Xin Ma and Tiwei Zhao

Resolving resolution dimensions in triangulated categories

Open Access
De Gruyter | Published online: May 5, 2021

Abstract

Let T be a triangulated category with a proper class ξ of triangles and X be a subcategory of T . We first introduce the notion of X -resolution dimensions for a resolving subcategory of T and then give some descriptions of objects having finite X -resolution dimensions. In particular, we obtain Auslander-Buchweitz approximations for these objects. As applications, we construct adjoint pairs for two kinds of inclusion functors and characterize objects having finite X -resolution dimensions in terms of a notion of ξ -cellular towers. We also construct a new resolving subcategory from a given resolving subcategory and reformulate some known results.

MSC 2010: 18G20; 18G25; 18G10

1 Introduction

Approximation theory is the main part of relative homological algebra and representation theory of algebras, and its starting point is to approximate arbitrary objects by a class of suitable subcategories. In particular, resolving subcategories play important roles in approximation theory (e.g., [13]). As an important example of resolving subcategories, Auslander and Buchweitz [4] studied the approximation theory of the subcategory consisting of maximal Cohen-Macaulay modules over an artin algebra, and Hernández et al. [5] developed an analogous theory for triangulated categories. Using the approximation triangles established by Hernández et al. [5, Theorem 5.4], Di and Wang [6] constructed additive functors (adjoint pairs) between additive quotient categories. On the other hand, Zhu [7] studied the resolution dimension with respect to a resolving subcategory in an abelian category, and Huang [8] introduced relative preresolving subcategories in an abelian category and defined homological dimensions relative to these subcategories, which generalized many known results (see [4,9,10]).

In analogy to relative homological algebra in abelian categories, Beligiannis [11] developed a relative version of homological algebra in a triangulated category T , that is, a pair ( T , ξ ) , in which ξ is a proper class of triangles (see Definition 2.4). Under this notion, a triangulated category is just equipped with a proper class consisting of all triangles. However, there are lots of non-trivial cases, for example, let T be a compactly generated triangulated category, then the class ξ consisting of pure triangles is a proper class ([12]), and the pair ( T , ξ ) is no longer triangulated in general. Later on, this theory has been paid more attentions and developed (e.g., [1317]). It is natural to ask how the approximation theory acts on this relative setting of triangulated categories. In [18], Ma et al., introduced the notions of (pre)resolving subcategories and homological dimensions relative to these subcategories in this relative setting, which gives a parallel theory analogy to that of abelian categories [8]. In this paper, we devote to further studying relative homological dimensions in triangulated categories with respect to a resolving subcategory. The paper is organized as follows:

In Section 2, we give some terminology and some preliminary results.

In Section 3, some homological properties of resolving subcategories are obtained. In particular, we obtain Auslander-Buchweitz approximation triangles (see Proposition 3.10) for objects having finite resolving resolution dimensions. Our main result is the following:

Theorem

Let X be a resolving subcategory of T and , a ξ x t -injective ξ -cogenerator of X . Assume that is closed under hokernels of ξ -proper epimorphisms or closed under direct summands. For any M T , if M X ^ , then the following statements are equivalent:

  1. (1)

    X - res.dim M m .

  2. (2)

    Ω n ( M ) X for all n m .

  3. (3)

    Ω X n ( M ) X for all n m .

  4. (4)

    ξ x t ξ n ( M , H ) = 0 for all n > m and all H .

  5. (5)

    ξ x t ξ n ( M , L ) = 0 for all n > m and all L ^ .

  6. (6)

    M admits a right X -approximation φ : X M , where φ is ξ -proper epic, such that K = Hoker φ satisfying - res.dim K m 1 .

  7. (7)

    There are two triangles

    W M X M M Σ W M
    and
    M W M X M Σ M
    in ξ such that X M and X M are in X and - res.dim W M m 1 , - res.dim W M = X - res.dim W M m .

In Section 4, we will further study objects having finite resolution dimensions with respect to a resolving subcategory X . We first construct adjoint pairs for two kinds of inclusion functors. Then we characterize objects having finite resolution dimensions in terms of a notion of ξ -cellular towers.

As an application, in Section 5, given a resolving subcategory X of T , we construct a new resolving subcategory GP X ( ξ ) with a ξ x t -injective ξ -cogenerator X X , which generalizes the Gorenstein projective subcategory GP ( ξ ) given by Asadollahi and Salarian [13]. Applying the obtained results to GP X ( ξ ) , we generalize some known results in [1315].

Throughout this paper, all subcategories are full, additive, and closed under isomorphisms.

2 Preliminaries

Let T be an additive category and Σ : T T an additive functor. One defines the category Diag ( T , Σ ) as follows:

  • An object of Diag ( T , Σ ) is a diagram in T of the form X u Y v Z w Σ X .

  • A morphism in Diag ( T , Σ ) between X i u i Y i v i Z i w i Σ X i , i = 1 , 2 , is a triple ( α , β , γ ) of morphisms in T such that the following diagram:

    commutes.

A triangulated category is a triple ( T , Σ , Δ ) , where T is an additive category and Σ : T T is an autoequivalence of T (called suspension functor), and Δ is a full subcategory of Diag ( T , Σ ) which is closed under isomorphisms and satisfies the axioms ( T 1 ) ( T 4 ) in [11, Section 2.1] (also see [19]), where the axiom ( T 4 ) is called the octahedral axiom. The elements in Δ are called triangles.

The following result is well known, which is an efficient tool in studying triangulated categories.

Remark 2.1

[11, Proposition 2.1] Let T be an additive category and Σ : T T an autoequivalence of T , and Δ a full subcategory of Diag ( T , Σ ) which is closed under isomorphisms. Suppose that the triple ( T , Σ , Δ ) satisfies all the axioms of a triangulated category except possibly of the octahedral axiom. Then, the following statements are equivalent:

  1. (1)

    Octahedral axiom. For any two morphisms u : X Y and v : Y Z , there exists a commutative diagram

    in which all rows and the third column are triangles in Δ .

  2. (2)

    Base change. For any triangle X u Y v Z w Σ X in Δ and any morphism α : Z Z , there exists the following commutative diagram:

    in which all rows and columns are triangles in Δ .

  3. (3)

    Cobase change. For any triangle X u Y v Z w Σ X in Δ and any morphism β : X X , there exists the following commutative diagram:

    in which all rows and columns are triangles in Δ .

Throughout this paper, T = ( T , Σ , Δ ) is a triangulated category.

Definition 2.2

[11] A triangle

X u Y v Z w Σ X
is called split if it is isomorphic to the triangle
X 1 0 X Z ( 0 , 1 ) Z 0 Σ X .

We use Δ 0 to denote the full subcategory of Δ consisting of all split triangles.

Definition 2.3

[11] Let ξ be a class of triangles in T .

  1. (1)

    ξ is said to be closed under base change (resp. cobase change) if for any triangle

    X u Y v Z w Σ X
    in ξ and any morphism α : Z Z (resp. β : X X ) as in Remark 2.1(2) (resp. Remark 2.1(3)), the triangle
    X u Y v Z w Σ X ( resp . X u Y v Z w Σ X )
    is in ξ .

  2. (2)

    ξ is said to be closed under suspension if for any triangle

    X u Y v Z w Σ X
    in ξ and any i Z (the set of all integers), the triangle
    Σ i X ( 1 ) i Σ i u Σ i Y ( 1 ) i Σ i v Σ i Z ( 1 ) i Σ i w Σ i + 1 X
    is in ξ .

  3. (3)

    ξ is called saturated if in the situation of base change as in Remark 2.1(2), whenever the third vertical and the second horizontal triangles are in ξ , then the triangle

    X u Y v Z w Σ X
    is in ξ .

Definition 2.4

[11] A class ξ of triangles in T is called proper if the following conditions are satisfied.

  1. (1)

    ξ is closed under isomorphisms, finite coproducts and Δ 0 ξ .

  2. (2)

    ξ is closed under suspensions and is saturated.

  3. (3)

    ξ is closed under base and cobase change.

Throughout this paper, we always assume that ξ is a proper class of triangles in T .

Definition 2.5

[11] Let

X u Y v Z w Σ X
be a triangle in ξ . Then, the morphism u (resp. v ) is called ξ - proper monic (resp. ξ - proper epic), and u (resp. v ) is called the hokernel of v (resp. the hocokernel of u ).

We use Hoker v to denote the hokernel of v : Y Z . Dually, we use Hocok u to denote the hocokernel of u : X Y . For any triangle,

X Y Z Σ X
in ξ . We say that X is closed under ξ - extensions if X , Z X , it holds that Y X . We say that X is closed under hokernels of ξ - proper epimorphisms (resp. hocokernels of ξ - proper monomorphisms) if Y , Z X (resp. X , Y X ), it holds that X X (resp. Z X ).

Definition 2.6

(see [11, 4.1]) An object P (resp. I ) in T is called ξ -projective (resp. ξ -injective) if for any triangle X Y Z Σ X in ξ , the induced complex

0 Hom T ( P , X ) Hom T ( P , Y ) Hom T ( P , Z ) 0
( resp. 0 Hom T ( Z , I ) Hom T ( Y , I ) Hom T ( X , I ) 0 )
is exact. We use P ( ξ ) (resp. ( ξ ) ) to denote the full subcategory of T consisting of ξ -projective (resp. ξ -injective) objects.

We say that T has enough ξ -projective objects if for any object M T , there exists a triangle K P M Σ K in ξ with P P ( ξ ) . Dually, we say that T has enough ξ -injective objects if for any object M T , there exists a triangle M I K Σ M in ξ with I ( ξ ) .

Remark 2.7

P ( ξ ) is closed under direct summands, hokernels of ξ -proper epimorphisms, and ξ -extensions. Dually, ( ξ ) is closed under direct summands, hocokernels of ξ -proper monomorphisms, and ξ -extensions.

Definition 2.8

Let be a subcategory of T .

  1. (1)

    A triangle

    X Y Z Σ X
    in ξ is called Hom T ( , ) - exact (resp. Hom T ( , ) - exact) if for any object E in , the induced complex
    0 Hom T ( E , X ) Hom T ( E , Y ) Hom T ( E , Z ) 0
    ( resp. 0 Hom T ( Z , E ) Hom T ( Y , E ) Hom T ( X , E ) 0 )
    is exact.

  2. (2)

    [13] A ξ -exact complex is a complex

    (2.1) X n + 1 d n + 1 X n d n X n 1
    in T such that for any n Z , there exists a triangle
    (2.2) K n + 1 g n X n f n K n h n Σ K n + 1
    in ξ and the differential d n is defined as d n = g n 1 f n . A ξ -exact complex as ( 2.1) is called Hom T ( , ) - exact (resp. Hom T ( , ) - exact) if the triangle (2.2) is Hom T ( , ) -exact (resp. Hom T ( , ) -exact) for any n Z .

Asadollahi and Salarian [13] introduced the notion of ξ -Gorenstein projective objects.

Definition 2.9

[13, Definition 3.6] Let T be a triangulated category with enough ξ -projective objects and X an object in T . A complete ξ -projective resolution is a Hom T ( , P ( ξ ) ) -exact ξ -exact complex

P 1 P 0 P 1
in T with all P i ξ -projective objects. The objects K n as in (2.2) are called ξ - Gorenstein projective objects. We use GP ( ξ ) to denote the full subcategory of T consisting of all ξ -Gorenstein projective objects.

Throughout this paper, we always assume that T is a triangulated category with enough ξ -projective objects and ξ -injective objects.

Let M be an object in T . Beligiannis [11] defined the ξ -extension groups ξ x t ξ n ( , M ) to be the n th right ξ -derived functor of the functor Hom T ( , M ) , that is,

ξ x t ξ n ( , M ) ξ n Hom T ( , M ) .

Remark 2.10

Let

X Y Z Σ X
be a triangle in ξ . By [ 11, Corollary 4.12], there exists a long exact sequence
0 ξ x t ξ 0 ( Z , M ) ξ x t ξ 0 ( Y , M ) ξ x t ξ 0 ( X , M )
ξ x t ξ 1 ( Z , M ) ξ x t ξ 1 ( Y , M ) ξ x t ξ 1 ( X , M )
of “ ξ x t ” functor. If T has enough ξ -injective objects and N is an object in T , then there exists a long exact sequence
0 ξ x t ξ 0 ( N , X ) ξ x t ξ 0 ( N , Y ) ξ x t ξ 0 ( N , Z )
ξ x t ξ 1 ( N , X ) ξ x t ξ 1 ( N , Y ) ξ x t ξ 1 ( N , Z )
of “ ξ x t ” functor.

Following Remark 2.10, we usually use the strategy of “dimension shifting,” which is an important tool in relative homological theory of triangulated categories.

Now, we set

X = { M T ξ x t ξ n 1 ( X , M ) = 0 for all X X } ,
X = { M T ξ x t ξ n 1 ( M , X ) = 0 for all X X } .

For two subcategories and X of T , we say X if X (equivalently, X ).

Taking C = = P ( ξ ) in [18, Definitions 3.1 and 3.2], we have the following definitions.

Definition 2.11

(cf. [18, Definition 3.1]) Let and X be two subcategories of T with X . Then, is called a ξ -cogenerator of X if for any object X in X , there exists a triangle

X H Z Σ X
in ξ with H an object in and Z an object in X . In particular, a ξ -cogenerator is called ξ x t - injective if X .

Definition 2.12

(cf. [18, Definition 3.2]) Let T be a triangulated category with enough ξ -projective objects and X a subcategory of T . Then, X is called a resolving subcategory of T if the following conditions are satisfied.

  1. (1)

    P ( ξ ) X .

  2. (2)

    X is closed under ξ -extensions.

  3. (3)

    X is closed under hokernels of ξ -proper epimorphisms.

3 Resolution dimensions with respect to a resolving subcategory

Taking = P ( ξ ) in [18, Definition 3.5], we first have the following definition.

Definition 3.1

Let X be a subcategory of T and M an object in T . The X -resolution dimension of M , written X - res.dim M , is defined by

X - res.dim M = inf { n 0 there exists a ξ -exact complex 0 X n X 1 X 0 M 0 in T with all X i objects in X } .
For a ξ -exact complex
f n + 1 X n f 2 X 1 f 1 X 0 f 0 M 0
with all X i X . The Hoker f n 1 is called an n th ξ - X -syzygy of M , denoted by Ω X n ( M ) . In case for X = P ( ξ ) , we write ξ - pd M X - res.dim M and Ω n ( M ) Ω P ( ξ ) n ( M ) . In case for X = GP ( ξ ) , X - res.dim M coincides with ξ - G pd M defined in [ 13] as ξ -Gorenstein projective dimension. We use X ^ to denote the full subcategory of T whose objects have finite X -resolution dimension.

Lemma 3.2

Let T be a triangulated category and X a resolving subcategory of T . For any object M T , if

0 X n X 1 X 0 M 0
and
0 Y n Y 1 Y 0 M 0
are ξ - exact complexes with all X i and Y i in X for 0 i n 1 , then X n X if and only if Y n X .

Proof

For M T , there exists a ξ -exact complex

0 K n P n 1 P 1 P 0 M 0
with P i P ( ξ ) for 0 i n 1 .

Consider the following triangle:

K 1 M X 0 M Σ K 1 M
in ξ . As a similar argument to that of [ 11, Proposition 4.11], we get the following ξ -exact complex
0 K n X n P n 1 X n 1 P n 2 X 2 P 1 X 1 P 0 X 0 0 .
Similarly, we have the following ξ -exact complex
0 K n Y n P n 1 Y n 1 P n 2 Y 2 P 1 Y 1 P 0 Y 0 0 .
Set
X Hoker ( X n 1 P n 2 X n 2 P n 3 )
and
Y Hoker ( Y n 1 P n 2 Y n 2 P n 3 ) .
Since X is resolving, we have that X and Y are objects in X . Consider the following triangles:
K n X n P n 1 X Σ K n
and
K n Y n P n 1 Y Σ K n
in ξ , we have that X n P n 1 X if and only if K n X if and only if Y n P n 1 X .

But from the following triangles in ξ

X n X n P n 1 P n 1 0 Σ X n and Y n Y n P n 1 P n 1 0 Σ Y n ,
we have that X n X if and only if X n P n 1 X , and Y n X if and only if Y n P n 1 X . Thus, X n X if and only if Y n X .□

Using the above, we can get:

Proposition 3.3

Let X be a resolving subcategory of T and M T . Then, the following statements are equivalent:

  1. (1)

    X - res.dim M m .

  2. (2)

    Ω n ( M ) X for n m .

  3. (3)

    Ω X n ( M ) X for n m .

Proof

Apply Lemma 3.2.□

Now we can compare resolution dimensions in a given triangle in ξ as follows.

Proposition 3.4

Let X be a resolving subcategory of T , and let

A B C Σ A
be a triangle in ξ . Then, we have the following statements:
  1. (1)

    X - res.dim B max { X - res.dim A , X - res.dim C } .

  2. (2)

    X - res.dim A max { X - res.dim B , X - res.dim C 1 } .

  3. (3)

    X - res.dim C max { X - res.dim A + 1 , X - res.dim B } .

Proof

For any A T , if X - res.dim A = m , by Proposition 3.3, we have the following ξ -exact complex

0 P m A P m 1 A P 1 A P 0 A A 0
in T with P i A P ( ξ ) for 0 i m 1 and P m A X .

  1. (1)

    Assume X - res.dim A = m and X - res.dim C = n . We proceed it by induction on m and n . The case m = n = 0 is trivial. Without loss of generality, we assume m n , then we can let P i A = 0 for i > m . As a similar argument to that of [11, Proposition 4.11], we get the following ξ -exact complex:

    0 P n A P n C P n 1 A P n 1 C P 0 A P 0 C B 0
    in T . Thus, X - res.dim B n and the desired assertion are obtained.

  2. (2)

    Assume X - res.dim B = m and X - res.dim C = n . We proceed it by induction on m and n . The case m = n = 0 is trivial. Without loss of generality, we assume m n 1 , then we can let P i B = 0 for i > m . By [18, Theorem 3.7], there exist a ξ -exact complex

    0 P n C P n 1 B P n 1 C P n 2 B P 2 C P 1 B K A 0
    and a triangle
    K P 1 C P 0 B P 0 C K [ 1 ]
    in ξ , it follows that K P ( ξ ) by Remark 2.7. Thus, X - res.dim A n 1 and the desired assertion is obtained.

  3. (3)

    Assume X - res.dim A = m and X - res.dim B = n . We proceed it by induction on m and n . The case m = n = 0 is trivial. Without loss of generality, we assume m + 1 n , then we can let P i A = 0 for i > m . By [18, Theorem 3.8], we have the following ξ -exact complex

    0 P n B P n 1 A P 2 B P 1 A P 1 B P 0 A P 0 B C 0
    in T , thus X - res.dim A n and the desired assertion is obtained.□

As direct results, we have the following closure properties for the subcategory X ^ .

Remark 3.5

If X is a resolving subcategory of T , then X ^ is closed under hokernels of ξ -proper epimorphisms, hocokernels of ξ -proper monomorphisms, and ξ -extensions.

Corollary 3.6

Let X be a resolving subcategory of T , and let

A B C Σ A
be a triangle in ξ . Then, we have the following statements:
  1. (1)

    (cf. [18, Proposition 3.11]) Assume that C is an object in X . Then, X - res.dim A = X - res.dim B .

  2. (2)

    Assume that B is an object in X . Then, either A X or else X - res.dim A = X - res.dim C 1 .

  3. (3)

    (cf. [18, Proposition 3.13]) Assume that A is an object in X and neither B nor C in X . Then, X - res.dim B = X - res.dim C .

Proposition 3.7

Let and X be two subcategories of T with X .

  1. (1)

    ^ X ^ .

  2. (2)

    If X is resolving, then for any M ^ , - res.dim M = X - res.dim M if and only if ^ X = .

    In particular, if X , and is closed under hokernels of ξ -proper epimorphisms or closed under direct summands, then ^ X = .

Proof

  1. (1)

    It is clear.

  2. (2)

    ( ) Clearly, ^ X . Let M ^ X . By the assumption, we have - res.dim M = X - res.dim M = 0 , then M , so ^ X . Thus, ^ X = .

( ) Let M ^ . Suppose - res.dim M = n and X - res.dim M = m . Clearly, m n . Consider the following ξ -exact complexes:

0 H n H 0 M 0
and
0 X m X 0 M 0
with H i and X j X for all 0 i n and 0 j m . Since X , we have Ω m ( M ) X by Lemma 3.2. Then, Ω m ( M ) ^ X = , and thus, - res.dim M m and the desired equality is obtained.

Now, we assume that X and is closed under hokernels of ξ -proper epimorphisms or closed under direct summands. Clearly, ^ X . Conversely, let M ^ X . There exists a ξ -exact complex

0 H n H n 1 H 0 M 0 .
Set K i = Hoker ( H i H i 1 ) for 0 i n 2 , where H 1 = M . Since X is resolving, we have K i X , and hence, K i ^ X . Consider the following triangle:
(1) H n H n 1 K n 2 Σ H n
in ξ . Since ξ x t ξ 1 ( K n 2 , H n ) = 0 by the assumption that X , we have that the triangle ( 1) is split. It follows that H n 1 H n K n 2 and there exists a triangle
K n 2 H n 1 H n 0 Σ K n 2
in ξ . Since is closed under hokernels of ξ -proper epimorphisms or closed under direct summands by assumption, we have K n 2 . Repeating this process, we can obtain each K i , hence, M and ^ X . Thus, ^ X = .□

Now we give the following definition.

Definition 3.8

Let X be a subcategory of T and M an object in T . A ξ -proper epimorphism X M is said to be a right X -approximation of M if Hom T ( X ˜ , X ) Hom T ( X ˜ , M ) 0 is exact for any X ˜ X . In this case, there is a triangle K X M Σ K in ξ .

We need the following easy and useful observation.

Lemma 3.9

Let and X be two subcategories of T .

  1. (1)

    If X , then X ^ . In particular, if , then ^ .

  2. (2)

    If M , then M ^ .

Proof

Apply Remark 2.10.□

The following is an analogous theory of Auslander-Buchweitz approximations (see [4,5]).

Proposition 3.10

Let X be a subcategory of T closed under ξ -extensions, and let be a subcategory of T such that is a ξ -cogenerator of X . Then, for each M T with X - res.dim M = n < , there exist two triangles

(2) K X M Σ K
and
(3) M W X Σ M
in ξ , where X , X X , - res.dim K n 1 and - res.dim W n ( if n = 0 , this should be interpreted as K = 0 ).

In particular, if X , then the ξ -proper epimorphism X M is a right X -approximation of M .

Proof

We proceed by induction on n . The case for n = 0 is trivial. If n = 1 , there exists a triangle

(4) X 1 X 0 M Σ X 1
in ξ with X 0 , X 1 X . Since is a ξ -cogenerator of X , there is a triangle
X 1 H X 1 Σ X 1
in ξ with H and X 1 X . Applying cobase change for the triangle ( 4) along the morphism X 1 H , we get the following commutative diagram:
Since ξ is closed under cobase changes, we obtain that the triangle
(5) H X 0 M Σ H
is in ξ with - res.dim H = 0 . Note that α u = α is ξ -proper epic, so we have that α is ξ -proper epic by [ 16, Proposition 2.7]; hence, the triangle
X 0 X 0 X 1 Σ X 0
is in ξ . Since X is closed under ξ -extensions by assumption, we have X 0 X . So, ( 5) is the first desired triangle.

For X 0 , there is a triangle

X 0 H 0 X 0 Σ X 0
in ξ with H 0 and X 0 X . Applying cobase change for the triangle ( 5) along the morphism X 0 H 0 , we get the following commutative diagram:
(6)
Note that u = β u is ξ -proper monic by [ 16, Proposition 2.6], so the third horizontal triangle is in ξ . Since γ v = γ is ξ -proper epic, γ is ξ -proper epic by [ 16, Proposition 2.7]. So the triangle
M U X 0 Σ M
is in ξ with - res.dim U 1 and X 0 X , which is the second desired triangle.

Now suppose n 2 . Then, there is a triangle

(7) K X 0 M Σ K
in ξ with X - res.dim K n 1 and X 0 X . For K , by the induction hypothesis, we get a triangle
K K X 2 Σ K
in ξ with - res.dim K n 1 and X 2 X . Applying cobase change for the triangle ( 7) along the morphism K K , we get the following commutative diagram:
Note that λ κ = λ is ξ -proper epic, then λ is ξ -proper epic by [ 16, Proposition 2.7], so the triangle
X 0 X X 2 Σ X 0
is in ξ . It follows that X X from the assumption that X is closed under ξ -extensions. Since ξ is closed under cobase changes, we obtain the first desired triangle
(8) K X M Σ K
in ξ with - res.dim K n 1 and X X .

For X , since is a ξ -cogenerator of X , we get the following triangle

X H 1 X Σ X
in ξ with H 1 and X X .

Applying cobase change for the triangle (8) along the morphism X H 1 , we get the following commutative diagram:

As a similar argument to that of the diagram (6), we obtain that the triangles
K H 1 W Σ K
and
(9) M W X Σ M
are in ξ . Thus, ( 9) is the second desired triangle in ξ with - res.dim W n and X X .

In particular, suppose X , by Lemma 3.9, we have X ^ . Then, ξ x t ξ 1 ( X ˜ , K ) = 0 for any X ˜ X , it follows that Hom T ( X ˜ , X ) Hom T ( X ˜ , M ) 0 is exact. Thus, the ξ -proper epimorphism X M is a right X -approximation of M .□

Proposition 3.11

Keep the notion as Proposition 3.10. Assume M X ^ with X - res.dim M = n < .

  1. (1)

    If X is resolving, then in the triangles (2) and (3), we have - res.dim K = n 1 and - res.dim W = X - res.dim W = n .

    In particular, if X , then the ξ -proper epimorphism X M in the triangle (2) is a right X -approximation of M , such that - res.dim K = n 1 (if n = 0 , it should be interpreted K = 0 ).

  2. (2)

    If X and X is resolving, then there is a triangle

    M M X Σ M
    in ξ with M X , X X and X - res.dim M = X - res.dim M .

  3. (3)

    1. (a)

      Let ω = . If ω is a ξ -cogenerator of and is closed under ξ -extensions, then X ω if and only if X ( ^ ) .

    2. (b)

      If X is a resolving and ω X = X X is a ξ -cogenerator of X and M X , then X - res.dim M = ω X - res.dim M .

  4. (4)

    Suppose that and X are resolving. If ω = is a ξ -cogenerator of and X ω , then M admits a right X -approximation X M such that K X M Σ K is a triangle in ξ , where - res.dim K = n 1 . In fact, we have ω - res.dim K = n 1 .

Proof

  1. (1)

    Suppose X is resolving. Applying Corollary 3.6(2) to the triangle (2) yields that X - res.dim K = n 1 . On the other hand, since X , we have n 1 = X - res.dim K - res.dim K n 1 . Thus, - res.dim K = n 1 .

    Moreover, applying Corollary 3.6(1) to the triangle (3) implies X - res.dim W = X - res.dim M = n . So, n = X - res.dim W - res.dim W n . Hence, - res.dim W = X - res.dim W = n .

    The last assertion follows from the above argument and Proposition 3.10.

  2. (2)

    Since X , we have X ^ by Lemma 3.9, and so the result immediately follows from (1).

  3. (3)

    1. (a)

      ( ) Suppose X ( ^ ) . Clearly, ω = ^ X , that is, X ω .

      ( ) Suppose X ω . Let L ^ . By Proposition 3.10, there exists a triangle

      K H 0 L Σ K
      in ξ with H 0 and ω - res.dim K - res.dim L 1 < . Note that K by Lemma 3.9, so L implies H 0 . Then, H 0 ω , and so, L ω ^ . Since X ω , we have L X by Lemma 3.9. Thus, X ( ^ ) .

    2. (b)

      Suppose X - res.dim M = n , by (1), there exists a triangle

      K X 0 M Σ K
      in ξ with X 0 X and ω X - res.dim K = n 1 . Note that M X and K X , so X 0 X , and hence, X 0 ω X . It follows that ω X - res.dim M n . But n = X - res.dim M ω X - res.dim M n , thus X - res.dim M = ω X - res.dim M .

  4. (4)

    Suppose X - res.dim M = n , by (1), there exists a triangle

    (10) K X 0 M Σ K
    in ξ with X 0 X and - res.dim K = n 1 . By (2), there is a triangle
    K K H Σ K
    in ξ with H , K and - res.dim K = - res.dim K . Then, K ^ . Applying cobase change for the triangle ( 10) along the morphism K K , we get the following commutative diagram:
    One can see that the triangle
    (11) K X M Σ K
    is in ξ and X X . Note that X ω , so X ^ by (3)(a). Then, ξ x t ξ 1 ( X ˜ , K ) = 0 for any X ˜ X , and so, Hom T ( X ˜ , X ) Hom T ( X ˜ , M ) 0 is exact. Thus, the ξ -proper epimorphism X M is a right X -approximation of M and - res.dim K = n 1 in the triangle ( 11). Note that K , so we have ω - res.dim K = - res.dim K = n 1 by (3)(b).□

Lemma 3.12

Let be a subcategory of T with . Assume that is closed under hokernels of ξ -proper epimorphisms or closed under direct summands. Then, = ^ .

Proof

Clearly, ^ .

Conversely, let M ^ . Consider the following ξ -exact complex:

0 H n H n 1 H 0 M 0 .
Set K i = Hoker ( H i H i 1 ) for 0 i n 2 , where H 1 = M . Then, M yields K i , and so the triangle
H n H n 1 K n 2 Σ H n
is split. It follows that H n 1 H n K n 2 and there exists a triangle
K n 2 H n 1 H n 0 Σ K n 2
in ξ . Since is closed under hokernels of ξ -proper epimorphisms or closed under direct summands by assumption, we have K n 2 . Repeating this process, we can obtain K i , hence M and ^ . Thus, ^ = .□

Proposition 3.13

Let X be a resolving subcategory and a ξ x t -injective ξ -cogenerator of X . Assume that is closed under hokernels of ξ -proper epimorphisms or closed under direct summands. Then, X = X ^ ^ = X ^ .

Proof

Clearly, X X ^ and X ^ ^ X ^ .

Now, let M X ^ . Then, by Lemma 3.9, we have M X ^ ^ , and hence, X ^ X ^ ^ .

On the other hand, by Proposition 3.10, there is a triangle

(12) K X M Σ K
in ξ with X X and - res.dim K < . Note that M implies K , and hence, K ^ = by Lemma 3.12. Note that ξ x t ξ 1 ( M , K ) = 0 , so the triangle ( 12) is split; hence, X K M . Consider the following triangle
M X K 0 Σ M
in ξ . It follows that M X from the assumption that X is resolving. Thus, X ^ X .□

Our main result is the following.

Theorem 3.14

Let X be a resolving subcategory of T and a ξ x t -injective ξ -cogenerator of X . Assume that is closed under hokernels of ξ -proper epimorphisms or closed under direct summands. For any M T , if M X ^ , then the following statements are equivalent:

  1. (1)

    X - res.dim M m .

  2. (2)

    Ω n ( M ) X for all n m .

  3. (3)

    Ω X n ( M ) X for all n m .

  4. (4)

    ξ x t ξ n ( M , H ) = 0 for all n > m and all H .

  5. (5)

    ξ x t ξ n ( M , L ) = 0 for all n > m and all L ^ .

  6. (6)

    M admits a right X -approximation φ : X M , where φ is ξ -proper epic, such that K = Hoker φ satisfying - res.dim K m 1 .

  7. (7)

    There are two triangles

    W M X M M Σ W M
    and
    M W M X M Σ M
    in ξ such that X M , X M X and - res.dim W M m 1 , - res.dim W M = X - res.dim W M m .

Proof

( 1 ) ( 2 ) ( 3 ) It follows from Proposition 3.3.

( 1 ) ( 6 ) It follows from Proposition 3.11(1).

( 1 ) ( 7 ) It follows from Proposition 3.11(1).

( 1 ) ( 4 ) Suppose X - res.dim M m . There is a ξ -exact complex

0 X m X 0 M 0
with all X i in X . Since is a ξ x t -injective ξ -cogenerator of X , we have ξ x t ξ k 1 ( X i , H ) = 0 for all H . So, ξ x t ξ n ( M , H ) ξ x t ξ n m ( X m , H ) = 0 for n > m .

( 4 ) ( 5 ) It follows from Lemma 3.9.

( 5 ) ( 4 ) It is clear.

( 4 ) ( 1 ) Since M X ^ , by Proposition 3.11(1), there is a triangle K X M Σ K in ξ with - res.dim K < and X X . Then, ξ x t ξ i ( K , H ) ξ x t ξ i + 1 ( M , H ) for H and i 1 since ξ x t ξ i 1 ( X , H ) = 0 . So, ξ x t ξ i m ( K , H ) = 0 . Note that - res.dim K < , so we have the following ξ -exact complex

0 H n H 0 K 0
with all H i . Then,
ξ x t ξ i ( Ω m 1 ( K ) , H ) ξ x t ξ i + m 1 ( K , H ) = 0
for i 1 and all H , which means Ω m 1 ( K ) . Note that - res.dim Ω m 1 ( K ) < , hence, Ω m 1 ( K ) ^ . It follows that Ω m 1 ( K ) from Lemma 3.12, so - res.dim K m 1 . Thus, X - res.dim M m .□

4 Additive quotient categories and ξ -cellular towers with respect to a resolving subcategory

In this section, we will further study objects having finite resolution dimension with respect to a resolving subcategory X . We first construct adjoint pairs for two kinds of inclusion functors. Then, we characterize objects having finite resolution dimension in terms of a notion of ξ -cellular towers.

4.1 Adjoint pairs

Suppose that D and X are two subcategories of T . Denote by [ D ] the ideal of X consisting of morphisms factoring through some object in D . Thus, we have a quotient category X / [ D ] , which is also an additive category.

Lemma 4.1

Let X be a resolving subcategory of T and a ξ x t -injective ξ -cogenerator of X . Assume that f : X M is a morphism in T with X X and M X ^ , then the following statements are equivalent:

  1. (1)

    f factors through an object in .

  2. (2)

    f factors through an object in ^ .

Proof

It suffices to show that ( 2 ) ( 1 ) . Suppose that f factors through an object L ^ . Then, f = g h , where h : X L and g : L M . Consider the following triangle

L H L Σ L
in ξ with H and L ^ . Note that is a ξ x t -injective ξ -cogenerator of X , by Lemma 3.9, we have ξ x t ξ 1 ( X , L ) = 0 . So, h factors through H , it follows that f factors through H .□

Lemma 4.2

Let X be a resolving subcategory of T and a ξ x t -injective ξ -cogenerator of X , and let M , N X ^ . Assume that f : M N is a morphism in T , consider two triangles

W M α X M p M Σ W M and W N β X N q N Σ W N
in ξ with X M , X N X and W M , W N ^ ( see Proposition 3.10), then we have the following statements:
  1. (1)

    There exists a morphism g : X M X N such that q g = f p .

  2. (2)

    If g , g : X M X N are two morphisms such that q g = f p and q g = f p , then [ g ] = [ g ] in Hom X / [ ] ( X M , X N ) .

Proof

  1. (1)

    Apply Proposition 3.10.

  2. (2)

    Suppose g , g : X M X N are two morphisms such that q g = f p and q g = f p , then q ( g g ) = q g q g = 0 , and so there exists a morphism h : X M W N such that g g = β h , that is, there is a commutative diagram as follows:

    Note that W N ^ , so g g : X M X N factors through an object in by Lemma 4.1. Thus, [ g ] = [ g ] in Hom X / [ ] ( X M , X N ) .□

By Lemma 4.2, there exists a well-defined additive functor

F : X ^ X / [ ] ,
which maps an object M X ^ to X M and a morphism f : M N Hom X ^ ( M , N ) to [ g ] Hom X / [ ] ( X M , X N ) as described in Lemma 4.2.

Clearly, we have F ( H ) = 0 for any object H . Hence, F factors through X ^ / [ ] . That is, there exists an additive functor μ : X ^ / [ ] X / [ ] making the following diagram commutes

where π is the canonical quotient functor.

Now we show that the additive functor μ defined above and the inclusion functor between additive quotients X / [ ] and X ^ / [ ] are adjoint.

Theorem 4.3

Let X be a resolving subcategory of T and a ξ x t -injective ξ -cogenerator of X . Then, the additive functor μ : X ^ / [ ] X / [ ] defined above is right adjoint to the inclusion functor X / [ ] X ^ / [ ] .

Proof

Let X X and N X ^ . By Proposition 3.10, there is a triangle

W N β X N q N Σ W N
in ξ with W N ^ and X N X . Note that the additive map
[ q ] : Hom X / [ ] ( X , μ ( N ) ) Hom X ^ / [ ] ( X , N )
is natural in both X and N by Lemma 4.2. We claim that [ q ] is an isomorphism.

Indeed, since is a ξ x t -injective ξ -cogenerator of X , by Lemma 3.9, we have ξ x t ξ 1 ( X , W N ) = 0 , and hence, Hom T ( X , X N ) Hom T ( X , N ) is an epimorphism, so [ q ] is still an epimorphism.

Now, assume that g : X X N is a morphism such that [ q g ] = [ q ] [ g ] = [ q ] [ g ] = [ 0 ] Hom X / [ ] ( X , N ) . Then, there exists an object H such that q g = t s as the following commutative diagram:

Note that ξ x t ξ 1 ( H , W N ) = 0 by assumption, so there exists a morphism θ : H X N such that t = q θ . Since q ( g θ s ) = q g q θ s = t s t s = 0 , so g θ s factors through W N . By Lemma 4.1, g θ s factors through an object in . It follows that [ g θ s ] = 0 Hom X / [ ] ( X , N ) . Since θ s = 0 Hom X / [ ] ( X , N ) , we have 0 = [ g ] Hom X / [ ] ( X , N ) . So [ q ] is a monomorphism, and thus, [ q ] is an isomorphism.□

Corollary 4.4

Let X be a resolving subcategory of T and a ξ x t -injective ξ -cogenerator of X . Assume that is closed under direct summands. For any N X ^ , the following statements are equivalent:

  1. (1)

    N ^ .

  2. (2)

    There is a triangle

    W N X N q N Σ W N
    in ξ with W N ^ and X N X such that [ q ] = [ 0 ] Hom X ^ / [ ] ( X , N ) .

Proof

The assertion ( 1 ) ( 2 ) follows from Lemma 4.1. It suffices to show ( 2 ) ( 1 ) . Note that the adjunction isomorphism established in Theorem 4.3 implies that the additive map

[ q ] : Hom X / [ ] ( X N , X N ) Hom X ^ / [ ] ( X N , N )
is isomorphic. Since [ q ] [ id X N ] = [ q id X N ] = [ q ] = [ 0 ] Hom X ^ / [ ] ( X N , N ) = 0 , so [ id X N ] = [ 0 ] Hom X ^ / [ ] ( X N , X N ) , and thus, id X N factors through an object H . It follows that X N is a direct summand of W N . Since is closed under direct summands, we have X N . Thus, N ^ .□

Next, we compare additive quotients ^ / [ X ] and X ^ / [ X ] .

Lemma 4.5

Let X be a resolving subcategory of T and a ξ x t -injective ξ -cogenerator of X , and let M , N X ^ . Assume that f : M N is a morphism in T , consider two triangles

M s W M l X M Σ M and N t W N r X N Σ N
in ξ with X M , X N X and W M , W N ^ ( see Proposition 3.10), then, we have the following statements:
  1. (1)

    There exists a morphism g : W M W N such that g s = t f .

  2. (2)

    If g , g : W M W N are two morphisms such that g s = t f and g s = t f , then [ g ] = [ g ] in Hom ^ / [ X ] ( X M , X N ) .

Proof

  1. (1)

    Since X by assumption, we have ξ x t ξ 1 ( X M , W N ) = 0 by Lemma 3.9. So, there exists a morphism g : W M W N such that g s = t f .

  2. (2)

    Suppose g , g : W M W N are two morphisms such that g s = t f and g s = t f , then ( g g ) s = g s g s = 0 , and so there exists a morphism h : X M W N such that g g = h l , that is, there is a commutative diagram as follows:

    Note that X M X , so g g : W M W N factors through an object in X . Thus, [ g ] = [ g ] in Hom ^ / [ X ] ( W M , W N ) .□

By Lemma 4.5, there exists a well-defined additive functor

G : X ^ ^ / [ X ] ,
which maps an object M X ^ to W M and a morphism f : M N Hom X ^ ( M , N ) to [ g ] Hom ^ / [ X ] ( W M , W N ) as described in Lemma 4.5.

Clearly, we have G ( X ) = 0 for any object X X . Hence, G factors through X ^ / [ X ] . That is, there exists an additive functor η : X ^ / [ X ] ^ / [ X ] making the following diagram commutes

where η is the canonical quotient functor.

Now we show that the additive functor η defined above and the inclusion functor between additive quotients ^ / [ X ] and X ^ / [ X ] are adjoint.

Theorem 4.6

Let X be a resolving subcategory of T and a ξ x t -injective ξ -cogenerator of X . Then, the additive functor η : X ^ / [ X ] ^ / [ X ] defined above is left adjoint to the inclusion functor ^ / [ X ] X ^ / [ X ] .

Proof

Let K be an object in ^ and M an object in X ^ . By Proposition 3.10, there is a triangle

M s W M l X M Σ M
in ξ with W M ^ and X M X . Note that the additive map
[ s ] : Hom ^ / [ X ] ( η ( M ) , K ) Hom X ^ / X ( M , K )
is natural in both M and K by Lemma 4.5. We claim that [ s ] is an isomorphism.

Indeed, since is a ξ x t -injective cogenerator of X , by Lemma 3.9, we have ξ x t ξ 1 ( X M , K ) = 0 , and hence, Hom T ( W M , K ) Hom T ( M , K ) is an epimorphism, so [ s ] is still an epimorphism.

Now, assume that g : W M K is a morphism such that [ g s ] = [ g ] [ s ] = [ s ] [ g ] = [ 0 ] Hom X ^ / [ X ] ( M , K ) . Then, there exists an object X X such that g s = k v . Since is a ξ x t -injective ξ -cogenerator of X , there exists a triangle

X H X Σ X
in ξ with H and X X . Note that ξ x t ξ 1 ( X M , H ) = 0 and ξ x t ξ 1 ( X , K ) = 0 , so we get the following commutative diagram:
It follows that [ v v ] = [ 0 ] Hom ^ / X ( W M , K ) as H X . Since v v s = k v = g s Hom ^ / [ X ] ( M , K ) , by Lemma 4.5(2), we have [ g ] = [ v v ] Hom ^ / [ X ] ( W M , K ) , and hence, [ g ] = 0 . So [ s ] is a monomorphism, and thus, [ s ] is an isomorphism.□

Corollary 4.7

Let X be a resolving subcategory of T and a ξ x t -injective ξ -cogenerator of X . Assume that X is closed under direct summands. For any N X ^ , the following statements are equivalent:

  1. (1)

    N X .

  2. (2)

    There is a triangle

    N s W N X N Σ N
    in ξ with W N ^ and X N X such that [ s ] = [ 0 ] Hom X ^ / [ X ] ( N , W N ) .

Proof

The assertion ( 1 ) ( 2 ) is obvious. It suffices to show ( 2 ) ( 1 ) . Note that the adjunction isomorphism established in Theorem 4.6 implies that the additive map

[ s ] : Hom ^ / [ X ] ( W N , W N ) Hom X ^ / X ( N , W N )
is isomorphic. Since [ s ] [ id W N ] = [ id W N s ] = [ s ] = [ 0 ] Hom X ^ / [ X ] ( N , W N ) = 0 , so [ id W N ] = [ 0 ] Hom ^ / [ X ] ( W N , W N ) , and thus, id W N factors through an object X X . It follows that W N is a direct summand of X . Since X is closed under direct summands, we have W N X . Thus, N X .□

4.2 A characterization of finite resolution dimension via ξ -cellular towers

For M X ^ , there exists a triangle

(13) K 1 f 0 X 0 g 0 M h 0 Σ K 1
in ξ with X 0 X and K 1 X ^ . Similarly, there exists a triangle
K 2 f 1 X 1 g 1 K 1 h 1 Σ K 2
in ξ with X 1 X and K 2 X ^ . Continuing the above procedure for K n , there exists a triangle
K n + 1 f n X n g n K n h n Σ K n + 1
in ξ with X n X and K n + 1 X ^ .

Applying cobase change for the triangle (13) along the morphism h 1 : K 1 Σ K 2 , we get the following commutative diagram:

where the triangle
(14) Σ K 2 u 2 C 2 v 2 M Σ 2 K 2
is in ξ . Next consider the triangle ( 14) along the morphism Σ h 2 : Σ K 2 Σ 2 K 3 , we get the following commutative diagram:
where the triangle Σ 2 K 3 u 3 C 3 v 3 M Σ 3 K 3 is in ξ .

Continuing in this manner, we obtain the following commutative diagram:

where all the horizontal triangles are in ξ .

Set C 0 = 0 and C 1 = X 0 . The above construction produces a tower

0 C 1 γ 1 C 2 γ 2 C n 1 γ n 1 C n ,
which we call the ξ -cellular tower of M with respect to X .

According to the above construction, one can obtain the following result by Proposition 3.3.

Theorem 4.8

Let X be a resolving subcategory of T . For any M T , if M X ^ , then the following statements are equivalent:

  1. (1)

    X - res.dim M n .

  2. (2)

    For each i > 0 , the morphisms v n + i : C n + i M of the ξ -cellular tower of M with respect to X constructed above are isomorphisms.

5 Applications

In this section, we will construct a new resolving subcategory from a given resolving subcategory, which generalizes the notion of ξ -Gorenstein projective objects given by Asadollahi and Salarian [13]. By applying the previous results to this subcategory, we obtain some known results in [1315].

Definition 5.1

Let X be a subcategory of T and M an object in T . A complete P ( ξ ) X -resolution of M is a Hom T ( , X ) -exact ξ -exact complex

P 1 P 0 X 0 X 1
in T with all P i P ( ξ ) , X i X X such that both
K 1 P 0 M Σ K 1 and M X 0 K 1 Σ M
are corresponding triangles in ξ . The GP X ( ξ ) - Gorenstein category is defined as
GP X ( ξ ) = { M T M admits a complete P ( ξ ) X -resolution } .

Remark 5.2

  1. (1)

    Since X is a resolving subcategory of T , we have P ( ξ ) X , so P ( ξ ) X X . Then, we have K 1 GP X ( ξ ) .

  2. (2)

    If M GP X ( ξ ) , then ξ x t ξ 0 ( M , X ) Hom T ( M , X ) and ξ x t ξ 1 ( M , X ) = 0 for any X X . In fact, the following ξ -exact complex:

    P 1 P 0 M 0
    is a ξ -projective resolution of M (see [ 11]), which is Hom T ( , X ) -exact.

    Evidently, M GP X ( ξ ) if and only if ξ x t ξ 0 ( M , X ) Hom T ( M , X ) and ξ x t ξ 1 ( M , X ) = 0 for any X X , and M admits a Hom T ( , X ) -exact ξ -exact complex

    0 M X 0 X 1
    with X i X X .

  3. (3)

    If X = P ( ξ ) , then we have X X = P ( ξ ) by Lemma 3.12, and thus, GP X ( ξ ) coincides with G P ( ξ ) defined in [13].

We have the following result.

Theorem 5.3

Let X be a resolving subcategory of T . Then, GP X ( ξ ) is a resolving subcategory of T .

Proof

Let P be a ξ -projective object. Consider the following ξ -exact complex:

0 0 P id P P 0 0
in T . Clearly, it is Hom T ( , X ) -exact. In particular,
0 0 P id P P 0 0 and P id P P 0 0 0 Σ P
are corresponding triangles in ξ . Since P X X by Remark 5.2(1). we have P ( ξ ) GP X ( ξ ) .

As a similar argument to the proof of [18, Theorem 4.3(1)], we obtain that GP X ( ξ ) is closed under ξ -extensions and hokernels of ξ -proper epimorphisms. Thus, GP X ( ξ ) is a resolving subcategory of T .□

Lemma 5.4

Let X be a resolving subcategory of T satisfying X X GP X ( ξ ) . Then, X X is a ξ x t -injective ξ -cogenerator of GP X ( ξ ) and is closed under hokernels of ξ -proper epimorphisms.

Proof

Let M GP X ( ξ ) . There is a Hom T ( , X ) -exact triangle

(15) M X 0 K 1 Σ M
in ξ with X 0 X X GP X ( ξ ) . For any X ˜ X , applying the functor Hom T ( , X ˜ ) to the triangle ( 15) yields the following commutative diagram:
where the two isomorphisms follow from the assumption that X 0 , M GP X ( ξ ) and Remark 5.2(2). It follows that ξ x t ξ 1 ( K 1 , X ˜ ) = 0 and ξ x t ξ 0 ( K 1 , X ˜ ) Hom T ( K 1 , X ˜ ) , so K 1 GP X ( ξ ) by Remark 5.2(2), then X X is a ξ -cogenerator of GP X ( ξ ) . Obviously, X X is a ξ x t -injective ξ -cogenerator of GP X ( ξ ) .

It is obvious that X X is closed under hokernels of ξ -proper epimorphisms.□

As an application of Theorem 3.14, we have:

Proposition 5.5

Let X be a resolving subcategory of T satisfying X X GP X ( ξ ) and M T . If M GP X ( ξ ) ^ , then the following statements are equivalent:

  1. (1)

    GP X ( ξ ) - res.dim M m .

  2. (2)

    Ω n ( M ) GP X ( ξ ) for all n m .

  3. (3)

    Ω GP X ( ξ ) n ( M ) GP X ( ξ ) for all n