Modules of finite Gorenstein flat dimension and approximations

We study approximations of modules of finite Gorenstein flat dimension by (projectively coresolved) Gorenstein flat modules and modules of finite flat dimension. These approximations determine the Gorenstein flat dimension and lead to descriptions of the corresponding relative homological dimensions, for such modules, in more classical terms. We also describe two hereditary Hovey triples on the category of modules of finite Gorenstein flat dimension, whose associated exact structures have homotopy categories equivalent to the stable category of projectively coresolved Gorenstein flat modules and the stable category of cotorsion Gorenstein flat modules, respectively.


INTRODUCTION
The concept of G-dimension for commutative Noetherian rings that was introduced by Auslander and Bridger in [1] has been extended to modules over any ring  through the notion of a Gorenstein projective module by Enochs and Jenda in [16].The class of Gorenstein injective modules was also defined in that paper, whereas Gorenstein flat modules were introduced in [18].The relative homological dimensions based on these modules were defined by Holm [23] in the standard way, by considering resolutions (or coresolutions) by modules in the given class.Gorenstein homological algebra has developed rapidly and found interesting applications in representation theory, algebraic geometry, and cohomological group theory.Many useful properties of the notion of Gorenstein projective dimension are consequences of the fact that the class () of Gorenstein projective modules contains all projective modules and is closed under extensions, kernels of epimorphisms, and direct summands.These properties of () were established by Holm in [loc.cit.],where the analogous (dual) properties of Gorenstein injective modules are also proved.It follows that modules of finite Gorenstein projective dimension admit approximations by Gorenstein projective modules and modules of finite projective dimension; cf.[13].Analogous approximations exist for modules of finite Gorenstein injective dimension.As the closure of the class () of Gorenstein flat modules under extensions was not known until recently, many properties of Gorenstein flat modules and modules of finite Gorenstein flat dimension were only available under additional assumptions on the ground ring.For example, Holm characterized in [23] the Gorenstein flat dimension of modules in terms of the vanishing of certain Tor-groups, in the case where the ring  is right coherent.Bennis realized the importance of knowing that () is closed under extensions and termed the rings for which this is true as being GF-closed; cf.[3].For modules over a GF-closed ring, he showed that the Gorenstein flat dimension can be characterized in terms of the vanishing of Tor-groups, extending the above-mentioned result by Holm.It was finally proved by Saroch and Stovicek in [26] that the class () is closed under extensions over any ring, so that all rings are GF-closed; in fact, ( (), () ⟂ ) is a complete hereditary cotorsion pair in the category of modules.Saroch and Stovicek have also introduced in [loc.cit.] a certain subclass of (), formed by the so-called projectively coresolved Gorentein flat modules (PGF-modules, for short), and showed that these modules are Gorenstein projective. 1 They showed that the class () of these modules forms the left-hand side of another complete hereditary cotorsion pair ( (), () ⟂ ) in the category of modules.
In this paper, we study the class () of modules of finite Gorenstein flat dimension and obtain approximations of modules therein (i) by PGF-modules and modules of finite flat dimension and (ii) by Gorenstein flat modules and cotorsion modules of finite flat dimension.It turns out that these approximations are really obtained by restricting the approximations resulting from the complete cotorsion pairs ( (), () ⟂ ) and ( (), () ⟂ ) to the exact subcategory () of the module category.These approximations determine the Gorenstein flat dimension Gfd   of any module .The following result illustrates this assertion, using the approximations that arise from the fact that both cotorsion pairs have enough injectives; it is extracted from parts of Theorem 2.1, Proposition 3.9, Theorem 4.2, and Proposition 5.7.
Then, Gfd   = fd  ; in particular,  has finite Gorenstein flat dimension if and only if  has finite flat dimension.
As a consequence of the existence of these approximations for modules in (), we obtain characterizations of the numerical values of the PGF-dimension (i.e., of the relative homological dimension, which is based on the class of PGF-modules [14]) and the Gorenstein flat dimension, in terms of the vanishing of certain Ext-groups.These characterizations are analogous to the corresponding characterization of the Gorenstein projective dimension for modules of finite Gorenstein projective dimension.We note that, in the case of the Gorenstein flat dimension, this characterization has been already established by Christensen et al. in [10].
It is easily seen that the exact categories () and () ∩ () of PGF-modules and cotorsion Gorenstein flat modules, respectively, are both Frobenius.The projective modules are the projective-injective objects of the former of these categories, whereas the projective-injective objects of the latter are the flat cotorsion modules.The proof of this claim for () is essentially identical to the proof of the corresponding claim for the class of Gorenstein projective modules, which can be found in [15].The claim for () ∩ () was noted by Gillespie in [21], in the special case where the ground ring is right coherent.We point out that cotorsion Gorenstein flat modules are particular examples of the Gorenstein flat cotorsion modules introduced in [11].In the sequel, we describe two hereditary Hovey triples ) and ( (), (), () ∩ () in the exact category (); here, we denote by () the class of all modules of finite flat dimension.The exact model structures associated with these Hovey triples have homotopy categories, which are equivalent to the stable categories of the Frobenious exact categories () and () ∩ (), respectively.In order to realize these stable categories as the homotopy categories of model structures, it is therefore sufficient to work on the subcategory ().We note that these Hovey triples can be obtained by restricting to the exact subcategory () the Hovey triples that were introduced by Saroch and Stovicek in the discussion following [26,Corollary 4.12]; the latter Hovey triples have been generalized to a relative setting in [19, section 3].Notations and terminology.We work over a fixed unital associative ring  and, unless otherwise specified, all modules are assumed to be left -modules.We denote by   the opposite ring of  and do not distinguish between right -modules and left   -modules.The classes of projective, flat and cotorsion modules are denoted by (), () and (), respectively.We say that a class  of modules is projectively resolving if () ⊆  and  is closed under extensions and kernels of epimorphisms.

PRELIMINARIES
In this section, we collect certain basic notions and preliminary results that will be used in the sequel.These involve the concept of a Hovey triple in exact additive categories, the basics on Gorenstein flat and PGF-modules, and the notion of relative injectivity of linear maps with respect to a class of modules.

Cotorsion pairs and Hovey triples
Let  be an exact additive category, in the sense of [9].Then, the Ext 1 -pairing induces an orthogonality relation between objects therein.If  is a class of objects in , then the left orthogonal ⟂  of  is the class consisting of those objects  ∈ , which are such that Ext 1  (, ) = 0 for all  ∈ .Analogously, the right orthogonal  ⟂ of  is the class consisting of those objects  ∈ , which are such that Ext 1  (, ) = 0 for all  ∈ .If ,  are two classes of objects in , then the pair (, ) is said to be a cotorsion pair in  if  = ⟂  and  ⟂ = ; cf.[17].The kernel of the cotorsion pair is the class  ∩ .The cotorsion pair is called hereditary if Ext   (, ) = 0 for all  > 0 and all objects  ∈  and  ∈ .The cotorsion pair is complete if for any object  ∈ , there exist short exact sequences (conflations) where ,  ′ ∈  and ,  ′ ∈ .An example of a complete hereditary cotorsion pair in the exact category of all modules is provided by the pair ((), ()); cf.[8].
A full subcategory  0 ⊆  is called exact if it has the 2-out-of-3 property for short exact sequences (conflations): If we are given a short exact sequence in  and two of the three objects involved are contained in  0 , then the third object is also contained in  0 .If  0 is such an exact subcategory and (, ) is a complete cotorsion pair in  with  ⊆  0 , then the pair (,  0 ∩ ) is easily seen to be a complete cotorsion pair in  0 ; it is the restriction of the original cotorsion pair to the exact subcategory  0 .
A Hovey triple on  is a triple (, ,  ) of subclasses of , which are such that the pairs (,  ∩  ) and ( ∩ ,  ) are complete cotorsion pairs and the class  is thick (i.e., it is closed under direct summands and satisfies the 2-out-of-3 property for short exact sequences in ).Based on the work of Hovey [24], Gillespie has shown that there is a bijection between Hovey triples on an idempotent complete exact category  and the so-called exact model structures on ; cf.[20,Theorem 3.3].In the context of Gillespie's bijection, it is proved in [20,Proposition 5.2] that for an exact model structure on , whose associated complete cotorsion pairs (,  ∩  ) and ( ∩ ,  ) are both hereditary, the exact category  ∩  is Frobenius with projective-injective objects equal to  ∩  ∩  .A result of Happel [22] implies that the stable category of  ∩  modulo its projective-objective objects is triangulated.As shown in [20,Proposition 4.4 and Corollary 4.8], the homotopy category of the given exact model structure is triangulated equivalent to the stable category of the Frobenius exact category  ∩  .

Basics on modules of finite Gorenstein flat dimension
Gorentein flat modules were defined in [18] as the syzygy modules of those acyclic complexes of flat modules that remain acyclic after applying the functor  ⊗ __ for any injective right module .It follows that the abelian groups Tor   (, ) are trivial if  > 0 for any injective right module  and any Gorenstein flat module .It is clear that the class () of Gorenstein flat modules contains all flat modules and is closed under direct sums.As shown in [26], the class () is also projectively resolving and closed under direct summands.In fact, ( (), () ⟂ ) is a complete hereditary cotorsion pair in the category of modules, whose kernel coincides with the class () ∩ () of flat cotorsion modules; cf.[26,Corollary 4.12].
The Gorenstein flat dimension of a module  was defined by Holm in [23] by the standard method, using resolutions by Gorenstein flat modules.Indeed, let  be a module and  a nonnegative integer.If are two exact sequences of modules with  0 , … ,  −1 ,  ′ 0 , … ,  ′ −1 ∈ (), then  ∈ () if and only if  ′ ∈ ().This follows from [1, Lemma 3.12], since the class () is projectively resolving, closed under direct sums and direct summands; see also [3,Lemma 2.9].The following result is an immediate consequence of this remark.
If the equivalent conditions in the above corollary are satisfied, we say that  has a Gorenstein flat resolution of length .The Gorenstein flat dimension Gfd   of  is the shortest length of a Gorenstein flat resolution of .Of course, if  has no Gorenstein flat resolution of finite length, then we write Gfd   = ∞.Since Gorenstein flat modules annihilate the functors Tor   (, __) for any  > 0 and any injective right module , a dimension shifting argument shows that Tor   (, ) = 0 for any  > Gfd   and any injective right module .Any flat module is Gorenstein flat and hence we always have Gfd   ≤ fd  .In fact, if  has finite flat dimension, then we have an equality Gfd   = fd  ; this is proved in [5,Theorem 2.2].In particular, any Gorenstein flat module of finite flat dimension is necessarily flat.
For future reference, we record some basic properties of the class () of all modules of finite Gorenstein flat dimension.

Proposition 1.2. Let (𝑀 𝑖 ) 𝑖 be a family of modules and 𝑀 =
⨁    the corresponding direct sum.Then, Gfd   = sup  Gfd    .In particular, the class () is closed under finite direct sums and direct summands.

PGF-modules
A variant of the Gorenstein flat modules, the so-called PGF-modules, were introduced by Saroch and Stovicek.The PGFmodules are the syzygy modules of those acyclic complexes of projective modules that remain acyclic after applying the functor  ⊗ __ for any injective right module .It is clear that PGF-modules are Gorenstein flat.As shown in [26, Theorem 4.9], the class () of PGF-modules is also projectively resolving, closed under direct sums and direct summands.In fact, ( (), () ⟂ ) is a complete hereditary cotorsion pair in the category of modules, whose kernel coincides with the class () of projective modules.Moreover, the right orthogonal () ⟂ is thick (i.e., it is closed under direct summands and satisfies the 2-out-of-3 property for short exact sequences).The inclusion () ⊆ () induces an inclusion () ⟂ ⊆ () ⟂ .It turns out that a module  is contained in () ⟂ if and only if  is a cotor-sion module contained in () ⟂ , that is, () ⟂ = () ∩ () ⟂ ; cf.[26,Theorem 4.12].The homological dimension theory, which is based on the class (), is studied in [14].

Relative injectivity of linear maps
Let  be a class of modules.Following [26], we say that a linear map  ∶  ⟶  is -injective provided that any linear map  ⟶  factors through  for any module  ∈ .If  is injective and coker  ∈ ⟂ , then  is -injective.On the other hand, if  is injective, -injective, and  ∈ ⟂ , then coker  ∈ ⟂ .It is easily seen that the notion of -injectivity has the following three properties with respect to a composable pair of morphisms  ∶  ⟶  and  ∶  ⟶ : In the sequel, we shall mostly use this notion of relative injectivity for the class  2 (()), which is obtained from the class () of projective modules by applying twice the Pontryagin duality functor  to it.We recall that the duality functor  maps any left (resp.right) module  onto the right (resp.left) module  = Hom(, ℚ∕ℤ).The class  2 (()) is defined analogously.
(ii)→(iii): The right module  is an injective cogenerator of the category of right modules and hence any injective right module is a direct summand of a direct product of copies of .Therefore, it suffices to consider the case where  is a direct power of .For any set Λ, we have () Λ = , where  =  (Λ) is the free module with basis Λ.Then, the standard Hom-tensor adjunction and our assumption on  imply that Tor  1 (, ) = Ext 1  (,  2 ) = 0. Since the abelian group ℚ∕ℤ cogenerates the category of abelian groups, it follows that Tor  1 (, ) = 0, as needed.(iii)→(i): If  is a flat module, then Lambek's criterion [25] implies that the right module  is injective and hence we have Ext  2 (()),  is  2 (())-injective, and fd   ′′ ≤ 1, then  ′′ is flat. Proof.

APPROXIMATIONS BY PGF-MODULES
In this section, we show that modules of finite Gorenstein flat dimension can be approximated by PGF-modules and modules of finite flat dimension, in a way that generalizes the exact sequences obtained in [26,Theorem 4.11].
Theorem 2.1.The following conditions are equivalent for a module  and an integer  ≥ 0: (ii) There exists a short exact sequence where  is a PGF-module and fd   = .(iii) There exists a short exact sequence where  is a PGF-module and (a) if  > 1, then fd   =  − 1, (b) if  = 1, then  is flat and  is not  2 (())-injective, (c) if  = 0, then  is flat and  is  2 (())-injective.(iv) There exists a projective module , such that the direct sum  ′ =  ⊕  fits into a short exact sequence where  is a PGF-module, fd   =  and  is () ⟂ -injective.(v) There exists a Gorenstein flat module , such that the direct sum  ′ =  ⊕  fits into a short exact sequence where  is a PGF-module and fd   = .If  = 1, we also require  to be  2 (())-injective.
Proof.(i)→(ii): We use induction on .If  = 0, this follows from [26,Theorem 4.11(4)].Assume that  ≥ 1 and the result is known for modules of Gorenstein flat dimension < .We consider a short exact sequence where  is Gorenstein flat and Gfd   =  − 1, and invoke the induction hypothesis to find a short exact sequence where  is a PGF-module and fd   =  − 1.We now form the pushout of that short exact sequence along the monomorphisms  ⟶  and obtain the commutative diagram with exact rows and columns Since both  and  are Gorenstein flat, the closure of () under extensions implies that  ′ is also Gorenstein flat.Therefore, there exists a short exact sequence where  is flat and  ′ ∈ ().Pushing out that short exact sequence along the epimorphism  ′ ⟶ , we obtain the commutative diagram with exact rows and columns Then, the rightmost vertical exact sequence is of the required type.Indeed, the horizontal exact sequence in the middle shows that fd   ′ ≤ .In fact, the latter inequality cannot be strict, since otherwise we would have Gfd   ′ ≤ fd   ′ ≤  − 1 and the rightmost vertical exact sequence would imply that Gfd   ≤ max{Gfd   ′ , Gfd   ′ } ≤  − 1; cf.Proposition 1.3(ii).
(ii)→(iii): We fix a short exact sequence as in (ii) and note that the module  fits into a short exact sequence where  is projective and fd   ′ =  − 1 (if  = 0, then  ′ is also flat).The pullback of that short exact sequence along the monomorphism  ⟶  induces a commutative diagram with exact rows and columns Since both  and  are PGF-modules, the closure of () under kernels of epimorphisms shows that  ′ is also a PGFmodule.Being a PGF-module,  is Gorenstein flat and hence Lemma 1.
(iii)→(iv): We fix a short exact sequence as in (iii) and note that the PGF-module  fits into a short exact sequence where  is projective and  ′ ∈ ().Then, the pushout of the latter short exact sequence along the epimorphism  ⟶  induces a commutative diagram with exact rows and columns Since  ′ is a PGF-module,  ′ is Gorenstein flat and hence  is  2 (())-injective (cf.Lemma 1.6(i)).It follows that  = • is  2 (())-injective if and only if  is  2 (())-injective.Now, the definition of the pushout and the injectivity of  imply that there is a short exact sequence If  > 1, the vertical short exact sequence in the middle shows that Tor   (__,  ′ ) = Tor  −1 (__, ) is nonzero and Tor  +1 (__,  ′ ) = Tor   (__, ) = 0, so that fd   ′ = .If  = 1, then  is flat and  is not  2 (())-injective.It follows that fd   ′ ≤ 1 and  is not  2 (())-injective.Then, Lemma 1.6(i) implies that  ′ is not flat and hence fd   ′ = 1.If  = 0, then  is flat and hence fd   ′ ≤ 1, as before.Since  is assumed to be  2 (())-injective in this case, it follows that  is also  2 (())-injective.The projective module  being obviously contained in ⟂  2 (()), Lemma 1.6(iii) implies that  ′ is flat.
In order to show that the short exact sequence (3) has the required additional property, we note that for any module  ∈ () ⟂ the two horizontal short exact sequences in diagram ( 2 (v)→(i): Consider an exact sequence as in (v) and note that Proposition 1.2 implies that Gfd   ′ = Gfd  .Therefore, it suffices to prove that Gfd   ′ = .Since the PGF-module  is Gorenstein flat and Gfd   ≤ fd   = , Proposition 1.3(i) implies that Gfd   ′ ≤ .It remains to show that the latter inequality cannot be strict.Indeed, let us assume that  ≥ 1 and Gfd   ′ ≤  − 1.
(i) In the case where  = 1, it is necessary to impose some restrictions on the short exact sequences appearing in Theorem 2.1(iii) and (v).Indeed, if  is a projective module and  ∈ (), then the split short exact sequence 0 ⟶  ⟶  ⊕  ⟶  ⟶ 0 is of the type appearing in Theorem 2.1(iii) for  = 1, but Gfd   = 0 ≠ 1.On the other hand, if  is a nonflat module with pd   = 1, then a projective resolution of  provides an exact sequence of the type appearing in Theorem 2.1(v) for  = 1, but Gfd   0 = 0 ≠ 1. (ii) The short exact sequence (4) is of the type appearing in Theorem 2.1(iii) for  = 0, but Gfd   ≠ 0. (Any Gorenstein flat module of finite flat dimension is necessarily flat.)We conclude that it is necessary to impose some additional restriction to the short exact sequence appearing in Theorem 2.1(iii), in the case where  = 0, in order to get that the third term of that exact sequence is Gorenstein flat. 2 (iii) The proof of the implication (v)→(i) in Theorem 2.1 uses only the weaker assumption that the module  in (v) is Gorenstein flat.
Recall that the (left) finitistic flat dimension fin.f.dim  of the ring is the supremum of the flat dimensions of all modules that have a finite flat dimension.Analogously, the (left) finitistic Gorenstein flat dimension fin.Gf.dim  is the supremum of the Gorenstein flat dimensions of all modules that have a finite Gorenstein flat dimension.The next result generalizes [23,Theorem 3.24].Proposition 2.3.For any ring , we have an equality fin.f.dim  = fin.Gf.dim .
Proof.Since the Gorenstein flat dimension is a refinement of the flat dimension, it is clear that fin.f.dim  ≤ fin.Gf.dim .In order to prove the reverse inequality, consider a module  of finite Gorenstein flat dimension, say with Gfd   = .Then, Theorem 2.1(ii) implies that there exists a module  with fd   = .It follows that  ≤ fin.f.dim .Since fin.Gf.dim  is the supremum of these s, we conclude that fin.Gf.dim  ≤ fin.f.dim , as needed.□

THE RELATION TO THE COTORSION PAIR ( 𝙿𝙶𝙵(𝑹), 𝙿𝙶𝙵(𝑹) ⟂ )
In this section, we use the approximation sequences obtained in Theorem 2.1 to characterize the PGF-modules and, more generally, the PGF-dimension of modules (which is introduced in [14]) within (), in terms of the vanishing of certain Ext-groups.We obtain a hereditary Hovey triple in (), such that the homotopy category of the associated exact model structure is triangulated equivalent to the stable category of PGF-modules.It will turn out that the exact sequences in Theorem 2.1 are precisely the approximation sequences of the complete cotorsion pair ( (), () ⟂ ) obtained in [26, Theorem 4.9], when applied to modules of finite Gorenstein flat dimension.
It follows from [26,Corollary 4.5] that Ext 1  (, ) = 0, whenever  is a PGF-module and  is flat.We elaborate on this result and provide a characterization of PGF-modules among modules in (), as an application of Theorem 2. where  ∈ () and  ∈ ().Then, our assumption implies that this sequence splits and hence  is a direct summand of the PGF-module .Since the class () is closed under direct summands, we conclude that  is a PGFmodule. □ The next result provides a characterization of the PGF-dimension for modules in () that extends [14, Proposition 3.6] and reduces to Lemma 3.1, in the case where  = 0. Proof.As we noted above, the Ext-group is trivial if  ∈ ().Conversely, assume that  is a module of finite Gorenstein flat dimension contained in () ⟂ .Invoking Theorem 2.1(ii), we obtain a short exact sequence where  ∈ () and  ∈ ().In view of our assumption on , this sequence splits and hence  is a direct summand of .Then, fd   ≤ fd   < ∞ and hence  ∈ (), as needed.□ The (left) Gorenstein weak global dimension Gwgl.dim  of  was introduced and studied in [4,6] and [7]; it is defined as the supremum of the Gorenstein flat dimensions of all modules.By considering the Gorenstein flat dimension of direct sums of modules, we can easily deduce that Gwgl.dim  < ∞ if and only if Gfd   < ∞ for any module .In order to characterize the finiteness of Gwgl.dim , the relevant homological invariants are sfli , the supremum of the flat lengths (dimensions) of injective modules, and its analog sfli   for the opposite ring   .By considering the flat dimension of products of injective modules, it is easily seen that sfli  < ∞ if and only if fd   < ∞ for any injective module .As shown in [12], the following conditions are equivalent: (i) Gwgl.dim  < ∞, (ii) Gwgl.dim   < ∞, (iii) the invariants sfli  and sfli   are finite.
If these conditions are satisfied, then Gwgl.dim  = Gwgl.dim  = sfli  = sfli   .For rings that satisfy these equivalent conditions, we may describe the class () of PGF-modules and its right orthogonal in classical terms.Corollary 3.4.Let  be a ring and assume that all injective modules (both left and right) have finite flat dimension.Then, () = ⟂ () and () ⟂ = ().
Proof.Our assumption implies that all modules have finite Gorenstein flat dimension, that is, that () = -Mod.Then, the two equalities in the statement follow from Lemma 3.1 and Lemma 3.3.□ Remark 3.5.
(i) Since the hypothesis in Corollary 3.4 is left-right symmetric, we also have (under the same assumptions) analogous conclusions for the ring   , that is, for the corresponding classes of right modules.(ii) Since injective modules are obviously contained in the right orthogonal () ⟂ and the same is true for right modules, the hypothesis in Corollary 3.4 is also necessary for the equalities () ⟂ = () and (  ) ⟂ = (  ) to hold.
The category () of modules of finite Gorenstein flat dimension is an extension closed subcategory of the abelian category of all modules (cf.Proposition 1.3(i)), which is also closed under direct summands (cf.Proposition 1.2).Therefore, () is an idempotent complete exact additive category [9].Proposition 3.6.The pair ) is a complete hereditary cotorsion pair in the exact category ().Proof.Since any flat module is Gorenstein flat, we always have Gfd   ≤ fd  .In order to prove the reverse inequality, it suffices to assume that Gfd   =  < ∞.Then, the truncation of a flat resolution of  provides an exact sequence where  0 , … ,  −1 are flat modules and  ∈ ().Since the class () ⟂ is thick and contains the flat modules, our assumption that  ∈ () ⟂ implies that  ∈ () ⟂ as well.Therefore,  ∈ () ∩ () ⟂ = ().We conclude that  admits a flat resolution of length  and hence fd   ≤  = Gfd  , as needed.□ We shall conclude this section by showing that the approximation sequences of Theorem 2.1 are precisely the approximation sequences of the complete cotorsion pair ( (), () ⟂ ) of [26, Theorem 4.9], applied to modules of finite Gorenstein flat dimension.To that end, we fix a module  and note that the completeness of the cotorsion pair provides two short exact sequences where ,  ′ ∈ () and ,  ′ ∈ () ⟂ .We may also consider a short exact sequence where  is projective, and consider its pullback along the monomorphism  ⟶ , in order to obtain a commutative diagram with exact rows and columns Since () is closed under kernels of epimorphisms, the horizontal short exact sequence in the middle shows that  ′′ is a PGF-module.Letting  ′′ = , we thereby obtain a third short exact sequence where  ′′ ∈ (),  ′′ ∈ () ⟂ , and  is projective.
Proof.We note that () is a thick subcategory, which contains all Gorenstein flat modules (and hence all PGFmodules).) in the exact category (), which is considered in [14, section 4], is precisely the restriction of the cotorsion pair ( (), () ⟂ ) therein.Moreover, the thickness of the category () can be used as in Proposition 3.9, to show that modules in () can be characterized by the assertion that one (and hence all) of the modules ,  ′ , and  ′′ in the short exact sequences ( 5) and ( 6) be of finite projective dimension.

APPROXIMATIONS BY GORENSTEIN FLAT MODULES
In this section, we show that modules of finite Gorenstein flat dimension can be approximated by Gorenstein flat modules and cotorsion modules of finite flat dimension, in a way analogous to the short exact sequences obtained in Section 2.

𝙶𝙵𝚕𝚊𝚝(𝑅).
We first state an auxiliary result we need.
Lemma 4.1.Let  be a Gorenstein flat module.Then, there exists a short exact sequence where  ′ is Gorenstein flat and  is flat cotorsion.If  is a cotorsion Gorenstein flat module, then  ′ is a cotorsion Gorenstein flat module as well.
Proof.We consider a short exact sequence where  ′ is Gorenstein flat and  is flat.Since the cotorsion pair ((), ()) is complete, we may also consider a short exact sequence where  ′ is flat and  is cotorsion.Of course,  is then flat cotorsion.Pushing out the latter short exact sequence along the epimorphism  ⟶  ′ , we obtain a commutative diagram with exact rows and columns Since  ′ and  ′ are Gorenstein flat, the closure of () under extensions implies that  ′′ is Gorenstein flat as well.
Then, the vertical exact sequence in the middle has the required properties.The final claim in the statement follows since the class of cotorsion modules is closed under cokernels of monomorphisms.□ Theorem 4.2.The following conditions are equivalent for a module  and a nonnegative integer : (i) Gfd   = .
(ii) There exists a short exact sequence where  is a Gorenstein flat module and  is cotorsion with fd   = .(iii) There exists a short exact sequence where  is a Gorenstein flat module and (a) if  > 1, then  is cotorsion with fd   =  − 1, (b) if  = 1, then  is flat cotorsion and  is not  2 (())-injective, (c) if  = 0, then  is flat cotorsion and  is  2 (())-injective.(iv) There exists a flat cotorsion module , such that the module  ′ =  ⊕  fits into a short exact sequence where  is Gorenstein flat,  is cotorsion with fd   = , and  is () ⟂ -injective.(v) There exists a Gorenstein flat module , such that the module  ′ =  ⊕  fits into a short exact sequence where  is Gorenstein flat and  is cotorsion with fd   = .If  = 1, we also require  to be  2 (())-injective.
Proof.(i)→(ii): In view of Theorem 2.1(ii), there exists a short exact sequence where  is a PGF-module and fd   = .The cotorsion pair ((), ()) is complete and hence there exists an exact sequence where  is cotorsion and  is flat.Then, the functors Tor   (__, ) and Tor   (__, ) are isomorphic if  > 0 and hence fd   = fd   = .Taking the pushout of the latter exact sequence along the epimorphism  ⟶ , we obtain a commutative diagram with exact rows and columns Since  is a PGF-module and  is flat, both of these modules are Gorenstein flat.The class () is closed under extensions and hence the rightmost vertical exact sequence shows that  is Gorenstein flat as well.Then, the horizontal sequence in the middle is of the required type.
(ii)→(iii): We fix a short exact sequence as in (ii) and use again the completeness of the cotorsion pair ((), ()) in order to find a short exact sequence where  is flat and  ′ cotorsion.Of course, we also have fd   ′ =  − 1 (if  = 0, then  ′ is also flat).The pullback of that short exact sequence along the monomorphism  ⟶  induces a commutative diagram with exact rows and columns Since both  and  are Gorenstein flat, the closure of () under kernels of epimorphisms shows that  ′ is also Gorenstein flat.Since  is Gorenstein flat, we can show that the leftmost vertical exact sequence is of the required type, as in the proof of the corresponding step in Theorem 2.1.
(iii)→(iv): We fix a short exact sequence as in (iii) and apply Lemma 4.1 to the Gorenstein flat module , in order to find a short exact sequence 0 ⟶  ⟶  ′ ⟶  ′ ⟶ 0, where  ′ is Gorenstein flat and  ′ is flat cotorsion.Then, the pushout of the latter short exact sequence along the epimorphism  ⟶  induces a commutative diagram with exact rows and columns Since both  and  ′ are cotorsion modules, it follows that  ′′ is cotorsion as well.Since  ′ is Gorenstein flat, the horizontal exact sequences in the diagram remain exact after applying the functor Hom(__, ) for any module  ∈ () ⟂ .We can now show that the induced short exact sequence 0 ⟶  ⟶  ⊕  ′ ⟶  ′′ ⟶ 0 is of the required type, as in the proof of the corresponding step in Theorem 2.1.
(v)→(i): This follows as in the proof of the corresponding step in Theorem 2.1; cf.Remark 2.2(iii).□ Remark 4.3.As with Theorem 2.1, it is necessary to impose some restrictions on the short exact sequences appearing in Theorem 4.2(iii) and (v), in the case where  = 1. 3 The same is also true for the short exact sequence appearing in Theorem 4.2(iii), in the case where  = 0.

THE RELATION TO THE COTORSION PAIR ( 𝙶𝙵𝚕𝚊𝚝(𝑹), 𝙶𝙵𝚕𝚊𝚝(𝑹) ⟂ )
In this section, we use the approximation sequences obtained in Theorem 4.2 to characterize the Gorenstein flat modules and, more generally, the Gorenstein flat dimension of modules within the class (), in terms of the vanishing of certain Ext-groups.We also obtain a hereditary Hovey triple in (), such that the homotopy category of the associated exact model structure is triangulated equivalent to the stable category of the Frobenius exact category of cotorsion Gorenstein flat modules.
As we have noted earlier, the very definition of Gorenstein flat modules (and modules of finite Gorenstein flat dimension) implies that these modules have trivial Tor-groups with injective right modules (in degrees exceeding the Gorenstein flat dimension).It was shown by Holm in [23] that, for modules in (), the triviality of these Tor-groups characterizes Gorenstein flat modules (and, more generally, the value of their Gorenstein flat dimension), provided that the ring is right coherent.The latter assumption on the ring was removed in [3,Theorem 2.8], where it was shown that Holm's characterization is actually valid over any ring, pending the proof of the closure of () under extensions, that was achieved in [26].For modules within (), we may also characterize the Gorenstein flat modules (and, more generally, the value of their Gorenstein flat dimension), in terms of the Ext-functors.
It follows from Lemma 3.1 that any module of finite flat dimension is contained in () ⟂ .Therefore, we conclude that () ∩ () ⊆ () ∩ () ⟂ = () ⟂ , that is, the group Ext 1  (, ) is trivial for any Gorenstein flat module  and any cotorsion module  with fd   < ∞.A proof of the next result may be also found in [10,Lemma 5.4]; we provide an alternative argument, which is based on the approximation sequence in Theorem 4.2(iii).Since () is closed under kernels of epimorphisms, the horizontal short exact sequence in the middle shows that  ′′ is Gorenstein flat.Letting  ′′ = , we thereby obtain a third short exact sequence 0 ⟶  ′′ ⟶  ⊕  ⟶  ′′ ⟶ 0, where  ′′ ∈ (),  ′′ ∈ () ⟂ , and  is projective.