proof, let us nevertheless sketch the argument in the case where D is an abelian category. The argument used to prove the result of Perry is parallel by translating all steps into the language of dg-categories.
Hence let D be an abelian category. By Proposition 3.2 and its proof we have where g acts on Hom(q, q) by sending t : q → q to the composition q ∼ = qg tg − → qg ∼ = q (the isomorphisms are provided by θ ). By Proposition 3.1 we have further where g ∈ G acts by sending t : pq → pq to pq ∼ = gpq gt − → gpq ∼ = pq. This yields where the embedding G → G × G is given by g → (g, g −1 ). This yields the claim by the adjunction with the restriction functor Res We show that for abelian groups to compute equivariant categories one can restrict to faithful actions (see Sect. 4.7 for the non-abelian case.) D with a dg-enhancement. Then there exists a quotient  G → G , and a faithful action (ρ , θ ) of G on D such that D G has the finite decomposition

Theorem 4.10 Let (ρ, θ ) be an action of a finite abelian group G on an indecomposable triangulated category
Proof Since D is indecomposable, G ∨ is finite, and we have (D G ) G ∨ ∼ = D by Proposition 3.5, for any orthogonal decomposition of D G in which the dual group G ∨ permutes the summands and acts transitively on the summands. Hence there are at most |G ∨ | many non-trivial summands in such a decomposition. Since any orthogonal decomposition can be refined to such a decomposition, this shows that there exists a finite decomposition into indecomposable triangulated categories D i on which G ∨ acts transitively. In particular, all D i are equivalent. Let D 0 be one of the components and let H ⊂ G ∨ be its stabilizer. By Lemma 4.4 we have The inclusion H ⊂ G ∨ yields a surjection G ∼ = (G ∨ ) ∨ H ∨ . Moreover, the dual group H ∨ acts on (D 0 ) H ∼ = D. Hence by Proposition 3.5 applied to (4.5) we get Hence the first claim follows by taking G := H ∨ . The second claim follows by the second part of Lemma 4.4.

A stronger version of Theorem 2.1
Using the techniques presented above, we can prove the following stronger version of part (c) of Theorem 2.1.

Corollary 4.11
Let D be an indecomposable triangulated category with a dgenhancement. Consider a subgroup Z n ⊂ Aut D for which there exists a Z n -invariant simple object in D.
Then there exists an action of Z n on D such that the induced map Z n → Aut D is the inclusion we started with.
Proof Using part (c) of Theorem 2.1, we have an action of Z n 2 on D such that the induced map Z n 2 → Aut D is the quotient map to the given subgroup Z n ⊂ Aut D. Applying Theorem 4.10 we hence find a faithful action by some Z m such that its image in Aut D is the subgroup Z n . We will show that m = n.
Let k = m/n and consider the short exact sequence The image of Z k in Aut D is the trivial group. Since H 2 (Z k , C * ) = 0 we see that the Z m -action on D restricts to the trivial action by Z k (or, more precisely, to an action which is isomorphic to the trivial action). We consider the induced action of Z n on the equivariant category where V i are the irreducible representations of Z k .
Since D Z m is indecomposable, Z n acts transitively on the summands (4.6). Let Z l ⊂ Z n be the stabilizer of the first summand. In particular, Z k = Z n /Z l . Applying Proposition 3.3 and Lemma 4.4 we obtain the equivalence Moreover, under this equivalence the forgetful functor p : D Z m → D is given by On the other hand, by the assumption that Z n ⊂ Aut D fixes some simple object, there is a simple object F ∈ D which is invariant under the Z m -action. By Lemma 3.7 the object F admits a linearization with respect to Z m and hence lies in the image of the forgetful functor from the equivariant category D Z m . In other words, F is of the form (4.7). This implies that Z n /Z l is trivial, so k = 1.

The equivariant category of Example 3.6
We return to Example 3.6. Recall that there we considered a subgroup Z 2 ⊂ Aut D generated by an involution g : D → D which does not act on D . Note that by Corollary 4.11 this implies that there is no simple object in D which is invariant under the action of Z 2 .
By part (c) of Theorem 2.1 the involution g defines a (unique) Z 4 -action on D . Here we determine the associated equivariant category. Let us assume that the Z 4 -action on D lifts to an action on a dg-enhancement and that Hom D (id, g) = 0.

Proof of Claim 2
We first show that the equivariant category D Z 4 is indecomposable: By Proposition 4.9 and assumption (4.8) this reduces to showing that However, we have seen in Example 3.6 that the generator of this vector space Since H 2 (Z 2 , C * ) = 0, the restriction of the Z 4 -action to the subgroup Z 2 ⊂ Z 4 is trivial. An application of Proposition 3.3 and Sect. 4.3 gives Since D Z 4 is indecomposable, Z 4 /Z 2 acts transitively on the two summands. The claim now follows from Lemma 4.4.
Recall the concrete example discussed at the end of Sect. 4.6. We have seen that the involution L b ⊗t * a (−) on D b (E ) does not define an action of Z 2 , but only a Z 4 -action. By the above claim we see that Another example is described in ([1], Section 7).

Faithful actions II
We extend Theorem 4.12 to the non-abelian case. We require slightly stronger assumptions. (ρ, θ ) be a G-action on a indecomposable triangulated category D which lifts to an action on a dg-enhancement of D. Assume that for all g ∈ G, one of the following two conditions holds:
Then there exists a finite decomposition and for every i a faithful action ρ i by a finite group K i on D such that D i ∼ = D K i .

Proof of Theorem 4.12
Recall from Proposition 4.9 that we have (4.9) Since G acts by conjugation on the right-hand side, we can consider the decomposition according to conjugacy classes c: For any non-trivial transformations t 1 : id → ρ g and t 2 : id → ρ g by assumption (ii) we can form the composition t −1 2 • t 1 , which (since D is indecomposable) is a multiple of the identity. We conclude that dim V c ∈ {0, 1} for all c.
Let H ⊂ G be the set of all elements whose conjugacy class c satisfies dim V c = 1. A direct check shows that H is a normal subgroup of G. Moreover for all h ∈ H there exist isomorphisms 10 (4.10) After modifying the action by the isomorphisms t h , we may assume ρ h = id and t h = id. Relation (4.10) is then equivalent to the condition that the composition is the identity for all g ∈ G and h ∈ H . Let V i , i ∈ I , be the projective irreducible representations of H for the cocycle defined by θ . We consider the decomposition of Sect. 4.3, and the isomorphism ( * ) is precisely ρ g applied to the inverse of (4.11). Since ( * ) is the identity and φ g −1 hg = id E ⊗ f i (g −1 hg), we find that This shows that for any j ∈ I /G (where G acts on the index set I by i → g(i)) the G-action on D H preserves and acts transitively on the subcategory Consider the quotient action G/H on D H as in Proposition 3.3. Then G/H acts also transitively on E j . Let K j ⊂ G/H be the subgroup which preserves a given (fixed) summand D ⊗ V i j of E j . Applying Lemma 4.4 yields the equivalence Hence by Proposition 3.3 we find that (4.12) By (4.11) all conjugacy classes of H consist of α-elements where α is the cocycle defined by θ , see footnote 9. Therefore the order of I equals the number of conjugacy classes of H , and the order of I /G equals the number of conjugacy classes of G which are contained in H . This latter number is by inspection the dimension of (4.9). We find that D G has at least dim many (not necessarily indecomposable) components. However, by Lemma 4.2 this is the maximal number of components, which shows that all of the summands (D ⊗ V i j ) K j in (4.12) are indecomposable. In particular, the action of K j on D ⊗ V i j ∼ = D is faithful. This proves the claim.

The Serre functor on equivariant categories
Let D be a C-linear triangulated category with finite-dimensional Hom's.
A Serre functor for D is an equivalence S : D → D together with a collection of bifunctorial isomorphisms  Hom(A , B), ψ : A → A and f ∈ Hom(B, S A). The functoriality in B is equivalent to for all f ∈ Hom(A, B), f ∈ Hom(B , S A) and ρ : B → B . The following is well known.  Hom(A, B) and f ∈ Hom(B, S A). Proof For any A, B ∈ D we have the chain of isomorphisms where we applied Serre duality in the first and third, and F and F −1 in the second and fourth step respectively. Since Serre duality and application of F is functorial in both arguments, the isomorphisms are functorial in both A and B. By the Yoneda lemma this gives the desired natural transformation t F . For the functoriality of t in F we need to show that for every A ∈ D we have the commutative diagram

S F G(A) F SG(A) F G S(A).
This is checked by applying Hom(B, −). The adjunctions used to define the composition yield precisely the adjunction to define the curved arrow. This shows (a). If we apply this chain of isomorphisms to η F A,F B (F f ), we obtain By definition, the resulting dual map f : On the other hand, we have by construction Let G be a finite group acting on D. Given an object (A, φ) in D G we claim that the collection The relation we need to check (given in (3.1)) is the commutativity of the outer triangle. Since t g is a natural transformation the upper right square commutes. The upper left triangle commutes since it is obtained by applying S to the diagram where the G-action on the left is defined by the linearizations φ, ψ and the G-action on the right is defined by the linearizations ψ and φ (as in (5.4)).
Proof The action of g ∈ G on Hom(B, S A) defined by ψ and φ is Hence for any f ∈ Hom(A, B) we obtain where we have used (5.1) and (5.2) in the second and Lemma 5.1 (b) and (a) in the third and fourth equality respectively. Hence the action on Hom(B, S A) with respect to ψ, φ is dual to the action on Hom(A, B) with respect to φ, ψ. This implies the claim.

Remark 5.3
We refer to [18] for more a advanced discussion of Serre functors on equivariant categories in the context of dg-categories. Our hands-on approach has the advantage to [18] in that we do not use dg-categories.

Hochschild cohomology
In this section we discuss how Hochschild cohomology can be used to describe equivariant categories. The definition requires us to work with enhanced categories, for which we take as model dg-categories (another, in characteristic 0 equivalent, choice would be stable ∞ categories as used in the work of Perry [18]).
Throughout let C be a pre-triangulated dg-category over C.

Definition
Let Func(C, C) be the dg-category of functors C → C where morphisms are natural transformations. The Hochschild cohomology of C with values in a functor φ : C → C is defined by The (absolute) Hochschild cohomology of C is

Equivariant category
Every equivalence F : C → C induces (functorially) a morphism on Hochschild cohomology by conjugation, In particular, a group action on C induces a group action on Hochschild cohomology. By work of Perry, we have the following description of the Hochschild cohomology of the equivariant category C G . Theorem 6.1 ([18], Theorem 4.4) Let a finite group G act on C via the equivalences ρ g : C → C for all g ∈ G. Then we have where G acts on the right by conjugation.
The cohomology group is called the categorical trace of g with respect to the given G-action [10]. If the element h ∈ G commutes with g, then we have an induced action of h on Tr C (g). The 2-characters of the representation are where the trace is taken in the supercommutative 11 sense. 12 Corollary 6.2 Assume HH • (C, ρ g ) is finite-dimensional for all g. Then Proof If a finite group G acts on a vector space V , then Applying this to the expression in Theorem 6.1 yields the claim.
In the decomposition (6.1) an element h acts by HH • (C, ρ g ) → HH • (C, ρ hgh −1 ). Hence we may rewrite 11 If g acts on the Z-graded vector space V = i V i , then we let Tr(g | V ) = i (−1) i Tr(g | V i ). 12 In physics language these are the h-twisted, g-twined characters of the representation. and c runs over all conjugacy classes of G.
Recall that the group of characters G ∨ acts on the equivariant category C G . The following describes the induced action on HH • (C G ).
Hence t lifts to the natural transformation Since t (A,φ) does not depend on the linearization, but only on the underlying object A, we find for all χ ∈ G ∨ that Hence, t is G ∨ -invariant and the claim is proven for the conjugacy class of the unit. More generally, consider an element h ∈ G in the center of G. As in the proof of Proposition 3.3 the functor ρ h : C → C lifts to the functor Let t : id C → ρ h [i] be a G-invariant natural transformation. As before, t lifts to a natural transformation t : Consider the natural transformation h : id C G → ρ h defined by Unlike before, this morphism depends on the linearization: For every character χ we Consider now the composition which under the isomorphism of Lemma 6.1 is precisely the Hochschild cohomology class corresponding to t. Then we have We finally consider the general case given by a conjugacy class c. By tracing through the isomorphism of Lemma 6.1 (compare also to the proof of Proposition 4.9) one checks that a G-invariant natural transformation The claim follows now by the same argument as before.

Hochschild homology and Serre functors
From now on we assume that the category C is equipped with a Serre functor S. The Hochschild homology of C is then defined by HH • (C) := HH • (C, S) By Lemma 5.1 (or, more precisely, its analogue for dg-categories) any equivalence F : C → C commutes with the Serre functor S in a canonical way. Hence, F acts on Hochschild homology by conjugation 14 Moreover, arguing as in Sect. 5 shows that the Serre functor S lifts to a Serre functor S on the equivariant category C G .

Remark 6.4
The existence of Serre functors on the equivariant category can also be argued more abstractly using the notions of 'smooth', 'proper' and 'dualizable' dgcategories for which we refer to Part I of [17]. This proceeds as follows.
Let C be a smooth and proper dg-category, for example the bounded derived dgcategory of coherent sheaves on a smooth and proper variety. Then C is dualizable and hence admits a Serre functor S, see ( [17], Section 4.6). If we have a finite group G acting on C, then since the morphisms in the equivariant category C G are precisely the G-invariant morphisms in C, it is immediate that C G is again proper. We claim that C G is also smooth: Indeed, we need to show that id Ind(C G ) ∈ Func(Ind(C G ), Ind(C G )) is a compact object, where Ind(C) denotes the Ind-completion of a category C. By ([1], Lemma 3.18) or ( [18], Lemma 3.6) we have Ind(C G ) = Ind(C) G and combining with ( [18], Lemma 4.7) this yields an isomorphism Func(Ind(C G ), Ind(C G )) ∼ = Func(Ind(C), Ind(C)) G×G .
By ( [18], Lemma 3.7) it suffices then to show that the image of id Ind(C G ) under the forgetful morphism Func(Ind(C), Ind(C)) G×G → Func(Ind(C), Ind(C)) is compact. But according to ([18], Lemma 4.7) this image is precisely g∈G ρ g and hence as a finite direct sum of compact objects compact (to show that ρ g is compact, use that id Ind(C) is compact, and since ρ g is invertible, left multiplication by it preserves compact objects, therefore also ρ g id Ind(C) = ρ g is compact).
Having that C G is both smooth and proper, we find that it is dualizable and hence admits a Serre functor S. Using that by Yoneda's Lemma every Serre functor is characterized by the defining isomorphism Hom(τ (−, −)) ∨ ∼ = Hom(−, S(−)), where τ is the functor that swaps th two factors (see [17], Proof of Lemma 4.19), one finds that the Serre functor S agrees with the one constructed in Sect. 5.

Calabi-Yau categories
Suppose now further that C is a Calabi-Yau category, i.e. that there exists a 2isomorphism for some integer n, called the dimension of C. In particular, we have We can ask under which conditions is the equivariant category C G again Calabi-Yau. The answer is very natural and given as follows. Similarly, the inverse of a also lifts. Hence, a is a 2-isomorphism.
Moreover, since a is a lift of a G-invariant class, arguing as in the proof of Lemma 6.3 shows that it is fixed by the action of G ∨ by conjugaction.

Hochschild and singular cohomology
Let X be a smooth projective variety and let D dg (X ) be the dg-enhancement of D b (X ). The Hochschild homology and cohomology of X are defined by HH • (X ) = HH • (D dg (X )), HH • (X ) = HH • (D dg (X )).
One has the Hochschild-Kostant-Rosenberg (HKR) isomorphism The kernel E also induces a morphism on singular cohomology where p, q are the projections of X × Y onto the factors and we let v(E) = ch(E) √ td X denote the Mukai vector. By the main result of [5] the action of FM E on Hochschild and singular cohomology are compatible under the HKR isomorphism, i.e. the following diagram commutes: HKR FM E, * HKR FM E, * In particular, in order to apply Lemma 6.5 to D dg (X ) for a Calabi-Yau variety X , it suffices to check the invariance of the Calabi-Yau form on singular cohomology, i.e. that the element in H 0 (X , ω X ) = H n,0 (X , C) corresponding to the isomorphism ω X ∼ = O X is preserved by G.

Equivariant categories of elliptic curves
We illustrate some of the results of the previous sections by applying them to the example of the derived category of coherent sheaves on an elliptic curve. Let E be a non-singular elliptic curve and let G be a finite group which acts on D b (E). By Lemma 2.4 the action is given by Fourier-Mukai transforms and hence induces an action on cohomology. We assume that each g ∈ G fixes the Calabi-Yau form, i.e. that the cohomological Fourier-Mukai transform FM E g , * acts trivially on H 1,0 (E). In this case we also say that the G-action on D b (E) is Calabi-Yau.
Our goal in this section is to prove the following: By a result of Orlov [16], see also ([11], Proposition 9.55), the subgroup of Aut D b (E) acting trivially on cohomology is isomorphic to Z × E × Pic 0 (E) (7.1) where the first summand is identified with the shift by [2], the second one with the pullback along translations and the last one with tensoring with degree 0 line bundles. We first show that any Calabi-Yau action acts trivially on cohomology.

Lemma 7.2 For any Calabi-Yau action by a finite group G on D b (E) the induced action on H * (E, Z) is trivial.
Proof Recall from for example ( [11], Section 9) that auto-equivalences of elliptic curves induce Hodge isometries on the integral cohomology lattice. Since H 1 (E, Z) is preserved by FM E g , * and by assumption the action on H 1,0 is trivial, it follows that FM E g , * acts trivially on H 1 (E, Z). The isometry FM E g , * induces an isometry of H 0 (E, Z) ⊕ H 2 (E, Z) which, as a lattice, is isometric to the hyperbolic plane U . We have Aut U = Z 2 × Z 2 . Moreover, only ±id can occur in our case, since the action of every auto-equivalence is orientationpreserving. Let us denote this isometry by τ .
Let ι : E → E denote the morphism given by multiplication with −1. Observe that the auto-equivalence F = [1] •ι fixes H 1 (E, Z) and acts as −id on H 0 (E) ⊕ H 2 (E), hence its cohomological Fourier-Mukai transform yields τ . Given any autoequivalence F : D b (E) → D b (E) which induces τ , the composition F • F acts trivial on cohomology and hence lies in the subgroup (7.1). It follows that every auto-equivalence inducing τ on cohomology has infinite order. Hence, it can never be contained in the image of G in Aut D.
Proof of Theorem 7.1 From the lemma and since G has finite order, it follows that the image of G in Aut D b (E) maps to E × Pic 0 (E). This shows that G preserves the category Coh(E) and hence lifts to an action of a dg-enhancement of D b (E). Moreover, also conditions (i) and (ii) of Theorem 4.12 are easily verified. By Theorem 4.12 we may hence assume that the action is faithful. Moreover, by Lemma 6.5 the equivariant derived category is again a 1-Calabi-Yau category. In particular, Coh(E) G is an indecomposable 1-Calabi-Yau abelian category.
To prove the claim we apply the classification ( [22], Theorem 1.1) of such categories. More precisely, the proof of ( [22], Theorem 4.7) shows that as soon as Coh(E) G has two non-isomorphic simple objects (i.e. objects T such that Hom(T , T ) = C), then Coh(E) G is equivalent to the category Coh(E ) for some elliptic curve E .
Hence we need to find two simple, non-isomorphic objects in Coh(E) G . Since the G-action preserves the canonical Bridgeland stability condition [3] on D b (E), we know by [14] (see also [1], Section 2.2) that D b (Coh(E) G ) has again a stability condition satisfying the support property. In particular, finite Jordan-Hölder filtrations of semistable objects exist. If, up to isomorphism, there would only exist one simple object, then there would be only one stable object T in D b (Coh(E) G ). Since pq acts by multiplication by |G| on the numerical K -group of Coh(E), this implies that p(T ) generates up to finite index the numerical K -group of E. However, clearly we have K num (E) = Z 2 which gives a contradiction.