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 
.
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 [
.The t-structure is a pair 
of full subcategories such that setting 
and
, satisfying:
; any object
is included in a triangle
where
, and
. In [
where 
is a projective space. Mutation can be regarded as a categorical realization of Coxeter or braid groups. In [
-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.
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 
[
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 of
if any 
has a left 
-approximation.
Let 
be a subcategory of
, we call 
-monic (resp. 
-epic) if
(resp.
) is exact.
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 of
are 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 in
is 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 
[
2) The pair 
is called a rigid torsion pair if 
is a torsion pair and 
for all 
[
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 [
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 all
then 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 
for
we 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
.
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 [
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 [
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 [
and
, there exists a triangle
in 
with 
and with 
being 
-epic.
Lemma 4.2 If 
is a torsion pair for
and
then
.
Proof: Noting that 
for
. Since
we have
, and then
or
. Therefore
. Since
we 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 any
there 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 [
-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
.
Supported by the National Natural Science Foundation of China (Grant No. 10971172, 11271119) and the Natural Science Foundation of Beijing (Grant No. 1122002).