ABSTRACT

We study the properties of torsion pairs in triangulated category by introducing the notions of d-Ext-projectivity and d-Ext-injectivity. In terms of -mutation of torsion pairs, we investigate the properties of torsion pairs in triangulated category under some conditions on subcategories and in.

1. Introduction

The notion of torsion theory (torsion pairs) in abelian categories was introduced by Dickson in 1966. Torsion theory plays an important role in the investigation of an abelian category. An abelian category is naturally embedded in a triangulated category like the bounded derived category. The analogous definition of torsion pairs in triangulated category is closely related to the notion of a t-structure. Beilinson, Bernstein and Deligne [1] introduce the definition of a t-structure in a triangulated category.The t-structure is a pair of full subcategories such that setting and

, satisfying:

; any object

is included in a triangle

where, and. In [2], Beligiannis and Reiten studied the torsion theory on pretriangulated, triangulated and stable categories. They discussed the connection between torsion theories in abelian and derived categories and indicated the relationship with tilting theory, they point out that the torsion pairs in triangulated category and t-structures essentially coincide. In 1987, Gorodentsev and Rudakov [3] made use of mutation when they classified the exceptional vector bundles on where is a projective space. Mutation can be regarded as a categorical realization of Coxeter or braid groups. In [4] and [5], Fomin and Zelevinsky introduced cluster algebras, these algebras give an algebraic and combinational framework for the positivity and canonical basis of quantum groups, which enjoy important combinational properties given in terms of the mutation for skew symmetric matrices. Cluster categories were introduced in [6], in which the mutation of cluster tilting objects was introduced. Recently, Geiss, Leclerc and Schroer [7] applied mutation to study rigid modules over preprojective algebras and the coordinate rings of maximal unipotent subgroups of semisimple Lie groups. Later Iyama and Yoshino [8] introduced the mutation of n-cluster tilting subcategories based on approximation theory. Recently, Zhou and Zhu [9] studied the notion of -mutation of torsion pairs in triangulated categories, and they proved that the -mutation of torsion pairs in triangulated categories is a torsion pair.They also studied its geometric meaning when the triangulated categories are the cluster categories of type or.

In this paper, we study the torsion pairs in triangulated categories and their properties in terms of -mutation pair. In a fixed triangulated category, we give the definition of torsion pairs in and study their properties with the notion of subcategory (resp.) whose objects are d-Ext-projective (resp. d-Extinjective). Under reasonable conditions on subcategories and of, we study the properties of torsion pairs in triangulated category in terms of - mutation pair.

2. Preliminaries

Through this paper, let be a triangulated category. We introduce some basic notions which will be used. Let and be subcategories of. We put

and.

We denote by the collection of objects in consisting of all such with the triangle

where By the octahedralaxiom, we have.

Definition 2.1 We call a pair of subcategories of a torsion pair if and

.

In this case, we can see that and.

Let be a morphism, we call a right approximation of [10] if and

is exact as functors on. We call a contravariantly finite subcategory of if any has a right - approximation. Dually, for a morphism , we call a left -approximation of if and

is exact. We call a covariantly finite subcategory ofif any has a left -approximation.

Let be a subcategory of, we call -monic (resp. -epic) if

(resp.

) is exact.

3. Torsion Pairs in Triangulated Categories

In this section, we introduce -cluster tilting torsion pairs, rigid torsion pairs and maximal rigid torsion pairs in a triangulated category, and study the properties of these torsion pairs.

Definition 3.1 Let be an extension-closed subcategory of. An object is called a d-Extprojective object of if for all . The d-Ext-injective objects ofare defined dually. An object is called a d-Ext-injective object of if for all. The subcategory of consisting of d-Ext-projective (or d-Ext-injective) objects inis denoted by (respectively).

Definition 3.2 Let and be subcategories of the triangulated category.

1) The pair is called a -cluster tilting torsion pair if is a torsion pair and satisfies the property: is functorially finite in and if and only if for all [11].

2) The pair is called a rigid torsion pair if is a torsion pair and for all [12].

3) The pair is called a maximal rigid torsion pair provided that is a torsion pair, is a rigid subcategory and satisfies the property:

if for any and all, then. In this case, the subcategory is called a maximal rigid subcategory [13].

Corollary 3.3 A pair is a maximal rigid torsion pair if and only if for all and for any rigid object in, we have

for all.

Proof: Now supposing is a maximal rigid torsion pair, by the definition we have

and

for all. It implies for all. For any rigid object in, take a triangle

where. Then we have the triangle

It follows that for all .

Conversely, suppose is a torsion pair with for all and

for any rigid object while

, then is rigid. If there exists an object in such that for any and all. Then is rigid. It follows that there is a triangle

for.

Then the above triangle splits. This implies that

for, i.e.,. is maximal rigid.

Corollary 3.4 A pair is a -cluster tilting torsion pair if and only if is functorially finite and for all.

Proof: By the definition, if we have

for allthen we obtain. On the other hand,

for all, it implys, i.e,. So.

Conversely, we only need to prove that if and only if for all. Supposing, we have

for

. Now, if forwe have that, since

. This implies.

Proposition 3.5 Let be a rigid torsion pairthen, for all

. Moreover, is covariantly finite in and is contravariantly finite in.

Proof: Let, we have that

for if and only if. Then if and only if

. For, let,

if and only if

. Since, we have that

if and only if, i.e., if and only if.

Now we prove that is covariantly finite in.

Since is a torsion pair, we have that

is a torsion pair for. For any object, take a triangle

where. Then we have a triangle

.

When, then

. Applying functor to the triangle above, we obtain

. Since, we have

. Then is a left -approximation of. Thus is covariantly finite in.

When, we have that. For any, take a triangle

where and. Then we have the triangle

Since is extension-closed, we obtain that

, and hence. It is easy to see that is covariantly finite in.

Finally, we prove is contravariantly finite in. In case, we have

.

Since is a torsion pair, for any, there exists a triangle

where and. Since and is closed under extensions, we have, and hence. It follows that is a right approximation of, and then is contravariantly finite in.

In case, we have

for and. Take a triangle

where and. Since and

for any and, we have that. Hence

, i.e., is a right -approximation of. It means that is contravariantly finite in.

Corollary 3.6 is a rigid torsion pair if and only if.

Proof: By proposition 3.5, we have that

. Since

for all if and only if for, hence in this case,.

Corollary 3.7 Let be a maximal rigid subcategory of, then 1) Every object is d-Ext-projective (or d-Extinjective) in.

2) An object is d-Ext-projective in if and only if.

Proof: 1) By Corollary 3.6, , (a) holds.

2) For any object

if and only if.

4. Torsion Pairs in

Let be a triangulated category and a subcategory of satisfying. For a subcategory of, put. Then consists of all such that there exists a triangle

with and a left -approximation.

Dually, for a subcategory of, put

.

Then consists of all such that there exists a triangle with and a right -approximation.

We call a pair of subcategories of a - mutation pair [5] if and .

Let be a subcategory of, we assume:

1) is extension closed;

2) forms a -mutation pair.

In the rest of this section, we assume that has a serre functor. We put We call a subcategory of an -subcategory of if it satisfies.

For an integer, we call a subcategory of if for.

Now we assume that is a functorially finite subcategory of and

.

It was proved in [8] that forms a triangulated category. The shift in is defined as follows: for any object, consider the left -approximation, and extend it to a triangle

where and. The is defined as the shift of in.

Then triangles in are defined as the complex

in, where and are the images of maps under the quotient functor respectively.

In the following, denotes the subcategory of consisting of objects, for the subcategory

satisfying.

Lemma 4.1 [8] For any and, there exists a triangle

in with and with being -epic.

Lemma 4.2 If is a torsion pair for

andthen.

Proof: Noting that for. Sincewe have, and then

or. Therefore

. Sincewe have that

. Thus we have

.

Lemma 4.3 Let X and be two objects in. Then for if and only if

for.

Proof: By Lemma 4.1, we have an exact sequence

where and is a right -approximation. Since

and

, we have

and for. Then if and only if for.

Theorem 4.4 Let be a subcategory of satisfying. Then is a torsion pair with in for if and only if is a torsion pair with in for

.

Proof: Noting that is a triangulated category with shift functor, we suppose that is a torsion pair. It follows from Lemma 4.2 that

. By Lemma 4.3, we have

for. For any, there is a triangle

where and as is a torsion pair in. Since all of are in, there is a triangle

in. Therefore,. Hence is a torsion pair in.

Conversely, we suppose is a torsion pair for. By Lemma 4.3, we have

for. For anythere is a triangle in:

where and for by Lemma 4.3. Then there is a triangle

in such that in. Hence in up to direct summands of. Thus is a subcategory of. Since there is a triangle in for any:

where and, we have

.

Therefore is a torsion pair in for .

Finally, we have.

Corollary 4.5 Let be a subcategory of satisfying, then we have the following:

1) is a rigid torsion pair in for if and only if is a rigid torsion pair in for.

2) is a -cluster tilting torsion pair in

for if and only if is a n-cluster tilting torsion pair in for.

3) is a maximal rigid torsion pair in

for if and only if is a maximal rigid torsion pair for.

Proof: 1) By Corollary 3.6, we only need to prove

if and only if. By Theorem 4.4 we have.

2) It follows from Theorem 4.9 in [8] that we have a one-one correspondence between -cluster tilting subcategories of containing and -cluster tilting subcategories of.

3) It is obvious that if and only if

for. Assume that,

while for any rigid object in. For any rigid object in, we have that is also rigid in. Then there is a triangle

in , where and. Thus there is a triangle

in with. Therefore,

.

It follows from Corollary 3.3 that is a maximal rigid torsion pair in for.

Conversely, while for any rigid object in. For any rigid object in, there is a triangle

in, where and. Applying

to this triangle we obtain that is a rigid object for. Then is a rigid object in for. And thus there is a triangle

in, where. Then there is a triangle

in such that is isomorphic to up to direct summands in. Since for

, it is easy to see that is closed under extensions for.

Therefore and then

because

by. So we have that

is a maximal rigid torsion pair in for

.

5. Acknowledgement

Supported by the National Natural Science Foundation of China (Grant No. 10971172, 11271119) and the Natural Science Foundation of Beijing (Grant No. 1122002).

REFERENCES