On Triangulated Orbit Categories

Dedicated to Claus Michael Ringel

on the occasion of his sixtieth birthday

Bernhard Keller

Received: June 8, 2005

Communicated by Peter Schneider

Abstract. We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by Aslak Buan, Robert Marsh and Idun Reiten which appeared in their study [8] with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (cf. also [16]) and a question by Hideto Asashiba about orbit categories. We observe that the resulting triangulated orbit categories provide many examples of triangulated categories with the Calabi-Yau property. These include the category of projective modules over a preprojective algebra of generalized Dynkin type in the sense of Happel-Preiser-Ringel [29], whose triangulated structure goes back to Auslander-Reiten’s work [6], [44], [7].

2000 Mathematics Subject Classification: Primary 18E30; Secondary 16G20. Keywords and Phrases: Derived category, Triangulated category, Orbit category, Calabi-Yau category

1 Introduction

Let T be an additive category and F : T → T an automorphism (a standard construction allows one to replace a category with autoequivalence by a category with automorphism). Let F^{Z} denote the group of automorphisms generated by F. By definition, the orbit category T /F = T /F^{Z} has the same objects as T and its morphisms from X to Y are in bijection with

⨁_{n∈Z} Hom_{T} (X, F^{n}Y).

The composition is defined in the natural way (cf. [17], where this category is called the skew category). The canonical projection functor π : T → T /F is endowed with a natural isomorphism π ∘ F →^{∼} π and 2-universal among such functors. Clearly T /F is still an additive category and the projection is an additive functor. Now suppose that T is a triangulated category and that F is a triangle functor. Is there a triangulated structure on the orbit category such that the projection functor becomes a triangle functor? One can show that in general, the answer is negative. A closer look at the situation even gives one the impression that quite strong assumptions are needed for the answer to be positive. In this article, we give a sufficient set of conditions. Although they are very strong, they are satisfied in certain cases of interest. In particular, one obtains that the cluster categories of [8], [16] are triangulated. One also obtains that the category of projective modules over the preprojective algebra of a generalized Dynkin diagram in the sense of Happel-Preiser-Ringel [29] is triangulated, which is also immediate from Auslander-Reiten’s work [6], [44], [7]. More generally, our method yields many easily constructed examples of triangulated categories with the Calabi-Yau property. Our proof consists in constructing, under quite general hypotheses, a ‘triangulated hull’ into which the orbit category T /F embeds. Then we show that under certain restrictive assumptions, its image in the triangulated hull is stable under extensions and hence equivalent to the hull. The contents of the article are as follows: In section 3, we show by examples that triangulated structures do not descend to orbit categories in general. In section 4, we state the main theorem for triangulated orbit categories of derived categories of hereditary algebras. We give a first, abstract, construction of the triangulated hull of an orbit category in section 5. This construction is based on the formalism of dg categories as developped in [32] [21] [57]. Using the natural t-structure on the derived category of a hereditary category we prove the main theorem in section 6. We give a more concrete construction of the triangulated hull of the orbit category in section 7. In some sense, the second construction is ‘Koszul-dual’ to the first: whereas the first construction is based on the tensor algebra

T_{A}(X) = ⊕_{n=0}^{∞}X^{⊗_{A}n}

of a (cofibrant) differential graded bimodule X over a differential graded algebra A, the second one uses the ‘exterior algebra’

A ⊕ X^{∧}[−1]

on its dual X^{∧} = RHom_{A}(X_{A}, A) shifted by one degree. In the cases considered by Buan et al. [8] and Caldero-Chapoton-Schiffler [16], this also yields an interesting new description of the orbit category itself in terms of the stable category [40] of a differential graded algebra. In section 8, we observe that triangulated orbit categories provide easily constructed examples of triangulated categories with the Calabi-Yau property. Finally, in section 9, we characterize our constructions by universal properties in the 2-category of enhanced triangulated categories. This also allows us to examine their functoriality properties and to formulate a more general version of the main theorem which applies to derived categories of hereditary categories which are not necessarily module categories.

2 Acknowledgments

The author thanks C. M. Ringel for his interest and encouragement throughout the years. He is indebted to A. Buan, R. Marsh and I. Reiten for the question which motivated the main theorem of this article and to H. Asashiba for suggesting a first generalization of its original formulation. He is grateful to H. Lenzing for pointing out the need to weaken its hypotheses to include orbit categories of general hereditary categories. He expresses his thanks to C. Geiss and J.-A. de la Peña for the hospitality he enjoyed at the Instituto de Matemáticas, UNAM, in August 2004 while the material of section 8 was worked out. He thanks R. Bocklandt, W. Crawley-Boevey, K. Erdmann, C. Geiss, A. Neeman, I. Reiten, A. Rudakov, J. Schröer, G. Tabuada, M. Van den Bergh, Feng Ye, Bin Zhu and an anonymous referee for helpful comments and corrections.

3 Examples

Let T be a triangulated category, F : T → T an autoequivalence and π : T → T /F the projection functor. In general, a morphism

u : X → Y

of T /F is given by a morphism

X → ⨁_{i=1}^{N} F^{n_{i}}Y

with N non vanishing components u_{1}, …, u_{N} in T . Therefore, in general, u does not lift to a morphism in T and it is not obvious how to construct a ‘triangle’

\xymatrix{X \ar[r]^{u} & Y \ar[r] & Z \ar[r] & SZ}

in T /F. Thus, the orbit category T /F is certainly not trivially triangulated. Worse, in ‘most’ cases, it is impossible to endow T /F with a triangulated structure such that the projection functor becomes a triangle functor. Let us consider three examples where T is the bounded derived category D^{b}(A) = D^{b}(modA) of the category of finitely generated (right) modules modA over an algebra A of finite dimension over a field k. Thus the objects of D^{b}(A) are the complexes

M = (… → M^{p} → M^{p+1} → …)

of finite-dimensional A-modules such that M^{p} = 0 for all large |p| and morphisms are obtained from morphisms of complexes by formally inverting all quasi-isomorphisms. The suspension functor S is defined by SM = M[1], where M[1]^{p} = M^{p+1} and d_{M[1]} = −d_{M}, and the triangles are constructed from short exact sequences of complexes. Suppose that A is hereditary. Then the orbit category D^{b}(A)/S^{2}, first introduced by D. Happel in [27], is triangulated. This result is due to Peng and Xiao [43], who show that the orbit category is equivalent to the homotopy category of the category of 2-periodic complexes of projective A-modules. On the other hand, suppose that A is the algebra of dual numbers k[X]/(X^{2}). Then the orbit category D^{b}(A)/S^{2} is not triangulated. This is an observation due to A. Neeman (unpublished). Indeed, the endomorphism ring of the trivial module k in the orbit category is isomorphic to a polynomial ring k[u]. One checks that the endomorphism 1 + u is monomorphic. However, it does not admit a left inverse (or else it would be invertible in k[u]). But in a triangulated category, each monomorphism admits a left inverse. One might think that this phenomenon is linked to the fact that the algebra of dual numbers is of infinite global dimension. However, it may also occur for algebras of finite global dimension: Let A be such an algebra. Then, as shown by D. Happel in [27], the derived category D^{b}(A) has Auslander-Reiten triangles. Thus, it admits an autoequivalence, the Auslander-Reiten translation τ, defined by

Hom(?, SτM) →^{∼} DHom(M,?) ,

where D denotes the functor Hom_{k}(?, k). Now let Q be the Kronecker quiver

\xymatrix{1 \ar[r] \ar[r] & 2}.

The path algebra A = kQ is finite-dimensional and hereditary so Happel’s theorem applies. The endomorphism ring of the image of the free module A_{A} in the orbit category D^{b}(A)/τ is the preprojective algebra Λ(Q) (cf. section 7.3). Since Q is not a Dynkin quiver, it is infinite-dimensional and in fact contains a polynomial algebra (generated by any non zero morphism from the simple projective P_{1} to τ^{−1}P_{1}). As above, it follows that the orbit category does not admit a triangulated structure.

4 The main theorem

Assume that k is a field, and T is the bounded derived category D^{b}(modA) of the category of finite-dimensional (right) modules modA over a finitedimensional k-algebra A. Assume that F : T → T is a standard equivalence [48], i.e. F is isomorphic to the derived tensor product

? ⊗^{L}_{A} X : D^{b}(modA) → D^{b}(modA)

for some complex X of A-A-bimodules. All autoequivalences with an ‘algebraic construction’ are of this form, cf. section 9.

Theorem 1 Assume that the following hypotheses hold:

1) There is a hereditary abelian k-category H and a triangle equivalence

D^{b}(modA) →^{∼} D^{b}(H)

In the conditions 2) and 3) below, we identify T with D^{b}(H).

2) For each indecomposable U of H, only finitely many objects F^{i}U, i ∈ Z, lie in H.

3) There is an integer N ≥ 0 such that the F -orbit of each indecomposable of T contains an object S^{n}U, for some 0 ≤ n ≤ N and some indecomposable object U of H.

Then the orbit category T /F admits a natural triangulated structure such that the projection functor T → T /F is triangulated.

The triangulated structure on the orbit category is (most probably) not unique. However, as we will see in section 9.6, the orbit category is the associated triangulated category of a dg category, the exact (or pretriangulated) dg orbit category, and the exact dg orbit category is unique and functorial (in the homotopy category of dg categories) since it is the solution of a universal problem. Thus, although perhaps not unique, the triangulated structure on the orbit category is at least canonical, insofar as it comes from a dg structure which is unique up to quasi-equivalence. The construction of the triangulated orbit category T /F via the exact dg orbit category also shows that there is a triangle equivalence between T /F and the stable category \underline{E} of some Frobenius category E. In sections 7.2, 7.3 and 7.4, we will illustrate the theorem by examples. In sections 5 and 6 below, we prove the theorem. The strategy is as follows: First, under very weak assumptions, we embed T /F in a naturally triangulated ambient category M (whose intrinsic interpretation will be given in section 9.6). Then we show that T /F is closed under extensions in the ambient category M. Here we will need the full strength of the assumptions 1), 2) and 3). If T is the derived category of an abelian category which is not necessarily a module category, one can still define a suitable notion of a standard equivalence T → T , cf. section 9. Then the analogue of the above theorem is true, cf. section 9.9.

5 Construction of the triangulated hull M

The construction is based on the formalism of dg categories, which is briefly recalled in section 9.1. We refer to [32], [21] and [57] for more background.

5.1 The dg orbit category

Let A be a dg category and F : A → A a dg functor inducing an equivalence in H^{0}A. We define the dg orbit category B to be the dg category with the same objects as A and such that for X, Y ∈ B, we have

B (X, Y) →^{∼} colim_{p} ⨁_{n≥0} A (F^{n}X, F^{p}Y) ,

where the transition maps of the colim are given by F. This definition ensures that H^{0}B is isomorphic to the orbit category (H^{0}A)/F.

5.2 The projection functor and its right adjoint

¿From now on, we assume that for all objects X, Y of A, the group

(H^{0}A) (X, F^{n}Y)

vanishes except for finitely many n ∈ Z. We have a canonical dg functor π : A → B. It yields an A-B-bimodule

(X, Y) ↦ B (πX, Y).

The standard functors associated with this bimodule are

- the derived tensor functor (=induction functor)

π_{∗} : DA → DB

- the derived Hom-functor (right adjoint to π_{∗}), which equals the restriction

along π:

π_{ρ} : DB → DA

For X ∈ A, we have

π_{∗}(X^{∧}) = (πX)^{∧} ,

where X^{∧} is the functor represented by X. Moreover, we have an isomorphism in DA

π_{ρ}π_{∗}(X^{∧}) = ⨁_{n∈Z} F^{n}(X^{∧}),

by the definition of the morphisms of B and the vanishing assumption made above.

5.3 Identifying objects of the orbit category

The functor π_{ρ} : DB → DA is the restriction along a morphism of dg categories. Therefore, it detects isomorphisms. In particular, we obtain the following: Let E ∈ DB, Z ∈ DA and let f : Z → π_{ρ}E be a morphism. Let g : π_{∗}Z → E be the morphism corresponding to f by the adjunction. In order to show that g is an isomorphism, it is enough to show that π_{ρ}g : π_{ρ}π_{∗}Z → π_{ρ}E is an isomorphism.

5.4 The ambient triangulated category

We use the notations of the main theorem. Let X be a complex of A-A-bimodules such that F is isomorphic to the total derived tensor product by X. We may assume that X is bounded and that its components are projective on both sides. Let A be the dg category of bounded complexes of finitely generated projective A-modules. The tensor product by X defines a dg functor from A to A. By abuse of notation, we denote this dg functor by F as well. The assumption 2) implies that the vanishing assumption of subsection 5.2 is satisfied. Thus we obtain a dg category B and an equivalence of categories

D^{b}(modA)/F →^{∼} H^{0}B.

We let the ambient triangulated category M be the triangulated subcategory of DB generated by the representable functors. The Yoneda embedding H^{0}B → DB yields the canonical embedding D^{b}(modA)/F → M. We have a canonical equivalence D(ModA) →^{∼} DA and therefore we obtain a pair of adjoint functors (π_{∗}, π_{ρ}) between D(ModA) and DB. The functor π_{∗} restricts to the canonical projection D^{b}(modA) → D^{b}(modA)/F.

6 The orbit category is closed under extensions

Consider the right adjoint π_{ρ} of π_{∗}. It is defined on DB and takes values in the unbounded derived category of all A-modules D(ModA). For X ∈ D^{b}(modA), the object π_{∗}π_{ρ}X is isomorphic to the sum of the translates F^{i}X, i ∈ Z, of X. It follows from assumption 2) that for each fixed n ∈ Z, the module H_{modA}^{n}F^{i}X vanishes for almost all i ∈ Z. Therefore the sum of the F^{i}X lies in D(modA). Consider a morphism f : π_{∗}X → π_{∗}Y of the orbit category T /F = D^{b}(modA)/F. We form a triangle

π_{∗}X → π_{∗}Y → E → Sπ_{∗}X

in M. We apply the right adjoint π_{ρ} of π_{∗} to this triangle. We get a triangle

π_{ρ}π_{∗}X → π_{ρ}π_{∗}Y → π_{ρ}E → Sπ_{ρ}π_{∗}X

in D(ModA). As we have just seen, the terms π_{ρ}π_{∗}X and π_{ρ}π_{∗}Y of the triangle belong to D(modA). Hence so does π_{ρ}E. We will construct an object Z ∈ D^{b}(modA) and an isomorphism g : π_{∗}Z → E in the orbit category. Using proposition 6.1 below, we extend the canonical t-structure on D^{b}(H) →^{∼} D^{b}(modA) to a t-structure on all of D(modA). Since H is hereditary, part b) of the proposition shows that each object of D(modA) is the sum of its H-homology objects placed in their respective degrees. In particular, this holds for π_{ρ}E. Each of the homology objects is a finite sum of indecomposables. Thus π_{ρ}E is a sum of shifted copies of indecomposable objects of H. Moreover, F π_{ρ}E is isomorphic to π_{ρ}E, so that the sum is stable under F. Hence it is a sum of F -orbits of shifted indecomposable objects. By assumption 3), each of these orbits contains an indecomposable direct factor of

⨁_{0≤n≤N}(H^{n}π_{ρ}E)[−n].

Thus there are only finitely many orbits involved. Let Z_{1}, …, Z_{M} be shifted indecomposables of H such that π_{ρ}E is the sum of the F -orbits of the Z_{i}. Let f be the inclusion morphism

Z = ⨁_{i=1}^{M} Z_{i} → π_{ρ}E

and g : π_{∗}Z → E the morphism corresponding to f under the adjunction. Clearly π_{ρ}g is an isomorphism. By subsection 5.3, the morphism g is invertible and we are done.

6.1 Extension of t-structures to unbounded categories

Let T be a triangulated category and U an aisle [35] in T . Denote the associated t-structure [10] by (U_{≤0}, U_{≥0}), its heart by U_{0}, its homology functors by H_{U}^{n} : T → U_{0} and its truncation functors by τ_{≤n} and τ_{>n}. Suppose that U is dominant, i.e. the following two conditions hold:

1) a morphism s of T is an isomorphism iff H_{U}^{n}(s) is an isomorphism for all n ∈ Z and

2) for each object X ∈ T , the canonical morphisms

Hom(?, X) → limHom(?, τ_{≤n}X) and Hom(X,?) → limHom(τ_{>n}X,?)

are surjective.

Let T^{b} be the full triangulated subcategory of T whose objects are the X ∈ T such that H_{U}^{n}(X) vanishes for all |n| ≫ 0. Let V^{b} be an aisle on T^{b}. Denote the associated t-structure on T^{b} by (V_{≤n}, V_{>n}), its heart by V_{0}, the homology functor by H_{V^{b}}^{n} : B → V_{0} and its truncation functors by (σ_{≤0}, σ_{>0}). Assume that there is an N ≫ 0 such that we have

H_{V^{b}}^{0} →^{∼} H_{V^{b}}^{0}τ_{>−n} and H_{V^{b}}^{0}τ_{≤n} →^{∼} H_{V^{b}}^{0}

for all n ≥ N. We define H_{V}^{0} : T → V_{0} by

H_{V}^{0}(X) = colimH_{V^{b}}^{0}τ_{>−n}τ_{≤m}X

and H_{V}^{n}(X) = H_{V}^{0}S^{n}X, n ∈ Z. We define V ⊂ T to be the full subcategory of T whose objects are the X ∈ T such that H_{V}^{n}(X) = 0 for all n > 0.

Proposition 1 a) V is an aisle in T and the associated t-structure is dominant.

b) If V^{b} is hereditary, i.e. each triangle

σ_{≤0}X → X → σ_{>0}X → Sσ_{≤0}X , X ∈ T^{b}

splits, then V is hereditary and each object X ∈ T is (non canonically) isomorphic to the sum of the S^{−n}H_{V}^{n}(X), n ∈ Z.

The proof is an exercise on t-structures which we leave to the reader.

7 Another construction of the triangulated hull of the orbit category

7.1 The construction

Let A be a finite-dimensional algebra of finite global dimension over a field k. Let X be an A-A-bimodule complex whose homology has finite total dimension. Let F be the functor

? ⊗^{L}_{A} X : D^{b}(modA) → D^{b}(modA).

We suppose that F is an equivalence and that for all L, M in D^{b}(modA), the group

Hom(L, F^{n}M)

vanishes for all but finitely many n ∈ Z. We will construct a triangulated category equivalent to the triangulated hull of section 5. Consider A as a dg algebra concentrated in degree 0. Let B be the dg algebra with underlying complex A ⊕ X[−1], where the multiplication is that of the trivial extension:

(a, x)(a′, x′) = (aa′, ax′ + xa′).

Let DB be the derived category of B and D^{b}(B) the bounded derived category, i.e. the full subcategory of DB formed by the dg modules whose homology has finite total dimension over k. Let per(B) be the perfect derived category of B, i.e. the smallest subcategory of DB containing B and stable under shifts, extensions and passage to direct factors. By our assumption on A and X, the perfect derived category is contained in D^{b}(B). The obvious morphism B → A induces a restriction functor D^{b}A → D^{b}B and by composition, we obtain a functor

D^{b}A → D^{b}B → D^{b}(B)/per(B)

Theorem 2 The category D^{b}(B)/per(B) is equivalent to the triangulated hull (cf. section 5) of the orbit category of D^{b}(A) under F and the above functor identifies with the projection functor.

Proof. If we replace X by a quasi-isomorphic bimodule, the algebra B is replaced by a quasi-isomorphic dg algebra and its derived category by an equivalent one. Therefore, it is no restriction of generality to assume, as we will do, that X is cofibrant as a dg A-A-bimodule. We will first compute morphisms in DB between dg B-modules whose restrictions to A are cofibrant. For this, let C be the dg submodule of the bar resolution of B as a bimodule over itself whose underlying graded module is

C = ∐_{n≥0} B ⊗_{A} X^{⊗_{A}n} ⊗_{A} B.

The bar resolution of B is a coalgebra in the category of dg B-B-bimodules (cf. e.g. [32]) and C becomes a dg subcoalgebra. Its counit is

ε : C → B ⊗_{A} B → B

and its comultiplication is given by

Δ(b_{0}, x_{1}, …, x_{n}, b_{n+1}) = ∑_{i=0}^{n}(b_{0}, x_{1}, …, x_{i}) ⊗ 1 ⊗ 1 ⊗ (x_{i+1}, …, b_{n+1}).

It is not hard to see that the inclusion of C in the bar resolution is an homotopy equivalence of left (and of right) dg B-modules. Therefore, the same holds for the counit ε : C → B. For an arbitrary right dg B-module L, the counit ε thus induces a quasi-isomorphism L ⊗_{B} C → L. Now suppose that the restriction of L to A is cofibrant. Then L⊗_{A}B⊗_{A}B is cofibrant over B and thus L⊗_{B} C → L is a cofibrant resolution of L. Let C_{1} be the dg category whose objects are the dg B-modules whose restriction to A is cofibrant and whose morphism spaces are the

Hom_{B}(L ⊗_{B} C, M ⊗_{B} C).

Let C_{2} be the dg category with the same objects as C_{1} and whose morphism spaces are

Hom_{B}(L ⊗_{B} C, M).

By definition, the composition of two morphisms f and g of C_{2} is given by

f ∘ (g ⊗ 1_{C}) ∘ (1_{L} ⊗ Δ).

We have a dg functor Φ : C_{2} → C_{1} which is the identity on objects and sends g : L → M to

(g ⊗ 1_{C}) ∘ (1_{L} ⊗ Δ) : L ⊗_{B} C → M ⊗_{B} C.

The morphism

Hom_{B}(L ⊗_{B} C, M) → Hom_{B}(L ⊗_{B} C, M ⊗_{B} C)

given by Φ is left inverse to the quasi-isomorphism induced by M ⊗_{B} C → M. Therefore, the dg functor Φ yields a quasi-isomorphism between C_{2} and C_{1} so that we can compute morphisms and compositions in DB using C_{2}. Now suppose that L and M are cofibrant dg A-modules. Consider them as dg Bmodules via restriction along the projection B → A. Then we have natural isomorphisms of complexes

Hom_{C_{2}}(L, M) = Hom_{B}(L ⊗_{B} C, M) →^{∼} Hom_{A}(∐_{n≥0} L ⊗_{A} X^{⊗_{A}n}, M).

Moreover, the composition of morphisms in C_{2} translates into the natural composition law for the right hand side. Now we will compute morphisms in the quotient category DB/per(B). Let M be as above. For p ≥ 0, let C_{≤p} be the dg subbimodule with underlying graded module

∐_{n=0}^{p} B ⊗_{A} X^{⊗_{A}n} ⊗_{A} B.

Then each morphism

P → M ⊗_{B} C

of DB from a perfect object P factors through

M ⊗_{B} C_{≤p} → M ⊗_{B} C

for some p ≥ 0. Therefore, the following complex computes morphisms in DB/per(B):

colim_{p≥1}Hom_{B}(L ⊗_{B} C, M ⊗_{B} C/C_{≤p−1}).

Now it is not hard to check that the inclusion

M ⊗ X^{⊗_{A}p} ⊗_{A} A → M ⊗_{B} (C/C_{≤p−1})

is a quasi-isomorphism of dg B-modules. Thus we obtain quasi-isomorphisms

Hom_{B}(L ⊗_{B} C, M ⊗ X^{⊗_{A}p} ⊗_{A} A) → Hom_{B}(L ⊗_{B} C, M ⊗_{B} C/C_{≤p−1})

and

∏_{n≥0} Hom_{A}(L ⊗_{A} X^{⊗_{A}n}, M ⊗_{A} X^{⊗_{A}p}) → Hom_{B}(L ⊗_{B} C, M ⊗_{B} C/C_{≤p−1}).

Moreover, it is not hard to check that if we define transition maps

∏_{n≥0} Hom_{A}(L⊗_{A}X^{⊗_{A}n}, M ⊗_{A}X^{⊗_{A}p}) → ∏_{n≥0} Hom_{A}(L⊗_{A}X^{⊗_{A}n}, M ⊗_{A}X^{⊗_{A}(p+1)})

by sending f to f ⊗_{A} 1_{X}, then we obtain a quasi-isomorphism of direct systems of complexes. Therefore, the following complex computes morphisms in DB/per(B):

colim_{p≥1} ∏_{n≥0} Hom_{A}(L ⊗_{A} X^{⊗_{A}n}, M ⊗_{A} X^{⊗_{A}p}).

Let C_{3} be the dg category whose objects are the cofibrant dg A-modules and whose morphisms are given by the above complexes. If L and M are cofibrant dg A-modules and belong to D^{b}(modA), then, by our assumptions on F, this complex is quasi-isomorphic to its subcomplex

colim_{p≥1} ∐_{n≥0} Hom_{A}(L ⊗_{A} X^{⊗_{A}n}, M ⊗_{A} X^{⊗_{A}p}).

Thus we obtain a dg functor

B → C_{3}

(where B is the dg category defined in 5.1) which induces a fully faithful functor H^{0}(B) → DB/per(B) and thus a fully faithful functor M → D^{b}(B)/per(B). This functor is also essentially surjective. Indeed, every object in D^{b}(B) is an extension of two objects which lie in the image of D^{b}(modA). The assertion about the projection functor is clear from the above proof.

7.2 The motivating example

Let us suppose that the functor F is given by

M ↦ τS^{−1}M ,

where τ is the Auslander-Reiten translation of D^{b}(A) and S the shift functor. This is the case considered in [8] for the construction of the cluster category. The functor F^{−1} is isomorphic to

M ↦ S^{−2}νM

where ν is the Nakayama functor

ν =? ⊗^{L}_{A} DA , DA = Hom_{k}(A, k).

Thus F^{−1} is given by the bimodule X = (DA)[−2] and B = A ⊕ (DA)[−3] is the trivial extension of A with a non standard grading: A is in degree 0 and DA in degree 3. For example, if A is the quiver algebra of an alternating quiver whose underlying graph is A_{n}, then the underlying ungraded algebra of B is the quadratic dual of the preprojective algebra associated with A_{n}, cf. [15]. The algebra B viewed as a differential graded algebra was investigated by Khovanov-Seidel in [36]. Here the authors show that D^{b}(B) admits a canonical action by the braid group on n+1 strings, a result which was obtained independently in a similar context by Zimmermann-Rouquier [52]. The canonical generators of the braid group act by triangle functors T_{i} endowed with morphisms ϕ_{i} : T_{i} → 1. The cone on each ϕ_{i} belongs to per(B) and per(B) is in fact equal to its smallest triangulated subcategory stable under direct factors and containing these cones. Thus, the action becomes trivial in D^{b}(B)/per(B) and in a certain sense, this is the largest quotient where the ϕ_{i} become invertible.

7.3 Projectives over the preprojective algebra

Let A be the path algebra of a Dynkin quiver, i.e. a quiver whose underlying graph is a Dynkin diagram of type A, D or E. Let C be the associated preprojective algebra [26], [20], [50]. In Proposition 3.3 of [7], Auslander-Reiten show that the category of projective modules over C is equivalent to the stable category of maximal Cohen-Macaulay modules over a representation-finite isolated hypersurface singularity. In particular, it is triangulated. This can also be deduced from our main theorem: Indeed, it follows from D. Happel’s description [27] of the derived category D^{b}(A) that the category projC of finite dimensional projective C-modules is equivalent to the orbit category D^{b}(A)/τ, cf. also [25]. Moreover, by the theorem of the previous section, we have an equivalence

projC →^{∼} D^{b}(B)/per(B)

where B = A ⊕ (DA)[−2]. This equivalence yields in fact more than just a triangulated structure: it shows that projC is endowed with a canonical Hochschild 3-cocycle m_{3}, cf. for example [11]. It would be interesting to identify this cocycle in the description given in [23].

7.4 Projectives over Λ(L_{n})

The category of projective modules over the algebra k[ε]/(ε^{2}) of dual numbers is triangulated. Indeed, it is equivalent to the orbit category of the derived category of the path algebra of a quiver of type A_{2} under the Nakayama autoequivalence ν. Thus, we obtain examples of triangulated categories whose Auslander-Reiten quiver contains a loop. It has been known since Riedtmann’s work [49] that this cannot occur in the stable category (cf. below) of a selfinjective finite-dimensional algebra. It may therefore seem surprising, cf. [58], that loops do occur in this more general context. However, loops already do occur in stable categories of finitely generated reflexive modules over certain non commutative generalizations of local rings of rational double points, as shown by Auslander-Reiten in [6]. These were completely classified by Reiten-Van den Bergh in [46]. In particular, the example of the dual numbers and its generalization below are among the cases covered by [46]. The example of the dual numbers generalizes as follows: Let n ≥ 1 be an integer. Following [30], the generalized Dynkin graph L_{n} is defined as the graph

1  2  ⋯  n − 1  n

Its edges are in natural bijection with the orbits of the involution which exchanges each arrow α with \overline{α} in the following quiver:

[figure: 1; ε=\overline{ε}; a_{1}; 2; \overline{a_{1}}; a_{2}; \overline{a_{2}}; ⋯; a_{n−2}; n − 1; \overline{a_{n−2}}; a_{n−1}; n; \overline{a_{n−1}}] .

The associated preprojective algebra Λ(L_{n}) of generalized Dynkin type L_{n} is defined as the quotient of the path algebra of this quiver by the ideal generated by the relators

r_{v} = ∑ α \overline{α} ,

where, for each 1 ≤ v ≤ n, the sum ranges over the arrows α with starting point v. Let A be the path algebra of a Dynkin quiver with underlying Dynkin graph A_{2n}. Using D. Happel’s description [27] of the derived category of a Dynkin quiver, we see that the orbit category D^{b}(A)/(τ^{n}S) is equivalent to the category of finitely generated projective modules over the algebra Λ(L_{n}). By the main theorem, this category is thus triangulated. Its Auslander-Reiten quiver is given by the ordinary quiver of Λ(L_{n}), cf. above, endowed with τ = 1: Indeed, in D^{b}(A), we have S^{2} = τ^{−(2n+1)} so that in the orbit category, we obtain

1 = (τ^{n}S)^{2} = τ^{2n}S^{2} = τ^{−1}.

8 On the Calabi-Yau property

8.1 Serre functors and localizations

Let k be a field and T a k-linear triangulated category with finite-dimensional Hom-spaces. We denote the suspension functor of T by S. Recall from [47] that a right Serre functor for T is the datum of a triangle functor ν : T → T together with bifunctor isomorphisms

DHom_{T} (X,?) →^{∼} Hom_{T} (?, νX) , X ∈ T ,

where D = Hom_{k}(?, k). If ν exists, it is unique up to isomorphism of triangle functors. Dually, a left Serre functor is the datum of a triangle functor ν′ : T → T and isomorphisms

DHom_{T} (?, X) →^{∼} Hom_{T} (ν′,?) , X ∈ T .

The category T has Serre duality if it has both a left and a right Serre functor, or equivalently, if it has a onesided Serre functor which is an equivalence, cf.

[47] [13]. The following lemma is used in [9].

Lemma 1 Suppose that T has a left Serre functor ν′. Let U ⊂ T be a thick triangulated subcategory and L : T → T /U the localization functor.

a) If L admits a right adjoint R, then Lν′R is a left Serre functor for T /U. b) More generally, if the functor ν′ : T → T admits a total right derived

functor Rν′ : T /U → T /U in the sense of Deligne [19] with respect to the localization T → T /U, then Rν′ is a left Serre functor for T /U.

Proof. a) For X, Y in T , we have

Hom_{T /U}(Lν′RX, Y) = Hom_{T} (ν′RX, RY)
= DHom_{T} (RY, RX) = DHom_{T /U}(Y, X).

Here, for the last isomorphism, we have used that R is fully faithful (L is a localization functor).

b) We assume that T is small. Let ModT denote the (large) category of functors from T^{op} to the category of abelian groups and let h : T → ModT denote the Yoneda embedding. Let L^{∗} be the unique right exact functor ModT → Mod(T /U) which sends hX to hLX, X ∈ T . By the calculus of (right) fractions, L^{∗} has a right adjoint R which takes an object Y to

colim_{Σ_{Y}} hY′ ,

where the colim ranges over the category Σ_{Y} of morphisms s : Y → Y′ which become invertible in T /U. Clearly L^{∗}R is isomorphic to the identity so that R is fully faithful. By definition of the total right derived functor, for each object X ∈ T /U, the functor

colim_{Σ_{X}} h(Lν′X′) = L^{∗}ν^{′∗}Rh(X)

is represented by Rν′(X). Therefore, we have

Hom_{T /U}(Rν′(X), Y) = Hom_{ModT /U}(L^{∗}ν^{′∗}Rh(X), h(Y))
= Hom_{ModT} (ν^{′∗}Rh(X), Rh(Y)).

Now by definition, the last term is isomorphic to

Hom_{ModT} (colim_{Σ_{X}} h(ν′X′), colim_{Σ_{Y}} hY′) = lim_{Σ_{X}} colim_{Σ_{Y}} Hom_{T} (ν′X′, Y′)

and this identifies with

lim_{Σ_{X}} colim_{Σ_{Y}} DHom_{T} (Y′, X′) = D(colim_{Σ_{X}} lim_{Σ_{Y}} Hom_{T} (Y′, X′))
= DHom_{T /U}(Y, X).

8.2 Definition of the Calabi-Yau property

Keep the hypotheses of the preceding section. By definition [39], the triangulated category T is Calabi-Yau of CY-dimension d if it has Serre duality and there is an isomorphism of triangle functors

ν →^{∼} S^{d}.

By extension, if we have ν^{e} →^{∼} S^{d} for some integer e > 0, one sometimes says that T is Calabi-Yau of fractional dimension d/e. Note that d ∈ Z is only determined up to a multiple of the order of S. It would be interesting to link the CY-dimension to Rouquier’s [51] notion of dimension of a triangulated category. The terminology has its origin in the following example: Let X be a smooth projective variety of dimension d and let ω_{X} = Λ^{d}T_{X}^{∗} be the canonical bundle. Let T be the bounded derived category of coherent sheaves on X. Then the classical Serre duality

DExt_{X}^{i}(F, G) →^{∼} Ext^{d−i}(G, F ⊗ ω_{X}) ,

where F, G are coherent sheaves, lifts to the isomorphism

DHom_{T} (F, G) →^{∼} Hom_{T} (G, F ⊗ ω_{X}[d]) ,

where F, G are bounded complexes of coherent sheaves. Thus T has Serre duality and ν =? ⊗ ω_{X}[d]. So the category T is Calabi-Yau of CY-dimension d iff ω_{X} is isomorphic to O_{X}, which means precisely that the variety X is Calabi-Yau of dimension d. If T is a Calabi-Yau triangulated category ‘of algebraic origin’ (for example, the derived category of a category of modules or sheaves), then it often comes from a Calabi-Yau A_{∞}-category. These are of considerable interest in mathematical physics, since, as Kontsevich shows [38], [37], cf. also [18], a topological quantum field theory is associated with each Calabi-Yau A_{∞}-category satisfying some additional assumptions^{1}.

8.3 Examples

(1) If A is a finite-dimensional k-algebra, then the homotopy category T of bounded complexes of finitely generated projective A-modules has a Nakayama functor iff DA is of finite projective dimension. In this case, the category T has Serre duality iff moreover A_{A} is of finite injective dimension, i.e. iff A is Gorenstein, cf. [28]. Then the category T is Calabi-Yau (necessarily of CYdimension 0) iff A is symmetric.

(2) If Δ is a Dynkin graph of type A_{n}, D_{n}, E_{6}, E_{7} or E_{8} and h is its Coxeter number (i.e. n + 1, 2(n − 1), 12, 18 or 30, respectively), then for the bounded derived category of finitely generated modules over a quiver with underlying graph Δ, we have isomorphisms

ν^{h} = (Sτ)^{h} = S^{h}τ^{h} = S^{(h−2)}.

1Namely, the associated triangulated category should admit a generator whose endomorphism A_{∞}-algebra B is compact (i.e. finite-dimensional), smooth (i.e. B is perfect as a bimodule over itself), and whose associated Hodge-de Rham spectral sequence collapses (this property is conjectured to hold for all smooth compact A_{∞}-algebras over a field of characteristic 0).

Documenta Mathematica 10 (2005) 551–581

Hence this category is Calabi-Yau of fractional dimension (h − 2)/h.

(3) Suppose that A is a finite-dimensional algebra which is selfinjective (i.e. A_{A} is also an injective A-module). Then the category modA of finite-dimensional A-modules is Frobenius, i.e. it is an abelian (or, more generally, an exact) category with enough projectives, enough injectives and where an object is projective iff it is injective. The stable category \underline{mod} A obtained by quotienting modA by the ideal of morphisms factoring through injectives is triangulated, cf. [27]. The inverse of its suspension functor sends a module M to the kernel ΩM of an epimorphism P → M with projective P. Let N M = M ⊗_{A} DA. Then \underline{mod} A has Serre duality with Nakayama functor ν = Ω ∘ N. Thus, the stable category is Calabi-Yau of CY-dimension d iff we have an isomorphism of triangle functors

Ω^{(d+1)} ∘ N = 1.

For this, it is clearly sufficient that we have an isomorphism

Ω_{A^{e}}^{d+1}(A) ⊗_{A} DA →^{∼} A ,

in the stable category of A-A-bimodules, i.e. modules over the selfinjective algebra A ⊗ A^{op}. For example, we deduce that if A is the path algebra of a cyclic quiver with n vertices divided by the ideal generated by all paths of length n − 1, then \underline{mod} A is Calabi-Yau of CY-dimension 3.

(4) Let A be a dg algebra. Let per(A) ⊂ D(A) be the subcategory of perfect dg A-modules, i.e. the smallest full triangulated subcategory of D(A) containing A and stable under forming direct factors. For each P in per(A) and each M ∈ D(A), we have canonical isomorphisms

DRHom_{A}(P, M) →^{∼} RHom_{A}(M, DRHom_{A}(P, A))

and

P ⊗^{L}_{A} DA →^{∼} DRHom_{A}(P, A).

So we obtain a canonical isomorphism

DRHom_{A}(P, M) →^{∼} RHom_{A}(M, P ⊗^{L}_{A} DA).

Thus, if we are given a quasi-isomorphism of dg A-A-bimodules

ϕ : A[n] → DA ,

we obtain

DRHom_{A}(P, M) →^{∼} RHom_{A}(M, P [n])

and in particular per(A) is Calabi-Yau of CY-dimension n.

(5) To consider a natural application of the preceding example, let B be the symmetric algebra on a finite-dimensional vector space V of dimension n and T ⊂ D(B) the localizing subcategory generated by the trivial B-module k (i.e. the smallest full triangulated subcategory stable under infinite sums and containing the trivial module). Let T^{c} denote its subcategory of compact objects. This is exactly the triangulated subcategory of D(B) generated by k, and also exactly the subcategory of the complexes whose total homology is finite-dimensional and supported in 0. Then T^{c} is Calabi-Yau of CY-dimension n. Indeed, if

A = RHom_{B}(k, k)

is the Koszul dual of B (thus, A is the exterior algebra on the dual of V concentrated in degree 1; it is endowed with d = 0), then the functor

RHom_{B}(k,?) : D(B) → D(A)

induces equivalences from T to D(A) and T^{c} to per(A), cf. for example [32]. Now we have a canonical isomorphism of A-A-bimodules A[n] →^{∼} DA so that per(A) and T are Calabi-Yau of CY-dimension n. As pointed out by I. Reiten, in this case, the Calabi-Yau property even holds more generally: Let M ∈ D(B) and denote by M_{T} → M the universal morphism from an object of T to M. Then, for X ∈ T^{c}, we have natural morphisms

Hom_{D(B)}(M, X[n]) → Hom_{T} (M_{T} , X[n]) →^{∼} DHom_{T} (X, M_{T})
→^{∼} DHom_{D(B)}(X, M).

The composition

(∗) Hom_{D(B)}(M, X[n]) → DHom_{D(B)}(X, M)

is a morphism of (co-)homological functors in X ∈ T^{c} (resp. M ∈ D(B)). We claim that it is an isomorphism for M ∈ per(B) and X ∈ T^{c}. It suffices to prove this for M = B and X = k. Then one checks it using the fact that

RHom_{B}(k, B) →^{∼} k[−n].

These arguments still work for certain non-commutative algebras B: If B is an Artin-Schelter regular algebra [2] [1] of global dimension 3 and type A and T the localizing subcategory of the derived category D(B) of non graded Bmodules generated by the trivial module, then T^{c} is Calabi-Yau and one even has the isomorphism (∗) for each perfect complex of B-modules M and each X ∈ T^{c}, cf. for example section 12 of [41].

8.4 Orbit categories with the Calabi-Yau property

The main theorem yields the following

Corollary 1 If d ∈ Z and Q is a quiver whose underlying graph is Dynkin of type A, D or E, then

T = D^{b}(kQ)/τ^{−1}S^{d−1}

is Calabi-Yau of CY-dimension d. In particular, the cluster category C_{kQ} is Calabi-Yau of dimension 2 and the category of projective modules over the preprojective algebra Λ(Q) is Calabi-Yau of CY-dimension 1.

The category of projective modules over the preprojective algebra Λ(L_{n}) of example 7.4 does not fit into this framework. Nevertheless, it is also Calabi-Yau of CY-dimension 1, since we have τ = 1 in this category and therefore ν = Sτ = S.

8.5 Module categories over Calabi-Yau categories

Calabi-Yau triangulated categories turn out to be ‘self-reproducing’: Let T be a triangulated category. Then the category modT of finitely generated functors from T^{op} to Modk is abelian and Frobenius, cf. [24], [42]. If we denote by Σ the exact functor modT → modT which takes Hom(?, X) to Hom(?, SX), then it is not hard to show [24] [25] that we have

Σ →^{∼} S^{3}

as triangle functors \underline{mod}T → \underline{mod}T . One deduces the following lemma, which is a variant of a result which Auslander-Reiten [7] obtained using dualizing R-varieties [4] and their functor categories [3], cf. also [5] [45]. A similar result is due to Geiss [25].

Lemma 2 If T is Calabi-Yau of CY-dimension d, then the stable category \underline{mod} T is Calabi-Yau of CY-dimension 3d − 1. Moreover, if the suspension of T is of order n, the order of the suspension functor of \underline{mod} T divides 3n.

For example, if A is the preprojective algebra of a Dynkin quiver or equals Λ(L_{n}), then we find that the stable category \underline{mod} A is Calabi-Yau of CYdimension 3 × 1 − 1 = 2. This result, with essentially the same proof, is due to Auslander-Reiten [7]. For the preprojective algebras of Dynkin quivers, it also follows from a much finer result due to Ringel and Schofield (unpublished). Indeed, they have proved that there is an isomorphism

Ω_{A^{e}}^{3}(A) →^{∼} DA

in the stable category of bimodules, cf. Theorems 4.8 and 4.9 in [15]. This implies the Calabi-Yau property since we also have an isomorphism

DA ⊗_{A} DA →^{∼} A

in the stable category of bimodules, by the remark following definition 4.6 in [15]. For the algebra Λ(L_{n}), the analogous result follows from Proposition 2.3 of [12].

9 Universal properties

9.1 The homotopy category of small dg categories

Let k be a field. A differential graded (=dg) k-module is a Z-graded vector space

V = ⨁_{p∈Z} V^{p}

endowed with a differential d of degree 1. The tensor product V ⊗ W of two dg k-modules is the graded space with components

⨁_{p+q=n} V^{p} ⊗ W^{q} , n ∈ Z ,

and the differential d ⊗ 1 + 1 ⊗ d, where the tensor product of maps is defined using the Koszul sign rule. A dg category [32] [21] is a k-category A whose morphism spaces are dg k-modules and whose compositions

A(Y, Z) ⊗ A(X, Y) → A(X, Z)

are morphisms of dg k-modules. For a dg category A, the category H^{0}(A) has the same objects as A and has morphism spaces H^{0}A (X, Y), X, Y ∈ A. A dg functor F : A → B between dg categories is a functor compatible with the grading and the differential on the morphism spaces. It is a quasi-equivalence if it induces quasi-isomorphisms in the morphism spaces and an equivalence of categories from H^{0}(A) to H^{0}(B). We denote by dgcat the category of small dg categories. The homotopy category of small dg categories is the localization Ho(dgcat) of dgcat with respect to the class of quasi-equivalences. According to [56], the category dgcat admits a structure of Quillen model category (cf. [22], [31]) whose weak equivalences are the quasi-equivalences. This implies in particular that for A, B ∈ dgcat, the morphisms from A to B in the localization Ho(dgcat) form a set.

9.2 The bimodule bicategory

For two dg categories A, B, we denote by rep(A, B) the full subcategory of the derived category D(A^{op} ⊗ B), cf. [32], whose objects are the dg A-B-bimodules X such that X(?, A) is isomorphic to a representable functor in D(B) for each object A of A. We think of the objects of rep(A, B) as ‘representations up to homotopy’ of A in B. The bimodule bicategory rep, cf. [32] [21], has as objects all small dg categories; the morphism category between two objects A, B is rep(A, B); the composition bifunctor

rep(B, C) × rep(A, B) → rep(A, B)

is given by the derived tensor product (X, Y) ↦ X ⊗^{L}_{B} Y. For each dg functor F : A → B, we have the dg bimodule

X_{F} : A ↦ B (?, F A) ,

which clearly belongs to rep(A, B). One can show that the map F ↦ X_{F} induces a bijection, compatible with compositions, from the set of morphisms from A to B in Ho(dgcat) to the set of isomorphism classes of bimodules X in rep(A, B). In fact, a much stronger result by B. Toën [57] relates rep to the Dwyer-Kan localization of dgcat.

9.3 Dg orbit categories

Let A be a small dg category and F ∈ rep(A, A). We assume, as we may, that F is given by a cofibrant bimodule. For a dg category B, define ẽff_{0}(A, F, B) to be the category whose objects are the pairs formed by an A-B-bimodule P in rep(A, B) and a morphism of dg bimodules

ϕ : P → P F.

Morphisms are the morphisms of dg bimodules f : P → P′ such that we have ϕ′ ∘f = (fF)∘ϕ in the category of dg bimodules. Define eff_{0}(A, F, B) to be the localization of ẽff_{0}(A, F, B) with respect to the morphisms f which are quasiisomorphisms of dg bimodules. Denote by eff(A, F, B) the full subcategory of eff_{0}(A, F, B) whose objects are the (P, ϕ) where ϕ is a quasi-isomorphism. It is not hard to see that the assignments

B ↦ eff_{0}(A, F, B) and B ↦ eff(A, F, B)

are 2-functors from rep to the category of small categories.

Theorem 3 a) The 2-functor eff_{0}(A, F,?) is 2-representable, i.e. there is small dg category B_{0} and a pair (P_{0}, ϕ_{0}) in eff(A, F, B_{0}) such that for each small dg category B, the functor

rep(B_{0}, B) → eff_{0}(A, F, B_{0}) , G ↦ G ∘ P_{0}

is an equivalence.

b) The 2-functor eff(A, F,?) is 2-representable.

c) For a dg category B, a pair (P, ϕ) is a 2-representative for eff_{0}(A, F,?) iff H^{0}(P) : H^{0}(A) → H^{0}(B) is essentially surjective and, for all objects A, B of A, the canonical morphism

⨁_{n∈N} A (F^{n}A, B) → B (P A, P B)

is invertible in D(k).

d) For a dg category B, a pair (P, ϕ) is a 2-representative for eff(A, F,?) iff H^{0}(P) : H^{0}(A) → H^{0}(B) is essentially surjective and, for all objects A, B of A, the canonical morphism

⨁_{c∈Z} colim_{r≫0}A (F^{p+r}A, F^{p+c+r}B) → B (P A, P B)

is invertible in D(k).

We define A/F to be the 2-representative of eff(A, F,?). For example, in the notations of 5.1, A/F is the dg orbit category B. It follows from part d) of the theorem that we have an equivalence

H^{0}(A)/H^{0}(F) → H^{0}(A/F).

Proof. We only sketch a proof and refer to [54] for a detailed treatment. Define B_{0} to be the dg category with the same objects as A and with the morphism spaces

B_{0} (A, B) = ⨁_{n∈N} A (F^{n}A, B).

We have an obvious dg functor P_{0} : A → B_{0} and an obvious morphism ϕ : P_{0} → P_{0}F. The pair (P_{0}, ϕ_{0}) is then 2-universal in rep. This yields a) and c). For b), one adjoins a formal homotopy inverse of ϕ to B_{0}. One obtains d) by computing the homology of the morphism spaces in the resulting dg category.

9.4 Functoriality in (A, F)

Let a square of rep

\xymatrix{A \ar[r]^{G} \ar[d]_{F} & A′ \ar[d]^{F′} \\ A \ar[r]^{G} & A′}

be given and an isomorphism

γ : F′G → GF

of rep(A, A′). We assume, as we may, that A and A′ are cofibrant in dgcat and that F, F′ and G are given by cofibrant bimodules. Then F′G is a cofibrant bimodule and so γ : F′G → GF lifts to a morphism of bimodules

γ̃ : F′G → GF.

If B is another dg category and (P, ϕ) an object of eff_{0}(A′, F′, B), then the composition

\xymatrix{P G \ar[r]^{ϕG} & P F′G \ar[r]^{P_{γ}} & P GF}

yields an object P G → P GF of eff_{0}(A, F, B). Clearly, this assignment extends to a functor, which induces a functor

eff(A′, F′, B) → eff(A, F, B).

By the 2-universal property of section 9.3, we obtain an induced morphism

\overline{G} : A/F → A′/F′.

One checks that the composition of two pairs (G, γ) and (G′, γ′) induces a functor isomorphic to the composition of \overline{G} with \overline{G′}.

9.5 The bicategory of enhanced triangulated categories

We refer to [33, 2.1] for the notion of an exact dg category. We also call these categories pretriangulated since if A is an exact dg category, then H^{0}(A) is triangulated. More precisely, E = Z^{0}(A) is a Frobenius category and H^{0}(A) is its associated stable category \underline{E} (cf. example (3) of section 8.3 for these notions). The inclusion of the full subcategory of (small) exact dg categories into Ho(dgcat) admits a left adjoint, namely the functor A ↦ pretr(A) which maps a dg category to its ‘pretriangulated hull’ defined in [14], cf. also [33, 2.2]. More precisely, the adjunction morphism A → pretr(A) induces an equivalence of categories

rep (pretr(A), B) → rep(A, B)

for each exact dg category B, cf. [55]. The bicategory enh of enhanced [14] triangulated categories, cf. [32] [21], has as objects all small exact dg categories; the morphism category between two objects A, B is rep(A, B); the composition bifunctor

rep(B, C) × rep(A, B) → rep(A, C)

is given by the derived tensor product (X, Y) ↦ X ⊗^{L}_{B} Y.

9.6 Exact dg orbit categories

Now let A be an exact dg category and F ∈ rep(A, A). Then A/F is the dg orbit category of subsection 5.1 and pretr(A/F) is an exact dg category such that H^{0}pretr(A/F) is the triangulated hull of section 5. In particular, we obtain that the triangulated hull is the stable category of a Frobenius category. From the construction, we obtain the universal property:

Theorem 4 For each exact dg category B, we have an equivalence of categories

rep(pretr(A/F), B) → eff(F, B).

9.7 An example

Let A be a finite-dimensional algebra of finite global dimension and T A the trivial extension algebra, i.e. the vector space A ⊕ DA endowed with the multiplication defined by

(a, f)(b, g) = (ab, ag + fb) , (a, f), (b, g) ∈ T A ,

and the grading such that A is in degree 0 and DA in degree 1. Let F : D^{b}(A) → D^{b}(A) equal τS^{2} = νS and let F̃ be the dg lift of F given by ? ⊗_{A} R[1], where R is a projective bimodule resolution of DA. Let D^{b}(A)_{dg} denote a dg category quasi-equivalent to the dg category of bounded complexes of finitely generated projective A-modules.

Theorem 5 The following are equivalent

(i) The k-category D^{b}(A)/F is naturally equivalent to its ‘triangulated hull’

H^{0}(pretr(D^{b}(A)_{dg}/F̃)).

(ii) Each finite-dimensional T A-module admits a grading.

Proof. We have a natural functor

modA → grmodT A

given by viewing an A-module as a graded T A-module concentrated in degree 0. As shown by D. Happel [27], cf. also [34], this functor extends to a triangle equivalence Φ from D^{b}(A) to the stable category \underline{grmod} T A, obtained from grmodT A by killing all morphisms factoring through projective-injectives. We would like to show that we have an isomorphism of triangle functors

Φ ∘ τS^{2} →^{∼} Σ ∘ Φ

where Σ is the grading shift functor for graded T A-modules: (ΣM)^{p} = M^{p+1}

for all p ∈ Z. ¿From [27], we know that τS →^{∼} ν, where ν =? ⊗^{L}_{A} DA. Thus it remains to show that

Φ ∘ νS →^{∼} Σ ∘ Φ.

As shown in [34], the equivalence Φ is given as the composition

D^{b}(modA) →  D^{b}(grmodT A) →
D^{b}(grmodT A)/per(grmodT A) → \underline{grmod} T A ,

where the first functor is induced by the above inclusion, the notation per(grmodT A) denotes the triangulated subcategory generated by the projective-injective T A-modules and the last functor is the ‘stabilization functor’ cf. [34]. We have a short exact sequence of graded T A-modules

0 → Σ^{−1}(DA) → T A → A → 0.

We can also view it as a sequence of left A and right graded T A-modules. Let P be a bounded complex of projective A-modules. Then we obtain a short exact sequence of complexes of graded T A-modules

0 → Σ^{−1}(P ⊗_{A} DA) → P ⊗_{A} T A → P → 0

functorial in P. It yields a functorial triangle in D^{b}(grmodA). The second term belongs to per(grmodT A). Thus in the quotient category

D^{b}(grmodT A)/per(grmodT A) ,

the triangle reduces to a functorial isomorphism

P →^{∼} SΣ^{−1}νP.

Thus we have a functorial isomorphism

Φ(P) →^{∼} SΣ^{−1}Φ(νP).

Since A is of finite global dimension, D^{b}(modA) is equivalent to the homotopy category of bounded complexes of finitely generated projective A-modules. Thus we get the required isomorphism

ΣΦ →^{∼} ΦSν.

More precisely, one can show that \underline{grmod} T A has a canonical dg structure and that there is an isomorphism

(D^{b}(A))_{dg} →^{∼} (\underline{grmod} T A)_{dg}

in the homotopy category of small dg categories which induces Happel’s equivalence and under which Σ corresponds to the lift F̃ of F = Sν. Hence the orbit categories D^{b}(modA)/τS^{2} and \underline{grmod} T A/Σ are equivalent and we are reduced to determining when \underline{grmod} T A/Σ is naturally equivalent to its triangulated hull. Clearly, we have a full embedding

\underline{grmod} T A/Σ → \underline{mod} T A

and its image is formed by the T A-modules which admit a grading. Now \underline{mod} T A is naturally equivalent to the triangulated hull. Therefore, condition (i) holds iff the embedding is an equivalence iff each finite-dimensional T Amodule admits a grading. In [53], A. Skowroński has produced a class of examples where condition (ii) does not hold. The simplest of these is the algebra A given by the quiver

\xymatrix{ & • \ar[dl]_{β} \ar[dd] \\ • \ar[dr]_{α} \\  & •}

with the relation αβ = 0. Note that this algebra is of global dimension 2.

9.8 Exact categories and standard functors

Let E be a small exact k-category. Denote by C^{b}(E) the category of bounded complexes over E and by Ac^{b}(E) its full subcategory formed by the acyclic bounded complexes. The categories with the same objects but whose morphisms are given by the morphism complexes are denoted respectively by C^{b}(E)_{dg} and Ac^{b}(E)_{dg}. They are exact dg categories and so is the dg quotient [33] [21]

D^{b}(E)_{dg} = C^{b}(E)_{dg}/Ac^{b}(E)_{dg}.

Let E′ be another small exact k-category. We call a triangle functor F : D^{b}(E) → D^{b}(E′) a standard functor if it is isomorphic to the triangle functor induced by a morphism

F̃ : D^{b}(E)_{dg} → D^{b}(E′)_{dg}

of Ho(dgcat). Slightly abusively, we then call F̃ a dg lift of F. Each exact functor E → E′ yields a standard functor; a triangle functor is standard iff it admits a lift to an object of rep(D^{b}(E)_{dg}, D^{b}(E′)_{dg}); compositions of standard functors are standard; an adjoint (and in particular, the inverse) of a standard functor is standard. If F : D^{b}(E) → D^{b}(E) is a standard functor with dg lift F̃, we have the dg orbit category D^{b}(E)_{dg}/F̃ and its pretriangulated hull

D^{b}(E)_{dg}/F̃ → pretr(D^{b}(E)_{dg}/F̃).

The examples in section 3 show that this functor is not an equivalence in general.

9.9 Hereditary categories

Now suppose that H is a small hereditary abelian k-category with the Krull-Schmidt property (indecomposables have local endomorphism rings and each object is a finite direct sum of indecomposables) where all morphism and extension spaces are finite-dimensional. Let

F : D^{b}(H) → D^{b}(H)

be a standard functor with dg lift F̃.

Theorem 6 Suppose that F satisfies assumptions 2) and 3) of the main theorem in section 4. Then the canonical functor

D^{b}(H)/F → H^{0}(pretr(D^{b}(H)_{dg}/F̃))

is an equivalence of k-categories. In particular, the orbit category D^{b}(H)/F admits a triangulated structure such that the projection functor becomes a triangle functor.

The proof is an adaptation, left to the reader, of the proof of the main theorem.

Suppose for example that D^{b}(H) has a Serre functor ν. Then ν is a standard functor since it is induced by the tensor product with bimodule

(A, B) ↦ DHom_{D^{b}(H)_{dg}}(B, A) ,

where D = Hom_{k}(?, k). The functor τ^{−1} = Sν^{−1} induces equivalences I →^{∼} SP and H_{ni} → H_{np}, where P is the subcategory of projectives, I the subcategory of injectives, H_{np} the subcategory of objects without a projective direct summand and H_{ni} the subcategory of objects without an injective direct summand. Now let n ≥ 2 and consider the autoequivalence F = S^{n}ν^{−1} = S^{n−1}τ^{−1} of D^{b}(H). Clearly F is standard. It is not hard to see that F satisfies the hypotheses 2) and 3) of the main theorem in section 4. Thus the orbit category

D^{b}(H)/F = D^{b}(H)/S^{n}ν^{−1}

is triangulated. Note that we have excluded the case n = 1 since the hypotheses 2) and 3) are not satisfied in this case, in general, as we see from the last example in section 3.

References

[1] M. Artin, J. Tate, and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 33–85.

[2] Michael Artin and William F. Schelter, Graded algebras of global dimension 3, Adv. in Math. 66 (1987), no. 2, 171–216.

[3] Maurice Auslander and Idun Reiten, Stable equivalence of Artin algebras, Proceedings of the Conference on Orders, Group Rings and Related Topics (Ohio State Univ., Columbus, Ohio, 1972) (Berlin), Springer, 1973, pp. 8– 71. Lecture Notes in Math., Vol. 353.

[4] , Stable equivalence of dualizing R-varieties, Advances in Math. 12 (1974), 306–366.

[5] , Stable equivalence of dualizing R-varieties. II. Hereditary dualizing R-varieties, Advances in Math. 17 (1975), no. 2, 93–121.

[6] , Almost split sequences for rational double points, Trans. Amer. Math. Soc. 302 (1987), no. 1, 87–97.

[7] , DTr-periodic modules and functors, Representation theory of algebras (Cocoyoc, 1994), CMS Conf. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 1996, pp. 39–50.

[8] Aslak Bakke Buan, Robert J. Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov, Tilting theory and cluster combinatorics, Advances in Mathematics, to appear.

[9] Aslak Bakke Buan, Robert J. Marsh, and Idun Reiten, Cluster-tilted algebras, Trans. Amer. Math. Soc., to appear.

[10] Alexander A. Beilinson, Joseph Bernstein, and Pierre Deligne, Analyse et topologie sur les espaces singuliers, Astérisque, vol. 100, Soc. Math. France, 1982 (French).

[11] David Benson, Henning Krause, and Stefan Schwede, Realizability of modules over Tate cohomology, Trans. Amer. Math. Soc. 356 (2004), no. 9, 3621–3668 (electronic).

[12] Jerzy Białkowski, Karin Erdmann, and Andrzej Skowroński, Deformed preprojective algebras of generalized Dynkin type, Preprint, 2004.

[13] A. Bondal and M. van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), no. 1, 1–36, 258.

[14] A. I. Bondal and M. M. Kapranov, Enhanced triangulated categories, Math. USSR Sbornik 70 (1991), 93–107.

[15] Sheila Brenner, Michael C. R. Butler, and Alastair D. King, Periodic algebras which are almost Koszul, Algebr. Represent. Theory 5 (2002), no. 4, 331–367.

[16] Philippe Caldero, Frédéric Chapoton, and Ralf Schiffler, Quivers with relations arising from clusters (A_{n} case), arXiv:math.RT/0401316.

[17] Claude Cibils and Eduardo Marcos, Skew category, Galois covering and smash product of a k-category, arXiv:math.RA/0312214, to appear in the Proceedings of the AMS.

[18] Kevin Costello, Topological conformal field theories and Calabi-Yau categories, arXiv:math.QA/0412149.

[19] Pierre Deligne, Cohomologie à supports propres, pp. vi+640, Springer-Verlag, Berlin, 1973, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat, Lecture Notes in Mathematics, Vol. 305.

[20] Vlastimil Dlab and Claus Michael Ringel, The preprojective algebra of a modulated graph, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 832, Springer, Berlin, 1980, pp. 216–231.

[21] Vladimir Drinfeld, DG quotients of DG categories, J. Algebra 272 (2004), no. 2, 643–691.

[22] W. G. Dwyer and J. Spaliński, Homotopy theories and model categories, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 73–126.

[23] Karin Erdmann and Nicole Snashall, On Hochschild cohomology of preprojective algebras. I, II, J. Algebra 205 (1998), no. 2, 391–412, 413–434.

[24] Peter Freyd, Stable homotopy, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), Springer, New York, 1966, pp. 121–172.

[25] Christof Geiss, The universal cover of a preprojective algebra, Talk at the 11th International Conference on Representations of Algebras, Pátzcuaro, August 2004.

[26] I. M. Gel′fand and V. A. Ponomarev, Model algebras and representations of graphs, Funktsional. Anal. i Prilozhen. 13 (1979), no. 3, 1–12.

[27] Dieter Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), no. 3, 339–389.

[28] Dieter Happel, On Gorenstein algebras, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 389–404.

[29] Dieter Happel, Udo Preiser, and Claus Michael Ringel, Binary polyhedral groups and Euclidean diagrams, Manuscripta Math. 31 (1980), no. 1-3, 317–329.

[30] , Vinberg’s characterization of Dynkin diagrams using subadditive functions with application to DTr-periodic modules, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 832, Springer, Berlin, 1980, pp. 280–294.

[31] Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999.

[32] Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102.

[33] , On the cyclic homology of exact categories, J. Pure Appl. Algebra 136 (1999), no. 1, 1–56.

[34] Bernhard Keller and Dieter Vossieck, Sous les catégories dérivées, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 6, 225–228.

[35] , Aisles in derived categories, Bull. Soc. Math. Belg. Sér. A 40 (1988), no. 2, 239–253.

[36] Mikhail Khovanov and Paul Seidel, Quivers, Floer cohomology, and braid group actions, J. Amer. Math. Soc. 15 (2002), no. 1, 203–271 (electronic).

[37] Maxim Kontsevich, Topological field theory for triangulated categories, Talk at the conference on K-Theory and Noncommutative Geometry, Institut Henri Poincaré, Paris, June 2004.

[38] , Feynman diagrams and low-dimensional topology, First European Congress of Mathematics, Vol. II (Paris, 1992), Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 97–121.

[39] , Triangulated categories and geometry, Course at the École Normale Supérieure, Paris, Notes taken by J. Bellaïche, J.-F. Dat, I. Marin, G. Racinet and H. Randriambololona, 1998.

[40] Henning Krause, The stable derived category of a Noetherian scheme, Compos. Math. 141 (2005), no. 5, 1128–1162.

[41] D. M. Lu, J. H. Palmieri, Q. S. Wu, and J. J. Zhang, A_{∞}-algebras for ring theorists, Proceedings of the International Conference on Algebra, vol. 11, 2004, pp. 91–128.

[42] Amnon Neeman, Triangulated Categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001.

[43] Liangang Peng and Jie Xiao, Root categories and simple Lie algebras, J. Algebra 198 (1997), no. 1, 19–56.

[44] Idun Reiten, Finite-dimensional algebras and singularities, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), Lecture Notes in Math., vol. 1273, Springer, Berlin, 1987, pp. 35–57.

[45] , Hereditary categories with Serre duality, Representations of algebra. Vol. I, II, Beijing Norm. Univ. Press, Beijing, 2002, pp. 109–121.

[46] Idun Reiten and Michel Van den Bergh, Two-dimensional tame and maximal orders of finite representation type, Mem. Amer. Math. Soc. 80 (1989), no. 408, viii+72.

[47] , Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), no. 2, 295–366 (electronic).

[48] Jeremy Rickard, Derived equivalences as derived functors, J. London Math. Soc. (2) 43 (1991), no. 1, 37–48.

[49] C. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980), no. 2, 199–224.

[50] Claus Michael Ringel, The preprojective algebra of a quiver, Algebras and modules, II (Geiranger, 1996), CMS Conf. Proc., vol. 24, Amer. Math. Soc., Providence, RI, 1998, pp. 467–480.

[51] Raphaël Rouquier, Dimensions of triangulated categories, arXiv:math.CT/0310134.

[52] Raphaël Rouquier and Alexander Zimmermann, Picard groups for derived module categories, Proc. London Math. Soc. (3) 87 (2003), no. 1, 197–225.

[53] Andrzej Skowroński, Generalization of Yamagata’s theorem on trivial extensions, Arch. Math. (Basel) 48 (1987), no. 1, 68–76.

[54] Gonçalo Tabuada, On the homotopy theory of dg categories, Ph. D. thesis in preparation.

[55] , Invariants additifs de dg-catégories, Int. Math. Res. Notices 53 (2005), 3309–3339.

[56] , Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories, C. R. Math. Acad. Sci. Paris 340 (2005), no. 1, 15–19.

[57] Bertrand Toën, The homotopy theory of dg-categories and derived Morita theory, arXiv:math.AG/0408337.

[58] Jie Xiao and Bin Zhu, Relations for the Grothendieck groups of triangulated categories, J. Algebra 257 (2002), no. 1, 37–50.

Bernhard Keller UFR de Mathématiques Institut de Mathématiques UMR 7586 du CNRS Case 7012 Université Paris 7 2, place Jussieu 75251 Paris Cedex 05 France keller@math.jussieu.fr 582

Documenta Mathematica 10 (2005)