Cluster Equivalence

and Graded Derived Equivalence

Claire Amiot, Steffen Oppermann^{1}

Received: March 7, 2011 Revised: March 6, 2012

Communicated by Wolfgang Soergel

Abstract. In this paper we introduce a new approach for organizing algebras of global dimension at most 2. We introduce the notion of cluster equivalence for these algebras, based on whether their generalized cluster categories are equivalent. We are particularly interested in the question how much information about an algebra is preserved in its generalized cluster category, or, in other words, how closely two algebras are related if they have equivalent generalized cluster categories. Our approach makes use of the cluster-tilting objects in the generalized cluster categories: We first observe that cluster-tilting objects in generalized cluster categories are in natural bijection with cluster-tilting subcategories of derived categories, and then prove a recognition theorem for the latter. Using this recognition theorem we give a precise criterion when two cluster equivalent algebras are derived equivalent. For a given algebra we further describe all the derived equivalent algebras which have the same canonical cluster tilting object in their generalized cluster category. Finally we show that in general, if two algebras are cluster equivalent, then (under certain conditions) the algebras can be graded in such a way that the categories of graded modules are derived equivalent. To this end we introduce mutation of graded quivers with potential, and show that this notion reflects mutation in derived categories.

2010 Mathematics Subject Classification: 16E35, 18E30, 16G20 Keywords and Phrases: cluster category, cluster tilting, derived equivalence, quiver mutation

1Work on this project has been done while both authors were working at NTNU, Trondheim, supported by NFR Storforsk grant no. 167130.

Contents

1. Introduction 1156 Notation 1159 2. Background 1160 2.1. Generalized cluster categories 1160 2.2. Cluster-tilting subcategories 1160 2.3. Basic results on graded algebras 1161 3. Cluster-tilting subcategories determine the derived category 1162 3.1. Bijection between cluster-tilting subcategories 1162 3.2. Recognition theorem 1164 4. Application to Iyama-Yoshino reduction 1168 4.1. Iyama-Yoshino reduction 1168 4.2. Reduction of the derived category 1170 5. Cluster equivalent algebras: the derived equivalent case 1173 5.1. Derived equivalence is graded equivalence 1173 5.2. Classification of tilting complexes 1179 6. Left (and right) mutation in the derived category and graded

quivers with potential 1181 6.1. Relation between left mutation and right mutation 1182 6.2. Jacobian algebras and cluster-tilting objects 1182 6.3. Left (and right) mutation of a graded quiver with potential 1184 6.4. Mutation and generalized cluster categories associated with QP 1186 6.5. Relation between mutation of graded QP and mutation in the

derived category 1187

7. Triangulated orbit categories 1190 7.1. Pretriangulated DG categories 1190 7.2. Universal property 1191 7.3. Iterated triangulated orbit categories 1194 8. Graded derived equivalence for cluster equivalent algebras 1196 8.1. Graded version of results of Section 6 1196 8.2. Graded derived equivalence 1198 References 1205

1. Introduction

Cluster Equivalence and Graded Derived Equivalence 1157

Tilting theory is an essential tool in representation theory of finite dimensional algebras. It permits to link algebras called derived equivalent, i.e. whose bounded derived categories are equivalent (see [Hap88] – also see [AHHK07] for a broader overview). The introduction of cluster algebras goes back to Fomin and Zelevinsky [FZ02]. The categorical interpretation of their combinatorics has been a crucial turn in tilting theory and has brought new perspectives into the field: cluster-tilting theory. The first step was the introduction of the cluster category C_{Q} associated with a finite acyclic quiver Q in [BMR^{+}06] (using [MRZ03]). The cluster category C_{Q} is defined as the orbit category D^{b}(kQ)/𝕊_{2} of the bounded derived category of the path algebra kQ under the action of the autoequivalence 𝕊_{2} := 𝕊[−2] where 𝕊 is the Serre functor of D^{b}(kQ). This category is triangulated (cf. [Kel05]), Hom-finite (the Hom-spaces are finite dimensional) and Calabi-Yau of dimension 2 (2-CY for short), that is there exists a functorial isomorphism Hom_{C_{Q}}(X, Y) ≃ DHom_{C_{Q}}(Y, X[2]) for any objects X and Y, where D is the duality over k. The category C_{Q} has certain special objects called clustertilting objects. From one cluster-tilting object it is possible to construct others using a procedure called mutation, whose combinatorics is very close to the combinatorics developed by Fomin and Zelevinsky for cluster algebras. The images of the tilting kQ-modules under the natural projection π_{Q}: D^{b}(kQ) → C_{Q} are cluster-tilting objects, and tilted algebras (= endomorphism algebras of tilting modules over a path algebra kQ) can be seen as specific quotients of cluster-tilted algebras (= endomorphism algebras of cluster-tilting objects in C_{Q}). The notion of cluster categories has been generalized in [Ami09] replacing the hereditary algebra kQ by an algebra Λ of global dimension at most two. In this case the orbit category D^{b}(Λ)/𝕊_{2} is in general neither triangulated nor Hom-finite. We restrict to the case where it is Hom-finite (the algebra Λ is then said to be τ_{2}-finite). The generalized cluster category is now defined to be the triangulated hull (D^{b}(Λ)/𝕊_{2})_{Δ} in the sense of [Kel05] of the orbit category D^{b}(Λ)/𝕊_{2}. The natural functor

\xymatrix{π_{Λ} : D^{b}(Λ) \ar[r] & D^{b}(Λ)/𝕊_{2} \ar@{^{(}->}[r] & (D^{b}(Λ)/𝕊_{2})_{Δ} =: C_{Λ}}

is triangulated. The aim of this paper is to study to what extent the derived categories are reflected by the cluster categories. More precisely we ask: If two algebras of global dimension at most two have the same cluster category, are they then automatically also derived equivalent? Since the answer to this question is negative in general, we further investigate how the two derived categories are related in case they are not equivalent. Our means towards this goal is to study cluster-tilting objects. It has been shown in [Ami09] that for any tilting complex T in D^{b}(Λ) such that the endomorphism algebra End_{D^{b}(Λ)}(T) is of global dimension ≤ 2, the object π_{Λ}(T) in C_{Λ} is cluster-tilting. Therefore cluster-tilting objects in cluster categories can be seen as analogs of tilting complexes in bounded derived categories. It is an important result in tilting theory that tilting objects determine (in some sense) the triangulated category ([Ric89], or, more generally, [Kel94]). Unfortunately, in cluster-tilting theory an analog of this theorem only exists for cluster categories coming from acyclic quivers (see [KR07]). The first main result of this paper is to provide a ‘cluster-tilting’ analog of the abovementioned theorem from tilting theory. One key observation is that the endomorphism algebra Λ̃ := End_{C_{Λ}}(π(Λ)) has a natural grading given by Λ̃ = ⨁_{p∈ℤ} Hom_{D^{b}(Λ)}(Λ, 𝕊_{2}^{−p}Λ). This graded algebra admits as ℤ-covering the locally finite subcategory

U_{Λ} := π_{Λ}^{−1}(π_{Λ}(Λ)) = add {𝕊_{2}^{p}Λ | p ∈ ℤ} ⊂ D^{b}(Λ),

which is a cluster-tilting subcategory of the derived category D^{b}(Λ).

Theorem 1.1 (Recognition theorem – Theorem 3.5). Let T be an algebraic triangulated category with a Serre functor and with a cluster-tilting subcategory V. Let Λ be a τ_{2}-finite algebra with global dimension ≤ 2. Assume that there is an equivalence of additive categories with 𝕊_{2}-action \xymatrix{f: U_{Λ} \ar[r]^{∼} & V} . Then there exists a triangle equivalence F: D^{b}(Λ) → T .

In contrast to Keller’s and Reiten’s theorem, this result is a recognition theorem for the derived category. That is, we use cluster-tilting theory for studying a classical problem in representation theory. Applying Theorem 1.1 helps us to study the notion of cluster-equivalent algebras, i.e. algebras of global dimension ≤ 2 with the same cluster category, which is the main subject of this paper. Using it, we give the following criterion on when two cluster equivalent algebras Λ_{1} and Λ_{2} with π_{Λ_{1}}Λ_{1} ≃ π_{Λ_{2}}Λ_{2} in the common cluster category are derived equivalent.

Theorem 1.2 (Theorem 5.6). In the above setup the cluster equivalent algebras Λ_{1} and Λ_{2} are derived equivalent if and only if the ℤ-graded algebras Λ̃_{1} and Λ̃_{2} are graded equivalent.

We give a characterization of the tilting complexes T of D^{b}(Λ_{1}) such that π_{Λ_{1}}(T) ≃ π_{Λ_{1}}(Λ_{1}) (Theorem 5.16), with which we can show that in case the equivalent conditions of Theorem 1.2 hold, the algebras Λ_{1} and Λ_{2} are iterated 2-APR tilts of one another. In case the equivalent conditions of Theorem 1.2 do not hold we then consider the case where the two different ℤ-gradings on Λ̃_{1} ≃ Λ̃_{2} are compatible (that is when they induce a ℤ^{2}-grading on Λ̃_{1}; an assumption that seems to be always satisfied in actual examples). In this case, the natural ℤ-grading on Λ̃_{2} induces a ℤ-grading on Λ_{1} and vice versa. We then obtain the following result.

Theorem 1.3 (Theorem 8.7). Let Λ_{1} and Λ_{2} be cluster equivalent, and assume we are in the setup described above. Then there is a triangle equivalence

D^{b}(gr Λ_{1}) ≃ D^{b}(gr Λ_{2})

(where gr Λ_{i} denotes the category of graded Λ_{i}-modules).

These results can be extended in the case where the cluster-tilting objects π_{Λ_{1}}(Λ_{1}) and π_{Λ_{2}}(Λ_{2}) are not isomorphic but are linked by a sequence of mutations. In order to extend them to this setup, we introduce a notion of left (and right) mutation. Indeed the mutation in the generalized cluster category can be lifted to a mutation in the derived category. Furthermore, by a result of Keller [Kel11], the cluster category C_{Λ} is equivalent to a cluster category associated to a quiver with potential. Hence, by a fundamental result of [KY11], endomorphism algebras of cluster-tilting objects related to the canonical one by mutations are Jacobian algebras, and are related to one another by mutations of quivers with potential introduced in [DWZ08]. These two observations lead us to introduce the notion of left (and right) mutation of graded quiver with potential in order to give a combinatorial description of the left (and right) mutation in the derived category. This graded mutation allows us to deduce a combinatorial way to prove that two algebras of global dimension at most 2 are derived equivalent. In particular, as a direct consequence of Theorem 1.2, we obtain the following result which is a generalization of a result due to Happel [Hap87] stating that two path algebras are derived equivalent if and only if they are iterated reflections from one another.

Theorem 1.4 (Corollary 6.14). Let Λ_{1} and Λ_{2} be two algebras of global dimension 2, which are τ_{2}-finite. Assume that one can pass from the graded quiver with potential associated with Λ̃_{1} to the graded quiver with potential associated with Λ̃_{2} using a finite sequence of left and right mutations. Then the algebras Λ_{1} and Λ_{2} are derived equivalent.

In [AO13a] we obtain the converse of Theorem 1.4 in the case where these algebras are cluster equivalent to hereditary algebras. There we also apply the results presented here to understand and describe algebras which are cluster equivalent to tame hereditary algebras.

The paper is organized as follows: Section 2 is devoted to recalling background on generalized cluster categories, cluster-tilting subcategories, and graded algebras. In Section 3 we prove Theorem 1.1. This result is applied to Iyama-Yoshino reduction of derived categories in Section 4. In Section 5 we prove Theorem 1.2, giving a criterion which determines which algebras are derived equivalent among cluster equivalent ones. We further classify derived equivalent algebras having the same canonical cluster-tilting object. We introduce mutation of graded quiver with potential in Section 6. Using a result of Keller and Yang, we show that this notion gives a combinatorial description of mutation in derived categories. In Section 7 we recall and apply to our setup some results on triangulated orbit categories due to Keller. Theorem 1.3, which exhibits gradings on cluster equivalent algebras making them graded derived equivalent, is shown in Section 8.

Notation. Throughout k is an algebraically closed field and all algebras are finite dimensional k-algebras. For a finite-dimensional k-algebra A, we denote by mod A the category of finite-dimensional right A-modules. For an additive k-linear category A we denote by mod A the category of finitely presented functors A^{op} → mod k. By triangulated category we mean k-linear triangulated category satisfying the Krull-Schmidt property. For all triangulated categories we denote the shift functor by [1].

2. Background

This section is devoted to recalling results that will be used in this paper. We first give the definition of generalized cluster categories, and then state some results on cluster-tilting subcategories, and graded algebras.

2.1. Generalized cluster categories. Let Λ be an algebra of global dimension ≤ 2. We denote by D^{b}(Λ) the bounded derived category of finitely generated Λ-modules. It has a Serre functor that we denote by 𝕊. We denote by 𝕊_{2} the composition 𝕊[−2], and by τ_{2} the composition H^{0}𝕊_{2}. The generalized cluster category C_{Λ} of Λ has been defined in [Ami09] as the triangulated hull of the orbit category D^{b}(Λ)/𝕊_{2}. We denote by π_{Λ} the triangle functor

\xymatrix{π_{Λ} : D^{b}(Λ) \ar@{->>}[r] & D^{b}(Λ)/𝕊_{2} \ar@{^{(}->}[r] & C_{Λ}} .

More details on triangulated hulls are given in Section 7 (Example 7.12).

Definition 2.1. An algebra Λ of global dimension ≤ 2 is said to be τ_{2}-finite if τ_{2} is nilpotent.

We set Λ̃ := End_{C_{Λ}}(πΛ) ≃ ⨁_{p≥0} Hom_{D^{b}(Λ)}(Λ, 𝕊_{2}^{−p}Λ). The algebra Λ is τ_{2}-finite if and only if the algebra Λ̃ is finite dimensional. In this case we have the following result:

Theorem 2.2 ([Ami09, Theorem 4.10]). Let Λ be an algebra of global dimension ≤ 2 which is τ_{2}-finite. Then the generalized cluster category C_{Λ} is a Hom-finite, 2-Calabi-Yau category.

2.2. Cluster-tilting subcategories.

Definition 2.3 (Iyama). Let T be a triangulated category, which is Hom-finite. A functorially finite subcategory V of T is cluster-tilting (or 2-cluster-tilting) if

V = {X ∈ T | Hom_{T} (X, V[1]) = 0} = {X ∈ T | Hom_{T} (V, X[1]) = 0}.

We will call an object T of T cluster-tilting if the category add (T) is clustertilting. If T is 2-Calabi-Yau, and T is a cluster-tilting object, the endomorphism algebra End_{T} (T) is called 2-Calabi-Yau-tilted. If the category T has a Serre functor 𝕊, then we have 𝕊_{2}V = V for any cluster-tilting subcategory V where 𝕊_{2} := 𝕊[−2].

Example 2.4. The following examples of cluster-tilting objects will be used in the rest of this paper:

(1) Let Q be an acyclic quiver. If T ∈ mod kQ is a tilting module, then π_{Q}(T) ∈ C_{Q} is a cluster-tilting object in the cluster category C_{Q} ([BMR^{+}06]).

(2) Let Λ be a τ_{2}-finite algebra of global dimension ≤ 2. Let T ∈ D^{b}(Λ) be a tilting complex such that End_{D}(T) has global dimension ≤ 2, then U_{T} = add {𝕊_{2}^{p}T | p ∈ ℤ} is cluster-tilting in D^{b}(Λ) ([Iya11, Theorem 1.22] or [Ami08, Proposition 5.4.2]).

(3) Let Λ be a τ_{2}-finite algebra of global dimension ≤ 2. Let T ∈ D^{b}(Λ) be a tilting complex such that End_{D}(T) has global dimension ≤ 2, then π_{Λ}(T) ∈ C_{Λ} is a cluster-tilting object in C_{Λ} ([Ami09, Theorem 4.10]).

Proposition 2.5 ([KR07]). Let T be a triangulated category with Serre functor 𝕊 and V ⊂ T be a cluster-tilting subcategory. Then for any X in T there exists a triangle called approximation triangle

\xymatrix{V_{1} \ar[r] & V_{0} \ar[r]^{v} & X \ar[r] & V_{1}[1]}

where V_{0} and V_{1} are objects in V and where \xymatrix{v: V_{0} \ar[r] & X} is a minimal right V-approxima-tion.

The following result explains how cluster-tilting subcategories can be mutated.

Theorem 2.6 ([IY08, Theorem 5.3]). Let T be a triangulated category with Serre functor 𝕊 and V ⊂ T be a cluster-tilting subcategory. Let X ∈ V be indecomposable, and set

V′ := add (ind (V) \ {𝕊_{2}^{p}X | p ∈ ℤ}),

where ind (V) denotes the indecomposable objects in V. Then there exists a unique cluster-tilting subcategory V^{∗} with V′ ⊆ V^{∗} ≠ V. Moreover

V^{∗} = add (V′ ∪ {𝕊_{2}^{p}X^{L} | p ∈ ℤ}) = add (V′ ∪ {𝕊_{2}^{p}X^{R} | p ∈ ℤ}),

where X^{L} and X^{R} are obtained via triangles

\xymatrix{X \ar[r]^{f} & B \ar[r] & X^{L} \ar[r] & X[1] & and & X^{R} \ar[r] & B′ \ar[r]^{g} & X \ar[r] & X^{R}[1]}

where f (resp. g) is a minimal left (resp. right) V′-approximation. These triangles are called left and right exchange triangles.

2.3. Basic results on graded algebras. Let G be an abelian group. (In this paper, G will always be ℤ or ℤ^{2}.) Let Λ := ⨁_{p∈G} Λ^{p} be a G-graded algebra. We denote by gr Λ the category of finitely generated graded modules over Λ with degree 0 morphisms. For a graded module M = ⨁_{p∈G} M^{p}, we denote by M⟨q⟩ the graded module ⨁_{p∈ℤ} M^{p+q} (that is, the degree p part of M⟨q⟩ is M^{p+q}). The locally bounded subcategory

Cov(Λ, G) := add {Λ⟨p⟩ | p ∈ G} ⊆ gr Λ

is called the G-covering of Λ.

Theorem 2.7 ([GM94]). Let Λ be a G-graded algebra. Then there is an equivalence

\xymatrix{mod Cov(Λ, G) \ar[r]^{∼} & gr Λ} .

Here is a consequence of [GG82, Theorem 5.3]:

Theorem 2.8 (Gordon-Green). Let Λ be an algebra with two different Ggradings. We denote by Cov(Λ_{1}, G) the G-covering corresponding to the first grading, and Cov(Λ_{2}, G) the G-covering corresponding to the second G-grading. Then the following are equivalent:

(1) There is an equivalence U: mod Cov(Λ_{1}, G) ⟶^{∼} mod Cov(Λ_{2}, G) such that the following diagram commutes.

\xymatrix{mod Cov(Λ_{1}, G) \ar[rr]^{U} \ar[dr] &  & mod Cov(Λ_{2}, G) \ar[dl] \\  & mod Λ}

(2) There exist r_{i} ∈ G and an isomorphism of G-graded algebras

\xymatrix{Λ_{2} \ar[r]^{∼}_{G} & ⨁_{p∈G} Hom_{Cov(Λ_{1},G)}(⨁_{i=1}^{n} P_{i}⟨r_{i}⟩, ⨁_{i=1}^{n} P_{i}⟨r_{i} + p⟩)}

where Λ_{1} = ⨁_{i=1}^{n} P_{i} in gr Λ_{1}. In this case we say that the gradings are equivalent.

3. Cluster-tilting subcategories determine the derived category

3.1. Bijection between cluster-tilting subcategories. In this subsection we show that, for a τ_{2}-finite algebra of global dimension ≤ 2, the projection functor induces a bijection between cluster-tilting subcategories of its cluster category and cluster-tilting subcategories of its derived category.

Proposition 3.1. Let Λ be a τ_{2}-finite algebra of global dimension ≤ 2. Let T ∈ C_{Λ} be a cluster-tilting object. Then π^{−1}(T) ⊂ D^{b}(Λ) is a cluster-tilting subcategory of D^{b}(Λ).

Proof. Since the functor π_{Λ} : D^{b}(Λ) → C_{Λ} is triangulated, we get the inclusions

π^{−1}(T) ⊂ {X ∈ D^{b}(Λ) | Hom_{D^{b}(Λ)}(X, π^{−1}(T)[1]) = 0}, and
π^{−1}(T) ⊂ {X ∈ D^{b}(Λ) | Hom_{D^{b}(Λ)}(π^{−1}(T), X[1]) = 0}

Let T ≃ T_{1} ⊕ ⋯ ⊕ T_{n} be the decomposition of T in indecomposable objects. For all i = 1, …, n, the object T_{i} is rigid. Hence, by [AO13b], there exists T′_{i} ∈ D^{b}(Λ) such that π(T′_{i}) = T_{i}. Now let X be in {X ∈ D^{b}(Λ) | Hom_{D^{b}(Λ)}(π^{−1}(T), X[1]) = 0}. Therefore for all i = 1, …, n and all p ∈ ℤ the space Hom_{D^{b}(Λ)}(𝕊_{2}^{p}T′_{i}, X[1]) vanishes. Then for all i = 1, …, n we have

Hom_{C}(T_{i}, π(X)[1]) = Hom_{C}(π(T′_{i}), π(X)[1]) ≃ ⨁_{p∈ℤ} Hom_{D}(𝕊_{2}^{p}T′_{i}, X[1]) = 0.

Since T is cluster-tilting, we have π(X) ∈ add (T), thus X ∈ π^{−1}(T). The last inclusion is shown similarly. □

The following proposition shows that the converse is also true.

Proposition 3.2. Let Λ be a τ_{2}-finite algebra of global dimension ≤ 2. Let V be a cluster-tilting subcategory of D^{b}(Λ). Then π_{Λ}(V) is a cluster-tilting subcategory of C_{Λ}.

In the proof we will need the following piece of notation:

Definition 3.3. For U and V subcategories of a triangulated category T we denote by U ∗V the full subcategory of T consisting of objects M of T appearing in a triangle \xymatrix{U \ar[r] & M \ar[r] & V \ar[r] & U[1]} with U ∈ U and V ∈ V.

One easily sees that U is a cluster-tilting subcategory of T if and only if T = U ∗ U[1] and Hom_{T} (U, U[1]) = 0.

Proof of Proposition 3.2. Let X and Y be objects in V. Then we have

Hom_{C_{Λ}}(π(X), π(Y)[1]) = Hom_{C_{Λ}}(π(X), π(Y [1])) =

= ⨁_{p∈ℤ} Hom_{D^{b}(Λ)}(X, 𝕊_{2}^{p}Y [1]) = 0

since 𝕊_{2}^{p}Y ∈ 𝕊_{2}^{p}V = V. Therefore we have Hom_{C_{Λ}}(π(V), π(V)[1]) = 0. Denote by U := U_{Λ} = π^{−1}(πΛ) the canonical cluster-tilting subcategory of D^{b}(Λ). Then since V is cluster-tilting in D^{b}(Λ) we have U ⊂ V[−1] ∗ V. Since π is a triangle functor, we have the inclusions

π(U) ⊂ π(V[−1] ∗ V) ⊂ π(V)[−1] ∗ π(V).

Now since π(U) is a cluster-tilting subcategory of C_{Λ} we have:

C_{Λ} = π(U) ∗ π(U)[1] = (π(V)[−1] ∗ π(V)) ∗ (π(V) ∗ π(V)[1]).

Since Hom_{C_{Λ}}(π(V), π(V)[1]) = 0, we have π(V) ∗ π(V) = π(V). Therefore, by associativity of ∗, we get

C_{Λ} = π(V)[−1] ∗ (π(V) ∗ π(V)) ∗ π(V)[1] = π(V)[−1] ∗ π(V) ∗ π(V)[1].

Now let X ∈ C_{Λ} such that Hom_{C_{Λ}}(X, π(V)[1]) = 0. There exists triangles

\xymatrix{π(V_{1})[−1] \ar[r] & π(V_{0})[−1] \ar[r] & Y \ar[r] & π(V_{1})}

and

\xymatrix{π(V_{2}) \ar[r] & Y \ar[r] & X \ar[r] & π(V_{2})[1]}

with V_{0}, V_{1}, V_{2} in V. Since Hom_{C_{Λ}}(X, π(V)[1]) = 0 the second triangle splits and we have X ⊕ π(V_{2}) ≃ Y. Then Hom_{C_{Λ}}(π(V)[−1], Y) = 0 and the first triangle splits. Hence we have π(V_{1}) ≃ Y ⊕ π(V_{0}) ≃ X ⊕ π(V_{2}) ⊕ π(V_{0}) and X ∈ π(V). Therefore π(V) is a cluster-tilting subcategory of C_{Λ}. □

3.2. Recognition theorem. The aim of this subsection is to prove Theorem 3.5. Since its setup is that of algebraic triangulated categories we recall the definition.

Definition 3.4. A triangulated category T is called algebraic if there is a Frobenius exact category E such that T = \underline{E}.

Theorem 3.5. Let T be an algebraic triangulated category with a Serre functor and with a cluster-tilting subcategory V. Let Λ be a τ_{2}-finite algebra with global dimension ≤ 2. Assume that there is an equivalence of additive categories with 𝕊_{2}-action \xymatrix{f: U_{Λ} \ar[r]^{∼} & V} (with U_{Λ} as in Example 2.4(2)). Then

there exists a triangle equivalence \xymatrix{F: D^{b}(Λ) \ar[r] & T} such that the following diagram commutes

\xymatrix{D^{b}(Λ) \ar[r]^{F} & T \\ U_{Λ} \ar[r]^{f} \ar@{^{(}->}[u] & V. \ar@{^{(}->}[u]}

Our strategy for the proof is as follows: We introduce the category of radical morphisms mor V (see Definition 3.6) for a cluster-tilting subcategory V. In the setup of Theorem 3.5 it follows that mor V and mor U_{Λ} are equivalent. We would like to complete the proof by saying that mor V is equivalent to T (and similarly for U_{Λ}), but unfortunately we do not have such an equivalence but just a bijection on objects (see Lemma 3.8). We will see that this bijection is nice enough to make the image of Λ a tilting object in T . Then the theorem follows from the tilting theorem.

Definition 3.6. Let U be a locally finite k-category. That is, for any indecomposable object X of U there are (up to isomorphism) only finitely many indecomposables Y such that U(X, Y) ⊕ U(Y, X) ≠ 0. We define the category mor U. Objects are radical maps U_{1} → U_{0} in U and morphisms are commutative squares.

Let u: U_{1} → U_{0} be an object in U. Since U is locally finite, the kernel M of the map

\xymatrix{0 \ar[r] & M \ar[r] & DU(U_{1}, −) \ar[r]^{u^{∗}} & DU(U_{0}, −)}

is in mod U the category of finitely presented functors U^{op} → mod k. Hence there exists a map h: H_{1} → H_{0} such that

\xymatrix{U(−, H_{1}) \ar[r]^{h_{∗}} & U(−, H_{0}) \ar[r] & M \ar[r] & 0}

is the minimal projective resolution of M. This map is uniquely defined up to isomorphism. Therefore we can define the map H: mor U → mor U as Hu = h. Similarly, we define H^{−}: mor U → mor U.

Lemma 3.7. If u ∈ mor U is left minimal then we have H^{−}Hu ≃ u. If u ∈ mor U is right minimal then we have HH^{−}u ≃ u.

Proof. The morphism u: U_{1} → U_{0} is left minimal if and only if the injective resolution

\xymatrix{0 \ar[r] & M \ar[r] & DU(U_{1}, −) \ar[r]^{u^{∗}} & DU(U_{0}, −)}

is minimal, hence we get the result. □

Let V ⊂ T and Λ be as in Theorem 3.5. Since Λ̃ is finite-dimensional, the category U = add {𝕊_{2}^{p}Λ | p ∈ ℤ} is locally finite, hence so is the category V. The autoequivalence 𝕊_{2} of V induces an autoequivalence of mor V that we denote also by 𝕊_{2}. Each map v: V_{1} → V_{0} decomposes in the direct sum of a left minimal map and a map of the form [0 → V_{2}]. Hence we can define a map Σ: mor V → mor V by

Σv = \{ \begin{matrix}H𝕊_{2}^{−}v & if v is left minimal \\ [V_{0} → 0] & if v = [0 → V_{0}]. \end{matrix}

This is clearly a bijection whose inverse is

Σ^{−}v = \{ \begin{matrix}𝕊_{2}H^{−}v & if v is right minimal \\ [0 → V_{1}] & if v = [V_{1} → 0]. \end{matrix}

Lemma 3.8. Let V ⊂ T and Λ be as in Theorem 3.5. Then the map

Cone: \xymatrix{mor V \ar[r] & T \\ v \ar@{|->}[r] & Cone(v)}

is a bijection on isomorphism classes of objects of the categories mor V and T . Moreover we have

(1) Cone(Σv) ≃ (Cone(v))[1];

(2) Cone(𝕊_{2}v) ≃ 𝕊_{2}(Cone(v)).

Proof. If two objects u and v of mor V are isomorphic, then their cones are isomorphic. Hence this map is well-defined on isomorphism classes of objects. By Proposition 2.5, for each T ∈ T there exists a triangle

\xymatrix{V_{1} \ar[r]^{v} & V_{0} \ar[r]^{w} & T \ar[r] & V_{1}[1]} .

The map v is in the radical if and only if w is a minimal right V-approximation. Since minimal right approximations exist and are unique up to isomorphism, the map Cone is bijective. Let v: 0 → V_{0} be in mor V. Then we have

Cone(Σv) = Cone([V_{0} → 0])
= V_{0}[1]
= (Cone(v))[1].

Let v: V_{1} → V_{0} be left minimal in mor V. Let h: H_{1} → H_{0} be H𝕊_{2}^{−}(v). Then we have an exact sequence in mod V:

\xymatrix{Hom_{T} (V, H_{1}) \ar[r]^{h_{∗}} & Hom_{T} (V,H_{0}) \ar[r] & DHom_{T} (𝕊_{2}^{−}V_{1},V) \ar[r]^{(𝕊_{2}^{−}v)^{∗}} & DHom_{T} (𝕊_{2}^{−}V_{0},V)}

By definition of 𝕊_{2} this sequence is isomorphic to

\xymatrix{Hom_{T} (V, H_{1}) \ar[r]^{h_{∗}} & Hom_{T} (V, H_{0}) \ar[r] & Hom_{T} (V, V_{1}[2]) \ar[r]^{v[2]^{∗}} & DHom_{T} (V, V_{0}[2])}

Since V is cluster-tilting, the space Hom_{T} (V, H_{1}[1]) vanishes and the cokernel of h_{∗} is isomorphic to Hom_{T} (V, Cone(h)). Since Hom_{T} (V, V_{0}[1]) vanishes, the kernel of the map v[2]^{∗} is Hom_{T} (V, Cone(v)[1]). Note that, by definition of H, the map h is right minimal. Hence Cone(h) does not have any non-zero direct summands in V[1]. Similarly, since v is left minimal, Cone(v) does not have any non-zero direct summands in V, and thus Cone(v)[1] does not have any non-zero summands in V[1]. By the equivalence Hom_{T} (V, −): T /(V[1]) → mod V (see [KR07]) the isomorphism Hom_{T} (V, Cone(h)) ≃ Hom_{T} (V, Cone(v)[1]) implies that also Cone(h) ≃ Cone(v)[1]. Hence we get

Cone(Σv) = Cone(H𝕊_{2}^{−}v)
= Cone(h)
≃ Cone(v)[1]

and we have (1). Assertion (2) is immediate. □

Lemma 3.9. In the setup of Theorem 3.5, let u be in mor U. Then for all p ∈ ℤ we have isomorphisms

Hom_{T} (fΛ, Cone(fu)[p]) ≃ Hom_{D^{b}(Λ)}(Λ, Cone(u)[p]).

Proof. Let \xymatrix{U_{1} \ar[r]^{u} & U_{0} \ar[r] & X \ar[r] & U_{0}[1]} be a triangle in D^{b}(Λ) with u ∈ mor U. Let

\xymatrix{U_{1}^{p} \ar[r]^{u^{p}} & U_{0}^{p} \ar[r] & X[p] \ar[r] & U_{0}^{p}[1]}

be a U-approximation triangle of X[p] in D^{b}(Λ) with u^{p} ∈ mor U. Then we have

Cone(f(u^{p})) ≃ Cone(f(Σ_{U}^{p}(u))) by Lemma 3.8(1),
≃ Cone(Σ_{V}^{p}(fu)) since f is an equivalence of 𝕊_{2}-categories
≃ Cone(fu)[p]  by Lemma 3.8(1).

Thus we have a triangle in T

\xymatrix{f(U_{1}^{p}) \ar[r]^{fu^{p}} & f(U_{0}^{p}) \ar[r] & Cone(fu)[p] \ar[r] & (fU_{1}^{p})[1]}

which gives an exact sequence

Hom_{T} (fΛ, fU_{1}^{p}) → Hom_{T} (fΛ, fU_{0}^{p}) → → Hom_{T} (fΛ, Cone(fu)[p]) → Hom_{T} (fΛ, fU_{1}^{p}[1]).

The space Hom_{T} (fΛ, fU_{1}^{p}[1]) vanishes since fU = V is a cluster-tilting subcategory of T . But since f is an equivalence we have

\xymatrix{Hom_{T} (fΛ, fU_{1}^{p}) \ar[r] & Hom_{T} (fΛ, fU_{0}^{p}) \ar[r] & Hom_{T} (fΛ, Cone(fu)[p]) \ar[r] & 0 \\ Hom_{D^{b}(Λ)}(Λ, U_{1}^{p}) \ar[u]^{≀} \ar[r] & Hom_{D^{b}(Λ)}(Λ, U_{1}^{p}) \ar[r] \ar[u]^{≀} & Hom_{D^{b}(Λ)}(Λ, Cone(u)[p]) \ar[r] & 0}

where the epimorphism in the second row follows from Hom_{D^{b}(Λ)}(Λ, U_{1}^{p}[1]) = 0. Hence we get

Hom_{T} (fΛ, Cone(fu)[p]) ≃ Hom_{D^{b}(Λ)}(Λ, Cone(u)[p]).  □

Proof of Theorem 3.5. Applying Lemma 3.9 to u = [0 → Λ] we get for each p ∈ ℤ

Hom_{T} (fΛ, fΛ[p]) ≃ Hom_{D^{b}(Λ)}(Λ, Λ[p]) = 0.

Therefore the object fΛ is a tilting object in the category T . We will use the following theorem which can be deduced from [Kel07, Theorem 8.5]:

Theorem 3.10 (Keller). Let T be a Hom-finite algebraic triangulated category. Let T ∈ T be a tilting object of T , i.e. for any i ≠ 0 the space Ext_{T}^{i} (T, T) vanishes. Denote by Λ the endomorphism algebra End_{T} (T) and assume it is of finite global dimension. Then there exists an algebraic equivalence F: D^{b}(Λ) → thick_{T} (T) sending the object Λ on T where thick_{T} (T) is the smallest triangulated subcategory of T containing T and stable under direct summands.

Hence we have an equivalence

\xymatrix{D^{b}(Λ) \ar[r]^{∼} & thick (fΛ) ⊂ T}

where thick (fΛ) is the thick subcategory of T generated by f(Λ). It remains to show that thick (fΛ) = T . Since Λ has finite global dimension it suffices to show that the only object Y ∈ T such that Hom_{T} (fΛ, Y [p]) = 0 ∀p ∈ ℤ is 0. So let Y ∈ T be such that for each p ∈ ℤ the space Hom_{T} (fΛ, Y [p]) vanishes. Form an approximation triangle in T

\xymatrix{V_{1} \ar[r]^{v} & V_{0} \ar[r] & Y \ar[r] & V_{1}[1]} .

Since f: U → V is an equivalence and since v is in mor V, there exists u: U_{0} → U_{1} ∈ mor U such that f(u) = v. Denote by X the cone of u. Then by Lemma 3.9 we have an isomorphism

Hom_{D^{b}(Λ)}(Λ, X[p]) ≃ Hom_{T} (fΛ, Y [p]) = 0 for all p ∈ ℤ.

Hence we have X = 0 and U_{0} = U_{1} = 0. Therefore we have Y = 0. Hence we have an equivalence \xymatrix{D^{b}(Λ) \ar[r]^{∼} & T} .

It remains to prove that the following diagram is commutative.

\xymatrix{D^{b}(Λ) \ar[r]^{∼}_{F} & T \\ U \ar[r]_{f}^{∼} \ar@{^{(}->}[u] & V \ar@{^{(}->}[u]}

Since F (Λ) = f(Λ) and since U is the cluster-tilting subcategory add {_{Λ}𝕊_{2}^{p}Λ | p ∈ ℤ}, it is enough to prove that the functor F commutes with 𝕊_{2}. This is clear by the uniqueness of the Serre functor in a triangulated category. □

4. Application to Iyama-Yoshino reduction

In this section, as an application of the recognition theorem (Theorem 3.5), we show that certain Iyama-Yoshino reductions of derived categories are derived categories again. For lightening the writing, in this section we denote by T (X, Y) the space of morphisms Hom_{T} (X, Y) in the category T . If U is a subcategory of T , we denote by [U](X, Y) the space of morphism in T between X and Y factorizing through an object in U. If T is a triangulated category with Serre functor _{T} 𝕊, we set _{T} 𝕊_{2} = _{T} 𝕊[−2], and simply write 𝕊_{2} for _{T} 𝕊_{2} when there is no danger of confusion.

4.1. Iyama-Yoshino reduction. This subsection is devoted to recalling some results of [IY08]. Let D be a triangulated k-category which is Hom-finite and with a Serre functor D𝕊. Let U be a full subcategory having the following properties:

• U is rigid, that is D(U, U[1]) = 0;

• U is functorially finite, that is any object of D has a right and a left U-approximation;

• U is stable under _{D}𝕊_{2} = _{D}𝕊[−2]. We define the full subcategory Z of D by

Z = {X ∈ D | D(U, X[1]) = 0}.

We denote by T the category Z/[U]. Its objects are those of Z and for X and Y in Z we have

T (X, Y) := D(X, Y)/[U](X, Y).

For X in Z, let X → U_{X} be a left U-approximation. We define X{1} to be the cone

\xymatrix{X \ar[r] & U_{X} \ar[r] & X\{1\} \ar[r] & X[1]} .

Remark 4.1. In [IY08], Iyama and Yoshino write X⟨1⟩ instead of X{1}. Here we deviate from their notation because in this paper pointy brackets are used to denote degree shifts.

Theorem 4.2 ([IY08]). The category T is triangulated, with shift functor {1} and Serre functor _{T} 𝕊 = _{D}𝕊_{2}{2}. Moreover there is a 1-1 correspondence between cluster-tilting subcategories of D containing U and cluster-tilting subcategories in T .

In this construction, for any triangle \xymatrix{X \ar[r] & Y \ar[r] & Z \ar[r] & X[1]} in D such that X, Y and Z are in Z, we have a morphism of triangles

[figure: X; u; Y; v; Z; w; X[1]; X; U_{X}; X{1}; X[1]]

Then the image of \xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & X\{1\}} in T is a triangle. Here we need the following version of Iyama-Yoshino reduction for the setup of algebraic triangulated categories.

Proposition 4.3. In the setup of Theorem 4.2, if D is algebraic triangulated then so is T .

Proof. Since D is algebraic triangulated we have D = \underline{E} for some Frobenius exact category E. We denote by F the preimage of Z in E. Since F is closed under extensions F is an exact category (whose exact sequences are those exact sequences in E which lie entirely in F). Let V be the preimage of U in E. Then clearly V ⊆ F. We claim that the objects in V are projective. Indeed if we have a short exact sequence

[figure: F_{1}; F_{2}; V]

in F with V ∈ V then, since Hom_{D}(V, F_{1}[1]) = 0 by definition of Z, the sequence splits. Similarly the objects in V are also injective. Since V contains all projective-injective objects in E one sees that for any F ∈ F the right V-approximation \xymatrix{V \ar[r] & F} is an admissible epimorphism in E. One easily checks that its kernel is again in F, so that the approximation is also an admissible epimorphism in F. Hence F has enough projectives, and these are precisely the objects in V. Dually F has enough injectives, which are again the objects in V. Thus F is Frobenius exact, and

T = Z/[U] = F/[V] = \underline{F}

is algebraic triangulated. □

Remark 4.4. The proof of Proposition 4.3 is also a simpler proof for Theorem 4.2 in the case when D is algebraic triangulated.

4.2. Reduction of the derived category. Let Λ = kQ/I be a τ_{2}-finite algebra of global dimension ≤ 2. Let i_{0} ∈ Q_{0} be a source of Q and e := e_{i_{0}} be the associated primitive idempotent of Λ. We apply Iyama-Yoshino’s construction for U_{e} = add {𝕊_{2}^{p}(eΛ) | p ∈ ℤ} ⊂ D^{b}(Λ). That is, we denote by Z the full subcategory of D := D^{b}(Λ)

Z := {X ∈ D | D(𝕊_{2}^{p}(eΛ), X[1]) = 0 ∀p ∈ ℤ}.

Then, by Proposition 4.3, the category T = Z/[U_{e}] is algebraic triangulated. Denote by Λ′ the algebra Λ/ΛeΛ ≃ (1 − e)Λ(1 − e). Since i_{0} is a source of the quiver Q, the algebra Λ is a one point extension of Λ′, namely

Λ = [\begin{matrix}Λ′ & (1 − e)Λe \\ 0 & k \end{matrix}] .

Then the projective Λ-modules are

(1 − e)Λ = [Λ′ (1 − e)Λe] and eΛ = [0 k] ,

and the injective Λ-modules are

(1 − e)DΛ = [DΛ′ 0] and eDΛ = [eDΛ(1 − e) k] .

Lemma 4.5. The algebra Λ′ is a k-algebra of global dimension ≤ 2.

Proof. Since i_{0} is a source of Q, for i ∈ Q_{0} with i ≠ i_{0} the minimal injective resolution of the simple S_{i} in mod Λ does not contain the injective module eDΛ. Therefore, using the description of injectives above, this injective resolution can be seen as an injective resolution in mod Λ′. □

The aim of this section is to prove the following theorem:

Theorem 4.6. There is a triangle equivalence D′ := D^{b}(Λ′) ≃ T := Z/[U_{e}].

We first prove several lemmas.

Lemma 4.7. We have an isomorphism

D′(𝕊_{2}Λ′, Λ′) ≃ D(𝕊_{2}((1 − e)Λ), (1 − e)Λ)/[eΛ].

Proof. We choose a projective resolution of DΛ′ in mod Λ′.

(∗)  \xymatrix{0 \ar[r] & P_{2} \ar[r] & P_{1} \ar[r] & P_{0} \ar[r] & DΛ′ \ar[r] & 0}.

Since Λ is a one point extension of Λ′ there is a short exact sequence

(†)  \xymatrix{0 \ar[r] & (1 − e)Λe \ar[r] & (1 − e)Λ \ar[r] & Λ′ \ar[r] & 0}

in mod (Λ^{′op} ⊗ Λ), where (1 − e)Λe is the Λ′-Λ-bimodule [0 (1 − e)Λe]. Note that as Λ-module this is just eΛ^{dim_{k}(1−e)Λe}. Applying P_{i} ⊗_{Λ′} − to (†) for i = 0, 1, 2 we obtain short exact sequences

[figure: 0; P_{i} ⊗_{Λ′} (1 − e)Λe; P_{i} ⊗_{Λ′} (1 − e)Λ; P_{i}; 0; P_{i}Λe; P_{i}Λ].

Inserting these in (∗) we obtain the following projective resolution of the Λmodule DΛ′:

\xymatrix{0 \ar[r] & P_{2}Λe \ar[r] & \begin{matrix}P_{2}Λ \\ ⊕ \\ P_{1}Λe \end{matrix} \ar[r] & \begin{matrix}P_{1}Λ \\ ⊕ \\ P_{0}Λe \end{matrix} \ar[r] & P_{0}Λ \ar[r] & DΛ′ \ar[r] & 0}

Since Λ is of global dimension ≤ 2, the map \xymatrix{P_{2}Λe \ar[r] & P_{1}Λe} is a split monomorphism, hence we can write

\xymatrix{0 \ar[r] & \begin{matrix}P_{2}Λ \\ ⊕ \\ eΛ^{m} \end{matrix} \ar[r] & \begin{matrix}P_{1}Λ \\ ⊕ \\ P_{0}Λe \end{matrix} \ar[r] & P_{0}Λ \ar[r] & DΛ′ \ar[r] & 0}

for some m ∈ ℕ. Since e is attached to a source of the quiver Q, the space [U_{e}]((1 − e)Λ, (1 − e)Λ) vanishes and we have

D(𝕊_{2}((1 − e)Λ), (1 − e)Λ)/[eΛ] = Coker(D(P_{1}Λ, (1 − e)Λ) → D(P_{2}Λ, (1 − e)Λ)).

Since i_{0} is a source of the quiver of Λ, if i, j ≠ i_{0} we have D(e_{i}Λ, e_{j}Λ) = D′(e_{i}Λ′, e_{j}Λ′). Hence we have

D(𝕊_{2}((1 − e)Λ),(1 − e)Λ)/[eΛ] ≃ Coker(D(P_{1}Λ, (1 − e)Λ) → D(P_{2}Λ, (1 − e)Λ))
≃ Coker(D(P_{1}, Λ′) → D(P_{2}, Λ′))
≃ D′(𝕊_{2}Λ′, Λ′).  □

Lemma 4.8. For any p ≥ 1 the composition map

T (𝕊_{2}^{−p+1}Λ, 𝕊_{2}^{−p}Λ) ⊗_{Λ} ⋯ ⊗_{Λ} T (𝕊_{2}^{−1}Λ, 𝕊_{2}^{−2}Λ)⊗_{Λ} T (Λ, 𝕊_{2}^{−1}Λ) → T (Λ, 𝕊_{2}^{−p}Λ)

is an isomorphism.

Proof. By definition of T for any X, Y ∈ T we have an exact sequence

\xymatrix{[U_{e}](X, Y) \ar[r] & D(X, Y) \ar[r] & T (X, Y) \ar[r] & 0.}

Hence we obtain the following diagram

\xymatrix{0 & 0 \\ T (𝕊_{2}^{−p+1}Λ, 𝕊_{2}^{−p}Λ) ⊗_{Λ} … ⊗_{Λ} T (Λ, 𝕊_{2}^{−1}Λ) \ar[r] \ar[u] & T (Λ, 𝕊_{2}^{−p}Λ) \ar[u] \\ D(𝕊_{2}^{−p+1}Λ, 𝕊_{2}^{−p}Λ) ⊗_{Λ} … ⊗_{Λ} D(Λ, 𝕊_{2}^{−1}Λ) \ar[u] \ar[r] & D(Λ, 𝕊_{2}^{−p}Λ) \ar[u] \\ ∗ \ar[r] \ar[u] & [U_{e}](Λ, 𝕊_{2}^{−p}Λ) \ar[u]}

Documenta Mathematica 19 (2014) 1155–1206

1172 Claire Amiot, Steffen Oppermann

with exact columns, and with

∗ = ⨁_{j=1}^{p} D(𝕊_{2}^{−p+1}Λ, 𝕊_{2}^{−p}Λ) ⊗_{Λ} ⋯ ⊗_{Λ} [U_{e}](𝕊_{2}^{−j+1}Λ, 𝕊_{2}^{−j}Λ) ⊗_{Λ} … ⊗_{Λ} D(Λ, 𝕊_{2}^{−1}Λ)

The surjectivity of the composition map

\xymatrix{T (𝕊_{2}^{−p+1}Λ, 𝕊_{2}^{−p}Λ) ⊗_{Λ} ⋯ ⊗_{Λ} T (Λ, 𝕊_{2}^{−1}Λ) \ar[r] & T (Λ, 𝕊_{2}^{−p}Λ)}

is now consequence of the following result: Lemma 4.9 ([Ami09]). The composition map

\xymatrix{D(𝕊_{2}^{−p+1}Λ, 𝕊_{2}^{−p}Λ) ⊗_{Λ} ⋯ ⊗_{Λ} D(Λ, 𝕊_{2}^{−1}Λ) \ar[r] & D(Λ, 𝕊_{2}^{−p}Λ)}

is an isomorphism. We now prove that the map

⨁_{j=1}^{p} D(𝕊_{2}^{−p+1}Λ, 𝕊_{2}^{−p}Λ) ⊗_{Λ} ⋯ ⊗_{Λ} [U_{e}](𝕊_{2}^{−j+1}Λ, 𝕊_{2}^{−j}Λ) ⊗_{Λ} … ⊗_{Λ} D(Λ, 𝕊_{2}^{−1}Λ)

[figure: [U_{e}](Λ, 𝕊_{2}^{−p}Λ)]

is surjective. Any morphism in [U_{e}](Λ, 𝕊_{2}^{−p}Λ) is a sum of morphisms factoring through various 𝕊_{2}^{−q}eΛ, with 0 ≤ q ≤ p. Since the right radical add {𝕊_{2}^{i}Λ | i ∈ ℤ}-approximation of 𝕊_{2}^{−q}eΛ lies in add 𝕊_{2}^{−q+1}Λ, and the left radical add {𝕊_{2}^{i}Λ | i ∈ ℤ}-approximation of 𝕊_{2}^{−q}eΛ lies in add 𝕊_{2}^{−q}Λ, we have that any map Λ → 𝕊_{2}^{−p}Λ factoring through 𝕊_{2}^{−q}eΛ lies in the image of

D(𝕊_{2}^{−p+1}Λ, 𝕊_{2}^{−p}Λ) ⊗_{Λ} ⋯ ⊗_{Λ} [U_{e}](𝕊_{2}^{−q+1}Λ, 𝕊_{2}^{−q}Λ) ⊗_{Λ} … ⊗_{Λ} D(Λ, 𝕊_{2}^{−1}Λ) →

→ [U_{e}](Λ, 𝕊_{2}^{−p}Λ).

Therefore, using the above diagram, the composition map

\xymatrix{T (𝕊_{2}^{−p+1}Λ, 𝕊_{2}^{−p}Λ) ⊗_{Λ} ⋯ ⊗_{Λ} T (Λ, 𝕊_{2}^{−1}Λ) \ar[r] & T (Λ, 𝕊_{2}^{−p}Λ)}

is an isomorphism. □ Lemma 4.10. For any p ∈ ℤ, we have D′(Λ′, 𝕊_{2}^{p}Λ′) ≃ T (Λ, 𝕊_{2}^{p}Λ). Proof. For p ≥ 0 both side vanishes since Λ and Λ′ are of global dimension at most 2. The case p = −1 is Lemma 4.7 since we have

[U_{e}]((1 − e)Λ, 𝕊_{2}^{−1}(1 − e)Λ) = [𝕊_{2}^{−1}eΛ]((1 − e)Λ, 𝕊_{2}^{−1}(1 − e)Λ). Using Lemmas 4.7 and 4.8 we show the assertion for any p ≤ −1 by an easy induction and using the fact that T (𝕊_{2}^{−1}Λ, 𝕊_{2}^{−2}Λ) ⊗_{Λ} T (Λ, 𝕊_{2}^{−1}Λ) ≃

≃ T (𝕊_{2}^{−1}(1 − e)Λ, 𝕊_{2}^{−2}(1 − e)Λ) ⊗_{Λ′} T ((1 − e)Λ, 𝕊_{2}^{−1}(1 − e)Λ).

□

Documenta Mathematica 19 (2014) 1155–1206

Remark 4.11. This lemma can also be proved using Theorem 6.2, but we think it is good to also have a direct proof.

Proof of Theorem 4.6. The strategy of the proof is to use the recognition theorem (Theorem 3.5). The category U_{Λ} = add {_{Λ}𝕊_{2}^{p}Λ | p ∈ ℤ} is a clustertilting subcategory of D which contains U_{e}. Therefore, by Theorem 4.2, its image under the natural functor Z → Z/[U_{e}] = T is a cluster-tilting subcategory of T . By Lemma 4.10 the category U_{Λ}/[U_{e}] is equivalent to the category U_{Λ′} = add {_{Λ′}𝕊_{2}^{p}Λ′ | p ∈ ℤ} ⊂ D′ = D^{b}(Λ′) as category with 𝕊_{2}-action. Therefore, by Theorem 3.5, we get an triangle equivalence D′ ≃ T . □

Remark 4.12. (1) Theorem 4.6 also holds if i_{0} is a sink of the quiver of Λ.

(2) This result is related to [Kel11, Theorem 7.4], where the author proves that the Iyama-Yoshino reduction of the generalized cluster category associated with a Jacobi-finite quiver with potential at a vertex is again a generalized cluster category associated with a Jacobi-finite quiver with potential.

5. Cluster equivalent algebras: the derived equivalent case

In this section we give a criterion for two cluster equivalent algebras to be derived equivalent. The main tool for proving this criterion is the recognition theorem (Theorem 3.5). Further we study derived equivalent algebras satisfying the assumption that the canonical cluster-tilting objects in the common cluster category are isomorphic. We show that these algebras are all iterated 2-APR tilts of one another.

5.1. Derived equivalence is graded equivalence.

Definition 5.1. Two τ_{2}-finite algebras Λ_{1} and Λ_{2} of global dimension ≤ 2 will be called cluster equivalent if there exists a triangle equivalence between their generalized cluster categories C_{Λ_{1}} and C_{Λ_{2}}.

Proposition 5.2. Let Λ_{1} and Λ_{2} be τ_{2}-finite algebras of global dimension ≤ 2. If Λ_{1} and Λ_{2} are derived equivalent, then there exists an equivalence F^{D}: D^{b}(Λ_{1}) → D^{b}(Λ_{2}) which induces an equivalence F: C_{Λ_{1}} → C_{Λ_{2}} as in the following diagram.

\xymatrix{D^{b}(Λ_{1}) \ar[r]^{F^{D}} \ar[d]^{π_{1}} & D^{b}(Λ_{2}) \ar[d]^{π_{2}} \\ C_{Λ_{1}} \ar[r]^{F} & C_{Λ_{2}}}

In particular derived equivalent algebras are cluster equivalent.

We refer to Section 7 for a formal proof of this proposition (Corollary 7.16).

Remark 5.3. As we will see in the examples later, algebras which are not derived equivalent can still be cluster equivalent.

As a consequence of Theorem 3.5 we have a first version of a criterion for cluster equivalent algebras to be derived equivalent.

Corollary 5.4. Let Λ_{1} and Λ_{2} be two τ_{2}-finite algebras of global dimension ≤ 2 which are cluster equivalent. Denote by π_{1} (resp. π_{2}) the canonical functor D^{b}(Λ_{1}) → C_{Λ_{1}} (resp. D^{b}(Λ_{2}) → C_{Λ_{2}}). Then the following are equivalent

(1) Λ_{1} and Λ_{2} are derived equivalent;

(2) there exists an 𝕊_{2}-equivalence between the categories π_{2}^{−1}(F π_{1}Λ_{1}) ⊂ D^{b}(Λ_{2}) and π_{1}^{−1}(π_{1}Λ_{1}) ⊂ D^{b}(Λ_{1}) for some triangle equivalence F: C_{Λ_{1}} → C_{Λ_{2}}.

Proof. (1) ⇒ (2): By Proposition 5.2 there exists a triangle equivalence F^{D} which induces a triangle equivalence \xymatrix{F: C_{Λ_{1}} \ar[r] & C_{Λ_{2}}} such that the diagram

\xymatrix{D^{b}(Λ_{1}) \ar[r]^{F^{D}} \ar[d]^{π_{1}} & D(Λ_{2}) \ar[d]^{π_{2}} \\ C_{Λ_{1}} \ar[r]^{F} & C_{Λ_{2}}}

commutes. Therefore we have 𝕊_{2}-equivalences

π_{2}^{−1}(F π_{1}Λ_{1}) = π_{2}^{−1}(π_{2}F^{D}Λ_{1})
= add {𝕊_{2}^{p}F^{D}Λ_{1} | p ∈ ℤ}
≃ add {𝕊_{2}^{p}Λ_{1} | p ∈ ℤ}  (by uniqueness of the Serre functor)
≃ π_{1}^{−1}(π_{1}Λ_{1}).

(2) ⇒ (1) Since F is a triangle equivalence, and since π_{1}Λ_{1} is cluster-tilting in C_{Λ_{1}}, F π_{1}Λ_{1} is cluster-tilting in C_{Λ_{2}}. Then by Proposition 3.1 the subcategory π_{2}^{−1}(F π_{1}Λ_{1}) is cluster-tilting in D^{b}(Λ_{2}). Hence, by Theorem 3.5, we get the result. □

Remark 5.5. It is not clear that the F^{D} constructed in the proof of (2) ⇒ (1) commutes with F, but it induces a (possibly different) triangle equivalence

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

Theorem 5.6. Let Λ_{1} and Λ_{2} be two τ_{2}-finite algebras of global dimension ≤ 2. For i = 1, 2 we denote by π_{i} the canonical functor D_{i} → C_{i}, where D_{i} := D^{b}(Λ_{i}) and C_{i} := C_{Λ_{i}}. Denote by Cov(Λ̃_{i}, ℤ) the ℤ-covering of the ℤ-graded algebra

Λ̃_{i} := ⨁_{p≥0} Hom_{D_{i}}(Λ_{i}, 𝕊_{2}^{−p}Λ_{i}).

Assume that we have an isomorphism of algebras \xymatrix{Λ̃_{1} \ar[r]^{∼}_{f} & Λ̃_{2}.} Then the fol-

lowing are equivalent

(1) there exists a derived equivalence \xymatrix{F^{D}: D_{1} \ar[r]^{∼} & D_{2}} such that the in-

duced triangle equivalence F: C_{1} → C_{2} satisfies F (π_{1}Λ_{1}) = π_{2}Λ_{2};

(2) there is a equivalence \xymatrix{f^{ℤ}: mod Cov(Λ̃_{1}, ℤ) \ar[r]^{∼} & mod Cov(Λ̃_{2}, ℤ)} extending f.

In this case the algebras Λ_{1} and Λ_{2} are cluster equivalent, and we have a commutative diagram:

\xymatrix{D_{1} \ar[d]_{π_{1}} \ar[r]^{∼}_{F^{D}} & D_{2} \ar[d]^{π_{2}} \\ C_{1} \ar[r]^{∼}_{F} \ar[d]_{Hom_{C_{1}} (π_{1}(Λ_{1}),−)} & C_{2} \ar[d]^{Hom_{C_{2}} (π_{2}(Λ_{2}),−)} \\ _{mod Λ̃_{1}} \ar[r]^{∼}_{f} & _{mod Λ̃_{2}} \\ mod Cov(Λ̃_{1}, ℤ) \ar[u] \ar[r]^{∼}_{f^{ℤ}} & mod Cov(Λ̃_{2}, ℤ) \ar[u]}

Proof. (1) ⇒ (2): Assume Condition (1) is satisfied. Then there exists a tilting complex T ∈ D^{b}(Λ_{1}) such that End_{D^{b}(Λ_{1})}(T) ≃ Λ_{2} and that the following diagram commutes

\xymatrix{D^{b}(Λ_{1}) \ar[d]^{π_{1}} \ar[r]^{RHom_{D^{b}(Λ_{1})}(T,−)} & D^{b}(Λ_{2}) \ar[d]^{π_{2}} \\ C_{Λ_{1}} \ar[r]^{∼}_{F} & C_{Λ_{2}}}

Since F π_{1}(Λ_{1}) is isomorphic to π_{2}(Λ_{2}) = π_{2}(RHom(T, T)), we have π_{1}(Λ_{1}) = π_{1}(T). So T can be written ⨁_{i=1}^{n} 𝕊_{2}^{−d_{i}}e_{i}Λ_{1} for certain d_{i} ∈ ℤ, where Λ_{1} = ⨁_{i=1}^{n} e_{i}Λ_{1} is the decomposition into indecomposable projective modules. Then we have the following isomorphisms of ℤ-graded algebras

Λ̃_{2} ≃_{ℤ} ⨁_{p∈ℤ} Hom_{D_{2}}(Λ_{2}, 𝕊_{2}^{−p}Λ_{2})

≃_{ℤ} ⨁_{p∈ℤ} Hom_{D_{1}}(T, 𝕊_{2}^{−p}T)

≃_{ℤ} ⨁_{p∈ℤ} Hom_{D_{1}}(⨁_{i=1}^{n} 𝕊_{2}^{−d_{i}}e_{i}Λ_{1}, ⨁_{i=1}^{n} 𝕊_{2}^{−d_{i}−p}e_{i}Λ_{1})

≃_{ℤ} ⨁_{p∈ℤ} Hom_{Cov(Λ̃_{1},ℤ)}(⨁_{i=1}^{n} e_{i}Λ̃_{1}⟨d_{i}⟩, ⨁_{i=1}^{n} e_{i}Λ̃_{1}⟨d_{i} + p⟩),

where e_{i}Λ̃_{1} is the projective Λ̃_{1}-graded module ⨁_{p∈ℤ} Hom_{D_{1}}(Λ_{1}, 𝕊_{2}^{−p}e_{i}Λ_{1}). Therefore, by Theorem 2.8, we have a commutative diagram

\xymatrix{mod Λ̃_{1} \ar[r]^{∼} & mod Λ̃_{2} \\ mod Cov(Λ̃_{1}, ℤ) \ar[r]^{∼} \ar[u] & mod Cov(Λ̃_{2}, ℤ) \ar[u]}

and we get (2).

(2) ⇒ (1): By assumption we have a commutative diagram

\xymatrix{mod Λ̃_{1} \ar[r]^{∼} & mod Λ̃_{2} \\ mod Cov(Λ̃_{1}, ℤ) \ar[r]^{∼} \ar[u] & mod Cov(Λ̃_{2}, ℤ) \ar[u]}

where the upper equivalence comes from the isomorphism of algebras Λ̃_{1} ≃ Λ̃_{2}. Then, by Theorem 2.8, there exists integers d_{i} such that Λ̃_{2} is isomorphic as graded algebra to

Λ̃_{2} ≃_{ℤ} ⨁_{p∈ℤ} Hom_{Cov(Λ̃_{1},ℤ)}(⨁_{i=1}^{n} e_{i}Λ̃_{1}⟨d_{i}⟩, ⨁_{i=1}^{n} e_{i}Λ̃_{1}⟨d_{i} + p⟩),

where Λ̃_{1} = ⨁_{i=1}^{n} e_{i}Λ̃_{1} is the decomposition of Λ̃_{1} into indecomposables. Thus we have e_{i}Λ̃_{1} ≃ ⨁_{p∈ℤ} Hom_{D_{1}}(Λ_{1}, 𝕊_{2}^{−p}e_{i}Λ_{1}). Therefore we have

Λ̃_{2} ≃_{ℤ} ⨁_{p∈ℤ} Hom_{D_{1}}(⨁_{i=1}^{n} 𝕊_{2}^{−d_{i}}e_{i}Λ_{1}, ⨁_{i=1}^{n} 𝕊_{2}^{−d_{i}−p}e_{i}Λ_{1}).

This isomorphism of graded algebras means that we have an equivalence of cluster-tilting subcategories with 𝕊_{2}-action

\xymatrix{U_{1} = add \{𝕊_{2}^{−d_{i}−p}e_{i}Λ_{1}, i = 1, …, n, p ∈ ℤ\} \ar[r]_{u}^{∼} & add \{𝕊_{2}^{−p}Λ_{2} | p ∈ ℤ\} = U_{2}}

sending ⨁_{i=1}^{n} 𝕊_{2}^{−d_{i}}e_{i}Λ_{1} to Λ_{2}. Therefore, by Theorem 3.5, we get a triangle equivalence F^{D} between D^{b}(Λ_{1}) and D^{b}(Λ_{2}) making the following square commutative.

\xymatrix{U_{1} \ar[r]^{u} \ar@{^{(}->}[d] & U_{2} \ar@{^{(}->}[d] \\ D^{b}(Λ_{1}) \ar[r]^{F^{D}} & D^{b}(Λ_{2})}

By Proposition 5.2, the functor F^{D} induces a triangle equivalence F: C_{Λ_{1}} → C_{Λ_{2}} such that the following diagram commutes,

\xymatrix{D^{b}(Λ_{1}) \ar[r]^{F^{D}} \ar[d]^{π_{1}} & D^{b}(Λ_{2}) \ar[d]^{π_{2}} \\ C_{Λ_{1}} \ar[r]^{F} & C_{Λ_{2}}}

and we have

F π_{1}Λ_{1} ≃ F π_{1}(⨁_{i=1}^{n} 𝕊_{2}^{−d_{i}}e_{i}Λ_{1}) ≃ π_{2}F^{D}(⨁_{i=1}^{n} 𝕊_{2}^{−d_{i}}e_{i}Λ_{1}) ≃ π_{2}Λ_{2}.

This completes the proof of the implication (2) ⇒ (1). Moreover the square

\xymatrix{C_{Λ_{1}} \ar[r]^{F} \ar[d]_{Hom_{C_{1}} (π_{1}Λ_{1},−)} & C_{Λ_{2}} \ar[d]^{Hom_{C_{2}} (π_{2}Λ_{2},−)} \\ _{mod Λ̃_{1}} \ar[r]^{f} & _{mod Λ̃_{2}}}

is commutative, and we get the commutative diagram of the theorem. □

Example 5.7. Let Λ_{1} = kQ^{1}/I^{1}, Λ_{2} = kQ^{2}/I^{2} and Λ_{3} = kQ^{3}/I^{3} be the algebras given by the following quivers:

Q^{1} = \xymatrix{ & 2 \\ 1 \ar@{-->}[ur]^{a} & & 3 \ar[ll]^{d} \ar[ll]_{c}} , Q^{2} = \xymatrix{ & 2 \ar@{-->}[dr]^{b} \\ 1 & & 3 \ar[ll]^{d} \ar[ll]_{c}} and Q^{3} = \xymatrix{ & 2 \ar@{-->}[dr]^{b} \\ 1 \ar@{-->}[ur]^{a} & & 3 \ar[ll]^{d}}

with relations I^{1} = ⟨ac⟩, I^{2} = ⟨cb⟩ and I^{3} = ⟨ba⟩. It is easy to check that the algebras Λ̃_{i}, for i = 1, 2, 3 are all isomorphic to the Jacobian algebra Jac(Q̃, W) (see Section 6.2 for definition) where

Q̃ = \xymatrix{ & 2 \ar@{-->}[dr]^{b} \\ 1 \ar@{-->}[ur]^{a} & & 3 \ar[ll]^{d} \ar[ll]_{c}} and W = cba.

The algebras Λ_{1} and Λ_{2} are derived equivalent since they are both derived equivalent to the path algebra of a quiver of type Ã_{2}. The quiver Q^{3} contains an oriented cycle, therefore the algebra Λ_{3} is not derived equivalent to a hereditary algebra. We compute the ℤ-coverings with respect to the different gradings:

[figure: 2; 1; a; 3; d; c; 2; b; Cov(Λ̃_{1}, ℤ) =; 1; a; 3; d; c; 2; b; 1; a; 3; d; c; 2; b; 1; a; 3; d; c; 2; b; Cov(Λ̃_{2}, ℤ) =; 1; a; 3; d; c; 2; b; 1; 3; d; c; 2; b; 1; a; 3; d; 2; b; Cov(Λ̃_{3}, ℤ) =; 1; a; 3; c; d; 2; b; 1; a; 3; c; d]

It is then clear that the first two locally finite categories are equivalent, but not to the third one.

Theorem 5.6 can be generalized to the case that Λ̃_{2} is isomorphic not necessarily to Λ̃_{1}, but to the endomorphism algebra of some cluster-tilting object in C_{1}.

Theorem 5.8. Let Λ_{1} and Λ_{2} be two τ_{2}-finite algebras of global dimension ≤ 2. Assume there is T ∈ D_{1} such that π_{1}(T) is basic cluster tilting in C_{1}, and

(1) there is an isomorphism \xymatrix{End_{C_{1}}(π_{1}T) \ar[r]^{∼} & End_{C_{2}}(π_{2}Λ_{2})}

(2) this isomorphism can be chosen in such a way that the two ℤ-gradings defined on Λ̃_{2}, given respectively by

⨁_{q∈ℤ} Hom_{D_{2}}(Λ_{2}, 𝕊_{2}^{−q}Λ_{2}) and ⨁_{p∈ℤ} Hom_{D_{1}}(T, 𝕊_{2}^{−p}T),

are equivalent. Then the algebras Λ_{1} and Λ_{2} are derived equivalent, and hence cluster equivalent.

Proof. The object π_{1}T is a cluster-tilting object in C_{1}. Hence the subcategory π_{1}^{−1}(π_{1}T) is a cluster-tilting subcategory of D_{1}. It is immediate to see that

π_{1}^{−1}(π_{1}T) = add {𝕊_{2}^{p}T | p ∈ ℤ}.

With the same argument used in the proof (2) ⇒ (1) of Theorem 5.6, we show that Condition (2) is equivalent to the fact that we have an equivalence of cluster-tilting subcategories with 𝕊_{2}-action:

U_{1} := add {𝕊_{2}^{p}T | p ∈ ℤ} ≃ add {𝕊_{2}^{q}Λ_{2}, q ∈ ℤ} =: U_{2}

Then we can apply Theorem 3.5 to get the result. □

Remark 5.9. In Section 6 we will introduce the notion of mutation of graded quivers with potential, which makes it possible to check the assumptions of Theorem 5.8 more easily. Therefore we give an example there (Example 6.15).

5.2. Classification of tilting complexes. In this subsection we classify tilting complexes giving rise to derived equivalent algebras with the same canonical cluster tilting object. Let Λ be a τ_{2}-finite algebra of global dimension ≤ 2. Let Λ = P_{1} ⊕ ⋯ ⊕ P_{n} the decomposition of the free module Λ into indecomposable projectives. Let T = ⨁_{i=1}^{n} 𝕊_{2}^{−d_{i}}P_{i} be a lift of the canonical cluster-tilting object π(Λ) (cf. Example 2.4(3)). Our aim is to determine when T is a tilting complex with gl.dim(End_{D}(T)) ≤ 2. First recall a result from [IO11].

Proposition 5.10 ([IO11, Theorem 4.5 and Proposition 4.7]). Let Λ be an algebra of global dimension ≤ 2. Let Λ = P_{0} ⊕ P_{R} be a decomposition such that

(1) Hom_{D}(P_{R}, P_{0}) = 0, and

(2) Ext_{Λ}^{1}(𝕊P_{R}, P_{0}) = 0 (recall that 𝕊 is the Serre functor, so 𝕊P_{R} is the

injective module corresponding to the projective module P_{R}). Then the complex T = 𝕊_{2}^{−1}P_{0}⊕P_{R} is a tilting complex with gl.dim End_{D}(T) ≤ 2.

In this case the complex T is called a 2-APR-tilt of Λ. We denote by U the cluster-tilting subcategory add {𝕊_{2}^{p}Λ | p ∈ ℤ} of D^{b}(Λ).

Definition 5.11. An object Σ in U is called a slice if the following holds: (1) Σ intersects the 𝕊_{2}-orbit of all P_{i} in exactly one point.

(2) Hom_{D}(Σ, 𝕊_{2}^{p}Σ) = 0 for all p > 0.

(3) Hom_{D}(Σ, 𝕊_{2}^{p}Σ[−1]) = 0 for all p ≥ 0.

Note that, if Σ is a slice, then 𝕊_{2}Σ and 𝕊_{2}^{−1}Σ are also slices. We define a partial order on slices. Let Σ = ⨁_{i=1}^{n} 𝕊_{2}^{−s_{i}}P_{i} and Σ′ = ⨁_{i=1}^{n} 𝕊_{2}^{−t_{i}}P_{i} be two slices. Then we write Σ ≤ Σ′ if for all i = 1, …, n we have s_{i} ≤ t_{i}. For two complexes Σ = ⨁_{i=1}^{n} 𝕊_{2}^{−s_{i}}P_{i} and Σ′ = ⨁_{i=1}^{n} 𝕊_{2}^{−t_{i}}P_{i}, we will denote by max(Σ, Σ′) the complex ⨁_{i=1}^{n} 𝕊_{2}^{−u_{i}}P_{i} where u_{i} = max(s_{i}, t_{i}), and by min(Σ, Σ′) the complex ⨁_{i=1}^{n} 𝕊_{2}^{−v_{i}}P_{i} where v_{i} = min(s_{i}, t_{i}).

Lemma 5.12. If Σ and Σ′ are slices, then max(Σ, Σ′) and min(Σ, Σ′) are slices.

Proof. By definition max(Σ, Σ′) intersects every 𝕊_{2}-orbit in exactly one point. Let S and S′ be indecomposable summands of max(Σ, Σ′). We can assume that S is a summand of Σ and S′ a summand of Σ′. Then there exists d ≥ 0 such that 𝕊_{2}^{d}S is a summand of Σ′. For p ≥ 0, the space Hom_{D}(S, 𝕊_{2}^{p}S′) ≃ Hom(𝕊_{2}^{d}S, 𝕊_{2}^{p+d}S′) vanishes since 𝕊_{2}^{d}S and S′ are summand of Σ′ and p + d ≥ 0. For the same reasons, the space Hom_{D}(S, 𝕊_{2}^{p}S′[−1]) ≃ Hom(𝕊_{2}^{d}S, 𝕊_{2}^{p+d}S′[−1]) vanishes. Hence max(Σ, Σ′) is a slice. The proof is similar for min(Σ, Σ′). □

Lemma 5.13. The object Λ is a slice.

Proof. Since the global dimension of Λ is ≤ 2, it is not hard to see (cf. [Ami09, Lemma 4.6]), that the cohomology of 𝕊_{2}^{p}Λ is in degree ≥ 2 for p ≥ 1. Therefore we immediately get (2) and (3). □

Lemma 5.14. Let Σ and Σ′ be two slices such that 𝕊_{2}Σ′ ≤ Σ ≤ Σ′. Assume that Σ is a tilting complex with endomorphism algebra of global dimension ≤ 2. Then Σ′ is a 2-APR-tilt of Σ.

Proof. The slices Σ′ and 𝕊_{2}Σ′ have no common summands, and since 𝕊_{2}Σ′ ≤ Σ ≤ Σ′, the slice Σ is a direct summand of 𝕊_{2}Σ′ ⊕ Σ′. Let P_{0} be the intersection Σ ∩ 𝕊_{2}Σ′ and P_{R} be the intersection Σ ∩ Σ′. Then we have 𝕊_{2}^{−1}P_{0} ⊕ P_{R} = Σ′. We shall prove that the decomposition Σ = P_{0} ⊕ P_{R} satisfies the properties (1) and (2) of Theorem 5.10. The space Hom_{D}(P_{R}, P_{0}) vanishes since P_{R} is in Σ′ and P_{0} is in 𝕊_{2}Σ′. The space Ext_{Λ}^{−1}(P_{R}, 𝕊_{2}^{−1}P_{0}) vanishes since P_{R} and 𝕊_{2}^{−1}P_{0} are both in Σ′. Therefore Σ′ is a 2-APR-tilt of Σ. □

Proposition 5.15. All slices are iterated 2-APR-tilts of Λ.

Proof. Let Σ be a slice. Let N be minimal with 𝕊_{2}^{N}Λ ≤ Σ, and M maximal with Σ ≤ 𝕊_{2}^{M}Λ. We prove the claim by induction on N − M. For N = M we have Σ = 𝕊_{2}^{N}Λ, thus Σ is an iterated 2-APR-tilt of Λ. So assume N > M. Let Σ′ = max(𝕊_{2}^{N−1}Λ, Σ). Then 𝕊_{2}Σ′ ≤ Σ ≤ Σ′, and by Lemma 5.14 Σ is a 2-APR-tilt of Σ′. Since 𝕊_{2}^{N−1}Λ ≤ Σ′ ≤ 𝕊_{2}^{M}Λ we know inductively that Σ′ is an iterated 2-APR-tilt of Λ, so Σ is also an iterated 2-APR-tilt of Λ. □

Theorem 5.16. For T = ⨁_{i=1}^{n} 𝕊_{2}^{−d_{i}}P_{i}, the following are equivalent: (1) T is a tilting complex with gl.dim End(T) ≤ 2;

(2) T is a slice;

(3) T is an iterated 2-APR-tilt of Λ;

(4) (a) Ext_{Λ}^{1}(S_{j}, S_{i}) ≠ 0 or Ext_{Λ}^{2}(S_{i}, S_{j}) ≠ 0 implies that d_{j} − d_{i} ∈ {0, 1}

and

(b) for any r we have Hom_{D}(P_{i}, 𝕊_{2}^{−r}P_{j}[−1]) ≠ 0 implies that d_{j} − d_{i} < r.

Proof. We have (2) ⇒ (3) ⇒ (1) by Proposition 5.15 and by Theorem 5.10. Assume now that T is a tilting complex whose endomorphism algebra Γ = End_{D^{b}(Λ)}(T) is of global dimension ≤ 2. Then by Lemma 5.13, Γ is a slice in D^{b}(Γ). Since T is a tilting complex, there is a triangle equivalence

\xymatrix{RHom_{D^{b}(Λ)}(T, −): D^{b}(Λ) \ar[r]^{∼} & D^{b}(Γ)} .

By uniqueness of the Serre functor, the functors _{Λ}𝕊_{2} and _{Γ}𝕊_{2} are isomorphic. Since T is of the form ⨁_{i=1}^{n} 𝕊_{2}^{−d_{i}}P_{i}, we have the following commutative square

\xymatrix{D^{b}(Λ) \ar[r]^{∼} & D(Γ) \\ U_{Λ} \ar[r]^{∼} \ar@{^{(}->}[u] & U_{Γ} \ar@{^{(}->}[u]}

where U_{Λ} is the additive subcategory add {𝕊_{2}^{p}Λ | p ∈ ℤ} of D^{b}(Λ). Therefore Σ is a slice in D^{b}(Λ) if and only if RHom_{D^{b}(Λ)}(T, Σ) is a slice in D^{b}(Γ). Hence T is a slice in D^{b}(Λ) and we have (1) ⇒ (2).

For the last equivalence, first note that Condition (4b) is clearly equivalent to Condition (3) of the definition of a slice. Let Λ̃ be the endomorphism algebra End_{C}(πΛ) = ⨁_{p∈ℤ} Hom_{D^{b}(Λ)}(Λ, 𝕊_{2}^{−p}Λ). This algebra is a graded algebra with positive grading generated in degrees 0 and 1. The arrows of its quiver QΛ̃ have degree 0 or 1. Moreover we have

♯{i → j ∈ QΛ̃} = dim Ext_{Λ}^{1}(S_{j}, S_{i}) + dim Ext_{Λ}^{2}(S_{i}, S_{j}),

where S_{i} (resp. S_{j}) is the simple module top of P_{i} (resp. P_{j}). Let T be a tilting complex of the form ⨁_{i=1}^{n} 𝕊_{2}^{−d_{i}}P_{i} = ⨁_{i=1}^{n} T_{i} and such that its endomorphism algebra Γ = End_{D^{b}(Λ)}(T) is of global dimension ≤ 2. The algebra

Γ̃ = End_{C}(π(T)) = ⨁_{p∈ℤ} Hom_{D^{b}(Λ)}(T, 𝕊_{2}^{−p}T)

is isomorphic to Λ̃. By the above remark, every arrow of the graded quiver QΓ̃ has degree 0 or 1. Assume that Ext_{Λ}^{1}(S_{j}, S_{i}) does not vanish. This means that there exists an irreducible map P_{i} → P_{j}, therefore there exists an irreducible map T_{i} → 𝕊_{2}^{−d_{i}+d_{j}}T_{j}. But since every arrow of the graded quiver QΓ̃ has degree 0 or 1, we have d_{i} −d_{j} ∈ {0, 1}. Assume now that Ext_{Λ}^{2}(S_{i}, S_{j}) does not vanish. This means that there exists an irreducible map P_{i} → 𝕊_{2}^{−1}P_{j}, therefore there exists an irreducible map T_{i} → 𝕊_{2}^{−d_{i}+d_{j}−1}T_{j}. Since every arrow of the graded quiver QΓ̃ has degree 0 or 1, we have d_{i} − d_{j} − 1 ∈ {0, 1}. Hence we have d_{j} − d_{i} ∈ {0, 1}. Therefore we have (1) ⇒ (4). Now assume that we have Condition (4) for the object T = ⨁_{i=1}^{n} 𝕊_{2}^{−d_{i}}P_{i}. We will prove that it is a slice. We obviously have Condition (1) and we have Condition (3) since we have (4b). The algebra

Γ̃ = End_{C}(π(T)) = ⨁_{p∈ℤ} Hom_{D^{b}(Λ)}(T, 𝕊_{2}^{−p}T)

is isomorphic to Λ̃. Condition (4a) exactly means, as we just saw above, that the graded quiver QΓ̃ only has arrows of degree 0 and 1. A non-zero morphism in Hom_{D^{b}(Λ)}(T, 𝕊_{2}^{p}T) implies that there exists a path of degree −p in the graded quiver QΓ̃. Hence p must be non-positive, and we have Condition (2) of the definition of a slice. Therefore we have (4) ⇒ (2). □

6. Left (and right) mutation in the derived category and graded quivers with potential

The aim of this section is to provide a combinatorial description of the mutation in the derived category. We first give a link between right and left mutation. Then we recall some results about quivers with potential (QP for short) and generalized cluster categories attached to them. Finally, we define the notion of mutation of a graded QP and state the main result which permits to compute explicitly the grading of the endomorphism ring of a cluster-tilting object in the cluster category.

6.1. Relation between left mutation and right mutation. Let T be a triangulated category with Serre functor 𝕊 and V ⊂ T be a cluster-tilting subcategory. Let X ∈ V be indecomposable, and set

V′ := add (ind (V) \ {𝕊_{2}^{p}X | p ∈ ℤ}),

where ind (V) denotes the indecomposable objects in V. Then by Theorem 2.6 there exists a unique cluster-tilting subcategory V^{∗} with V′ ⊆ V^{∗} ≠ V.

Proposition 6.1. In the setup of Theorem 2.6 (left and right exchange triangles), if for any p ≠ 0 any map X → 𝕊_{2}^{p}X factors through V′, then we have

X^{R} ≃ 𝕊_{2}X^{L}.

Proof. Let \xymatrix{X^{L}[−1] \ar[r]^{f} & X \ar[r]^{u} & B \ar[r] & X^{L}} be the left exchange triangle. Let g: X^{L}[−1] → U be a non-zero morphism with U an indecomposable object in V. Then U is not in V′ since Hom(X^{L}[−1], V′) = 0 (Theorem 2.6). Hence there exists a p such that U = 𝕊_{2}^{p}X. Since f is a left V-approximation g factors through f. Since for p ≠ 0 all maps X → 𝕊_{2}^{p}X factor through V′, hence through u, we have p = 0. Therefore we have

Hom_{T} (X^{L}[−1], V) = Hom_{T} (X^{L}[−1], X).

But now we have isomorphisms in mod V

DHom_{T} (V, 𝕊_{2}X^{L}[1]) ≃ Hom_{T} (X^{L}[−1], V) = = Hom_{T} (X^{L}[−1], X) ≃ DHom_{T} (X, 𝕊_{2}X^{L}[1]).

Hence the only indecomposable object in V admitting non-zero maps to 𝕊_{2}X^{L}[1] is X. Now let

\xymatrix{𝕊_{2}X^{L} \ar[r] & B′ \ar[r] & X′ \ar[r] & 𝕊_{2}X^{L}[1]}

be the triangle obtained from a left V′-approximation of 𝕊_{2}X^{L}. By Theorem 2.6 we have X′ = 𝕊_{2}^{q}X for some q ∈ ℤ. By the above observation we have q = 0, so X′ = X. Now, since the above map \xymatrix{B′ \ar[r] & X′ = X} is a right V′-approximation, this triangle is precisely the triangle defining X^{R}. So we see X^{R} = 𝕊_{2}X^{L}. □

6.2. Jacobian algebras and cluster-tilting objects. Quivers with potentials and the associated Jacobian algebras have been studied in [DWZ08]. Let Q be a finite quiver. For each arrow a in Q, the cyclic derivative ∂_{a} with respect to a is the unique linear map

∂_{a}: kQ → kQ

which sends a path p to the sum ∑_{p=uav} vu taken over all decompositions of the path p (where u and v are possibly idempotent elements e_{i} associated to a vertex i). A potential on Q is any (possibly infinite) linear combination W of cycles in Q. The associated Jacobian algebra is

Jac(Q, W) := kQ̂/⟨∂_{a}W; a ∈ Q_{1}⟩,

where kQ̂ is the completed path algebra, that is the completion of kQ with respect to the ideal generated by the arrows, and ⟨∂_{a}W; a ∈ Q_{1}⟩ is the closure of the ideal generated by ∂_{a}W for a ∈ Q_{1}. Associated with any quiver with potential (Q, W), a cluster category C_{(Q,W)} is constructed in [Ami09]. It uses the notion of Ginzburg dg algebra. We refer the reader to [Ami09] for an explicit construction. When the associated Jacobian algebra is finite dimensional, the category C_{(Q,W)} is 2-Calabi-Yau and endowed with a canonical cluster-tilting object T_{(Q,W)} (that is an object such that add (T_{(Q,W)}) is a cluster-tilting subcategory) whose endomorphism algebra is isomorphic to Jac(Q, W). The next result gives a link between cluster categories associated with algebras of global dimension ≤ 2 and cluster categories associated with QP.

Theorem 6.2 ([Kel11, Theorem 6.11 a)]). Let Λ = kQ/I be a τ_{2}-finite algebra of global dimension ≤ 2, such that I is generated by a finite minimal set of relations {r_{i}}. (By this we mean that the set {r_{i}} is the disjoint union of sets representing a basis of the Ext_{Λ}^{2}-space between any two simple Λ-modules.) The relation r_{i} starts at the vertex s(r_{i}) and ends at the vertex t(r_{i}). Then there is a triangle equivalence:

C_{Λ} ≃ C_{(Q̃,W)},

where the quiver Q̃ is the quiver Q with additional arrows a_{i}: t(r_{i}) → s(r_{i}), and the potential W is ∑_{i} a_{i}r_{i}. This equivalence sends the cluster-tilting object π(Λ) on the cluster-tilting object T_{(Q̃,W)}. As a consequence we have an isomorphism of algebras:

End_{C}(πΛ) ≃ Jac(Q̃, W).

Here is an immediate observation which introduces the natural grading of End_{C}(πΛ) to Theorem 6.2. It gives a motivation for introducing the left gradedmutation of a quiver.

Proposition 6.3. Let Λ = kQ/I be a τ_{2}-finite algebra of global dimension ≤ 2. Denote by (Q̃, W) the quiver with potential defined in Theorem 6.2. Then there exists a unique ℤ-grading on Q̃ such that

(1) the potential W is homogeneous of degree 1;

(2) there is an isomorphism of quivers \xymatrix{Q̃^{\{0\}} \ar[r]^{∼} & Q} , where Q̃^{\{0\}} is the

subquiver of Q̃ of arrows of degree 0. This grading on Q̃ yields a grading on Jac(Q̃, W) and we have an isomorphism of ℤ-graded algebras

\xymatrix{Jac(Q̃, W) \ar[r]^{∼}_{ℤ} & ⨁_{p∈ℤ} Hom_{D^{ℤ}}(Λ, 𝕊_{2}^{−p}Λ)} .

Proof. This is achieved by giving the arrows in Q_{1} degree zero, and the arrows in Q̃_{1} \ Q_{1} (that is the arrows corresponding to minimal relations) degree one.

□

6.3. Left (and right) mutation of a graded quiver with potential. Extending Fomin and Zelevinsky mutations of quivers [FZ02], Derksen, Weyman, and Zelevinsky have introduced the notion of mutation of quivers with potential in [DWZ08]. We adapt this notion to G-graded quivers with potential homogeneous of degree r ∈ G. In the following subsections of this section we will use this definition for G = ℤ and r = 1, and in Section 8 for G = ℤ^{2} and r = (1, 1).

Definition 6.4. Let (Q, W, d) be a G-graded quiver with potential homogeneous of degree r (G-graded QP for short). Let i ∈ Q_{0} be a vertex, such that there are neither loops nor 2-cycles incident to i. We define μ̃_{i}^{L}(Q, W, d) = (Q′, W′, d′). The quiver Q′ is defined as in [DWZ08] as follows

• for any subquiver \xymatrix{u \ar[r]^{a} & i \ar[r]^{b} & v} with i, u and v pairwise different vertices, we add an arrow [ba]: u → v;

• we replace all arrows a incident with i by an arrow a^{∗} in the opposite

direction. The potential W′ is also defined as in [DWZ08] by the sum [W] + W^{∗} where [W] is formed from the potential W replacing all compositions ba through the vertex i by the new arrows [ba], and where W^{∗} is the sum ∑ b^{∗}a^{∗}[ba]. The new degree d′ is defined as follows:

• d′(a) = d(a) if a is an arrow of Q and Q′;

• d′([ba]) = d(b) + d(a) if ba is a composition passing through i;

• d′(a^{∗}) = −d(a) + r if the target of a is i;

• d′(b^{∗}) = −d(b) if the source of b is i.

Similarly, we can define μ̃_{i}^{R}(Q, W, d) = (Q′, W′, d′) by setting d′(a^{∗}) = −d(a) for arrows a such that t(a) = i, and d′(b^{∗}) = −d(b) + r for arrows b with s(b) = i. As in [DWZ08], it is possible to define trivial and reduced graded quivers with potential. A G-graded QP (Q, W, d) is trivial if the potential W is in the space kQ_{2} spanned by paths of length 2, and if the Jacobian algebra Jac(Q, W, d) is isomorphic to the semisimple algebra kQ_{0}. A G-graded QP (Q, W, d) is reduced if W ∩ kQ_{2} is zero.

Definition 6.5. Two G-graded QP (Q, W, d) and (Q′, W′, d′) are graded right equivalent if there exists an isomorphism of G-graded algebras \xymatrix{φ: (kQ, d) \ar[r]^{∼}_{G} & (kQ′, d′)} such that φ|_{kQ_{0}} = id and such that φ(W) is

cyclically equivalent to W′ in the sense of [DWZ08].

The Splitting Theorem of [DWZ08] still holds in the graded case. Indeed the right equivalence constructed in the proof of [DWZ08, Lemma 4.8], is graded.

Theorem 6.6 (Graded Splitting Theorem – compare [DWZ08, Theorem 4.6]). Let (Q, W, d) be a G-graded QP. Then there exists a (unique up to graded right equivalence) reduced graded QP (Q^{red}, W^{red}, d^{red}) and a (unique up to graded right equivalence) trivial QP (Q^{triv}, W^{triv}, d^{triv}) such that (Q, W, d) is graded right equivalent to the direct sum

(Q^{red}, W^{red}, d^{red}) ⊕ (Q^{triv}, W^{triv}, d^{triv}).

(The direct sum means that the vertices of both summands and the sum coincide, and the arrows of the sum are the disjoint union if the arrows of the summands.)

Therefore, we can deduce the following.

Proposition 6.7. Let (Q, W, d) be a G-graded quiver with potential homogeneous of degree r and let i be vertex of Q without incident loops or 2cycles. Then the reduction (Q^{′red}, W^{′red}, d′) in the sense of [DWZ08] of μ̃_{i}^{L}(Q, W, d) = (Q′, W′, d′) has potential homogeneous of degree r with respect to the grading d′.

Proof. It is an easy computation to check that the potential W′ is homogeneous of degree r with respect to d′. Then the graded splitting theorem implies that the reduction process does not change the homogeneity of the potential. □

Definition 6.8. The Graded Splitting Theorem and the above proposition allow us to defined the left mutation at vertex i μ_{i}^{L}(Q, W, d) of a G-graded quiver with potential (Q, W, d) as the reduction of the graded quiver with potential μ̃_{i}^{L}(Q, W, d). Similarly we define the right mutation at vertex i μ_{i}^{R}(Q, W, d) of a graded QP (Q, W, d) as the reduction of the graded quiver with potential μ̃_{i}^{R}(Q, W, d).

One immediately checks the following.

Lemma 6.9. Let (Q, W, d) be a G-graded QP with potential homogeneous of degree r, and let i be a vertex without incident loops or 2-cycles. Then the Ggraded QP (Q, W, d), μ_{i}^{L}μ_{i}^{R}(Q, W, d) and μ_{i}^{R}μ_{i}^{L}(Q, W, d) are graded right equivalent.

The following lemma gives a direct link between left mutation and right mutation of a graded quiver.

Lemma 6.10. Let (Q, W, d) be a G-graded quiver with potential homogeneous of degree r and let i be a vertex of Q without incident loops or 2-cycles. Then there is a graded equivalence

mod Jac(μ_{i}^{L}(Q, W, d)) → mod Jac(μ_{i}^{R}(Q, W, d)).

Proof. The algebras Jac(μ_{i}^{L}(Q, W, d)) and Jac(μ_{i}^{R}(Q, W, d)) are isomorphic. Let us decompose the Jacobian algebra Jac(μ_{i}^{L}(Q, W, d)) as a direct sum of graded projective Jac(μ_{i}^{L}(Q, W, d))-modules.

Jac(μ_{i}^{L}(Q, W, d)) = ⨁_{j∈Q_{0}} P_{j}.

One can check that the graded endomorphism algebra of

( ⨁_{j∈Q_{0},j≠i} P_{j}) ⊕ P_{i}⟨−r⟩

is then isomorphic as graded algebra to Jac(μ_{i}^{R}(Q, W, d)). □

6.4. Mutation and generalized cluster categories associated with QP. Let (Q, W) be a quiver with potential, such that the Jacobian algebra Jac(Q, W) is finite dimensional, and T_{(Q,W)} ∈ C_{(Q,W)} be the canonical clustertilting object of the generalized cluster category C_{(Q,W)}. There is a canonical bijection between the vertices of Q and the indecomposable objects of T_{(Q,W)}. Let i be in Q_{0}, T_{i} be the corresponding summand of T_{(Q,W)} ≃ T_{i} ⊕ T′, and

\xymatrix{T_{i} \ar[r] & B \ar[r] & T_{i}^{L} \ar[r] & T_{i}[1] & and & T_{i}^{R} \ar[r] & B′ \ar[r] & T_{i} \ar[r] & T_{i}^{R}[1]}

be the approximation triangles as defined in Theorem 2.6. Then, we have T_{i}^{R} ≃ T_{i}^{L} since the category C_{(Q,W)} is 2-Calabi-Yau. We denote by μ_{i}(T_{(Q,W)}) the new cluster-tilting object T′ ⊕ T_{i}^{L}. The following fundamental result links the [DWZ08]-mutation of QP and the [IY08]-mutation of the cluster-tilting object T_{(Q,W)}.

Theorem 6.11 (Keller-Yang [KY11]). Let (Q, W) be a quiver with potential whose Jacobian algebra is finite dimensional, and i ∈ Q_{0} a vertex such that there is no 2-cycles nor loops at vertex i in Q. Then there exists a triangle equivalence

C_{μ_{i}(Q,W)} ≃ C_{(Q,W)}

sending the cluster-tilting object T_{μ_{i}(Q,W)} ∈ C_{μ_{i}(Q,W)} on the cluster-tilting object μ_{i}(T_{(Q,W)}) ∈ C_{(Q,W)}, where T_{i} is the indecomposable summand of T_{(Q,W)} associated with the vertex i of Q, and where μ_{i}(Q, W) is the mutation of the quiver with potential (Q, W) at vertex i.

This triangle equivalence is compatible with vertices in the following sense: the bijection between the vertices of Q and the indecomposable direct summands of T_{(Q,W)} together with this equivalence induce a bijection between the vertices of μ̃_{i}(Q, W) and the indecomposable summands of T_{μ_{i}(Q,W)}, which is the canonical one.

The compatibility between [DWZ08]-mutation of quivers with potential and [IY08]-mutation of cluster tilting objects, given by Theorem 6.11, can be understood more precisely in the following way: Let T_{(Q,W)} ≃ T_{i} ⊕ T′ as above. If the map T_{i} → B is a minimal left (add T′)- approximation, then B is isomorphic to ⨁_{j,a: i→j} T_{j}. So for any arrow a: i → j in Q, there is a non-zero map T^{j} → T_{i}^{L}. This map corresponds to the new arrow a^{∗}: j → i in μ̃_{i}(Q, W). In the same way, for any arrow b: j → i in Q, there is a non-zero map T_{i}^{R} ≃ T_{i}^{L} → T_{j} which corresponds to the new arrow b^{∗} in μ̃_{i}(Q, W). Furthermore, arrows [ba]: j → ℓ in μ_{i}(Q), where a: j → i and b: i → ℓ are in Q, correspond to the composition of the maps associated to b and a.

6.5. Relation between mutation of graded QP and mutation in the derived category. Let Λ be a τ_{2}-finite algebra of global dimension ≤ 2. Let T ≃ T_{1} ⊕ … ⊕ T_{n} be an object in D^{b}(Λ) such that π(T) is a (basic) clustertilting object in C_{Λ}. By Proposition 3.1, the category U_{T} = add {𝕊_{2}^{p}T | p ∈ ℤ} = π^{−1}(π(T)) is a cluster-tilting subcategory of D^{b}(Λ). Let i ∈ {1, …, n}, and T′ = ⊕_{j≠i}T_{j}. We denote by U_{i} the additive subcategory add {𝕊_{2}^{p}T′ | p ∈ ℤ} of D^{b}(Λ). Consider the left exchange triangle associated to T_{i} (defined in Theorem 2.6) in D^{b}(Λ)

\xymatrix{T_{i} \ar[r]^{u} & B \ar[r]^{v} & T_{i}^{L} \ar[r]^{w} & T_{i}[1]}

where u: T_{i} → B is a minimal left U_{i}-approximation of T_{i}. We denote by μ_{i}^{L}(T) the object T′ ⊕ T_{i}^{L} ∈ D^{b}(Λ). Then, by Theorem 2.6, the category add {𝕊_{2}^{p}(μ_{i}^{L}(T)) | p ∈ ℤ} is a cluster-tilting subcategory of D^{b}(Λ), and by Proposition 3.2 the object π(μ_{i}^{L}(T)) is cluster-tilting in C_{Λ}. The endomorphism algebras of π(T) and π(μ_{i}^{L}(T)) are naturally ℤ-graded since we have

End_{C}(π(T)) ≃ ⨁_{p∈ℤ} Hom_{D}(T, 𝕊_{2}^{p}T)

and

End_{C}(π(μ_{i}^{L}(T))) ≃ ⨁_{p∈ℤ} Hom_{D}(μ_{i}^{L}(T), 𝕊_{2}^{−p}(μ_{i}^{L}(T))).

The following theorem links the gradings of End_{C}(π(Λ)) and End_{C}(π(T)), when T is obtained from Λ by iterated mutations and is mainly a consequence of Theorem 6.11.

Theorem 6.12. Let Λ = kQ/I be a τ_{2}-finite algebra of global dimension ≤ 2, and denote by (Q̃, W, d) the graded QP defined in Proposition 6.3. Assume that there exists a sequence i_{1}, i_{2}, …, i_{l} of vertices of Q̃ such that for any j = 0, …, l there is no 2-cycle on the vertex i_{j+1} in the quiver Q^{j} where (Q^{j}, W^{j}) := μ_{i_{j}} ∘ ⋯∘μ_{i_{1}}(Q̃, W). Denote by T the object in D^{b}(Λ) defined by T := μ_{i_{l}}^{L}∘⋯∘μ_{i_{1}}^{L}(Λ). Then there is an isomorphism of ℤ-graded algebras

\xymatrix{⨁_{p∈ℤ} Hom_{D}(T, 𝕊_{2}^{−p}(T)) \ar[r]^{∼}_{ℤ} & Jac(μ_{i_{l}}^{L} ∘ ⋯ ∘ μ_{i_{1}}^{L}(Q̃, W, d)).}

Proof. For j = 1, …, l denote by T^{j} the object μ_{i_{j}}^{L} ∘ ⋯ ∘ μ_{i_{1}}^{L}(Λ), and by (Q^{j}, W^{j}, d^{j}) the graded QP μ_{i_{j}}^{L} ∘ ⋯ ∘ μ_{i_{1}}^{L}(Q̃, W, d). We put T^{0} := Λ and (Q^{0}, W^{0}, d^{0}) := (Q̃, W, d). The object π(T^{j}) is a cluster-tilting object of the cluster category C_{Λ}. By Theorem 6.2, there is a triangle equivalence C_{Λ} ≃ C_{(Q^{0},W^{0})} sending the cluster-tilting π(Λ) on the cluster-tilting object T_{(Q^{0},W^{0})} ∈ C_{(Q^{0},W^{0})}. Now note that the quiver Q^{0} does not have loops. Indeed, since Λ is of finite global dimension, its Gabriel quiver does not contain any loop [Len69, Igu90]. Moreover since Λ̃ is finite dimensional, it is easy to see that there is no loop of degree 1 in the quiver Q^{0}. Furthermore, if a QP does not contain any loops, after one mutation at a vertex not on a 2-cycle, it still does not contain any loops. Therefore by an immediate induction, one checks that (Q^{j}, W^{j}, d^{j}) does not have loops for any j = 0, …, l. Hence we can apply iteratively Theorem 6.11 and we obtain an equivalence of triangulated categories

C_{(Q^{0},W^{0})} ≃ C_{(Q^{j},W^{j})}

sending μ_{i_{j}} ∘⋯∘μ_{i_{1}}(T_{(Q^{0},W^{0})}) onto T_{(Q^{j},W^{j})}. Hence we obtain an isomorphism of algebras

\xymatrix{⨁_{p∈ℤ} Hom_{D}(T^{j}, 𝕊_{2}^{−p}(T^{j})) \ar[r]^{∼} & Jac(Q^{j}, W^{j}).}

The only thing to check is that this isomorphism preserves the grading. We proceed by induction on the number of mutations j. For j = 0, we have

⨁_{p∈ℤ} Hom_{D}(T^{0}, 𝕊_{2}^{−p}(T^{0})) ≃_{ℤ} ⨁_{p∈ℤ} Hom_{D}(Λ, 𝕊_{2}^{−p}(Λ)) ≃_{ℤ} Jac(Q^{0}, W^{0}, d^{0})

by Proposition 6.3. So assume the result for j ≥ 0, we have by induction hypothesis

\xymatrix{⨁_{p∈ℤ} Hom_{D}(T^{j}, 𝕊_{2}^{−p}(T^{j})) \ar[r]^{∼}_{ℤ} & Jac(Q^{j}, W^{j}, d^{j}).}

Let (T^{j})′ be the direct sum of the summands of T^{j} not corresponding to the vertex i_{j+1}. Then the left exchange triangle is of the form

T_{i_{j+1}}^{j} → ⨁_{a: i_{j+1}→s} 𝕊_{2}^{−d(a)}T_{s}^{j} → (T_{i_{j+1}}^{j})^{L} → T_{i_{j+1}}^{j}[1],

where the sum in the second term runs over all arrows of Q^{j} starting in i_{j+1}. An arrow a^{∗}: s → i_{j+1} of the mutated quiver corresponds to the component map 𝕊_{2}^{−d(a)}T_{s}^{j} → (T_{i_{j+1}}^{j})^{L}, and therefore has degree −d(a). Similarly we see that, for an arrow b: s → i_{j+1}, the new arrow b^{∗} corresponds to the map (T_{i_{j+1}}^{j})^{R} → 𝕊_{2}^{d(b)}T_{s}^{j}, and thus, by Proposition 6.1, to a

map 𝕊_{2}(T_{i_{j+1}}^{j})^{L} → 𝕊_{2}^{d(b)}T_{s}^{j}. Therefore the degree of b^{∗} is 1 − d(b). Finally, any arrow of type [ab] in the quiver of Jac(μ_{i_{j+1}}^{L}(Q^{j}, W^{j}, d^{j})) corre-

sponds to a non-zero composition \xymatrix{𝕊_{2}^{d(b)}T_{r}^{j} \ar[r] & T_{i_{j+1}}^{j} \ar[r] & 𝕊_{2}^{−d(a)}T_{s}^{j}} , there-

fore its degree in End_{C}(π(μ_{i_{j+1}}^{L}(T^{j})) is d(a) + d(b). Thus using T^{j+1} = μ_{i_{j+1}}^{L}(T^{j}) we obtain an isomorphism of graded algebras

⨁_{p∈ℤ} Hom_{D}(T^{j+1}, 𝕊_{2}^{−p}T^{j+1}) ≃_{ℤ} Jac(μ_{i_{j+1}}^{L}(Q^{j}, W^{j}, d^{j})) = Jac(Q^{j+1}, W^{j+1}, d^{j+1}).

□

Remark 6.13. There exists a similar ‘right version’ of this theorem.

Graded quivers with potential and Theorem 6.12 permit to see the existence of a T as in Theorem 5.8 without explicitly constructing it, and thus to show that certain algebras are derived equivalent. More precisely we have the following consequence.

Corollary 6.14. Let Λ_{1} and Λ_{2} be two algebras of global dimension 2 , which are τ_{2}-finite. Assume that one can pass from the graded QP associated with Λ̃_{1} to the graded QP associated with Λ̃_{2} using a finite sequence of left and right mutations at vertices not on 2-cycles. Then the algebras Λ_{1} and Λ_{2} are derived equivalent.

Example 6.15. Let Λ_{1} and Λ_{2} be the following algebras:

Λ_{1} = 1 2 3 4 5 6 Λ_{2} = 1 2 3 4 5 6

One can easily compute the quiver with potential associated with the algebras Λ̃_{1} and Λ̃_{2}. Their graded quivers are

(Q̃_{1}, d_{1}) = 1 2 3 4 5 6 (Q̃_{2}, d_{2}) = 1 2 3 4 5 6 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1

Now applying the left graded mutations μ_{6}^{L} ∘ μ_{3}^{L} to the graded quiver with potential (Q̃_{1}, W_{1}, d_{1}) one can check that we obtain

μ_{6}^{L} ∘ μ_{3}^{L}(Q̃_{1}, d_{1}) = 1 2 3 4 5 6 0 0 0 -1 1 0 1 2

By Theorem 6.12, we have an isomorphism of ℤ-graded algebras

Jac(μ_{6}^{L} ∘ μ_{3}^{L}(Q̃_{1}, W_{1}, d_{1})) ≃_{ℤ} End_{C_{1}}(π_{1}(μ_{P_{6}}^{L} ∘ μ_{P_{3}}^{L}(Λ_{1}))).

It is immediate to see that the ℤ-graded algebras Jac(μ_{6}^{L} ∘ μ_{3}^{L}(Q̃_{1}, W_{1}, d_{1})) and Jac(Q̃_{2}, W_{2}, d_{2}) are graded equivalent. Therefore, by Corollary 6.14, the algebras Λ_{1} and Λ_{2} are derived equivalent. Now if we apply the left graded mutations μ_{6}^{L} ∘ μ_{2}^{L} ∘ μ_{4}^{L} to the graded quiver with potential (Q̃_{1}, W_{1}, d_{1}) one can check that we obtain the acyclic graded quiver (Q′, d′) := μ_{6}^{L} ∘ μ_{2}^{L} ∘ μ_{4}^{L}(Q̃_{1}, d_{1}) = 1 2 3 4 5 6 0 1 0 1 1 0

It is easy to see that (Q′, d′) is not graded equivalent to (Q′, 0). Therefore Theorem 5.8 does not yield a derived equivalence between Λ_{1} and kQ′. In fact, since there is an oriented cycle in the quiver of Λ_{2}, we know that Λ_{2}, and hence also Λ_{1}, is not piecewise hereditary.

7. Triangulated orbit categories

This section is devoted to recalling some results of Keller [Kel91, Kel05, Kel06] (see also the appendix of [IO13]), and to applying them to our setup.

7.1. Pretriangulated DG categories.

Definition 7.1. A DG category is a ℤ-graded category (i.e. morphism spaces are ℤ-graded, and composition of morphisms respects this grading) with a differential d of degree 1 satisfying the Leibniz rule. For a DG category X we denote by H^{0}X the category with the same objects as X and with

Hom_{H^{0}X} (X, Y) := H^{0}(Hom_{X}^{•} (X, Y)).

Example 7.2. Let A be an additive category. Then the class C(A)_{dg} of complexes over A becomes a DG category if we set:

Hom_{A}^{n}(X, Y) := ∐_{i∈ℤ} Hom_{A}(X^{i}, Y^{i+n}), and

d((f_{i})_{i∈ℤ}) = (f_{i}d_{Y} − (−1)^{n}d_{X}f_{i+1}), for (f_{i})_{i∈ℤ} ∈ Hom_{A}^{n}(X, Y).

Then it is not hard to check that Z^{0}(C(A)_{dg}) ≃ C(A) (where Z^{0} is the kernel of the differential d^{0}), the category of complexes, and H^{0}(C(A)_{dg}) ≃ H(A) the homotopy category of complexes over A.

The opposite category of a DG category and the tensor product of two DG categories are DG categories again (see [Kel91] – one has to be careful with the signs).

Definition 7.3. Let X be a DG category. A DG X -module is a DG functor X^{op} → C(Mod k)_{dg}. The DG X -modules form a DG category again, that we also denote by C(X)_{dg} by abuse of notation. Whenever we say two DG X - modules are isomorphic, we mean they are isomorphic in Z^{0}(C(X)_{dg}). We denote by DX the category obtained from H^{0}(C(X)_{dg}) by inverting quasiisomorphisms.

A DG X -module is representable, if it is isomorphic to a DG X -module of the form Hom_{X}^{•} (−, X) for some object X in X. We denote by pretr X the pretriangulated hull of X, i.e. the smallest subcategory of C(X)_{dg} which contains the representable DG X -modules, and which is closed under mapping cones (of morphisms in Z^{0}(C(X)_{dg})) and translations.

Note that by the Yoneda lemma, the natural DG functor Hom_{X}^{•} (−,?): X → pretr X is fully faithful. We call a DG category X pretriangulated if the Yoneda functor is dense.

Remark 7.4. If F: X → Y is a DG functor between DG categories, it induces an induction functor F^{∗}: C(X)_{dg} → C(Y)_{dg}. It sends representable functors to representable functors, and hence it induces a DG functor F^{∗}: pretr X → pretr Y.

Proposition 7.5 (Keller [Kel06]). Let X be a pretriangulated DG category. Then H^{0}(X) is an algebraic triangulated category. Moreover any algebraic triangulated category comes up in this construction.

Example 7.6. Let Λ be an algebra of finite global dimension.

• Let X := C^{b}(proj Λ)_{dg} be the DG category of bounded complexes of finitely generated projective Λ-modules. Then X is pretriangulated and the triangulated category H^{0}(X) is equivalent to D^{b}(Λ).

• Similarly, assume that Λ is G-graded, where G is an abelian group. Let Y := C^{b}(proj Cov(Λ, G))_{dg} be the DG category of bounded complexes of finitely generated projective Cov(Λ, G)-modules. Then Y is pretriangulated and the triangulated category H^{0}(Y) is equivalent to D^{b}(Cov(Λ, G)) ≃ D^{b}(gr Λ).

Note that the G grading on Λ and the homological grading of complexes are completely independent. In particular we emphasize that we only consider degree-preserving morphisms with respect to G, while we consider morphisms of arbitrary homological degree.

Definition 7.7. Let X and Y be DG categories. We denote by rep (X , Y) the full subcategory of D(X^{op} ⊗ Y) formed by the objects R, such that for all X ∈ X, the object R(X ⊗ −) is isomorphic to a representable DG Y-module in D(Y).

Example 7.8. Let F: X → Y be a DG functor. Then F induces an object R_{F} in D(X^{op} ⊗ Y) given by R_{F} (X ⊗ Y) := Hom_{Y}^{•}(Y, F X). Since R_{F} (X ⊗ −) is represented by F X, R_{F} is in rep (X , Y).

7.2. Universal property.

Definition 7.9. Let X be a DG category, and S: X → X a DG functor inducing an equivalence on H^{0}(X). Then the DG orbit category X /S has the same objects as X, and

Hom_{X /S}^{•}(X, Y) := colim_{p≫0} ⨁_{i≥0} Hom_{X}^{•} (S^{p}X, S^{p+i}Y) ≃

≃ ⨁_{i∈ℤ} colim_{p≫0} Hom_{X}^{•} (S^{p}X, S^{p+i}Y).

Definition 7.10. Let T := H^{0}(X) be an algebraic triangulated category, and S: X → X be a DG functor inducing an equivalence on T . Then the triangulated orbit category of T modulo S is defined to be

(T /S)_{Δ} := H^{0}(pretr (X /S)).

There is a natural DG functor π_{X} : X → X /S which induces a triangle functor

\xymatrix{π_{T} := H^{0}(π_{X}): T \ar[r] & (T /S)_{Δ}} .

Remark 7.11. The notation (T /S)_{Δ} is not strictly justified. Indeed the triangulated category H^{0}(pretr (X /S)) depends on X and S: X → X and not only on T and H^{0}S. But in this paper the triangulated categories that we use have canonical enhancement in DG categories, and the auto-equivalences have canonical lifts.

Example 7.12. We can now give a more precise definition of the generalized cluster category given in Section 2. Let Λ be a τ_{2}-finite algebra of global dimension ≤ 2. Let X := C^{b}(proj Λ)_{dg} be the DG category of bounded complexes of finitely generated projective Λmodules, and let S: X → X be the DG functor S := − ⊗_{Λ} p_{Λ^{op}⊗Λ}DΛ[−2] where p_{Λ^{op}⊗Λ}DΛ is a projective resolution of DΛ as a Λ-Λ-bimodule. Then we have

H^{0}(X /S) ≃ D^{b}(Λ)/𝕊_{2} and H^{0}(pretr (X /S)) ≃ (D^{b}(Λ)/𝕊_{2})_{Δ} =: C_{Λ}.

We are now ready to state a consequence of the universal property of the triangulated orbit category.

Proposition 7.13 ([Kel05]). Let T := H^{0}(Y) and T′ := H^{0}(X) be two algebraic triangulated categories, and S: Y → Y be a DG functor inducing an equivalence on T . Let F: Y → X be a DG functor, and assume that there is an isomorphism in rep (Y, X)

F ∘ S ≃ F.

Then F induces a triangulated functor (T /S)_{Δ} → T′ such that the following diagram commutes

\xymatrix{T \ar[r]^{H^{0}F} \ar[d]_{π_{T}} & T′ \\ (T /S)_{Δ} \ar[ur]} .

Corollary 7.14. Let Λ be a ℤ-graded algebra of finite global dimension. Then we have a triangle equivalence:

(D^{b}(Cov(Λ, ℤ))/⟨1⟩)_{Δ} ≃ D^{b}(Λ).

Proof. Let X := C^{b}(proj Λ)_{dg} and Y := C^{b}(proj Cov(Λ, ℤ))_{dg} be the DG categories defined in Example 7.6. The autoequivalence ⟨1⟩ of D^{b}(Cov(Λ, ℤ)) is induced by tensoring with the Y-Y-bimodule _{id}Cov(Λ, ℤ)_{⟨1⟩} (that is, the right Y-action is twisted by ⟨1⟩). The forgetful functor D^{b}(Cov(Λ, ℤ)) → D^{b}(Λ) is induced by the Y-X -bimodule Cov(Λ, ℤ) (obtained from the Y-Y-bimodule Cov(Λ, ℤ) by applying the forgetful functor on the right side). It is clear that we have an isomorphism of Cov(Λ, ℤ)-Λ-bimodules

idCov(Λ, ℤ)_{⟨1⟩} ≃ Cov(Λ, ℤ)

Therefore, by Proposition 7.13, there exists a triangle functor G

\xymatrix{D^{b}(Cov(Λ, ℤ)) \ar[rr] \ar[dr] &  & D^{b}(Λ) \\  & (D^{b}(Cov(Λ, ℤ))/⟨1⟩)_{Δ} \ar@{-->}[ur]^{G}}

The functor \xymatrix{D^{b}(Cov(Λ, ℤ))/⟨1⟩ \ar[r] & D^{b}(Λ)} is fully faithful, hence so is G. Moreover D^{b}(Λ) is generated as triangulated category by the simples, which are in the image of G. Therefore the functor G is an equivalence of triangulated categories. □

Proposition 7.15. Let T := H^{0}(X) and T′ := H^{0}(Y) be two algebraic triangulated categories, and S: X → X (resp. S′: Y → Y) be a DG functor inducing an equivalence on T (resp. on T′). Let F: X → Y be a DG functor, and assume that there is an isomorphism in rep (X , Y)

F ∘ S ≃ S′ ∘ F.

Then F induces a triangulated functor (T /S)_{Δ} → (T′/S′)_{Δ} such that the following diagram commutes.

\xymatrix{T \ar[r]^{H^{0}(F)} \ar[d]_{π_{T}} & T′ \ar[d]^{π_{T ′}} \\ (T /S)_{Δ} \ar[r] & (T′/S′)_{Δ}}

Proof. Let π_{Y} be the DG functor Y → pretr (Y/S′). Then we have π_{Y} ∘ S′ ≃ π_{Y} in rep (Y, pretr (Y/S′)) by definition. Therefore we have isomorphism in rep (X , pretr (Y/T)):

(π_{Y} ∘ F) ∘ S ≃ π_{Y} ∘ S′ ∘ F ≃ π_{Y} ∘ F.

Hence by Proposition 7.13, we get a commutative diagram:

\xymatrix{T = H^{0}(X) \ar[r]^{H^{0}(F)} \ar[d]_{H^{0}(π_{X})} & T′ = H^{0}(Y) \ar[d]^{H^{0}(π_{Y})} \\ H^{0}(pretr (X /S)) \ar[r]^{f} & H^{0}(pretr (Y/S′))} .

where f is a triangle functor. □

Corollary 7.16. Let Λ_{1} and Λ_{2} be derived equivalent algebras of global dimension ≤ 2 which are τ_{2}-finite. Then they are cluster equivalent.

Proof. Since Λ_{1} and Λ_{2} are derived equivalent, there exists T ∈ D^{b}(Λ_{1}^{op} ⊗ Λ_{2}) such that

\xymatrix{D^{b}(Λ_{1}) \ar[r]^{−⊗^{L}_{Λ_{1}} T} & D^{b}(Λ_{2})}

is an equivalence. By the previous proposition it is enough to check that there

exists an isomorphism in D^{b}(Λ_{1}^{op} ⊗ Λ_{2}) between DΛ_{1} ⊗^{L}_{Λ_{1}} T and T ⊗^{L}_{Λ_{2}} DΛ_{2}. Using the isomorphisms in D^{b}(Λ_{1}^{op} ⊗ Λ_{2})

DΛ_{1} ⊗^{L}_{Λ_{1}} T ≃ DHom_{Λ_{2}}(T, T) ⊗^{L}_{Λ_{1}} T

≃ Hom_{Λ_{2}}(T, T ⊗^{L}_{Λ_{2}} DΛ_{2}) ⊗^{L}_{Λ_{1}} T,

we get a natural morphism DΛ_{1} ⊗^{L}_{Λ_{1}} T → T ⊗^{L}_{Λ_{2}} DΛ_{2} in D^{b}(Λ_{1}^{op} ⊗ Λ_{2}) induced by the evaluation morphism. This morphism is clearly an isomorphism in D^{b}(Λ_{2}) by uniqueness of the Serre functor. Hence it is an isomorphism in D^{b}(Λ_{1}^{op} ⊗ Λ_{2}). □

7.3. Iterated triangulated orbit categories.

Proposition 7.17. Let T := H^{0}(X) be an algebraic triangulated category, and S, T: X → X be DG functors inducing equivalences on T , such that there is a natural isomorphism S ∘ T ≃ T ∘ S. Then there is an equivalence of DG categories

pretr (pretr (X /S)/T) ≃ pretr (pretr (X /T)/S).

Therefore, there is a triangle equivalence:

((T /S)_{Δ}/T)_{Δ} ≃ ((T /T)_{Δ})/S)_{Δ}.

Proof. We divide the proof into two lemmas.

Lemma 7.18. Under the hypothesis of Proposition 7.17 there is an equivalence of DG categories

(X /S)/T ≃ (X /T)/S.

Proof. The functor colim is a left adjoint, hence it commutes with colimits. Therefore, since S and T commute, we have

Hom_{(X /S)/T}^{•} (X, Y) = colim_{q} ⨁_{j≥0} colim_{p} ⨁_{i≥0} Hom_{X}^{•} (T^{q}S^{p}X, T^{q+j}S^{p+i}Y)

≃ colim_{p} ⨁_{i≥0} colim_{q} ⨁_{j≥0} Hom_{X}^{•} (T^{q}S^{p}X, T^{q+j}S^{p+i}Y)

≃ colim_{p} ⨁_{i≥0} colim_{q} ⨁_{j≥0} Hom_{X}^{•} (S^{p}T^{q}X, S^{p+i}T^{q+j}Y)

≃ Hom_{(X /T)/S}^{•}(X, Y)  □

We denote by X /{S, T } the DG category (X /S)/T ≃ (X /T)/S.

Lemma 7.19. Under the hypothesis of Proposition 7.17 there is an equivalence of DG categories

\xymatrix{pretr (X /\{S, T \}) \ar[r]^{∼} & pretr (pretr (X /S)/T)} .

Proof. We have a fully faithful DG functor

\xymatrix{X /S \ar@{^{(}->}[r] & pretr (X /S)} .

It induces a fully faithful DG functor

\xymatrix{X /\{S, T \} \ar@{^{(}->}[r] & pretr (X /S)/T \ar@{^{(}->}[r] & pretr (pretr (X /S)/T)} .

Since the category pretr (pretr (X /S)/T) is a pretriangulated DG category, we get a fully faithful DG functor

\xymatrix{pretr (X /\{S, T \}) \ar@{^{(}->}[r] & pretr (pretr (X /S)/T)} .

Now the objects of pretr (pretr (X /S)/T) are iterated cones of objects in X, hence this DG functor is also dense. Thus we get an equivalence of DG categories

\xymatrix{pretr (X /\{S, T \}) \ar[r]^{∼} & pretr (pretr (X /S)/T)} .  □

We can now prove Proposition 7.17. Using Lemma 7.19 and its symmetric version (interchanging S and T), we immediately get a DG equivalence

pretr (pretr (X /S)/T) ≃ pretr (pretr (X /T)/S).

By taking the H^{0} of each side we get the triangle equivalence

((T /S)_{Δ})/T)_{Δ} ≃ ((T /T)_{Δ})/S)_{Δ}.  □

Corollary 7.20. Let Λ be a ℤ-graded algebra of global dimension ≤ 2 which

is τ_{2}-finite. Let Cov(Λ, ℤ) be the ℤ-covering of Λ, and 𝕊_{2} := − ⊗^{L}_{Cov(Λ,ℤ)} DCov(Λ, ℤ)[−2]. Then there is a triangle equivalence

((D^{b}(Cov(Λ, ℤ))/𝕊_{2})_{Δ}/⟨1⟩)_{Δ} ≃ C_{Λ}.

Proof. We apply Proposition 7.17 for X := C^{b}(proj Cov(Λ, ℤ))_{dg}, S := − ⊗_{Cov(Λ,ℤ)} P, and T := ⟨1⟩, where P is a bimodule projective resolution of DCov(Λ, ℤ)[−2]. Then we clearly have S ∘ T ≃ T ∘ S. Thus we get

((D^{b}(Cov(Λ, ℤ))/𝕊_{2})_{Δ}/⟨1⟩)_{Δ} ≃ H^{0}(pretr (pretr (X /S)/T)) ≃

≃ H^{0}(pretr (pretr (X /T)/S)) ≃ C_{Λ}.

□

8. Graded derived equivalence for cluster equivalent algebras

8.1. Graded version of results of Section 6. In this section we generalize the previous results to the case of a graded algebra Λ. Let Λ be a ℤ-graded algebra of global dimension ≤ 2 which is τ_{2}-finite. We

denote by Cov(Λ, ℤ) its ℤ-covering. The functor − ⊗^{L}_{Cov(Λ,ℤ)} DCov(Λ, ℤ)[−2] is an autoequivalence of D^{b}(Cov(Λ, ℤ)) that we will denote by 𝕊_{2} by abuse of notation. We denote by π_{Λ}^{ℤ} the composition

\xymatrix{π_{Λ}^{ℤ}: D^{b}(Cov(Λ, ℤ)) \ar[r] & D^{b}(Λ) \ar[r]^{π_{Λ}} & C_{Λ}} .

This graded version of Proposition 3.1 is not hard to check:

Proposition 8.1. Let Λ be a ℤ-graded algebra which is τ_{2}-finite and of global dimension ≤ 2. Let T be a cluster-tilting object in C_{Λ}. The subcategory (π_{Λ}^{ℤ})^{−1}(T) is a cluster-tilting subcategory of D^{b}(Cov(Λ, ℤ)). In particular add {𝕊_{2}^{p}Λ⟨q⟩ | p, q ∈ ℤ} = (π_{Λ}^{ℤ})^{−1}(π_{Λ}(Λ)) is a cluster-tilting subcategory of D^{b}(Cov(Λ, ℤ)).

Here is the graded version of Proposition 6.3.

Proposition 8.2. Let Q be a ℤ-graded quiver, and I ⊂ kQ be an admissible ideal which is generated by homogeneous elements and such that the algebra Λ = kQ/I is of global dimension ≤ 2 and τ_{2}-finite. Denote by (Q̃, W) the quiver with potential defined in Theorem 6.2. Then there exists a unique ℤ^{2}- grading on Q̃ such that

(1) the potential W is homogeneous of degree (1, 1);

(2) there is an isomorphism of ℤ-graded quivers \xymatrix{Q̃^{\{0\}×ℤ} \ar[r]^{∼}_{ℤ} & Q} .

This grading on Q̃ yields a grading on Jac(Q̃, W) and we have an isomorphism of ℤ^{2}-graded algebras

\xymatrix{Jac(Q̃, W) \ar[r]^{∼}_{ℤ^{2}} & ⨁_{p,q∈ℤ} Hom_{D^{ℤ}}(Λ⟨0⟩, 𝕊_{2}^{−p}Λ⟨−p + q⟩)} .

Proof. There are two kinds of arrows in the quiver Q̃: arrows of Q and arrows coming from minimal relations. By (2) any arrow a coming from an arrow of Q has to be of degree (0, deg(a)). Let r be a minimal relation in I. By definition deg(r) is well defined. Then by Condition (1) since ra_{r} is a term of W, the degree of a_{r} has to be (1, 1 − deg(r)). Hence we have existence and uniqueness of such a grading. We have the following isomorphisms:

Λ̃ := End_{C_{Λ}}(π(Λ)) ≃ End_{C_{Λ}}(π^{ℤ}(Λ⟨0⟩)) ≃ ⨁_{p,q∈ℤ} Hom_{D^{ℤ}}(Λ⟨0⟩, 𝕊_{2}^{−p}Λ⟨−p + q⟩).

Since Jac(Q̃, W) ≃ Λ̃ by Theorem 6.2, we just have to check that it respects the gradings previously defined. Let a be an arrow of the quiver Q. It can be seen as an element of Hom_{D^{ℤ}}(Λ⟨0⟩, Λ⟨deg(a)⟩) so as an element of degree (0, deg(a)) of the algebra Λ̃ = ⨁_{p,q∈ℤ} Hom_{D^{ℤ}}(Λ⟨0⟩, 𝕊_{2}^{−p}Λ⟨−p + q⟩). Now let r be a minimal relation in I, and a_{r} be the corresponding arrow in Q̃. The minimal relation r corresponds to an element of Ext_{Λ}^{2}(S_{i}, S_{j}⟨−deg(r)⟩) where s(r) = j and t(r) = i. Hence it is an element in

Hom_{D^{ℤ}}(Λ⟨0⟩, 𝕊_{2}^{−1}Λ⟨−deg(r)⟩) = Hom_{D^{ℤ}}(Λ⟨0⟩, 𝕊_{2}^{−1}Λ⟨−1 + (−deg(r) + 1)⟩)

so an element of degree (1, −deg(r) + 1) in Λ̃. Hence the isomorphism Jac(Q̃, W, d) ≃ Λ̃ given by Theorem 6.2 is an isomorphism of ℤ^{2}-graded algebras. □

Let Λ be a ℤ-graded algebra of global dimension ≤ 2 which is τ_{2}-finite. Let T be an object in D^{b}(Cov(Λ, ℤ)) such that π^{ℤ}(T) is a (basic) cluster-tilting object in C_{Λ}. The endomorphism algebra

End_{C}(π^{ℤ}(T)) = ⨁_{p,q∈ℤ} Hom_{D^{ℤ}}(T, 𝕊_{2}^{−p}T ⟨−p + q⟩)

is naturally ℤ^{2}-graded. Let T_{i} be an indecomposable summand of T ≃ T_{i} ⊕ T′. We denote by U_{i} the additive subcategory add {𝕊_{2}^{p}T′⟨q⟩ | p, q ∈ ℤ} of D^{b}(Cov(Λ, ℤ)). Consider a triangle in D^{b}(Cov(Λ, ℤ))

\xymatrix{T_{i} \ar[r]^{u} & B \ar[r]^{v} & T_{i}^{L} \ar[r]^{w} & T_{i}[1]}

where u: T_{i} → B is a minimal left U_{i}-approximation of T_{i}. We call this triangle the graded left exchange triangle associated with T_{i}. We write μ_{i}^{L}(T) := T′ ⊕ T_{i}^{L}. Since the image of the graded left exchange triangle in D^{b}(Cov(Λ, ℤ)) is an exchange triangle in C_{Λ}, the object π^{ℤ}(T′ ⊕ T_{i}^{L}) is cluster-tilting in C_{Λ}. It is also possible to consider the graded right exchange triangle associated with T_{i}:

\xymatrix{T_{i}^{R} \ar[r]^{u′} & B′ \ar[r]^{v′} & T_{i} \ar[r] & T_{i}^{R}[1]} .

Theorem 8.3. Let Λ = kQ/I be a ℤ-graded τ_{2}-finite algebra of global dimension ≤ 2, and denote by (Q̃, W, d) the ℤ^{2}-graded QP defined in Proposition 8.2. Assume that there exists a sequence i_{1}, i_{2}, …, i_{l} of vertices of Q̃ such that for any j = 0, …, l there is no 2-cycle on the vertex i_{j+1} in the quiver Q^{j} where (Q^{j}, W^{j}) := μ_{i_{j}} ∘ ⋯ ∘ μ_{i_{1}}(Q̃, W). Denote by T the object in D^{ℤ} := D^{b}(Cov(Λ, ℤ)) defined by T := μ_{i_{l}}^{L} ∘ ⋯ ∘ μ_{i_{1}}^{L}(Λ⟨0⟩). Then there is an isomorphism of ℤ^{2}-graded algebras

\xymatrix{⨁_{(p,q)∈ℤ^{2}} Hom_{D^{ℤ}}(T, 𝕊_{2}^{−p}T ⟨−p + q⟩) \ar[r]^{∼}_{ℤ^{2}} & Jac(μ_{i_{l}}^{L} ∘ ⋯ ∘ μ_{i_{1}}^{L}(Q̃, W, d)).}

Proof. The proof is very similar to the proof of Theorem 6.12. We briefly outline the last step, which is a bit more technical. Let b: r → i_{j+1} be an arrow in Q^{j} and denote by (x, y) the degree of b. We consider the graded right exchange triangle in D^{b}(Cov(Λ, ℤ))

\xymatrix{(T_{i_{j+1}}^{j})^{R} \ar[r]^{u′} & B′ \ar[r]^{v′} & T_{i_{j+1}}^{j} \ar[r] & (T_{i_{j+1}}^{j})^{R}[1]} ,

where T_{i_{j+1}}^{j} denotes the summand of T^{j} := μ_{i_{j}}^{L} ∘…∘μ_{i_{1}}^{L}(Λ⟨0⟩) ∈ D^{b}(Cov(Λ, ℤ)) corresponding to the vertex i_{j+1}. Then 𝕊_{2}^{x}T_{r}^{j}⟨x−y⟩ is a direct summand of B′, and the reverse arrow b^{∗} corresponds to the component (T_{i_{j+1}}^{j})^{R} → 𝕊_{2}^{x}T_{r}^{j}⟨x−y⟩. Since, by Proposition 6.1, we have (T_{i_{j+1}}^{j})^{R} ≃ 𝕊_{2}(T_{i_{j+1}}^{j})^{L} we obtain that b^{∗} corresponds to a map

\xymatrix{(T_{i_{j+1}}^{j})^{L} \ar[r] & 𝕊_{2}^{x−1}T_{r}^{j}⟨x − 1 − (y − 1)⟩} ,

thus to a map of degree (1 − x, 1 − y). □

8.2. Graded derived equivalence. In this section we generalize Theorem 5.6 to the setup where the algebras are not graded equivalent. In this setup, we will not get a derived equivalence between the algebras Λ_{1} and Λ_{2}, but a derived equivalence between their coverings Cov(Λ_{1}, ℤ) and Cov(Λ_{2}, ℤ), for suitable gradings on them. In order to do that, we will use a graded version of the recognition theorem (Theorem 3.5). The proof of this theorem is very similar to the proof of Theorem 3.5.

Theorem 8.4. Let T be an algebraic triangulated category with a Serre functor and with a cluster-tilting subcategory V. Let Λ be a τ_{2}-finite algebra with global dimension ≤ 2, and with a ℤ-grading. Denote by U the cluster-tilting subcategory add {𝕊_{2}^{−p}Λ⟨−p + q⟩ | p, q ∈ ℤ} of D^{b}(Cov(Λ, ℤ)). Assume that there is an equivalence of additive categories with 𝕊_{2}-action \xymatrix{f: U \ar[r]^{∼} & V} . Then there exists a triangle equivalence F: D^{b}(Cov(Λ, ℤ)) → T such that the following diagram commutes

\xymatrix{D^{b}(Cov(Λ, ℤ)) \ar[r]^{F} & T \\ U \ar[r]^{f} \ar@{^{(}->}[u] & V \ar@{^{(}->}[u]}

For the statement of the main theorem of this section, we will need this technical definition.

Definition 8.5. Let Λ_{1} and Λ_{2} be two algebras of global dimension ≤ 2 which are τ_{2}-finite. For j = 1, 2, we denote by π_{j} the canonical functor π_{j}: D_{j} → C_{j} where D_{j} := D^{b}(Λ_{j}) and C_{j} = C_{Λ_{j}}. We will say that Λ_{1} and Λ_{2} satisfy the compatibility condition if there exists a sequence i_{1}, i_{2}, …, i_{l} (called compatible sequence) and a ℤ-graded QP (Q̃^{1}, W^{1}, d^{1}) with W^{1} homogeneous of degree 1 such that:

(1) we have an isomorphism of ℤ-graded algebras

Jac(Q̃^{1}, W^{1}, d^{1}) ≃_{ℤ} ⨁_{p∈ℤ} Hom_{D_{1}}(Λ_{1}, 𝕊_{2}^{−p}Λ_{1})

(2) for any 0 ≤ j ≤ l, the quiver of End_{C_{1}}(T_{j}) has neither loops nor 2-cycles at the vertex i_{j+1}, where T_{j} := μ_{i_{j}} ∘ ⋯ ∘ μ_{i_{2}} ∘ μ_{i_{1}}(π_{1}Λ_{1});

(3) there exists an isomorphism End_{C_{1}}(T_{l}) ≃ End_{C_{2}}(π_{2}Λ_{2});

(4) the isomorphisms in (1) and (3) can be chosen in such a way that there exists a ℤ-grading d^{2} on (Q̃^{2}, W^{2}) := μ_{i_{l}} ∘ ⋯ ∘ μ_{i_{2}} ∘ μ_{i_{1}}(Q̃^{1}, W^{1}) such that we have an isomorphism of ℤ-graded algebras

Jac(Q̃^{2}, W^{2}, d^{2}) ≃_{ℤ} ⨁_{q∈ℤ} Hom_{D_{2}}(Λ_{2}, 𝕊_{2}^{−q}Λ_{2}).

Remark 8.6. (1) The grading d^{2} will typically not be the grading obtained

by mutation on graded quivers with potential.

(2) In this definition, (1), (2), (3) mean that the quivers with potential of Λ̃_{1} and Λ̃_{2} (see Theorem 6.2) are linked by a sequence of mutations, and that neither loops nor 2-cycles occur at any intermediate step of this sequence of mutations. (The condition of not having loops or 2-cycles is automatic if the QP is rigid.)

(3) If Λ̃_{1} ≃ Λ̃_{2}, then conditions (1), (2) and (3) hold. We then have two (possibly) different ℤ-gradings on Λ̃_{1} = Λ̃_{2}. Condition (4) means that these two gradings yield a ℤ^{2}-grading on Λ̃_{1} = Λ̃_{2}.

(4) When conditions (1), (2) and (3) are satisfied, we do not know of any counterexamples to condition (4) being satisfied.

Theorem 8.7. Let Λ_{1} and Λ_{2} be two algebras of global dimension ≤ 2 which are τ_{2}-finite and which satisfy the compatibility condition. Then

(1) there are ℤ-gradings on Λ_{1} and on Λ_{2}, such that there exists a derived equivalence

\xymatrix{F^{ℤ}: D_{1}^{ℤ} \ar[r]^{∼} & D_{2}^{ℤ}} ,

where D_{j}^{ℤ} := D^{b}(Cov(Λ_{j}, ℤ)).

(2) the equivalence F^{ℤ} induces a triangle equivalence \xymatrix{F: C_{1} \ar[r]^{∼} & C_{2}} such that the following diagram commutes:

\xymatrix{D_{1}^{ℤ} \ar[d] \ar[dd]_{π_{1}^{ℤ}} \ar[r]^{∼}_{F^{ℤ}} & D_{2}^{ℤ} \ar[d] \ar[dd]^{π_{2}^{ℤ}} \\ D_{1} \ar[d]^{π_{1}} & D_{2} \ar[d]_{π_{2}} \\ C_{1} \ar[r]^{∼}_{F} & C_{2}}

Proof of (1). Take (Q̃^{1}, W^{1}, d^{1}) and the isomorphism (1) of the compatibility condition

\xymatrix{Jac(Q̃^{1}, W^{1}, d^{1}) \ar[r]^{∼}_{ℤ} & ⨁_{p∈ℤ} Hom_{D_{1}}(Λ_{1}, 𝕊_{2}^{−p}Λ_{1})} .

Let s := i_{1}, …, i_{l} be a compatible sequence and define T^{1} as the object

T^{1} := μ_{i_{l}}^{L} ∘ ⋯ ∘ μ_{i_{2}}^{L} ∘ μ_{i_{1}}^{L}(Λ_{1}) ∈ D^{b}(Λ_{1}).

By Condition (2) of the compatible sequence we can apply Theorem 6.12 and we get an isomorphism:

\xymatrix{Jac(μ_{s}^{L}(Q̃^{1}, W^{1}, d^{1})) \ar[r]^{∼}_{ℤ} & ⨁_{p∈ℤ} Hom_{D_{1}}(T^{1}, 𝕊_{2}^{−p}T^{1})} ,

where μ_{s}^{L} is the composition μ_{i_{l}}^{L} … μ_{i_{1}}^{L}. By Condition (4) of the compatibility condition, there exists a ℤ-grading d^{2} on (Q̃^{2}, W^{2}) := μ_{i_{l}} … μ_{i_{1}}(Q̃^{1}, W^{1}) such that W^{2} is homogeneous of degree 1 and such that we have an isomorphism

\xymatrix{Jac(Q̃^{2}, W^{2}, d^{2}) \ar[r]^{∼}_{ℤ} & ⨁_{q∈ℤ} Hom_{D_{2}}(Λ_{2}, 𝕊_{2}^{−p}Λ_{2})} .

Therefore the ℤ^{2}-grading (μ_{s}^{L}(d^{1}), d^{2}) on Q̃^{2} makes W^{2} homogeneous of degree (1, 1). Moreover we have an isomorphism

Jac(Q̃^{2}, W^{2}, (μ_{s}^{L}(d^{1}), d^{2}))^{ℤ×{0}} ≃ Λ_{2},

hence we get a ℤ-grading on Λ_{2}. By the uniqueness of the ℤ^{2}-grading of Proposition 8.2 we have an isomorphism of ℤ^{2}-graded algebras:

\xymatrix{Jac(Q̃_{2}, W^{2}, (μ_{s}^{L}(d^{1}), d^{2})) \ar[r]^{∼}_{ℤ^{2}} & ⨁_{p,q∈ℤ} Hom_{D_{2}^{ℤ}}(Λ_{2}⟨0⟩, 𝕊_{2}^{−q}Λ_{2}⟨−q + p⟩)} .

We define T^{2} ∈ D^{b}(Cov(Λ_{2}, ℤ)) as T^{2} := μ_{i_{1}}^{R} ⋯ μ_{i_{l}}^{R}(Λ_{2}⟨0⟩). By Condition (2) of the compatible sequence we can apply Theorem 8.3 and we get an isomorphism:

\xymatrix{Jac(μ_{s}^{R}(Q̃^{2}, W^{2}, μ_{s}^{L}(d^{1}), d^{2})) \ar[r]^{∼}_{ℤ^{2}} & ⨁_{p,q∈ℤ} Hom_{D_{2}^{ℤ}}(T^{2}, 𝕊_{2}^{−q}T^{2}⟨−q + p⟩)} ,

where μ_{s}^{R} is the composition μ_{i_{1}}^{R} … μ_{i_{l}}^{R}. By Lemma 6.9 the graded QP μ_{s}^{R}(Q̃^{2}, W^{2}, μ_{s}^{L}(d^{1})) = μ_{s}^{R}(μ_{s}^{L}(Q̃^{1}, W^{1}, d^{1})) is graded right equivalent to (Q̃^{1}, W^{1}, d^{1}). Therefore we have an isomorphism

(∗) \xymatrix{Jac(Q̃^{1}, W^{1}, (d^{1}, μ_{s}^{R}(d^{2}))) \ar[r]^{∼}_{ℤ^{2}} & ⨁_{p,q∈ℤ} Hom_{D_{2}^{ℤ}}(T^{2}, 𝕊_{2}^{−q}T^{2}⟨−q + p⟩)}

By definition of d^{1} there is an isomorphism

Λ_{1} ≃ Jac((Q̃^{1}, W^{1}, d^{1}, μ_{s}^{R}(d^{2})))^{{0}×ℤ}.

Therefore we get a ℤ-grading on Q^{1}. The ideal I^{1} is generated by {∂_{a}W^{1}, d^{1}(a) = 1}, hence is generated by elements which are homogeneous with respect to the grading μ_{s}^{R}(d^{2}). By the uniqueness of the ℤ^{2}-grading of Proposition 8.2 we have

(†) \xymatrix{Jac(Q̃_{1}, W^{1}, (d^{1}, μ_{s}^{R}(d^{2}))) \ar[r]^{∼}_{ℤ^{2}} & ⨁_{p,q∈ℤ} Hom_{D_{1}^{ℤ}}(Λ_{1}⟨0⟩, 𝕊_{2}^{−p}Λ_{1}⟨−p + q⟩)} .

Finally, combining (∗) and (†) we get

(‡) \xymatrix{⨁_{p,q∈ℤ} Hom_{D_{2}^{ℤ}}(T^{2}, 𝕊_{2}^{−q}T^{2}⟨−q + p⟩) \ar[r]^{∼}_{ℤ^{2}} & ⨁_{p,q∈ℤ} Hom_{D_{1}^{ℤ}}(Λ_{1}⟨0⟩, 𝕊_{2}^{−p}Λ_{1}⟨−p + q⟩)} .

The category add {𝕊_{2}^{p}T^{2}⟨q⟩ | p, q ∈ ℤ} = (π_{2}^{ℤ})^{−1}(μ_{S}π_{2}Λ_{2}) is cluster-tilting in D^{b}(Cov(Λ_{2}, ℤ)) by Proposition 8.1. The isomorphism (‡) implies that we have an equivalence of ℤ^{2}-categories

\xymatrix{add \{𝕊_{2}^{−p}Λ_{1}⟨−p + q⟩ | p, q ∈ ℤ\} \ar[r]^{∼}_{f} & add \{𝕊_{2}^{−q}T^{2}⟨−q + p⟩ | p, q ∈ ℤ\}} .

In order to apply Theorem 8.4, we have to check that f commutes with 𝕊_{2}:

f(𝕊_{2}(𝕊_{2}^{−p}Λ_{1}⟨−p + q⟩)) = f(𝕊_{2}^{−p+1}Λ_{1}⟨−p + 1 + (q − 1)⟩)
= 𝕊_{2}^{−q+1}T^{2}⟨−q + 1 + (p − 1)⟩
= 𝕊_{2}(𝕊_{2}^{−q}T^{2}⟨−q + p⟩)
= 𝕊_{2}f(𝕊_{2}^{−p}Λ_{1}⟨−p + q⟩)

Therefore, by Theorem 8.4, we get a triangle equivalence F: D^{b}(Cov(Λ_{1}, ℤ)) → D^{b}(Cov(Λ_{2}, ℤ)) which extends f.

Proof of (2). Now for i = 1, 2 let X_{i} := C^{b}(proj Cov(Λ_{i}, ℤ))_{dg} be the DG category of bounded complexes of projective Cov(Λ_{i}, ℤ)-modules. The functor

\xymatrix{F: D^{b}(Cov(Λ_{1}, ℤ)) = H^{0}(X_{1}) \ar[r] & D^{b}(Cov(Λ_{2}, ℤ)) = H^{0}(X_{2})}

can be seen as H^{0}(F_{dg}) where F_{dg} := − ⊗_{Cov(Λ_{1},ℤ)} P and P is a projective resolution of ⨁_{p∈ℤ} 𝕊_{2}^{−p}T^{2}⟨−p⟩ as Cov(Λ_{1}, ℤ)-Cov(Λ_{2}, ℤ)-bimodule. For i = 1, 2 we set S_{i} := − ⊗_{Cov(Λ_{i},ℤ)} X_{i} where X_{i} is a projective resolution of DCov(Λ_{i}, ℤ)[−2] as Cov(Λ_{i}, ℤ)-bimodule.

In order to prove that we have an isomorphism

F_{dg} ∘ S_{1} ≃ S_{2} ∘ F_{dg} in rep (X_{1}, X_{2}),

it is enough to prove that we have an isomorphism

X_{1} ⊗_{Cov(Λ_{1},ℤ)} P ≃ P ⊗_{Cov(Λ_{2},ℤ)} X_{2} in D^{b}((Cov(Λ_{1}, ℤ))^{op} ⊗ Cov(Λ_{2}, ℤ)),

and the proof is similar to the proof of Corollary 7.16. Now, by Proposition 7.15, we get a DG functor F_{S}: pretr (X_{1}/S_{1}) → pretr (X_{2}/S_{2}) such that H^{0}(F_{S}) is an equivalence and such that the following diagram commutes

\begin{matrix}D^{b}(Cov(Λ_{1}, ℤ)) \\ = H^{0}(X_{1}) \end{matrix}

H^{0}(π_{X1} )

[figure]

F =H^{0}(F_{dg}) [figure] \begin{matrix}D^{b}(Cov(Λ_{2}, ℤ)) \\ = H^{0}(X_{2}) \end{matrix}

H^{0}(π_{X2} )

[figure]

(D^{b}(Cov(Λ_{1}, ℤ))/𝕊_{2})_{Δ}

= H^{0}(pretr (X_{1}/S_{1}))
H^{0}(F_{S})

[figure] (D^{b}(Cov(Λ_{2}, ℤ))/𝕊_{2})_{Δ} = H^{0}(pretr (X_{2}/S_{2}))

For i = 1, 2 we set T_{i} := ⟨1⟩: X_{i} → X_{i}. It is immediate to check that we have

S_{i} ∘ T_{i} ≃ T_{i} ∘ S_{i} in rep (X_{i}, X_{i}).

Moreover, the functor T_{i}, as DG equivalence of X_{i} induces a DG functor T_{i} on pretr (X_{i}/S_{i}) such that π_{X_{i}} ∘ T_{i} ≃ T_{i} ∘ π_{X_{i}}. We have the following isomorphisms in D((Cov(Λ_{1}, ℤ))^{op} ⊗ Cov(Λ_{2}, ℤ)),

idCov(Λ_{1}, ℤ)_{⟨−1⟩} ⊗_{Cov(Λ_{1},ℤ)} P ≃ _{id}Cov(Λ_{1}, ℤ)_{⟨−1⟩} ⊗^{L}_{Cov(Λ_{1},ℤ)} ⨁_{q∈ℤ} 𝕊_{2}^{−q}T^{2}⟨−q⟩

≃ ⨁_{q∈ℤ} 𝕊_{2}^{−q+1}T^{2}⟨−q + 1⟩

≃ 𝕊_{2}(⨁_{q∈ℤ} 𝕊_{2}^{−q}T^{2}⟨−q⟩)⟨1⟩

≃ P ⟨1⟩ ⊗_{Cov(Λ_{2},ℤ)} X_{2}

and hence an isomorphism

F_{dg} ∘ T_{1}^{−1} ≃ S_{2} ∘ T_{2} ∘ F_{dg}, in rep (X_{1}, X_{2}).

Therefore we have isomorphisms in rep (X_{1}, pretr (X_{2}/S_{2}))

F_{S} ∘ T_{1}^{−1} ∘ π_{X_{1}} ≃ F_{S} ∘ π_{X_{1}} ∘ T_{1}^{−1}
≃ π_{X_{2}} ∘ F_{dg} ∘ T_{1}^{−1}
≃ π_{X_{2}} ∘ S_{2} ∘ T_{2} ∘ F_{dg}
≃ π_{X_{2}} ∘ T_{2} ∘ F_{dg}
≃ T_{2} ∘ F_{s} ∘ π_{X_{1}},

and we deduce an isomorphism

F_{S} ∘ T_{1}^{−1} ≃ T_{2} ∘ F_{S} in rep (pretr (X_{1}/S_{1}), pretr (X_{2}, S_{2})).

Applying again Proposition 7.15 we get a DG functor

F_{S,T} : pretr (pretr (X_{1}/S_{1})/T_{1}) → pretr (pretr (X_{2}/S_{2})/T_{2})

such that H^{0}(F_{S,T}) is a triangle equivalence and such that the following diagram commutes

\xymatrix{pretr (X_{1}/S_{1}) \ar[r]^{F_{S}} \ar[d]^{π_{X_{1}/S_{1}}} & pretr (X_{2}/S_{2}) \ar[d]^{π_{X_{2}/S_{2}}} \\ pretr (pretr (X_{1}/S_{1})/T_{1})) \ar[r]^{F_{S,T}} & pretr (pretr (X_{2}/S_{2})/T_{2})} .

Applying H^{0} and using Corollary 7.20 we obtain a commutative diagram

\xymatrix{D^{b}(Cov(Λ_{1}, ℤ)) \ar[d]^{H^{0}(π_{X_{1}} )} \ar[r]^{F =H^{0}(F_{dg})} & D^{b}(Cov(Λ_{2}, ℤ)) \ar[d]^{H^{0}(π_{X_{2}} )} \\ (D^{b}(Cov(Λ_{1}, ℤ))/𝕊_{2})_{Δ} \ar[r]_{H^{0}(F_{S})} \ar[d]^{H^{0}(π_{X_{1}/S_{1}} )} & (D^{b}(Cov(Λ_{2}, ℤ))/𝕊_{2})_{Δ} \ar[d]^{H^{0}(π_{X_{2}/S_{2}} )} \\ C_{Λ_{1}} ≃ ((D^{b}(Cov(Λ_{1}, ℤ))/𝕊_{2})_{Δ}/⟨1⟩)_{Δ} \ar[r]^{H^{0}(F_{S,T})} & ((D^{b}(Cov(Λ_{2}, ℤ))/𝕊_{2})_{Δ}/⟨1⟩)_{Δ} ≃ C_{Λ_{2}}}

□

Example 8.8. Let H = kQ and Λ_{3} = kQ^{3}/I^{3} be the algebras given by the following quivers (we keep the notation of Example 5.7):

Q = \xymatrix{ & 2 \ar[dl]_{α} \\ 1 & & 3 \ar[ll]^{γ} \ar@{-->}[ul]_{β}} , and Q^{3} = [figure: 2; b; 1; a; 3; d]

with relations I^{3} = ⟨ba⟩. The graded QP associated with these algebras of global dimension ≤ 2 are (Q, 0, d) and (Q̃_{3}, W_{3}, d_{3}) given by

Q = [figure: 2; 0; α; 1; 3; 0; γ; 0; β] and Q̃^{3} = [figure: 2; 0; b; 1; 0; a; 3; 0; d; 1; c]

with W_{3} = cba. It is immediate to see that there is an isomorphism μ_{2}(Q, 0) ≃ (Q̃^{3}, W_{3}) which sends α^{∗} on a, β^{∗} on b, [αβ] on c and γ on d. If we compute μ_{2}^{L}(Q, 0, d) we get the following grading on μ_{2}(Q):

μ_{2}(Q) = [figure: 2; 1; β^{∗}; 1; 0; α^{∗}; 3; 0; γ; 0; [αβ]]

Then we get a ℤ^{2}-grading on Q̃^{3} given by (μ_{2}^{L}(d), d_{3}):

Q̃^{3} = [figure: 2; (1,0); b; 1; (0,0); a; 3; (0,0); d; (0,1); c]

which makes the potential W_{3} homogeneous of degree (1, 1) and which induces a grading on Λ_{3}

Q^{3} = [figure: 2; 1; b; 1; 0; a; 3.; 0; d]

Now μ_{2}^{R}(Q̃_{3}, W_{3}, d_{3}) yields a ℤ-grading on Q

Q = [figure: 2; 1; a^{∗}; 1; 3.; 0; d; 0; b^{∗}]

Here are the ℤ-coverings of the ℤ-graded algebras H and Λ_{3}

[figure: 2; 1; 3; γ; β; 2; α; Cov(H, ℤ) =; 1; 3; β; γ; 2; α; 1; 3; β; γ; 2; 1; a; 3; d; 2; b; and Cov(Λ_{3}, ℤ) =; 1; a; 3; d; 2; b; 1; a; 3; d]

By Theorem 8.7 we have a derived equivalence D^{b}(Cov(H, ℤ)) ≃ D^{b}(Cov(Λ_{3}, ℤ)).

Remark 8.9. (1) The algebras given by the quivers

[figure: 1; 2; α; 1; 3; β; γ; and; 1; 3; d; 2; b; 1; a]

and the relation ba = 0 are derived equivalent. One can pass from one to the other by doing the left mutation in the derived category at vertex 2. Using this repeatedly one can also directly check that we have a derived equivalence D^{b}(Cov(H, ℤ)) ≃ D^{b}(Cov(Λ_{3}, ℤ)).

(2) In the paper [AO13a], we use Theorem 8.7 to deduce the shape of the AR-quiver of the derived category D^{b}(Λ_{3}), and of any algebra which is cluster equivalent to the path algebra of a quiver of type Ã_{n}.

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Claire Amiot Institut Fourier 100 rue des mathématiques 38402 Saint Martin d’Hères France claire.amiot@ujf-grenoble.fr Steffen Oppermann Insitutt for matematiske fag Norges Teknisk-Naturvitenskapelige Universitet N-7491 Trondheim Norway steffen.oppermann@math.ntnu.no