On the Structure of Calabi-Yau Categories

with a Cluster Tilting Subcategory

Gonçalo Tabuada

Received: July 18, 2006 Revised: March 9, 2007

Communicated by Stefan Schwede

Abstract. We prove that for d ≥ 2, an algebraic d-Calabi-Yau triangulated category endowed with a d-cluster tilting subcategory is the stable category of a DG category which is perfectly (d+1)-Calabi-Yau and carries a non degenerate t-structure whose heart has enough projectives.

2000 Mathematics Subject Classification: Primary 18E30, 18E35, 18D20; Secondary 16G30, 16G70. Keywords and Phrases: Triangulated category, Calabi-Yau property, t-structure, DG category, Verdier’s quotient, Brown representability theorem.

Contents

1. Introduction 194 2. Acknowledgments 194 3. Preliminaries 194 4. Embedding 196 5. Determination of the image of G 199 6. Alternative description 203 7. The main theorem 206 Appendix A. Extension of t-structures 208 References 211

1. Introduction

In this article, we propose a description of a class of Calabi-Yau categories using the formalism of DG-categories and the notion of ‘stabilization’, as used for the description of triangulated orbit categories in section 7 of [21]. For d ≥ 2, let C be an algebraic d-Calabi-Yau triangulated category endowed with a d-cluster tilting subcategory T , cf. [23] [18] [19], see also [3] [13] [14]. Such categories occur for example,

- in the representation-theoretic approach to Fomin-Zelevinsky’s cluster algebras [12], cf. [6] [9] [15] and the references given there,

- in the study of Cohen-Macaulay modules over certain isolated singularities, cf. [17] [23] [16], and the study of non commutative crepant resolutions [36], cf. [17].

From C and T we construct an exact dg category B, which is perfectly (d + 1)- Calabi-Yau, and a non-degenerate aisle U, cf. [25], in H^{0}(B) whose heart has enough projectives. We prove, in theorem 7.1, how to recover the category C from B and U using a general procedure of stabilization defined in section 7. This extends previous results of [24] to a more general framework. It follows from [30] that for d = 2, up to derived equivalence, the category B only depends on C (with its enhancement) and not on the choice of T . In the appendix, we show how to naturally extend a t-structure, cf. [2], on the compact objects of a triangulated category to the whole category.

Example Let k be a field, A a finite-dimensional hereditary k-algebra and C = C_{A} the cluster category of A, see [7] [8], i.e. the quotient of the bounded derived category of finitely generated modules over A by the functor F = τ^{−1}[1], where τ denotes the AR-translation and [1] denotes the shift functor. Then B is given by the dg algebra, see section 7 of [21],

B = A ⊕ (DA)[−3]

and theorem 7.1 reduces to the equivalence

D^{b}(B)/per(B) ⟶^{∼} C_{A}

of section 7.1 of [21].

2. Acknowledgments

This article is part of my Ph. D. thesis under the supervision of Prof. B. Keller. I deeply thank him for countless useful discussions and for his perfect guidance and generous patience. This work was supported by FCT-Portugal, scholarship SFRH/BD/14035/2003.

3. Preliminaries

Let k be a field. Let E be a k-linear Frobenius category with split idempotents. Suppose that its stable category C = \underline{E}, with suspension functor S, has finitedimensional Hom-spaces and admits a Serre functor Σ, see [4]. Let d ≥ 2 be an integer. We suppose that C is Calabi-Yau of CY-dimension d, i.e. [27] there is an isomorphism of triangle functors

S^{d} →^{∼} Σ .

We fix such an isomorphism once and for all. See section 4 of [23] for several examples of the above situation. For X, Y ∈ C and n ∈ ℤ, we put

Ext^{n}(X, Y) = Hom_{C}(X, S^{n}Y) .

We suppose that C is endowed with a d-cluster tilting subcategory T ⊂ C, i.e. a) T is a k-linear subcategory,

b) T is functorially finite in C, i.e. the functors Hom_{C}(?, X)|T and Hom_{C}(X,?)|T are finitely generated for all X ∈ C,

c) we have Ext^{i}(T, T′) = 0 for all T, T′ ∈ T and all 0 < i < d and

d) if X ∈ C satisfies Ext^{i}(T, X) = 0 for all 0 < i < d and all T ∈ T , then

T belongs to T . Let M ⊂ E be the preimage of T under the projection functor. In particular, M contains the subcategory P of the projective-injective objects in M. Note that T equals the quotient \underline{M} of M by the ideal of morphisms factoring through a projective-injective. We dispose of the following commutative square:

\xymatrix{M \ar@{^{(}->}[r] \ar@{->>}[d] & E \ar@{->>}[d] \\ T \ar@{^{(}->}[r] & \underline{E} = C .}

We use the notations of [20]. In particular, for an additive category A, we denote by C(A) (resp. C^{−}(A), C^{b}(A), …) the category of unbounded (resp. right bounded, resp. bounded, …) complexes over A and by H(A) (resp. H^{−}(A), H^{b}(A), …) its quotient modulo the ideal of nullhomotopic morphisms. By [26], cf. also [31], the projection functor E → \underline{E} extends to a canonical triangle functor H^{b}(E)/H^{b}(P) → \underline{E}. This induces a triangle functor H^{b}(M)/H^{b}(P) → \underline{E}. It is shown in [30] that this functor is a localization functor. Moreover, the projection functor H^{b}(M) → H^{b}(M)/H^{b}(P) induces an equivalence from the subcategory H_{E-ac}^{b}(M) of bounded E-acyclic complexes with components in M onto its kernel. Thus, we have a short exact sequence of triangulated categories

0 ⟶ H_{E-ac}^{b}(M) ⟶ H^{b}(M)/H^{b}(P) ⟶ C ⟶ 0 .

Let B be the dg (=differential graded) subcategory of the category C^{b}(M)_{dg} of bounded complexes over M whose objects are the E-acyclic complexes. We denote by G : H^{−}(M) → D(B^{op})^{op} the functor which takes a right bounded complex X over M to the dg module

B ↦ Hom_{M}^{•}(X, B) ,

where B is in B.

Remark 3.1. By construction, the functor G restricted to H_{E-ac}^{b}(M) establishes an equivalence

G : H_{E-ac}^{b}(M) ⟶^{∼} per(B^{op})^{op} .

Recall that if P is a right bounded complex of projectives and A is an acyclic complex, then each morphism from P to A is nullhomotopic. In particular, the complex Hom_{M}^{•}(P, A) is nullhomotopic for each P in H^{−}(P). Thus G takes H^{−}(P) to zero, and induces a well defined functor (still denoted by G)

G : H^{b}(M)/H^{b}(P) ⟶ D(B^{op})^{op} .

4. Embedding

Proposition 4.1. The functor G is fully faithful.

For the proof, we need a number of lemmas. It is well-known that the category H^{−}(E) admits a semiorthogonal decomposition, cf. [5], formed by H^{−}(P) and its right orthogonal H_{E-ac}^{−}(E), the full subcategory of the right bounded E-acyclic complexes. For X in H^{−}(E), we write

pX → X → a_{p}X → SpX

for the corresponding triangle, where pX is in H^{−}(P) and a_{p}X is in H_{E-ac}^{−}(E). If X lies in H^{−}(M), then clearly a_{p}X lies in H_{E-ac}^{−}(M) so that we have an induced semiorthogonal decomposition of H^{−}(M).

Lemma 4.1. The functor Υ : H^{b}(M)/H^{b}(P) ⟶ H_{E-ac}^{−}(M) which takes X to a_{p}X is fully faithful.

Proof. By the semiorthogonal decomposition of H^{−}(M), the functor X ↦ a_{p}X induces a right adjoint of the localization functor

H^{−}(M) ⟶ H^{−}(M)/H^{−}(P)

and an equivalence of the quotient category with the right orthogonal H_{E-ac}^{−}(M).

\xymatrix{ & H^{−}(P) \ar@{^{(}->}[d] \\  & H^{−}(M) \ar[d] \ar[u] & H_{E-ac}^{−}(M) = H(P)^{⊥} \ar@{^{(}->}[l] \\ H^{b}(M)/H^{b}(P) \ar@{^{(}->}[r] & H^{−}(M)/H^{−}(P) \ar@{^{(}->}[u] \ar@{-->}[ur]^{∼} & .}

Moreover, it is easy to see that the canonical functor

H^{b}(M)/H^{b}(P) ⟶ H^{−}(M)/H^{−}(P) is fully faithful so that we obtain a fully faithful functor

H^{b}(M)/H^{b}(P) ⟶ H_{E-ac}^{−}(M)

taking X to a_{p}X. √

Remark 4.1. Since the functor G is triangulated and takes H^{−}(P) to zero, for X in H^{b}(M), the adjunction morphism X → a_{p}X yields an isomorphism

G(X) ⟶^{∼} G(a_{p}X) = G(ΥX) .

Let D− ; \underline{M}(M) be the full subcategory of the derived category D(M) formed by the right bounded complexes whose homology modules lie in the subcategory Mod \underline{M} of Mod M. The Yoneda functor M → Mod M, M ↦ M^{∧}, induces a full embedding

Ψ : H_{E-ac}^{−}(M) ↪ D− ; \underline{M}(M) .

We write V for its essential image. Under Ψ, the category H_{E-ac}^{b}(M) is identified with per\underline{M}(M). Let Φ : D− ; \underline{M}(M) → D(B^{op})^{op} be the functor which takes X to the dg module

B ↦ Hom^{•}(X_{c}, Ψ(B)) ,

where B is in H_{E-ac}^{b}(M) and X_{c} is a cofibrant replacement of X for the projective model structure on C(M). Since for each right bounded complex M with components in M, the complex M^{∧} is cofibrant in C(M), it is clear that the functor G : H^{b}(M)/H^{b}(P) → D(B^{op})^{op} is isomorphic to the composition Φ ∘ Ψ ∘ Υ. We dispose of the following commutative diagram

[figure: H^{b}(M)/H^{b}(P); Υ; H_{E-ac}^{−}(M); Ψ; ∼; D− ; \underline{M}(M); Φ; D(B^{op})^{op}; V; H_{E-ac}^{b}(M); H_{E-ac}^{b}(M); ∼; per\underline{M}(M); ∼; per(B^{op})^{op}]

Lemma 4.2. Let Y be an object of D− ; \underline{M}(M).

a) Y lies in per\underline{M}(M) iff H^{p}(Y) is a finitely presented \underline{M}-module for all p ∈ ℤ and vanishes for all but finitely many p.

b) Y lies in V iff H^{p}(Y) is a finitely presented \underline{M}-module for all p ∈ ℤ and vanishes for all p ≫ 0.

Proof. a) Clearly the condition is necessary. For the converse, suppose first that Y is a finitely presented \underline{M}-module. Then, as an M-module, Y admits a resolution of length d + 1 by finitely generated projective modules by theorem 5.4 b) of [23]. It follows that Y belongs to per\underline{M}(M). Since per\underline{M}(M) is triangulated, it also contains all shifts of finitely presented \underline{M}-modules and all extensions of shifts. This proves the converse. b) Clearly the condition is necessary. For the converse, we can suppose without loss of generality that Y^{n} = 0, for all n ≥ 1 and that Y^{n} belongs to proj M, for n ≤ 0. We now construct a sequence

⋯ → P_{n} → ⋯ → P_{1} → P_{0}

of complexes of finitely generated projective M-modules such that P_{n} is quasiisomorphic to τ_{≥−n}Y for each n and that, for each p ∈ ℤ, the sequence of Mmodules P_{n}^{p} becomes stationary. By our assumptions, we have τ_{≥0}Y →^{∼} H^{0}(Y). Since H^{0}(Y) belongs to mod \underline{M}, we know by theorem 5.4 c) of [23] that it belongs to per(M) as an M-module. We define P_{0} to be a finite resolution of H^{0}(Y) by finitely generated M-modules. For the induction step, consider the following truncation triangle associated with Y

S^{i+1}H^{−i−1}(Y) → τ_{≥−i−1}Y → τ_{≥−i}Y → S^{i+2}H^{−i−1}(Y) ,

for i ≥ 0. By the induction hypothesis, we have constructed P_{0}, …, P_{i} and we dispose of a quasi-isomorphism P_{i} →^{∼} τ_{≥−i}Y. Let Q_{i+1} be a finite resolution of S^{i+2}H^{−i−1}(Y) by finitely presented projective M-modules. We dispose of a morphism f_{i} : P_{i} → Q_{i+1} and we define P_{i+1} as the cylinder of f_{i}. We define P as the limit of the P_{i} in the category of complexes. We remark that Y is quasi-isomorphic to P and that P belongs to V. This proves the converse. √

Let X be in H_{E-ac}^{−}(M).

Remark 4.2. Lemma 4.2 shows that the natural t-structure of D(M) restricts to a t-structure on V. This allows us to express Ψ(X) as

Ψ(X) ⟶^{∼} holim_{i} τ_{≥−i}Ψ(X) ,

where τ_{≥−i}Ψ(X) is in per\underline{M}(M).

Lemma 4.3. We dispose of the following isomorphism

Φ(Ψ(X)) = Φ(holim_{i} τ_{≥−i}Ψ(X)) ⟶^{∼} holim_{i} Φ(τ_{≥−i}Ψ(X)) .

Proof. It is enough to show that the canonical morphism induces a quasiisomorphism when evaluated at any object B of B. We have

Φ(holim_{i} τ_{≥−i}Ψ(X))(B) = Hom^{•}(holim_{i} τ_{≥−i}Ψ(X), B) ,

but since B is a bounded complex, for each n ∈ Z, the sequence

i ↦ Hom^{n}(τ_{≥−i}Ψ(X), B)

stabilizes as i goes to infinity. This implies that

Hom^{•}(holim_{i} τ_{≥−i}Ψ(X), B) ⟵^{∼} holim_{i} Φ(τ_{≥−i}Ψ(X))(B) .
√

Lemma 4.4. The functor Φ restricted to the category V is fully faithful.

Proof. Let X, Y be in H_{E-ac}^{−}(M). The following are canonically isomorphic :

Hom_{D(B^{op})^{op}}(ΦΨX, ΦΨY)
Hom_{D(B^{op})}(ΦΨY, ΦΨX)

(4.1) Hom_{D(B^{op})}(hocolim_{i} Φτ_{≥−i}ΨY, hocolim_{j} Φτ_{≥−j}ΨX)

holim_{i} Hom_{D(B^{op})}(Φτ_{≥−i}ΨY, hocolim_{j} Φτ_{≥−j}ΨX)
(4.2)  holim_{i} hocolim_{j} Hom_{D(B^{op})}(Φτ_{≥−i}ΨY, Φτ_{≥−j}ΨX)
holim_{i} hocolim_{j} Hom_{per\underline{M}(M)}(τ_{≥−j}ΨX, τ_{≥−i}ΨY)
(4.3)  holim_{i} Hom_{V}(holim_{j} τ_{≥−j}ΨX, τ_{≥−i}ΨY)
Hom_{V}(Ψ(X), Ψ(Y)) .

Here (4.1) is by the lemma 4.3 seen in D(B^{op}), (4.2) is by the fact that Φτ_{≥−i}ΨY is compact and (4.3) is by the fact that τ_{≥−i}ΨY is bounded. √

It is clear now that lemmas 4.1, 4.3 and 4.4 imply the proposition 4.1.

5. Determination of the image of G

Let L_{ρ} : D^{−}(\underline{M}) → D− ; \underline{M}(M) be the restriction functor induced by the projection functor M → \underline{M}. L_{ρ} admits a left adjoint L : D− ; \underline{M}(M) → D^{−}(\underline{M}) which takes Y to Y ⊗_{M}^{𝕃} \underline{M}. Let B^{−} be the dg subcategory of C^{−}(Mod M)_{dg} formed by the objects of D− ; \underline{M}(M) that are in the essential image of the restriction of Ψ to H_{E-ac}^{b}(M). Let B′ be the DG quotient, cf. [11], of B^{−} by its quasiisomorphisms. It is clear that the dg categories B′ and B are quasi-equivalent, cf. [22], and that the natural dg functor M → C^{−}(Mod M)_{dg} factors through B^{−}. Let R′ : D(B^{op})^{op} → D(\underline{M}^{op})^{op} be the restriction functor induced by the dg functor \underline{M} → B′. Let Φ′ : D− ; \underline{M}(M) → D(B^{′op})^{op} be the functor which takes X to the dg module

B′ ↦ Hom^{•}(X_{c}, B′) ,

where B′ is in B′ and X_{c} is a cofibrant replacement of X for the projective model structure on C(Mod M). Finally let Γ : D(\underline{M}) → D(\underline{M}^{op})^{op} be the functor that sends Y to

M ↦ Hom^{•}(Y_{c}, \underline{M}(?, M)) ,

where Y_{c} is a cofibrant replacement of Y for the projective model structure on C(Mod \underline{M}) and M is in \underline{M}.

We dispose of the following diagram :

\xymatrix{ &  &  & D(B^{op})^{op} & B \ar[d] \\ H^{b}(M)/H^{b}(P) \ar@{^{(}->}[r]^{Υ} & H_{E-ac}^{−}(M) \ar[r]^{Ψ} & D− ; \underline{M}(M) \ar[d]_{L} \ar[r]^{Φ′} \ar@{-->}[ur]^{Φ} & D(B^{′op})^{op} \ar[d]^{R′} \ar[u]_{∼} & B′ \\  &  & D^{−}(\underline{M}) \ar[r]_{Γ} & D(\underline{M}^{op})^{op} & \underline{M} \ar[u]}

Lemma 5.1. The following square

\xymatrix{D− ; \underline{M}(M) \ar[r]^{Φ′} \ar[d]_{L} & D(B^{′op})^{op} \ar[d]^{R′} & B′ \\ D^{−}(\underline{M}) \ar[r]_{Γ} & D(\underline{M}^{op})^{op} & \underline{M} \ar[u]}

is commutative.

Proof. By definition (R′ ∘ Φ′)(X)(M) equals Hom^{•}(X_{c}, \underline{M}(?, M)). Since \underline{M}(?, M) identifies with L_{ρ}M^{∧} and by adjunction, we have

Hom^{•}(X_{c}, \underline{M}(?, M)) ⟶^{∼} Hom^{•}(X_{c}, L_{ρ}M^{∧}) ⟶^{∼} Hom^{•}((LX)_{c}, \underline{M}(?, M)) ,

where the last member equals (Γ ∘ L)(X)(M). √

Lemma 5.2. The functor L reflects isomorphisms .

Proof. Since L is a triangulated functor, it is enough to show that if L(Y) = 0, then Y = 0. Let Y be in D− ; \underline{M}(M) such that L(Y) = 0. We can suppose, without loss of generality, that H^{p}(Y) = 0 for all p > 0. Let us show that H^{0}(Y) = 0. Indeed, since H^{0}(Y) is an \underline{M}-module, we have H^{0}(Y) ≅ L^{0}H^{0}(Y), where L^{0} : Mod M → Mod \underline{M} is the left adjoint of the inclusion Mod \underline{M} → Mod M. Since H^{p}(Y) vanishes in degrees p > 0, we have

L^{0}H^{0}(Y) = H^{0}(LY) .

By induction, one concludes that H^{p}(Y) = 0 for all p ≤ 0. √

Proposition 5.1. An object Y of D− ; \underline{M}(M) lies in the essential image of the functor Ψ ∘ Υ : H^{b}(M)/H^{b}(P) → D− ; \underline{M}(M) iff τ_{≥−n}Y is in per\underline{M}(M), for all n ∈ ℤ and L(Y) belongs to per(\underline{M}).

Proof. Let X be in H^{b}(M)/H^{b}(P). By lemma 4.2 a), τ_{≥−n}ΨΥ(X) is in per\underline{M}(M), for all n ∈ ℤ. Since X is a bounded complex, there exists an s ≪ 0 such that for all m < s the m-components of Υ(X) are in P, which implies that LΨΥ(X) belongs to per(\underline{M}). Conversely, suppose that Y is an object of D− ; \underline{M}(M) which satisfies the conditions. By lemma 4.2, Y belongs to V. Thus we have Y = Ψ(Y′) for some Y′ in H_{E-ac}^{−}(M). We now consider Y′ as an object of H^{−}(M) and also write Ψ for the functor H^{−}(M) → D^{−}(M) induced by the Yoneda functor. We can express Y′ as

Y′ ⟵^{∼} hocolim_{i} σ_{≥−i}Y′ ,

where the σ_{≥−i} are the naive truncations. By our assumptions on Y′, σ_{≥−i}Y′ belongs to H^{b}(M)/H^{b}(P), for all i ∈ ℤ. The functors Ψ and L clearly commute with the naive truncations σ_{≥−i} and so we have

L(Y) = L(ΨY′) ⟵^{∼} hocolim_{i} L(σ_{≥−i}ΨY′) ⟶^{∼} hocolim_{i} σ_{≥−i}L(ΨY′) .

By our hypotheses, L(Y) belongs to per(\underline{M}) and so there exists an m ≫ 0 such that

L(Y) = L(ΨY′) ⟵^{∼} σ_{≥−m}L(ΨY′) = L(σ_{≥−m}ΨY′) .

By lemma 5.2, the inclusion

Ψ(σ_{≥−m}Y)′ = σ_{≥−m}ΨY′ ⟶ Ψ(Y′) = Y

is an isomorphism. But since σ_{≥−m}Y′ belongs to H^{b}(M)/H^{b}(P), Y identifies with Ψ(σ_{≥−m}Y′). √

Remark 5.1. It is clear that if X belongs to per(\underline{M}), then Γ(X) belongs to per(\underline{M}^{op})^{op}. We also have the following partial converse.

Lemma 5.3. Let X be in D_{mod \underline{M}}^{−}(\underline{M}) such that Γ(X) belongs to per(\underline{M}^{op})^{op}. Then X is in per(\underline{M}).

Proof. By lemma 4.2 b) we can suppose, without loss of generality, that X is a right bounded complex with finitely generated projective components. Applying Γ, we get a perfect complex Γ(X). In particular Γ(X) is homotopic to zero in high degrees. But since Γ is an equivalence

proj \underline{M} ⟶^{∼} (proj \underline{M}^{op})^{op} ,

it follows that X is already homotopic to zero in high degrees. √

Remark 5.2. The natural right aisle on D(M) is the full subcategory of the objects X such that H^{n}(X) = 0 for all n < 0. The associated truncation functor τ_{≥0} takes per\underline{M}(M) to itself. Therefore, the natural right aisle on D(M) restricts to a natural right aisle U^{op} on per\underline{M}(M).

Definition 5.1. Let U be the natural left aisle in per\underline{M}(M)^{op} associated with U^{op}.

Lemma 5.4. The natural left aisle U on per\underline{M}(M)^{op} →^{∼} per(B^{op}) satisfies the conditions of proposition A.1 b).

Proof. Clearly the natural left aisle U in per\underline{M}(M)^{op} is non-degenerate. We need to show that for each C ∈ per\underline{M}(M)^{op}, there is an integer N such that Hom(C, S^{N}U) = 0 for each U ∈ U. We dispose of the following isomorphism

Hom_{per\underline{M}(M)^{op}}(C, S^{N}U) →^{∼} Hom_{per\underline{M}(M)}(S^{−N}U^{op}, C) ,

where U^{op} denotes the natural right aisle on per\underline{M}(M). Since by theorem 5.4 c) of [23] an \underline{M}-module admits a projective resolution of length d + 1 as an M-module and C is a bounded complex, we conclude that for N ≫ 0

Hom_{per\underline{M}(M)}(S^{−N}U^{op}, C) = 0 .

This proves the lemma. √

We denote by τ_{≤n} and τ_{≥n}, n ∈ ℤ, the associated truncation functors on D(B^{op})^{op}.

Lemma 5.5. The functor Φ : D− ; \underline{M}(M) → D(B^{op})^{op} restricted to the category V is exact with respect to the given t-structures.

Proof. We first prove that Φ(V_{≤0}) ⊂ D(B^{op})_{≤0}^{op}. Let X be in V_{≤0}. We need to show that Φ(X) belongs to D(B^{op})_{≤0}^{op}. The following have the same classes of objects :

D(B^{op})_{≤0}^{op}
D(B^{op})_{>0}

(per(B^{op})_{≤0})^{⊥} (5.1) ⊥(per(B^{op})^{op})_{>0} , (5.2)

where in (5.1) we consider the right orthogonal in D(B^{op}) and in (5.2) we consider the left orthogonal in D(B^{op})^{op}. These isomorphisms show us that Φ(X) belongs to D(B^{op})_{≤0}^{op} iff

Hom_{D(B^{op})^{op}}(Φ(X), Φ(P)) = 0 ,

for all P ∈ per\underline{M}(M)_{>0}. Now, by lemma 4.4 the functor Φ is fully faithful and so

Hom_{D(B^{op})^{op}}(Φ(X), Φ(P)) ⟶^{∼} Hom_{per\underline{M}(M)}(X, P) .

Since X belongs to V_{≤0} and P belongs to per\underline{M}(M)_{>0}, we conclude that

Hom_{per\underline{M}(M)}(X, P) = 0 ,

which implies that Φ(X) ∈ D(B^{op})_{≤0}^{op}. Let us now consider X in V. We dispose of the truncation triangle

τ_{≤0}X → X → τ_{>0}X → Sτ_{≤0}X .

The functor Φ is triangulated and so we dispose of the triangle

Φτ_{≤0}X → X → Φτ_{>0}X → SΦτ_{≤0}X ,

where Φτ_{≤0}X belongs to D(B^{op})_{≤0}^{op}. Since Φ induces an equivalence between per\underline{M}(M) and per(B^{op})^{op} and Hom(P, τ_{>0}X) = 0, for all P in V_{≤0}, we conclude that Φτ_{>0}X belongs to D(B^{op})_{>0}^{op}. This implies the lemma. √

Definition 5.2. Let D(B^{op})_{f}^{op} denote the full triangulated subcategory of D(B^{op})^{op} formed by the objects Y such that τ_{≥−n}Y is in per(B^{op})^{op}, for all n ∈ ℤ, and R(Y) belongs to per(\underline{M}^{op})^{op}.

Proposition 5.2. An object Y of D(B^{op})^{op} lies in the essential image of the functor G : H^{b}(M)/H^{b}(P) → D(B^{op})^{op} iff it belongs to D(B^{op})_{f}^{op}.

Proof. Let X be in H^{b}(M)/H^{b}(P). It is clear that the τ_{≥−n}G(X) are in per(B^{op})^{op} for all n ∈ ℤ. By proposition 5.1 we know that LΨΥ(X) belongs to per(\underline{M}). By lemma 5.1 and remark 5.1 we conclude that RG(X) belongs to per(\underline{M}^{op})^{op}. Let now Y be in D(B^{op})_{f}^{op}. We can express it, by the dual of lemma A.2 as the homotopy limit of the following diagram

⋯ → τ_{≥−n−1}Y → τ_{≥−n}Y → τ_{≥−n+1}Y → ⋯,

where τ_{≥−n}Y belongs to per(B^{op})^{op}, for all n ∈ ℤ. But since Φ induces an equivalence between per\underline{M}(M) and per(B^{op})^{op}, this last diagram corresponds to a diagram

⋯ → M_{−n−1} → M_{−n} → M_{−n+1} → ⋯

in per\underline{M}(M). Let p ∈ ℤ. The relations among the truncation functors imply that the image of the above diagram under each homology functor H^{p}, p ∈ ℤ, is stationary as n goes to +∞. This implies that

H^{p} holim_{n} M_{−n} ⟶^{∼} lim_{n} H^{p} M_{−n} ≅ H^{p} M_{j} ,

for all j < p. We dispose of the following commutative diagram

\xymatrix{holim_{n} M_{−n} \ar[r] \ar[d] & holim_{n} τ_{≥−i} M_{−n} ≅ M_{−i} \\ τ_{≥−i} holim_{n} M_{−n} \ar@{-->}[ur]^{∼}}

which implies that

τ_{≥−i} holim_{n}M_{−n} ⟶^{∼} M_{−i} ,

for all i ∈ ℤ. Since holim_{n} M_{−n} belongs to V, lemma 4.3 allows us to conclude that Φ(holim_{n} M_{−n}) ≅ Y. We now show that holim_{n} M_{−n} satisfies the conditions of proposition 5.1. We know that τ_{≥−i} holim_{n} M_{−n} belongs to per\underline{M}(M), for all i ∈ ℤ. By lemma 5.1 (Γ ∘ L)(holim_{n} M_{−n}) identifies with R(Y), which is in per(\underline{M}^{op})^{op}. Since holim_{n} M_{−n} belongs to V, its homologies lie in mod \underline{M} and so we are in the conditions of lemma 5.1, which implies that L(holim_{n} M_{−n}) belongs to per\underline{M}(M). This finishes the proof. √

6. Alternative description

In this section, we present another characterization of the image of G, which was identified as D(B^{op})_{f}^{op} in proposition 5.2. Let M denote an object of M and also the naturally associated complex in H^{b}(M). Since the category H^{b}(M)/H^{b}(P) is generated by the objects M ∈ M and the functor G is fully faithful, we remark that D(B^{op})_{f}^{op} equals the triangulated subcategory of D(B^{op})^{op} generated by the objects G(M), M ∈ M. The rest of this section is concerned with the problem of characterizing the objects G(M), M ∈ M. We denote by P_{M} the projective \underline{M}-module \underline{M}(?, M) associated with M ∈ M and by X_{M} the image of M under Ψ ∘ Υ.

Lemma 6.1. We dispose of the following isomorphism

Hom_{D− ; \underline{M}(M)}(X_{M}, Y) ⟵^{∼} Hom_{mod \underline{M}}(P_{M}, H^{0}(Y)) ,

for all Y ∈ D− ; \underline{M}(M).

Proof. Clearly X_{M} belongs to D\underline{M}(M)_{≤0} and is of the form

⋯ → P_{n}^{∧} → ⋯ → P_{1}^{∧} → P_{0}^{∧} → M^{∧} → 0 ,

where P_{n} ∈ P, n ≥ 0. Now Yoneda’s lemma and the fact that H^{m}(Y)(P_{n}) = 0, for all m ∈ ℤ, n ≥ 0, imply the lemma. √

Remark 6.1. Since the functor Φ restricted to V is fully faithful and exact, we have

Hom_{D(B^{op})^{op}}(G(M), Φ(Y)) ⟵^{∼} Hom_{per(B^{op})^{op}}(Φ(P_{M}), H^{0}(Φ(Y))) ,

for all Y ∈ V.

We now characterize the objects G(M) = Φ(X_{M}), M ∈ M, in the triangulated category D(B^{op}). More precisely, we give a description of the functor

R_{M} := Hom_{D(B^{op})}(?, Φ(X_{M})) : D(B^{op})^{op} → Mod k

using an idea of M. Van den Bergh, cf. lemma 2.13 of [10]. Consider the following functor

F_{M} := Hom_{per(B^{op})}(H^{0}(?), Φ(P_{M})) : per(B^{op})^{op} → mod k .

Remark 6.2. Remark 6.1 shows that the functor R_{M} when restricted to per(B^{op}) coincides with F_{M}.

Let DF_{M} be the composition of F_{M} with the duality functor D = Hom(?, k). Note that DF_{M} is homological.

Lemma 6.2. We dispose of the following isomorphism of functors on per(B^{op})

DF_{M} ⟶^{∼} Hom_{D(B^{op})}(Φ(X_{M}),?[d + 1]) .

Proof. The following functors are canonically isomorphic to DF Φ :

DHom_{per(B^{op})}(H^{0}Φ(?), Φ(P_{M}))

(6.1) DHom_{per(B^{op})}(ΦH^{0}(?), Φ(P_{M}))

DHom_{per\underline{M}(M)}(P_{M}, H^{0}(?)) (6.2)

DHom_{D− ; \underline{M}(M)}(X_{M},?) (6.3)

Hom_{D− ; \underline{M}(M)}(?[−d − 1], X_{M}) (6.4)

Hom_{D(B^{op})^{op}}(Φ(?)[−d − 1], Φ(X_{M})) (6.5) Hom_{D(B^{op})^{op}}(Φ(X_{M}), Φ(?)[d + 1]) (6.6)

Step (6.1) follows from the fact that Φ is exact. Step (6.2) follows from the fact that Φ is fully faithful and we are considering the opposite category. Step (6.3) is a consequence of lemma 6.1. Step (6.4) follows from the (d + 1)-Calabi-Yau property and remark 4.2. Step (6.5) is a consequence of Φ being fully faithful and step (6.6) is a consequence of working in the opposite category. Since the functor Φ^{op} establish an equivalence between per\underline{M}(M)^{op} and per(B^{op}) the lemma is proven. √

Now, since the category Mod k is cocomplete, we can consider the left Kan extension, cf. [28], E_{M} of DF_{M} along the inclusion per(B^{op}) ↪ D(B^{op}). We dispose of the following commutative square :

\xymatrix{per(B^{op}) \ar[r]^{DF_{M}} \ar@{^{(}->}[d] & mod k \ar@{^{(}->}[d] \\ D(B^{op}) \ar@{-->}[r]_{E_{M}} & Mod k .}

For each X of D(B^{op}), the comma-category of morphisms P → X from a perfect object P to X is filtered. Therefore, the functor E_{M} is homological. Moreover, it preserves coproducts and so DE_{M} is cohomological and transforms coproducts into products. Since D(B^{op}) is a compactly generated triangulated category, the Brown representability theorem, cf. [29], implies that there is a Z_{M} ∈ D(B^{op}) such that

DE_{M} ⟶^{∼} Hom_{D(B^{op})}(?, Z_{M}) .

Remark 6.3. Since the duality functor D establishes and anti-equivalence in mod k, the functor DE_{M} restricted to per(B^{op}) is isomorphic to F_{M}.

Theorem 6.1. We dispose of an isomorphism

G(M) ⟶^{∼} Z_{M} .

Proof. We now construct a morphism of functors from R_{M} to DE_{M}. Since R_{M} is representable, by Yoneda’s lemma it is enough to construct an element in DE_{M}(Φ(X_{M})). Let C be the category per(B^{op}) ↓ Φ(X_{M}), whose objects are the morphisms Y′ → Φ(X_{M}) and let C′ be the category X_{M} ↓ per\underline{M}(M), whose objects are the morphisms X_{M} → X′. The following are canonically isomorphic :

DE_{M}(Φ(X_{M}))

(6.7) D colim_{C}Hom_{D(B^{op})}(Φ(X_{M}), Y′[d + 1])

(6.8) D colim_{C′}Hom_{D− ; \underline{M}(M)}(X′[−d − 1], X_{M})

(6.9) D colim_{i} Hom_{D− ; \underline{M}(M)}((τ_{≥−i}X_{M})[−d − 1], X_{M})

lim_{i} DHom_{D− ; \underline{M}(M)}((τ_{≥−i}X_{M})[−d − 1], X_{M})

lim_{i} Hom_{D− ; \underline{M}(M)}(X_{M}, τ_{≥−i}X_{M}) (6.10)

Step (6.7) is a consequence of the definition of the left Kan extension and lemma 6.2. Step (6.8) is obtained by considering the opposite category. Step (6.9) follows from the fact that the system (τ_{≥−i}X_{M})_{i∈ℤ} forms a cofinal system for the index system of the colimit. Step (6.10) follows from the (d+1)-Calabi-Yau property. Now, the image of the identity by the canonical morphism

Hom_{D− ; \underline{M}(M)}(X_{M}, X_{M}) ⟶ lim_{i} Hom_{D− ; \underline{M}(M)}(X_{M}, τ_{≥−i}X_{M}) ,

gives us an element of (DE_{M})(Φ(X_{M})) and so a morphism of functors from R_{M} to DE_{M}. We remark that this morphism is an isomorphism when evaluated at the objects of per(B^{op}). Since both functors R_{M} and DE_{M} are cohomological, transform coproducts into products and D(B^{op}) is compactly generated, we conclude that we dispose of an isomorphism

G(M) ⟶^{∼} Z_{M} .

√

7. The main theorem

Consider the following commutative square as in section 3:

\xymatrix{M \ar@{^{(}->}[r] \ar@{->>}[d] & E \ar@{->>}[d] \\ T \ar@{^{(}->}[r] & \underline{E} = C .}

In the previous sections we have constructed, from the above data, a dg category B and a left aisle U ⊂ H^{0}(B), see [25], satisfying the following conditions :

- B is an exact dg category over k such that H^{0}(B) has finite-dimensional Hom-spaces and is Calabi-Yau of CY-dimension d + 1,

- U ⊂ H^{0}(B) is a non-degenerate left aisle such that :

- for all B ∈ B, there is an integer N such that Hom_{H^{0}(B)}(B, S^{N}U) = 0 for each U ∈ U,

- the heart H of the t-structure on H^{0}(B) associated with U has

enough projectives.

Let now A be a dg category and W ⊂ H^{0}(A) a left aisle satisfying the above conditions. We can consider the following general construction : Let Q denote the category of projectives of the heart H of the t-structure on H^{0}(A) associated with W. We claim that the following inclusion

Q ↪ H ↪ H^{0}(A) ,

lifts to a morphism Q →^{j} A in the homotopy category of small dg categories, cf. [20] [32] [33] [34] [35]. Indeed, recall the following argument from section 7 of [22]: Let Q̃ be the full dg subcategory of A whose objects are the same as those of Q. Let τ_{≤0}Q̃ denote the dg category obtained from Q̃ by applying the truncation functor τ_{≤0} of complexes to each Hom-space. We dispose of the following diagram in the category of small dg categories

[figure: Q̃; A; τ_{≤0}Q̃; Q; H^{0}(Q̃); .]

Let X, Y be objects of Q. Since X and Y belong to the heart of a t-structure in H^{0}(A), we have

Hom_{H^{0}(A)}(X, Y [−n]) = 0 ,

for n ≥ 1. The dg category A is exact, which implies that

H^{−n}Hom• ; Q̃(X, Y) ⟶^{∼} Hom_{H^{0}(A)}(X, Y [−n]) = 0 ,

for n ≥ 1. This shows that the dg functor τ_{≤0}Q̃ → H^{0}(Q̃) is a quasi-equivalence and so we dispose of a morphism Q →^{j} A in the homotopy category of small dg categories. We dispose of a triangle functor j^{∗} : D(A) → D(Q) given by restriction. By proposition A.1, the left aisle W ⊂ H^{0}(A) admits a smallest extension to a left aisle D(A^{op})_{≤0}^{op} on D(A^{op})^{op}. Let D(A^{op})_{f}^{op} denote the full triangulated subcategory of D(A^{op})^{op} formed by the objects Y such that τ_{≥−n}Y is in per(A^{op})^{op}, for all n ∈ ℤ, and j^{∗}(Y) belongs to per(Q^{op})^{op}.

Definition 7.1. The stable category of A with respect to W is the triangle quotient

stab(A, W) = D(A^{op})_{f}^{op}/per(A^{op})^{op} .

We are now able to formulate the main theorem. Let B be the dg category and U ⊂ H^{0}(B) the left aisle constructed in sections 1 to 5.

Theorem 7.1. The functor G induces an equivalence of categories

G̃ : C ⟶^{∼} stab(B, U) .

Proof. We dispose of the following commutative diagram :

\xymatrix{C \ar@{-->}[r]^{G̃}_{∼} & stab(B, U) \\ H^{b}(M)/H^{b}(P) \ar[u] \ar[r]^{G}_{∼} & D(B^{op})_{f}^{op} \ar[u] \\ H_{E-ac}^{b}(M) \ar[u] \ar[r]_{∼} & per(B^{op})^{op} . \ar[u]}

Documenta Mathematica 12 (2007) 193–213

The functor G is an equivalence since it is fully faithful by proposition 4.1 and essentially surjective by proposition 5.2. Since we dispose of an equivalence H_{E-ac}^{b}(M) ⟶^{∼} per(B^{op})^{op} by construction of B and the columns of the above diagram are short exact sequences of triangulated categories, the theorem is proved. √

Appendix A. Extension of t-structures

Let T be a compactly generated triangulated category with suspension functor S. We denote by T_{c} the full triangulated sub-category of T formed by the compact objects, see [29]. We use the terminology of [25]. Let U ⊆ T_{c} be a left aisle on T_{c}, i.e. a full additive subcategory U of T_{c} which satisfies:

a) SU ⊂ U,

b) U is stable under extensions, i.e. for each triangle

X → Y → Z → SX

of T_{c}, we have Y ∈ U whenever X, Z ∈ U and

c) the inclusion functor U ↪ T_{c} admits a right adjoint. As shown in [25], the concept of aisle is equivalent to that of t-structure.

Proposition A.1. a) The left aisle U admits a smallest extension to a

left aisle T_{≤0} on T .

b) If U ⊆ T_{c} is non-degenerate ( i.e., f : X → Y is invertible iff H^{p}(f) is invertible for all p ∈ ℤ) and for each X ∈ T_{c}, there is an integer N such that Hom(X, S^{N}U) = 0 for each U ∈ U, then T_{≤0} is also non-degenerate.

Proof. a) Let T_{≤0} be the smallest full subcategory of T that contains U and is stable under infinite sums and extensions. It is clear that T_{≤0} is stable by S since U is. We need to show that the inclusion functor T_{≤0} ↪ T admits a right adjoint. For completeness, we include the following proof, which is a variant of the ‘small object argument’, cf. also [1]. We dispose of the following recursive procedure. Let X = X_{0} be an object in T . For the initial step consider all morphisms from any object P in U to X_{0}. This forms a set I_{0} since T is compactly generated and so we dispose of the following triangle

[figure: ∐_{f∈I_{0}}P; X_{0}; X_{1}; ∐_{f∈I_{0}}P .]

For the induction step consider the above construction with X_{n}, n ≥ 1, in the place of X_{n−1} and I_{n} in the place of I_{n−1}. We dispose of the following diagram

[figure: X = X_{0}; X_{1}; X_{2}; X_{3}; ⋯; X′; ∐_{f∈I_{0}}P; ∐_{f∈I_{1}}P; ∐_{f∈I_{2}}P; ∐_{f∈I_{3}}P; ,]

where X′ denotes the homotopy colimit of the diagram (X_{i})_{i∈ℤ}. Consider now the following triangle

S^{−1}X′ → X′′ → X → X′ ,

where the morphism X → X′ is the transfinite composition in our diagram. Let P be in U. We remark that since P is compact, Hom_{T} (P, X′) = 0. This also implies, by construction of T_{≤0}, that Hom_{T} (R, X′) = 0, for all R in T_{≤0}. The long exact sequence obtained by applying the functor Hom_{T} (R,?) to the triangle above shows that

Hom(R, X′′) ⟶^{∼} Hom(R, X) .

Let X′′_{n−1}, n ≥ 1, be an object as in the following triangle

X = X_{0} → X_{n} → X′′_{n−1} → S(X) .

A recursive application of the octahedron axiom implies that X′′_{n−1} belongs to S(T_{≤0}), for all n ≥ 1. We dispose of the isomorphism

hocolim_{n} X′′_{n−1} ⟶^{∼} S(X′′) .

Since hocolim_{n} X′′_{n−1} belongs to S(T_{≤0}), we conclude that X′′ belongs to T_{≤0}. This shows that the functor that sends X to X′′ is the right adjoint of the inclusion functor T_{≤0} ↪ T . This proves that T_{≤0} is a left aisle on T . We now show that the t-structure associated to T_{≤0}, cf. [25], extends, from T_{c} to T , the one associated with U. Let X be in T_{c}. We dispose of the following truncation triangle associated with U

X_{U} → X → X^{U^{⊥}} → SX_{U} .

Clearly X_{U} belongs to T_{≤0}. We remark that U^{⊥} = T_{≤0}^{⊥}, and so X^{U^{⊥}} belongs to T_{>0} := T_{≤0}^{⊥}. We now show that T_{≤0} is the smallest extension of the left aisle U. Let V be an aisle containing U. The inclusion functor V ↪ T commutes with sums, because it admits a right adjoint. Since V is stable under extensions and suspensions, it contains T_{≤0}.

b) Let X be in T . We need to show that X = 0 iff H^{p}(X) = 0 for all p ∈ ℤ. Clearly the condition is necessary. For the converse, suppose that H^{p}(X) = 0 for all p ∈ ℤ. Let n be an integer. Consider the following truncation triangle

H^{n+1}(X) → τ_{>n}X → τ_{>n+1}X → SH^{n+1}(X) .

Since H^{n+1}(X) = 0 we conclude that

τ_{>n}X ∈ ⋂_{m∈ℤ}T_{>m} ,

for all n ∈ ℤ. Now, let C be a compact object of T . We know that there is a k ∈ ℤ such that C ∈ T_{≤k}. This implies that

Hom_{T} (C, τ_{>n}X) = 0

for all n ∈ ℤ, since τ_{>n}X belongs to (T_{≤k})^{⊥}. The category T is compactly generated and so we conclude that τ_{>n}X = 0, for all n ∈ ℤ. The following truncation triangle

τ_{≤n}X → X → τ_{>n}X → Sτ_{≤n}X ,

implies that τ_{≤n}X is isomorphic to X for all n ∈ ℤ. This can be rephrased as saying that

X ∈ ⋂_{n∈ℕ}T_{≤−n} .

Now by our hypothesis there is an integer N such that

Hom_{T} (C, U_{≤−N}) = 0 .

Since C is compact and by construction of T_{≤−N}, we have

Hom_{T} (C, T_{≤−N}) = 0 .

This implies that Hom_{T} (C, X) = 0, for all compact objects C of T . Since T is compactly generated, we conclude that X = 0. This proves the converse. √

Lemma A.1. Let (Y_{p})_{p∈ℤ} be in T . We dispose of the following isomorphism

H^{n} (∐_{p}Y_{p}) ⟵^{∼} ∐_{p} H^{n}(Y_{p}) ,

for all n ∈ ℤ.

Proof. By definition H^{n} := τ_{≥n} τ_{≤n} , n ∈ ℤ. Since τ_{≥n} admits a right adjoint, it is enough to show that τ_{≤n} commute with infinite sums. We consider the following triangle

∐_{p} τ_{≤n}Y_{p} → ∐_{p} Y_{p} → ∐_{p} τ_{>n}Y_{p} → S(∐_{p} τ_{≤n}Y_{p}) .

Here ∐_{p} τ_{≤n}Y_{p} belongs to T_{≤n} since T_{≤n} is stable under infinite sums. Let P

be an object of S^{n}U. Since P is compact, we have

Hom_{T} (P, ∐_{p} τ_{>n}Y_{p}) ⟵^{∼} ∐_{p} Hom_{T} (P, τ_{>n}Y_{p}) = 0 .

Since T_{≤n} is generated by S^{n}U, ∐_{i} τ_{>n}Y_{p} belongs to T_{>n}. Since the truncation

triangle of ∐_{p} Y_{p} is unique, this implies the following isomorphism

∐_{p} τ_{≤n}Y_{p} ⟶^{∼} τ_{≤n}(∐_{p} Y_{p}) .

This proves the lemma. √

Proposition A.2. Let X be an object of T . Suppose that we are in the conditions of proposition A.1 b). We dispose of the following isomorphism

hocolim_{i} τ_{≤i}X ⟶^{∼} X .

Proof. We need only show that

H^{n}(hocolim_{i} τ_{≤i}X) ⟶^{∼} H^{n}(X) ,

for all n ∈ ℤ. We dispose of the following triangle, cf. [29],

∐_{p} τ_{≤p}X → ∐_{q} τ_{≤q}X → hocolim_{i} τ_{≤i}X → S(∐_{p} τ_{≤p}X) .

Since the functor H^{n} is homological, for all n ∈ ℤ and it commutes with infinite sums by lemma A.1, we obtain a long exact sequence

⋯ → ∐_{p} H^{n}(τ_{≤p}X) → ∐_{q} H^{n}(τ_{≤q}X) → H^{n} (hocolim_{i} τ_{≤i}X) →

→ ∐_{p} H^{n} S(τ_{≤p}X) → ∐_{q} H^{n} S(τ_{≤q}X) → ⋯

We remark that the morphism ∐_{p} H^{n} S(τ_{≤p}X) → ∐_{q} H^{n} S(τ_{≤q}X) is a split

monomorphism and so we obtain

H^{n}(X) = colim_{i} H^{n}(τ_{≤i}X) ⟶^{∼} H^{n}(hocolim_{i} τ_{≤i}X) .
√

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Gonçalo Tabuada Université Paris 7

– Denis Diderot UMR 7586 du CNRS, case 7012 2 Place Jussieu 75251 Paris cedex 05 France tabuada@math.jussieu.fr 214

Documenta Mathematica 12 (2007)