On cluster-tilting objects in a triangulated category
with Serre duality
Wuzhong Yang 1 , Jie Zhang 2 and Bin Zhu 3
Abstract
Let D be a Krull-Schmidt, Hom-finite triangulated category with a Serre functor and a
cluster-tilting object T . We introduce the notion of an FΛ -stable support τ-tilting module,
induced by the shift functor and the Auslander-Reiten translation, in the cluster-tilted algebra
Λ = Endop
D (T ). We show that there exists a bijection between basic cluster-tilting objects
in D and basic FΛ -stable support τ-tilting Λ-modules. This generalizes a result of AdachiIyama-Reiten [AIR]. As a consequence, we obtain that all cluster-tilting objects in D have
the same number of non-isomorphic indecomposable direct summands.
Key words: cluster-tilting object; Serre functor; support τ-tilting module; F-stable.
2000 Mathematics Subject Classification: 18E30;16G20;16G70.
1
Introduction
The concept of cluster algebras was introduced by S. Fomin and A. Zelevinsky in [FZ]. In a series
of papers they developed a theory of cluster algebras, which has influential connections with
representation theory of finite dimensional algebras. In order to model some essential ingredients
in the definition of cluster algebras by similar concepts in a category with additional structure,
cluster categories were invented in [BMRRT] (see also [CCS] for type An ). An important class
of objects in these categories are the cluster-tilting objects, which play the role as clusters do in
cluster algebras. The endomorphism algebras of cluster-tilting objects are called cluster-tilted
algebras [BMR], whose module categories are closely connected with cluster categories [BMR,
KR, KZ, IY].
In [Smi], the author investigated the connections between cluster-tilting objects in a cluster category D and tilting modules in the module category of a cluster-tilted algebra Λ. It was shown
that a tilting Λ-module can be lifted to a cluster-tilting object in D. However, cluster-tilting
objects in D do not always correspond to tilting Λ-modules. Actually, this fact also holds for
any 2-Calabi-Yau (2-CY for short) triangulated category with cluster-tilting objects [FL, HJ].
Recently, T. Adachi, O. Iyama and I. Reiten introduced the τ-tilting theory [AIR]. They established a bijection between cluster-tilting objects in D and so-called support τ-tilting modules in
modΛ when D is 2-CY (see also [CZZ]). This bijection answers the question why the modules
in modΛ corresponding to the cluster-tilting objects in D are not necessarily tilting modules. It
is then natural to ask what kind of modules in modΛ correspond to the cluster-tilting objects in
D if D is n-Calabi-Yau, or more generally, D is a triangulated category with a Serre functor.
1
Supported by the NSF of China (Grants No. 11131001)
Corresponding author and supported by BIT basic scientific research grant (Grant No. 3170012211408) and NSF
of China (Grants No. 11401022).
3
Supported by the NSF of China (Grants No. 11131001)
2
1
Let D be a triangulated category with a Serre functor and T be a cluster-tilting object in D. Then
D has an auto-equivalence F = τ−1
D [1], where τD is the Auslander-Reiten translation in D. S.
Koenig and B. Zhu in [KZ] proved that there also exists an equivalence
op
HomD (T, −) : D/addT [1] −→ modΛ = mod EndD (T ),
where modΛ is the category of finitely generated left Λ-modules. It leads us to consider the
following questions:
Question 1.1. Do cluster-tilting objects in D correspond to support τ-tilting modules in modΛ
bijectively again?
Unfortunately, the answer is negative. We give the following counterexample. Let Q be the
quiver as follows:
α
1
/2
β
/3.
We consider the repetitive cluster category D = Db (kQ)/hτ−2
Q [2]i introduced by B. Zhu in [Z11],
whose objects are the same in Db (kQ), and whose morphisms are given by
M
i
HomDb (kQ)/hτ−2 [2]i (X, Y) =
HomDb (kQ) (X, (τ−2
Q [2]) Y).
Q
i∈Z
It is shown in [Z11] that D is a triangulated category with a Serre functor S, where τQ is the
Auslander-Reiten translation in Db (kQ). Note that it is not 2-CY (but it is a fractional CalabiYau category with CY-dimension 24 ). The AR-quiver of D is as follows:
P3 [3]
I1D [1]
PD2 [2]
P3 [2]
P1 [1]
I1
P3D [3]
PD1 [2]
I2D [1]
PD2 [1]
D I2
S2
S D2 [1]
P
D 2
P3
P3D [1]
P
D 1
I2D [2]
S 2 [2]
P
D 2
P3
Figure 1
where Pi (respectivly, Ii , S i ) is the indecomposable projective (respectively, injective, simple)
module corresponding to vertex i. Note that the direct sum T 1 = S 2 ⊕ P3 [1] ⊕ P1 [1] ⊕ I1 [1] ⊕
S 2 [2] ⊕ P3 [3] of the encircled indecomposable objects gives a cluster-tilting object, the clusterop
tilted algebra Λ1 = EndD (T 1 ) is given by the following quiver with rad2 = 0.
3o
4
2X
F1
/6
5
2
The AR-quiver of modΛ1 is as follows:
2
4
6
1
o
D5
6o
6
D3
D5
4o
D3
5
6
D1
o
2o
D1
3
4
1
2
Figure 2
2
It is easy to see that M1 = 3
4 ⊕ 3 ⊕ 3 is a support τ-tilting Λ1 -module, but the object in D
corresponding to M1 is P1 [1] ⊕ I2 [1] ⊕ I1 [1] ⊕ I1 ⊕ S 2 [1] ⊕ P1 , which is not a cluster-tilting object
since it has self-extensions.
The counterexample forces us to consider the weak version of the above question.
Question 1.2. Can tilting modules in modΛ be lifted to cluster-tilting objects in D?
Let us reconsider the above example again. We take another cluster-tilting object T 2 = P3 ⊕
op
P2 ⊕ P1 ⊕ S 2 [1] ⊕ I2 [1] ⊕ P3 [2] (see Figure 1), the cluster-tilted algebra Λ2 = EndD (T 2 ) is not
connected, it is given by the disconnected quiver:
/2
1
/3
/5
4
/6
with no relations. The AR-quiver of modΛ2 is given as follows
1
6o
D2
D5
3
2o
D3
3o
1
D2
2o
o
D4
5o
6
4
D5
4
5
6
1
1
5⊕ 4
Clearly, M2 = 2 ⊕ 1
⊕
1
⊕
6
⊕
is a tilting Λ2 -module, but the object in D corresponding
6 5
3 2
6
to M2 is P1 ⊕ I2 ⊕ I1 ⊕ S 2 [1] ⊕ I2 [1] ⊕ P3 [2], which is not a cluster-tilting object, and not even a
rigid object.
In order to explain why the answers to Questions 1.1 and 1.2 are negative, we need the following
definition for an additive category C with an auto-equivalence G. We call an object M in C
G-stable if G(M) M. We introduce, in this paper, the auto-equivalence FΛ of modΛ induced
by the auto-equivalence F of D. The main result of this paper is a generalization of a bijection
in [AIR], and our result shows that a properly modified version of Questions 1.1 and 1.2 have
positive answers.
3
Theorem 1.3. (see Theorem 2.14 for details). Let D be a triangulated category with a Serre
op
functor and a cluster-tilting object T , and let Λ = EndD (T ) be the corresponding cluster-tilted
algebra. Then there is a bijection between the set of isomorphism classes of basic cluster-tilting
objects in D and the set of isomorphism classes of basic FΛ -stable support τ-tilting Λ-modules.
We have the following direct consequence, which generalizes some results in [DK, ZZ].
Corollary 1.4. Let D be a triangulated category with a Serre functor and a cluster-tilting object
T , and F = τ−1
D [1]. Assume that U is a basic object in D, then the following are equivalent.
• U is cluster-tilting.
• U is F-stable and maximal rigid.
• U is an F-stable rigid object and |U| = |T |, where |U| denotes the number of nonisomorphic indecomposable direct summands of U.
In particular, all basic cluster-tilting objects in D have the same number of indecomposable
direct summands.
We end this section with some conventions. Throughout this article, k is an algebraically closed
field. By Λ, we denote a finite dimensional basic k-algebra. All modules we consider in this paper
are left modules. For any triangulated category D, we assume that it satisfies the Krull-RemakSchmidt property. In D, we denote the shift functor by [1] and define ExtiD (X, Y) BHomD (X,
Y[i]) for any objects X and Y. If T is a subcategory of D, then we always assume that T is a full
subcategory which is closed under taking isomorphisms, direct sums and direct summands. The
quotient category of D by T denoted by D/T , is a category with the same objects as D and the
space of morphisms from X to Y is the quotient of group of morphisms from X to Y in D by the
subgroup consisting of morphisms factor through an object in T . For an object T in D, addT
denotes the full subcategory consisting of direct summands of direct sum of finitely many copies
of T . For two morphisms f : M → N and g : N → L, the composition of f and g is denoted by
g ◦ f : M → L.
2
Cluster-tilting objects and FΛ -stable support τ-tilting modules
Assume that D is a k-linear, Hom-finite, triangulated category. Recall from [BK] a Serre functor
S : D → D is a k-linear equivalence with bifunctorial isomorphisms
HomD (A, B) DHomD (B, SA)
for any A, B ∈ D, where D is the duality over k. In [RVdB], I. Reiten and M. Van den Bergh
proved that if D admits a Serre functor S, then D has Auslander-Reiten triangles. Moreover, if
τD is the Auslander-Reiten translation in D, then S τD [1].
In the following, we always assume the category D has a Serre functor S. Recall that an important
class of objects in D are rigid objects: An object A in D is called rigid if Ext1D (A, A) = 0. With
a collection of rigid objects, one can define maximal rigid objects and cluster-tilting objects as
follows:
Definition 2.1. (1) An object M in D is called maximal rigid if it is rigid and maximal with
respect to the property: addM = {X ∈ D | Ext1D (M ⊕ X, M ⊕ X) = 0}.
4
(2) We call a rigid object T in D cluster-tilting if
addT = {X ∈ D | Ext1D (T, X) = 0} = {X ∈ D | Ext1D (X, T ) = 0}.
op
In this situation, the algebra EndD (T ) is called a cluster-tilted algebra.
Let τD be the Auslander-Reiten translation in D, set F = τ−1
D [1], then we have the following
results, which will be used frequently in this paper.
Theorem 2.2 ([KZ]). Let T be a basic cluster-tilting object in D, then FT = T and the functor
HomD (T, −) induces an equivalence
op
D/addT [1] −→ modEndD (T ).
We recall in the following some definitions related to τ-tilting theory in [AIR] introduced by T.
Adachi, O. Iyama and I. Reiten. Let Λ be a finite dimensional k-algebra and τ := τΛ be the
Auslander-Reiten translation in modΛ. We denote by projΛ the category of finitely generated
projective Λ-modules.
Definition 2.3. Let (X, P) be a pair with X ∈ modΛ and P ∈projΛ.
• The pair (X, P) is said to be basic if X and P are basic.
• X is called τ-rigid if HomΛ (X, τX) = 0. We say the pair (X, P) is a τ-rigid pair if X is
τ-rigid and HomΛ (P, X) = 0.
• We say X is τ-tilting if X is τ-rigid and |X| = |Λ|. A τ-rigid pair (X, P) is said to be a
support τ-tilting pair if |X| + |P| = |Λ|. In this case, X is called a support τ-tilting module.
We denote by τ-tiltΛ (respectively, sτ-tiltΛ) the set of isomorphism classes of basic τ-tilting
(respectively, support τ-tilting) modules in modΛ. The following proposition gives some characterizations of a τ-rigid pair being a support τ-tilting pair.
Proposition 2.4 ([AIR]). Let (T, P) be a τ-rigid pair for Λ. Then the following are equivalent.
• (T, P) is a support τ-tilting pair.
• If (T ⊕ X, P) is τ-rigid for some Λ-module X, then X ∈ addT .
• The condition HomΛ (T, τX) = 0, HomΛ (X, τT ) = 0 and HomΛ (P, X) = 0 implies that
X ∈ addT .
Throughout this article, let D be a k-linear Hom-finite triangulated category with a cluster-tilting
op
object T , and Λ = EndD (T ) the corresponding cluster-tilted algebra. By Theorem 2.2, there is
an equivalence of categories:
∼
(−) B HomD (T, −) : D/addT [1] −→ modΛ,
Moreover, we get a k-algebra automorphism of Λ
∼
F∗ : HomD (T, T ) = Λ −→ Λ = HomD (FT, FT )
f
7−→ F( f ).
This allows us to define a functor
FΛ : modΛ → modΛ
as follows.
5
(1)
• If M ∈ modΛ, we set by FΛ (M) the vector space M endowed with the structure of left
Λ-module given by
a · m B F∗−1 (a)m
for all m ∈ M and a ∈ Λ.
• If f : M → M 0 is a Λ-homomorphism, then we define FΛ ( f ) by
FΛ ( f )(m) B f (m),
for all m ∈ FΛ (M).
It is easy to show that FΛ : modΛ → modΛ is a functor. Dually, we can define another functor
GΛ : modΛ → modΛ as follows: If M ∈ modΛ, we denote by GΛ (M) the vector space M
endowed with the structure of left Λ-module given by
a · m B F∗ (a)m,
for all m ∈ M and a ∈ Λ. If f : M → M 0 is a Λ-homomorphism, then we define GΛ ( f ) by
GΛ ( f )(m) B f (m),
for all m ∈ GΛ (M). Moreover, we have the following observation.
Proposition 2.5. The functor FΛ : modΛ → modΛ is an equivalence.
Proof. It is straightforward to check that GΛ is the quasi-inverse of FΛ .
From the constructions of FΛ and GΛ , we have the following direct observation.
Proposition 2.6. For any object X in D, we have FX FΛ (X) and F −1 X GΛ (X) in modΛ.
Proof. Since F is an equivalence in D, we get a k-linear isomorphism of vector spaces
2.2
FΛ (X) = X = HomD (T, X) HomD (FT, FX) ==== HomD (T, FX) = FX.
For any f ∈ FΛ (X) = FΛ (HomD (T, X)), a ∈ Λ = HomD (T, T ),
F(a · f ) = F( f ◦ F∗−1 (a)) = F( f ) ◦ F(F∗−1 (a)) = F( f ) ◦ a = aF( f ),
where ◦ is the composition of morphisms in D. Hence F : FΛ (X) → FX is a Λ-module isomorphism. Similarly, we can show that F −1 X GΛ (X).
The following lemma which was proved in [P] for the case D is a 2-CY category, can be easily
generalized to our setting by the same approach.
Lemma 2.7. Let D be a triangulated category with a Serre functor S and a cluster-tilting object
T . Then for any objects X and Y in D, there is a bifunctorial isomorphism
HomD/addT (τ−1
D Y, X) D[T ](X[−1], Y),
where [T ](X, Y) denotes the subgroup of HomD (X, Y) consisting of morphisms which factor
through objects in addT .
6
Proof. Since T is a cluster-tilting object, by Theorem 3.1 in [IY], we know there exists a triangle
η
T 1 −→ T 0 −→ X −→ T 1 [1]
in D with T 0 and T 1 in addT . Consider the morphism
ϕ : HomD (T 1 , Y) −→ HomD (X[−1], Y)
f
7−→
f ◦ η[−1].
We have
D[T ](X[−1], Y) DImϕ ImDϕ.
Since the category D has a Serre functor S τD [1],
DHomD (T 1 , Y) HomD (S−1 Y, T 1 ) HomD (τ−1
D Y, T 1 [1]),
DHomD (X[−1], Y) HomD (S−1 Y, X[−1]) HomD (τ−1
D Y, X).
Thus, Dϕ is isomorphic to
−1
φ : HomD (τ−1
D Y, X) −→ HomD (τD Y, T 1 [1])
g 7−→ η ◦ g.
Note that Kerφ = [T ](τ−1
D Y, X). Hence, we have isomorphisms
−1
D[T ](X[−1], Y) Imφ HomD (τ−1
D Y, X)/Kerφ HomD/addT (τD Y, X).
By the above lemma, we can express Ext1D (X, Y) in terms of the images X and Y in modΛ. For
two Λ-modules M and N, from now on, we use hX, Yi to denote dimk HomΛ (X, Y) for simplicity.
We first consider the following case:
Lemma 2.8. Let X and Y be objects in D such that there are no non-zero indecomposable direct
summands of T [1] for X and Y. Then X[1] τFΛ (X). Moreover,
dimk Ext1D (X, Y) = hX, τFΛ (Y)i + hFΛ (Y), τFΛ (X)i.
Proof. We first prove X[1] τFΛ (X). By Proposition 4.7 in [KZ], the residue class of any sink
(respectively, source) map in D is again a sink (respectively, source) map in modΛ. Combining
this with Proposition 2.6, we obtain
X[1] = τD FX τFX τFΛ (X).
Consider the following exact sequence,
0 → [T [1]](X, Y[1]) → HomD (X, Y[1]) → HomD/addT [1] (X, Y[1]) → 0.
By the equivalence (1), we get
HomD/addT [1] (X, Y[1]) HomΛ (X, Y[1]) HomΛ (X, τFΛ (Y)).
7
(2)
Using Lemma 2.7 and (2), we have
[T [1]](X, Y[1])
DHomD/addT [1] (τ−1
D Y[1], X[1])
=
DHomD/addT [1] (FY, X[1])
2.6
DHomΛ (FΛ (Y), τFΛ (X)).
Together, 0 → DHomΛ (FΛ (Y), τFΛ (X)) → Ext1D (X, Y) → HomΛ (X, τFΛ (Y)) → 0 is an exact
sequence. Hence, dimk Ext1D (X, Y) = hX, τFΛ (Y)i + hFΛ (Y), τFΛ (X)i.
For the general case, we have the following proposition.
Proposition 2.9. Let X = X 0 ⊕ X 00 and Y = Y 0 ⊕ Y 00 be objects in D such that X 00 and Y 00 are the
maximal direct summands of X and Y respectively, which belong to addT [1]. Then
dimk Ext1D (X, Y) = hX 0 , τFΛ (Y 0 )i + hFΛ (Y 0 ), τFΛ (X 0 )i + hX 00 [−1], Y 0 i + hY 00 [−1], GΛ (X 0 )i.
Proof. Since X 00 and Y 00 belong to addT [1], we have
Ext1D (X, Y) Ext1D (X 0 , Y 0 ) ⊕ Ext1D (X 00 , Y 0 ) ⊕ Ext1D (X 0 , Y 00 ).
(3)
By Lemma 2.8, we get
dimk Ext1D (X 0 , Y 0 ) = hX 0 , τFΛ (Y 0 )i + hFΛ (Y 0 ), τFΛ (X 0 )i.
(4)
Moreover we have
Ext1D (X 00 , Y 0 ) HomD (X 00 [−1], Y 0 )
HomD/addT [1] (X 00 [−1], Y 0 )
HomΛ (X 00 [−1], Y 0 ).
(5)
Since D has a Serre functor S and similarly as for equation (5) we obtain
Ext1D (X 0 , Y 00 ) DHomD (Y 00 [1], SX 0 ) DHomΛ (Y 00 [−1], SX 0 [−2]).
(6)
By Proposition 2.6, we have
SX 0 [−2] F −1 X 0 = GΛ (X 0 ).
Thus, (6) can be written as
Ext1D (X 0 , Y 00 ) DHomΛ (Y 00 [−1], SX 0 [−2]) DHomΛ (Y 00 [−1], GΛ (X 0 )).
Thus the assertion follows from (3), (4), (5) and (7) immediately.
(7)
In order to state our main result in this paper, we need the following definition.
Definition 2.10. Let C be an additive category with an auto-equivalence G. Assume that M and
P are two objects in C.
(1) An object M is said to be G-stable if G(M) M.
(2) We call a pair (M, P) is G-stable if G(M) M and G(P) P.
8
We denote by isoD the set of isomorphism classes of objects in D. By the equivalence (1), we
get a bijection
f : isoD ←→ iso(modΛ) × iso(projΛ)
(−)
e B (X 0 , X 00 [−1]),
X = X 0 ⊕ X 00 7−→ X
(8)
where X 00 is a maximal direct summand of X which belongs to addT [1].
From now on, we denote by F-rigidD the set of isomorphism classes of basic F-stable rigid
objects in D and by FΛ -τ-rigidΛ the set of isomorphism classes of basic FΛ -stable τ-rigid pairs
for Λ.
For the proof of our main result, the following lemma is needed.
f be the bijection in (8). Then X = X 0 ⊕ X 00 is
Lemma 2.11. Let X be an object in D and (−)
e = (X 0 , X 00 [−1]) is FΛ -stable in modΛ.
F-stable in D if and only if X
Proof. If X is F-stable in D, then FX 0 ⊕ FX 00 X 0 ⊕ X 00 . By Theorem 2.2, we have
FX 00 ∈ addT [1] and addFX 0 ∩ addT [1] = 0.
Thus, FX 0 X 0 and FX 00 X 00 . Therefore,
FΛ (X 0 ) FX 0 X 0 and FΛ (X 00 [−1]) FX 00 [−1] X 00 [−1].
e = (X 0 , X 00 [−1]) is FΛ -stable in modΛ, then FX 0 FΛ (X 0 ) X 0 and FX 00 [−1] Conversely, if X
00
FΛ (X [−1]) X 00 [−1]. Thus FX 0 X 0 and FX 00 X 00 in D, which implies that FX = FX 0 ⊕
FX 00 X 0 ⊕ X 00 = X.
Theorem 2.12. Let D be a k-linear Hom-finite triangulated category with a Serre functor S and
op
f in (8) induces a bijection
a cluster-tilting object T , and Λ = EndD (T ). Then the bijection (−)
F-rigidD ←→ FΛ -τ-rigidΛ
Proof. Let X be a basic F-stable rigid object in D, then by Proposition 2.9, we have
HomΛ (X 0 , τFΛ (X 0 )) = HomΛ (X 00 [−1], X 0 ) = 0.
(9)
e is FΛ -stable. Thus, (9) implies that X
e is a τ-rigid pair for Λ.
Using Lemma 2.11, we know X
e be a basic FΛ -stable τ-rigid pair for Λ, then
Conversely, let X
GΛ (X 0 ) GΛ ◦ FΛ (X 0 ) X 0 .
Similarly, by Proposition 2.9 and Lemma 2.11, we can easily check that X is a basic F-stable
rigid object in D.
We denote by FΛ -sτ-tpairΛ the set of isomorphism classes of basic FΛ -stable support τ-tilting
pairs for Λ and by FΛ -sτ-tiltΛ the set of isomorphism classes of basic FΛ -stable support τ-tilting
Λ-modules. For a basic support τ-tilting pair (X, P), the following observation shows that if X is
FΛ -stable, then so is P.
9
Lemma 2.13. There exists a bijection
ξ : FΛ -sτ-tpairΛ ←→ FΛ -sτ-tiltΛ
7−→
(X, P)
X.
Proof. Clearly, ξ is well-defined. For any basic FΛ -stable support τ-tilting Λ-module X, by
Proposition 2.3 in [AIR] there exists a unique basic Λ-module P ∈ projΛ such that (X, P) is a
basic support τ-tilting pair for Λ. We may assume that P T 1 , where T 1 ∈ addT . Thus,
FΛ (P) = FΛ (T 1 ) = FT 1 ∈ projΛ.
Moreover we have
HomΛ (FΛ (P), X) HomΛ (FΛ (P), FΛ (X)) HomΛ (P, X) = 0,
and
|FΛ (P)| + |X| = |P| + |X| = |Λ|.
Therefore, (X, FΛ (P)) is also a basic support τ-tilting pair. By the uniqueness of P, we have
FΛ (P) P and (X, P) is a basic FΛ -stable support τ-tilting pair for Λ.
We denote by F-m-rigidD (respectively, c-tiltD) the set of isomorphism classes of basic Fstable maximal rigid (respectively, cluster-tilting) objects in D and by FΛ -sτ-tiltΛ the set of
isomorphism classes of basic FΛ -stable support τ-tilting Λ-modules. The following result gives
a close relationship between the cluster-tilting objects in D and support τ-tilting Λ-modules.
Theorem 2.14. Let D be a k-linear Hom-finite triangulated category with a Serre functor S and
op
f in (8) induces the following
a cluster-tilting object T , and Λ = EndD (T ). Then the bijection (−)
bijections
(a)
F-m-rigidD ←→ FΛ -sτ-tiltΛ
(b)
c-tiltD ←→ FΛ -sτ-tiltΛ
f induces a bijection
Proof. (a) By Lemma 2.13, it suffices to show that (−)
F-m-rigidD ←→ FΛ -sτ-tpairΛ.
Let X = X 0 ⊕ X 00 be a basic F-stable maximal rigid object in D, where X 00 is the maximal direct
e is a basic FΛ -stable
summands of X which belong to addT [1]. By Theorem 2.12, we know that X
τ-rigid pair.
If (X 0 ⊕ M, X 00 [−1]) is τ-rigid for some Λ-module M, then X ⊕ M is rigid by Theorem 2.12.
e is a support τ-tilting pair by
Since X is maximal rigid, we have M ∈ addX = addX 0 . Thus X
Proposition 2.4.
e is a basic FΛ -stable support τ-tilting pair. By Theorem 2.12 we
Conversely, we assume that X
need to show that X is maximal rigid. If X ⊕ M = (X 0 ⊕ M 0 ) ⊕ (X 00 ⊕ M 00 ) is a rigid object, then
(X 0 ⊕ M 0 , X 00 [−1] ⊕ M 00 [−1]) is a τ-rigid pair. Thus,
M 0 ∈ addX 0 and M 00 [−1] ∈ addX 00 [−1],
which imply that M ∈ addX.
10
f induces a bijection c-tiltD ↔ FΛ -sτ-tpairΛ. If X is
(b) Similarly, we only need to show that (−)
e
a basic cluster-tilting object in D, then X is a basic FΛ -stable support τ-tilting pair by Theorem
2.2 and (a).
e is a basic FΛ -stable support τ-tilting pair, then X is F-stable and maximal rigid
Conversely, if X
by (a). Thus we only need to show that
{Y ∈ D | Ext1D (X, Y) = 0} ⊆ addX ⊇ {Z ∈ D | Ext1D (Z, X) = 0}.
(1) Assume that Ext1D (X, Y) = 0 for some object Y ∈ D. Then by Proposition 2.9, we have
hX 0 , τFΛ (Y 0 )i = hFΛ (Y 0 ), τFΛ (X 0 )i = hX 00 [−1], Y 0 i = hY 00 [−1], GΛ (X 0 )i = 0.
e is FΛ -stable and FΛ is an auto-equivalence of modΛ, we get
Since X
HomΛ (X 0 , τFΛ (Y 0 )) = HomΛ (FΛ (Y 0 ), τX 0 ) = HomΛ (X 00 [−1], FΛ (Y 0 )) = 0,
HomΛ (Y 00 [−1], X 0 ) HomΛ (Y 00 [−1], GΛ ◦ FΛ (X 0 )) HomΛ (Y 00 [−1], GΛ (X 0 )) = 0.
(10)
(11)
By Proposition 2.4 and (10), we obtain FΛ (Y 0 ) ∈ addX 0 = addFΛ (X 0 ), which implies that Y 0 ∈
addX 0 . From (11), we have Y 00 [−1] ∈ addX 00 [−1], then Y 00 ∈ addX 00 . Therefore, Y ∈ addX.
(2) If Ext1D (Z, X) = 0 for some object Z ∈ D, then
Ext1D (X, τD Z[−1]) = HomD (X[1], SZ) DExt1D (Z, X) = 0.
By (1), we know that τD Z[−1] = F −1 Z ∈ addX, then Z ∈ add(FX) = addX.
Together, X is a cluster-tilting object in D.
As an application of Theorem 2.14, we give the following corollary to characterize cluster-tilting
objects in D.
Corollary 2.15. In a triangulated category D with a Serre functor and a cluster-tilting object T ,
a basic object U is cluster-tilting if and only if it is F-stable and maximal rigid if and only if it is
an F-stable rigid object and |U| = |T |.
Proof. The first “if and only if” follows from bijections in Theorem 2.14 immediately. Let X be
an object in D, we define
e = |(X 0 , X 00 [−1])| B |X 0 | + |X 00 [−1]|.
|X|
e = |X|. If X is a basic cluster-tilting object in D, then X is an F-stable rigid object and
Clearly, |X|
e is a basic FΛ -stable support τ-tilting pair by Theorem 2.14. Hence
X
e = |Λ| = |T |.
|X| = |X|
e ∈ FΛ -τ-rigidΛ. Since |U|
e = |U| = |T | = |Λ|,
Conversely, if U ∈ F-rigidD and |U| = |T |, then U
e
we know that U is a basic FΛ -stable support τ-tilting pair. Hence U is a cluster-tilting object in
D by Theorem 2.14.
11
Remark 2.16. (a) When D is 2-CY, then F = τ−1
D [1] = Id. Thus we have
c-tiltD = m-rigidD = {U ∈ rigidD | |U| = |T |},
this was proved in [AIR, DK, ZZ].
(b) This corollary implies that all basic cluster-tilting objects have the same number of indecomposable direct summands for a triangulated category D with a Serre functor.
We denote by c-tiltT D the set of isomorphism classes of basic cluster-tilting objects in D which
do not have non-zero direct summands in addT [1] and by FΛ -τ-tiltΛ the set of isomorphism
classes of basic FΛ -stable τ-tilting Λ-modules. Immediately, we have the following result.
Corollary 2.17. Let D be a k-linear Hom-finite triangulated category with a Serre functor S and
op
f in (8) induces a bijection
a cluster-tilting object T , and Λ = EndD (T ). Then the bijection (−)
c-tiltT D ←→ FΛ -τ-tiltΛ
Remark 2.18. Theorem 2.12, Theorem 2.14 and Corollary 2.17 generalize a result of AdachiIyama-Reiten [AIR], They proved this theorem in case D is a 2-CY category.
β
α
/2
/ 3 and D = Db (kQ)/τ−1 [3] be the 3-cluster
Example 2.19. Let Q be the quiver 1
Q
category of type A3 , where τQ is the Auslander-Reiten translation in Db (kQ). Then D is a
4-Calabi-Yau triangulated category (see [K1, K2, T, Z08] for details) and its AR-quiver is as
follows:
1
1
The direct sum T = 2 ⊕ 3[1] ⊕ 2[1] ⊕ 1[1] ⊕ 2[2] ⊕ 3[3] ⊕ 2[3] gives a cluster-tilting object, the
3
3
op
cluster-tilted algebra Λ = EndD (T ) is given by the following quiver with rad2 = 0.
3
2g
w
1S
B7
4
/6
5
The AR-quiver of modΛ is as follows:
12
1
2
4
6
E7
1o
o
E7
2
E5
6o
E3
o
E5
7
1
4o
o
E3
5
6
2
3
4
1
2
For an indecomposable module M in modΛ, the smallest positive integer n such that FΛn (M) M
is called the FΛ -period of M, denoted by o(M). In this example, it is easy to see that any
indecomposable Λ-module M is τ-rigid and satisfies o(M) = 7 = |Λ|. Thus, FΛ -sτ-tiltΛ = {0, Λ}.
Using our results, we obtain c-tiltD = {T [1], T }.
Example 2.20. Let A = kQ/I be a self-injective algebra given by the quiver Q: 1 o
α
/
2 and
β
I =< αβαβ, βαβα >. Let D be the stable module category modA of A, then it is a triangulated
category with a Serre functor and it is not 2-CY. We describe the AR-quiver of modA in the
following picture:
1
2
A
1
2
2
1
A
2
1
1
2
1o
2
2
A2
o
A1
1
A
2o
1
A
o
2
1o
2
1
2
1
Figure 3
where the leftmost and rightmost columns are identified. Thus, we also get the AR-quiver of
2
modA by deleting the first row in Figure 3. The direct sum T = 2 ⊕ 1 is a cluster-tilting object in
2
op
D. The cluster-tilted algebra Λ = EndD (T ) = kQ0 /I 0 is given by the quiver Q0 : a o
I 0 =< γδ, δγ >. The AR-quiver of modΛ is
13
γ
δ
/
b and
a
b
a
Eb
ao
Eb
b
a
Hence, FΛ -sτ-tiltΛ = {0, Λ} and c-tiltD = {T [1], T }.
Finally, We want to mention that one can also investigate the connections between cluster-tilting
objects in D, functorially finite torsion classes in modΛ and two-term silting complexes for Λ
(see [AIR, M]). Note that the auto-equivalence FΛ of modΛ induces an auto-equivalence of
Kb (projΛ), and we denote it by FΛ again. By using a similar approach in [M], we get the
following
Remark 2.21. Let D be a triangulated category with a Serre functor and a cluster-tilting object
op
T , and let Λ = EndD (T ) be the corresponding cluster-tilted algebra. Then we have bijective
correspondences between the set of isomorphism classes of basic cluster-tilting objects in D, the
set of isomorphism classes of functorially finite torsion classes in modΛ which are stable under
FΛ , and the set of isomorphism classes of two-term silting complexes in Kb (projΛ) which are
stable under FΛ .
Acknowledgments
The first author would like to thank Wen Chang for helpful discussions on the subject.
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Department of Mathematical Sciences, Tsinghua University, 100084, Beijing, P. R. China
E-mail address: [email protected]
School of Mathematics and Statistics, Beijing Institute of Technology, 100081, Beijing, P. R.
China
E-mail address: [email protected]
Department of Mathematical Sciences, Tsinghua University, 100084, Beijing, P. R. China
E-mail address: [email protected]
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