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Research article

A new method to construct model structures from left Frobenius pairs in extriangulated categories

  • Received: 25 July 2021 Revised: 12 March 2022 Accepted: 20 March 2022 Published: 23 May 2022
  • Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. In this paper, we first introduce the concept of left Frobenius pairs on an extriangulated category C, and then establish a bijective correspondence between left Frobenius pairs and certain cotorsion pairs in C. As an application, some new admissible model structures are established from left Frobenius pairs under certain conditions, which generalizes a result of Hu et al. (J. Algebra 551 (2020) 23–60).

    Citation: Yajun Ma, Haiyu Liu, Yuxian Geng. A new method to construct model structures from left Frobenius pairs in extriangulated categories[J]. Electronic Research Archive, 2022, 30(8): 2774-2787. doi: 10.3934/era.2022142

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  • Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. In this paper, we first introduce the concept of left Frobenius pairs on an extriangulated category C, and then establish a bijective correspondence between left Frobenius pairs and certain cotorsion pairs in C. As an application, some new admissible model structures are established from left Frobenius pairs under certain conditions, which generalizes a result of Hu et al. (J. Algebra 551 (2020) 23–60).



    The notion of extriangulated categories, whose extriangulated structures are given by E-triangles with some axioms, was introduced by Nakaoka and Palu in [1] as a simultaneous generalization of exact categories and triangulated categories. They gave a bijective correspondence between Hovey twin cotorsion pairs and admissible model structures which unified the work of Hovey, Gillespie and Yang (see [2,3,4]). Exact categories and triangulated categories are extriangulated categories, while there exist some other examples of extriangulated categories which are neither exact nor triangulated, see [1,5,6].

    Motivated by the ideas of projective covers and injective envelopes, Auslander and Buchweitz analyzed the framework in which the theory of maximal Cohen-Macaulay approximation can be developed. They systematically established their theory in abelian categories, which is known as Auslander-Buchweitz approximation theory. Up to now, Auslander-Buchweitz approximation theory has many important applications, see for example [7,8,9,10]. In particular, Becerril and coauthers [7] have revisited Auslander-Buchweitz approximation theory. From the notions of relative generators and cogenerators in approximation theory, they introduced the concept of left Frobenius pairs in an abelian category, established a bijective correspondence between left Frobenius pairs and relative cotorsion pairs, and showed how to construct an exact model structure from a strong left Frobenius pair, as a result of Hovey-Gillespie correspondence applied to two complete cotorsion pairs on an exact category (see [2,3]).

    The aim of this paper is to introduce the concept of left Frobenius pairs in an extriangulated category and give a method to construct more admissible model structures from strong left Frobenius pairs. For this purpose, we need to establish a bijective correspondence between left Frobenius pairs and cotorsion pairs in an extriangulated category under certain conditions.

    The paper is organized as follows. In Section 2, we recall the definition of an extriangulated category and outline some basic properties that will be used later. In Section 3, we first introduce the concept of left Frobenius pairs (see Definition 3.4), and then study relative resolution dimension and thick subcategories with respect to a given left Frobenius pair. As a result, we give a bijective correspondence between left Frobenius pairs and cotorsion pairs in an extriangulated category under certain conditions (see Theorem 3.12). In Section 4, we give a method to construct the admissible model structure from a strong left Frobenius pair under certain conditions (see Theorem 4.4), which generalizes a main result of Hu et al. in [5]. This is based on the bijective correspondence established in Section 3.

    Throughout this paper, C denotes an additive category. By the term "subcategory" we always mean a full additive subcategory of an additive category closed under isomorphisms and direct summands. We denote by C(A,B) the set of morphisms from A to B in C.

    Let X and Y be two subcategories of C, a morphism f:XC in C is said to be an X-precover of C if XX and C(X,f):C(X,X)C(X,C) is surjective for all XX. If any CY admits an X-precover, then X is called a precovering class in Y. By dualizing the definitions above, we get notions of an X-preenvelope of C and a preenveloping class in Y. For more details, we refer to [23].

    Let us briefly recall some definitions and basic properties of extriangulated categories from [1]. We omit some details here, but the reader can find them in [1].

    Assume that E:Cop×CAb is an additive bifunctor, where C is an additive category and Ab is the category of abelian groups. For any objects A,CC, an element δE(C,A) is called an E-extension. Let s be a correspondence which associates an equivalence class to any E-extension δE(C,A). This s is called a realization of E, if it makes the diagram in [1,Definition 2.9] commutative. A triplet (C,E,s) is called an extriangulated category if it satisfies the following conditions.

    1. E:Cop×CAb is an additive bifunctor.

    2. s is an additive realization of E.

    3. E and s satisfy certain axioms in [1,Definition 2.12].

    In particular, we recall the following axioms which will be used later:

    (ET4) Let δE(D,A) and δE(F,B) be E-extensions realized by

    respectively. Then there exists an object EC, a commutative diagram

    in C, and an E-extension δ realized by which satisfy the following compatibilities.

    realizes ,

    ,

    .

    (ET4) Dual of (ET4).

    Remark 2.1. Note that both exact categories and triangulated categories are extriangulated categories see [1,Example 2.13] and extension closed subcategories of extriangulated categories are again extriangulated see [1,Remark 2.18]. Moreover, there exist extriangulated categories which are neither exact categories nor triangulated categories see [1,Proposition 3.30], [6,Example 4.14] and [5,Remark 3.3].

    Lemma 2.2. [1,Corollary 3.12] Let be an extriangulated category and

    an -triangle. Then we have the following long exact sequences:

    where natural transformations and are induced by -extension via Yoneda's lemma.

    Let be as above, we use the following notation:

    A sequence is called a conflation if it realizes some -extension . In this case, is called an inflation, is called a deflation, and we write it as

    We usually do not write this "" if it is not used in the argument.

    Given an -triangle we call the CoCone of and the Cone of .

    An -triangle sequence in [11] is displayed as a sequence

    over such that for any , there are -triangles and the differential .

    An object is called projective if for any -triangle and any morphism , there exists satisfying . Injective objects are defined dually. We denote the subcategory consisting of projective (resp., injective) objects in by (resp., ).

    We say has enough projectives (resp., enough injectives) if for any object (resp., ), there exists an -triangle (resp., satisfying ).

    Remark 2.3. (1) If is an exact category, then the definitions of having enough projectives and having enough injectives agree with the usual definitions.

    (2) If is a triangulated category, then and consist of zero objects.

    Definition 2.4. [1,Definition 4.2] Let , be two subcategories of . Define full subcategories and of as follows.

    (1) belongs to if and only if it admits a conflation satisfying and ;

    (2) belongs to if and only if it admits a conflation satisfying and . $

    Suppose that is an extriangulated categories with enough projectives and injectives. For a subcategory , put , and for , we define inductively by

    We call the -th syzygy of (see [12,Section 5]). Dually we define the -th cosyzygy by and for

    Let be any object in . It admits an -triangle

    where (resp., ). In [12] the authors defined higher extension groups in an extriangulated category having enough projectives and injectives as for , and they showed the following result:

    Lemma 2.5. [12,Proposition 5.2] Let be an -triangle. For any object , there are long exact sequences

    From now on to the end of the paper, we always suppose that is an extriangulated categories with enough projectives and injectives.

    In this section, we introduce the concept of Frobenius pairs and show that it has very nice homological properties, which are necessary to construct cotorsion pairs from Frobenius pairs. At first, we need introduce the following definitions.

    Definition 3.1. Let be a subcategory of .

    For any non-negative integer , we denote by (resp., ) the class of objects such that there exists an -triangle sequence

    (resp., )

    with each . Moreover, we set , .

    For any , the -resolution dimension of is defined as

    resdim : = min.

    If for any , then resdim.

    For a subcategory of , define . Similarly, we can define

    Definition 3.2. Let and be two subcategories of . We say that

    (1) is a cogenerator for , if and for each object , there exists an -triangle with and . The notion of a generator is defined dually.

    (2) is -injective if . The notion of an -projective subcategory is defined dually.

    (3) is an -injective cogenerator for if is a cogenerator for and . The notion of an -projective generator for is defined dually.

    (4) is a thick subcategory if it is closed under direct summand and for any -triangle

    in and two of are in , then so is the third.

    The following theorem unifies some results of [13] and [9]. It shows that any object in admits two -triangles: one giving rise to an -precover and the other to a -preenvelope.

    Theorem 3.3. Let and be two subcategories of . Suppose is closed under extensions and is a cogenerator for . Consider the following conditions:

    (1) is in

    (2) There exists an -triangle with and

    (3) There exists an -triangle with and

    Then, . If is also closed under CoCone of deflations, then , and hence all three conditions are equivalent. If is -injective, then is an -precover of and is a -preenvelope of .

    Proof. The proof is dual to that of [14,Proposition 3.6].

    Definition 3.4. A pair is called a left Frobenius pair in if the following holds:

    is closed under extensions and CoCone of deflations,

    is an -injective cogenerator for .

    If in addition is also an -projective generator for , then we say is a strong left Frobenius pair.

    Example 3.5. (1) Assume that - is the category of left -modules for a ring . A left -module is called Gorenstein projective [23,24] if there is an exact sequence of projective left -modules

    with such that is exact for any projective left -module . Let be the full subcategory of - consisting of all Gorenstein projective modules and the subcategory of - consisting of all projective modules. Then is a strong left Frobenius pair.

    (2) Let be a triangulated category with a proper class of triangles. Asadollahi and Salarian [15] introduced and studied -Gprojective and -Ginjective objects, and developed a relative homological algebra in . Let denotes the full subcategory of -Gprojective objects and denotes the full subcategory of -projective objects. Then is a strong left Frobenius pair.

    (3) Let be a triangulated category, and let be a silting subcategory of with , where is the smallest full subcategory of which contains and which is closed under taking isomorphisms, finite direct sums, and direct summands. Then is a left Frobenius pair by [14,Corollary 3.7] and [16,Proposition 2.7], where .

    (4) In [17], the authors showed that if is a complete and hereditary cotorsion pair in an abelian category and is closed under kernels of epimorphisms, then is a strong left Frobenius pair, where is the class of objects in satisfying that there exists an exact sequence

    with each term in such that and is exact for any object in .

    Lemma 3.6. Let be a left Frobenius pair in .Given an -triangle with , then if and only if .

    Proof. The proof is dual to that of [14,Lemma 3.8].

    Proposition 3.7. Let be a left Frobenius pair in . The following statements are equivalent for any and non-negative integer .

    (1) resdim.

    (2) If is an -triangle sequence with for , then .

    Proof. is trivial.

    Let be in . Then by Theorem 3.3, we have an -triangle sequence with and for . Since , it is easy to see that . Thus we have for all and . If is an -triangle sequence with for , then we have for all and . Note that by Lemma 3.6. Hence there exists an -triangle with and by Theorem 3.3. It follows that the above -triangle splits. Hence .

    If is a subcategory of , then we denote by the smallest thick subcategory that contains . The following result shows that for a left Frobenius pair in , is an extriangulated category. In particular, if is a triangulated category, then is the smallest triangulated subcategory of containing and is closed under direct summands and isomorphisms.

    Proposition 3.8. Let be a left Frobenius pair in . Then .

    Proof. For any -triangle we need to check that if any two of and are in , then the third one is in . Since is closed under extensions by the dual of [14,Corollary 3.7], it suffices to show that if , then if and only if . We first show that if and are in , then . Since , we have an -triangle with . By , we obtain a commutative diagram

    It follows that as and are in . Therefore

    Suppose now and are in . It follows from Lemma 3.6 that . Applying the just established result to the -triangle one has that .

    Suppose . We proceed by induction on resdim. If , then and are in .

    Suppose . There is an -triangle with and resdim. By , we obtain the following commutative diagrams:

    Hence there is an -triangle

    By Lemma 3.6, , and Proposition 3.7 shows that resdim. By the induction hypothesis, and are in . It follows that and are in . Hence is closed under direct summands. Thus .

    Definition 3.9. [1,Definition 4.1] Let , be a pair of full additive subcategories, closed under isomorphisms and direct summands. The pair (, ) is called a cotorsion pair on if it satisfies the following conditions:

    (1) ;

    (2) For any , there exists a conflation satisfying and ;

    (3) For any , there exists a conflation satisfying and . $

    Lemma 3.10. Let and be two subcategories of such that is -injecive. Then the following statements hold.

    (1) If is a cogenerator for , then .

    (2) If is a cogenerator for , then .

    Proof. The proof is dual to that of [14,Proposition 4.2].

    The following result gives a method to construct cotorsion pairs on extriangulated categories.

    Proposition 3.11. Let be a left Frobenius pair in . Then is a cotorsion pair on the extriangulated category .

    Proof. Note that is an extriangulated category by [1,Remark 2.18]. It suffices to show that is closed under direct summands by Theorem 3.3. Note that by Proposition 3.10. Since = is closed under direct summands by Proposition 3.8, so is . This completes the proof.

    Now we are in a position to state and prove the main result of this section.

    Theorem 3.12. Let be an extriangulated category. The assignments

    give mutually inverse bijections between the following classes:

    (1) Left Frobenius pairs in .

    (2) Cotorsion pairs on the extriangulated category with .

    Proof. Let be a left Frobenius pair. Then is a cotorsion pair on the extriangulated category by Proposition 3.11. Note that and . Then is a cotorsion pair on the extriangulated category with .

    Assume is a cotorsion pair on the extriangulated category with . For , we have an -triangle with and . Thus as is a thick subcategory. Since is a cotorsion pair on the extriangulated category , it follows from [1,Remark 4.6] that is closed under extensions in It implies that Thus . Note that . It follows that is an -injective cogenerator. Let be an -triangle with . Then we have an exact sequence for any . Since , . Note that as is a thick subcategory. Thus there exists an -triangle $ with and as is a cotorsion pair on the extriangulated category . Therefore the above -triangle splits by . Hence . So is closed under CoCone of deflations. Note that is closed under extensions in It follows that is closed under extensions in . Thus is a left Frobenius pair in .

    Based on the above argument, it is enough to check that the compositions

    and

    are identities. Since is an -injective cogenerator for , where the first equality is due to Proposition 3.10 and the second equality is due to Proposition 3.8. It follows from [1,Remark 4.4] that Thus This completes the proof.

    As a consequence of Theorem 3.12 and Remark Remark 2.3, we have the following result.

    Corollary 3.13. [7,Throrem 5.4] Let be an abelian category with enough projectives and injectives. The assignments

    give mutually inverse bijections between the following classes:

    (1) Left Frobenius pairs in .

    (2) Cotorsion pairs on the exact category with .

    As an application, we have the following result in [10].

    Corollary 3.14. [10,Theorem 3.11] Let be a triangulated category. The assignments

    give mutually inverse bijections between the following classes:

    (1) Left Frobenius pairs in .

    (2) Co-t-structures on the triangulated category

    Proof. Note that any triangulated category can be viewed as an extriangulated category, and its projective objects and injective objects consist of zero objects by Remark Remark 2.3.

    Let be a left Frobenius pair. By Theorem 3.12, is a cotorsion pair on the triangulated category . Since is closed under CoCone of deflations and extensions, it is easy to see that . Hence is a co-t-structure on the triangulated category .

    Assume is a co-t-structure on the triangulated category . It is easy to see that is a cotorsion pair on with . Hence the corollary follows from Theorem 3.12.

    Definition 3.15. [18,Definition 2.1] Let and be rings. An --bimodule is semidualizing if:

    (1) admits a degreewise finite -projective resolution.

    (2) admits a degreewise finite -projective resolution.

    (3) The homothety map is an isomorphism.

    (4) The homothety map is an isomorphism.

    (5) .

    Definition 3.16. [18,Definition 3.1] A semidualizing bimodule is faithfully semidualizing if it satisfies the following conditions for all modules and .

    (1) If then .

    (2) If then .

    Definition 3.17. [18,Definition 4.1] The Bass class with respect to consists of all -modules satisfying

    (1) .

    (2) The natural evaluation homomorphism is an isomorphism.

    Remark 3.18. Let be a faithfully semidualizing module. Then Bass class is an exact category by [18,Theorem 6.2] and has enough projectives and injectives by [20,Remark 3.13].

    By [18], the class of -projective left -modules, denoted by the collection of the left -modules of the form for some projective left -module . Recall from [20] that a left -module is called -Gorenstein projective if there is an exact sequence of left -modules

    with each term in such that and both and are exact for any object in . It should be noted that -Gorenstein projectives defined here are different from those defined in [19] when is a commutative Noetherian ring (see [20,Proposition 3.6]).

    For convenience, we write - for the classes of -Gorenstein projective left -modules. By [20,Proposition 3.5], one has that - . As a consequence of Theorem 3.12, we have the following result.

    Corollary 3.19. Let be a faithfully semidualizing module. Then

    (1) - is a strong left Frobenius pair in .

    (2) - is a cotorsion pair on .

    Proof. Since is projectively resolving and by [18,Corollary 6.4] and [18,Theorem 6.4], - is closed under kernels of epimorphisms and direct summand by [21,Theorem 4.12] and [21,Proposition 4.11]. Hence - is a strong left Frobenius pair in . (2) follows from Theorem 3.12.

    In this section, we shall use our results in Section 3 to construct more admissible model structures in extriangulated categories. At first, we need to recall the following definition.

    Definition 4.1. [1,Definition 5.1] Let (, ) and (, ) be cotorsion pairs on . Then , ), (, )) is called a twin cotorsion pair if it satisfies . Moreover, is called a Hovey twin cotorsion pair if it satisfies = . $

    In [1] Nakaoka and Palu gave a correspondence between admissible model structures and Hovey twin cotorsion pairs on . Essentially, an admissible model structure on is a Hovey twin cotorsion pair , ), (, )) on . For more details, we refer to [1,Section 5]. By a slight abuse of language we often refer to a Hovey twin cotorsion pair as an admissible model structure.

    Lemma 4.2. Let be a strong left Frobenius pair in . Then is a cotorsion pair on the extriangulated category .

    Proof. Since , one has . For any , there exists an -triangle

    with and by Theorem 3.3. Since is a generator for , we have an -triangle with and . By , we obtain a commutative diagram

    It follows that as and are in . Note that . The second column and show that is a cotorsion pair on .

    Proposition 4.3. Let be a strong left Frobenius pair in . Then is an admissible model structure on the extriangulated category

    Proof. By Theorem 3.12 and Lemma 4.2, we only need to check that . It is obvious that Let . Then we have an -triangle with and . By Theorem 3.3, one has that Since , it follows that is a direct summand of . Note that is closed under direct summand by Proposition 3.11. Thus . Hence the equality holds.

    Theorem 4.4. Let be a strong left Frobenius pair in . If is a non-negative integer, then the following statements are equivalent:

    (1) .

    (2) is an admissible model structure on .

    Proof. . If , then is a cotorsion pair on by Lemma 4.2 and is a cotorsion pair on by Theorem 3.12. To prove , we only need to check that Note that is obvious. Let . Then there is an -triangle with and by Theorem 3.3. Since is closed under extensions, . Hence implies .

    . Since is a cotorsion pair on , one has that by Theorem 33.

    As an application, we have the following result in [3].

    Corollary 4.5. [3,Theorem 8.6] Suppose is a Gorenstein ring. Let be the subcategory of - consisting of Gorenstein projective modules and the subcategory of - consisting of projective modules. Then - is an admissible model structure on -.

    Proof. It follows from Example 3.5 and Proposition 4.3.

    Let be a non-negative integer. In the following, we denote by (resp., the class of modules with -Gorenstein projective (resp., -projective) dimension at most

    Corollary 4.6. Let be a faithfully semidualizing module. Then the following statements are equivalent:

    (1) .

    (2) is an admissible model structure on .

    Proof. It follows from Corollary 3.19 and Theorem 4.4.

    Let be an extriangulated category and a proper class of -triangles. By [5], an object is called -projective if for any -triangle

    in , the induced sequence of abelian groups $ is exact. We denote the class of -projective objects of . Recall from [5] that an object is called -projective if there exists a diagram

    in satisfying that: (1) is -projective for each integer ; (2) there is a -exact -triangle in and for each integer such that for some . We denote by the class of -projective objects in . Specializing Theorem 4.4 to the case , we have the following result in [5].

    Corollary 4.7. [5,Theorem 5.9] Let be an extriangulated category satisfying Condition (see [1,Condition 5.8]). Assume that is a proper class in .Set , that is,

    for any , and .If is a non-negative integer, then the following conditions are equivalent:

    -.

    is an admissible model structure on , where -.

    Proof. It is easy to check that is a strong left Frobenius pair in . Thus the corollary follows from Theorem 4.4.

    The authors are grateful to the referees for reading the paper carefully and for many suggestions on mathematics and English expressions. Yajun Ma was supported by NSFC (Grant No. 12171230). Haiyu Liu and Yuxian Geng were supported by the NSFC (Grants Nos. 12171206 and 12126424) and the Natural Science Foundation of Jiangsu Province (Grant No. BK20211358).

    The authors declare there is no conflicts of interest.



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