Homological dimension based on a class of Gorenstein flat modules

In this paper, we study the relative homological dimension based on the class of projectively coresolved Gorenstein flat modules (PGF-modules), that were introduced by Saroch and Stovicek. The resulting PGF-dimension of modules has several properties in common with the Gorenstein projective dimension, the relative homological theory based on the class of Gorenstein projective modules. In particular, there is a hereditary Hovey triple in the category of modules of finite PGF-dimension, whose associated homotopy category is triangulated equivalent to the stable category of PGF-modules. Studying the finiteness of the PGF global dimension reveals a connection between classical homological invariants of left and right modules over the ring, that leads to generalizations of certain results by Jensen, Gedrich and Gruenberg that were originally proved in the realm of commutative Noetherian rings.


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 R through the notion of a Gorenstein projective module.Such a module is, by definition, a syzygy of an acyclic complex of projective modules which remains acyclic when applying the functor Hom R ( , P ) for any projective module P .The modules of finite Gorenstein projective dimension are defined in the standard way, using resolutions by Gorenstein projective modules.A Gorenstein flat module is a syzygy of an acyclic complex of flat modules which remains acyclic when applying the functor I ⊗ R for any injective right module I.The modules of finite Gorenstein flat dimension are then defined using resolutions by Gorenstein flat modules.The standard reference for these notions is Holm's paper [22].The relation between Gorenstein projective and Gorenstein flat modules remains somehow mysterious in general.As shown in [loc.cit.],all Gorenstein projective modules are Gorenstein flat if the ground ring is right coherent and Research supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the "1st Call for H.F.R.I.Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment grant", project number 4226.
has finite left finitistic dimension (i.e. if there is an upper bound on the projective dimension of all modules that have finite projective dimension).
The projectively coresolved Gorenstein flat modules (PGF-modules, for short) were defined by Saroch and Stovicek in [26]; these are the syzygies of the acyclic complexes of projective modules that remain acyclic when applying the functor I ⊗ R for any injective right module I.It is clear that PGF-modules are Gorenstein flat.As shown in [26,Theorem 4.4], PGFmodules are also Gorenstein projective.A schematic presentation of the classes GProj(R), GFlat(R) and PGF(R) of Gorenstein projective, Gorenstein flat and PGF-modules respectively is given below Here, Proj(R) and Flat(R) denote the classes of projective and flat modules respectively and all arrows are inclusions.Moreover, the class Proj(R) of projective modules is the intersection PGF(R) ∩ Flat(R) and all classes pictured above are projectively resolving; in fact, GFlat(R) is the smallest projectively resolving class of modules that contains both PGF(R) and Flat(R); these assertions are proved in [26].
In this paper, we study the relative homological dimension which is based on the class PGF(R) and define the PGF-dimension PGF-dim R M of a module M as the minimal length of a resolution of M by PGF-modules (provided that such a resolution exists).The resulting class PGF(R) of modules of finite PGF-dimension has many of the standard properties that one would expect.In particular, it is closed under direct sums, direct summands and has the 2-out-of-3 property for short exact sequences of modules.The PGF-dimension is a refinement of the ordinary projective dimension, whereas the Gorenstein projective dimension is a refinement of the PGF-dimension.In other words, if M is a module of finite projective dimension (resp. of finite PGF-dimension), then M has finite PGF-dimension (resp.finite Gorenstein projective dimension) and PGF-dim R M = pd R M (resp.Gpd R M = PGF-dim R M).When restricted to the class Flat(R) of flat modules, the PGF-dimension coincides with the projective dimension.The modules of finite PGF-dimension can be approximated by modules of finite projective dimension and PGF-modules, in analogy with the case of modules of finite Gorenstein projective dimension.In particular, this leads to a description, up to triangulated equivalence, of the stable category of PGF-modules modulo projective modules, as the homotopy category of the exact model structure which is associated with a Hovey triple in the category PGF(R).Using the analogous approximations of Gorenstein flat modules by PGFmodules and flat modules, that were obtained by Saroch and Stovicek in [26], we describe a similar Hovey triple in the category GFlat(R).Therefore, in order to realize the stable category of PGF-modules as the homotopy category of a Quillen model structure, it is sufficient to work on either subcategory PGF(R) or GFlat(R) of the module category.
In order to present an application of the notion of PGF-dimension studied in this paper, we consider the invariants silp R and spli R, which are defined as the suprema of the injective lengths of projective modules and the projective lengths of injective modules, respectively.It is easily seen that these invariants are equal, if they are both finite.Nevertheless, as Gedrich and Gruenberg point out in [17], it is not clear whether the finiteness of one of these implies the finiteness of the other, i.e. whether we always have an equality silp R = spli R. In the special case where R is an Artin algebra, the equality silp R = spli R is equivalent to the Gorenstein Symmetry Conjecture in representation theory; cf.[2, Conjecture 13], [4, §11] and [5,Chapter VII].
The study of the finiteness of the PGF global dimension reveals a connection between the silp and spli invariants for left and right modules over any ring, which may be itself used in order to show that: If both spli R and spli R op are finite, then silp R = spli R and silp R op = spli R op .Using the Hopf algebra structure of the group algebra kG of a group G with coefficients in a commutative ring k, Gedrich and Gruenberg proved in [17] that silp kG ≤ spli kG, in the special case where the commutative ring k is Noetherian of finite self-injective dimension.It follows from the result displayed above that we actually have an inequality silp R ≤ spli R for any ring R which is isomorphic with its opposite ring R op .In particular, the inequality holds for group algebras of groups over any commutative coefficient ring.On the other hand, Jensen has proved in [24, 5.9] that the equality silp R = spli R holds for any commutative Noetherian ring R. The result displayed above, combined with earlier work in [14], shows that the equality silp R = spli R actually holds for any commutative ℵ 0 -Noetherian ring R, i.e. for any commutative ring R all of whose ideals are countably generated.
Notations and terminology.We work over a fixed unital associative ring R and, unless otherwise specified, all modules are left R-modules.We denote by R op the opposite ring of R and do not distinguish between right R-modules and left R op -modules.If λ(R) is an invariant, which is defined in terms of a certain class of left R-modules, then we denote by λ(R op ) the corresponding invariant, which is defined for R in terms of the appropriate class of right R-modules.Finally, we say that a class C of modules is projectively resolving if Proj(R) ⊆ C and C is closed under extensions and kernels of epimorphisms.

Preliminary notions
In this section, we collect certain basic notions and preliminary results that will be used in the sequel.These involve basic concepts related to Gorenstein homological algebra in module categories and the theory of Hovey triples in exact additive categories.
I. Gorenstein projective and Gorenstein flat modules.An acyclic complex P * of projective modules is said to be a complete projective resolution if the complex of abelian groups Hom R (P * , Q) is acyclic for any projective module Q.Then, a module is called Gorenstein projective if it is a syzygy of a complete projective resolution.Holm's paper [22] is the standard reference in Gorenstein homological algebra.The class GProj(R) of Gorenstein projective modules is projectively resolving; it is also closed under direct sums and direct summands.The Gorenstein projective dimension Gpd R M of a module M is the length of a shortest resolution of M by Gorenstein projective modules.If no such resolution of finite length exists, then we write Gpd R M = ∞.If M is a module of finite projective dimension, then M has finite Gorenstein projective dimension as well and Gpd R M = pd R M.
An acyclic complex F * of flat modules is said to be a complete flat resolution if the complex of abelian groups I ⊗ R F * is acyclic for any injective right module I.We say that a module is Gorenstein flat if it is a syzygy of a complete flat resolution.We let GFlat(R) be the class of Gorenstein flat modules.The Gorenstein flat dimension Gfd R M of a module M is the length of a shortest resolution of M by Gorenstein flat modules.If no such resolution of finite length exists, then we write Gfd R M = ∞.If M is a module of finite flat dimension, then M has finite Gorenstein flat dimension as well and Gfd R M = fd R M.
Even though the relation between Gorenstein projective and Gorenstein flat modules is not fully understood, the notion of a projectively coresolved Gorenstein flat module (for short, PGF-module) defined in [26] sheds some light in and helps clarifying that relation.A PGFmodule is a syzygy of an acyclic complex of projective modules P * , which is such that the complex of abelian groups I ⊗ R P * is acyclic for any injective right module I.It is clear that the class PGF(R) of PGF-modules is contained in GFlat(R).The inclusion PGF(R) ⊆ GProj(R) is proved in [26,Theorem 4.4]; in fact, it is shown that Ext 1 R (M, F ) = 0 for any PGF-module M and any flat module F .It is also proved in [loc.cit.] that the classes PGF(R) and GFlat(R) are both projectively resolving, closed under direct sums and direct summands.
II. Gorenstein global dimensions.The existence of complete projective resolutions of modules (i.e. of complete projective resolutions that coincide in sufficiently large degrees with an ordinary projective resolution of the module) has been studied by Gedrich and Gruenberg [17], Cornick and Kropholler [12], in connection with the existence of complete cohomological functors in the category of modules.Even though they were mainly interested in the case where R is the integral group ring of a group, they were able to characterize those rings over which all modules admit complete projective resolutions, in terms of the finiteness of the invariants spli R and silp R. Here, spli R is the supremum of the projective lengths (dimensions) of injective modules and silp R is the supremum of the injective lengths (dimensions) of projective modules.As shown by Holm [22], the existence of a complete projective resolution for a module M is equivalent to the finiteness of the Gorenstein projective dimension Gpd R M of M. From this point of view, the above result by Cornick and Kropholler was alternatively proved by Bennis and Mahbou in [7], where the notion of the Gorenstein global dimension of the ring was introduced, in analogy with the classical notion of global dimension defined in [9, Chapter VI, §2]; see also [15, §4].More precisely, the (left) Gorenstein global dimension Ggl.dim R of the ring R is defined by letting Then, the following conditions are equivalent: (i) Ggl.dim R < ∞, (ii) Gpd R M < ∞ for any module M, (iii) any module M admits a complete projective resolution, (iv) the invariants spli R and silp R are finite.If these conditions are satisfied, then Ggl.dim R = spli R = silp R.
The corresponding characterization of the finiteness of the Gorenstein weak global dimension Gwgl.dimR of the ring R, which is defined by letting Gwgl.dimR = sup{Gfd R M : M a left R-module}, turned out to be more difficult to achieve.The relevant homological invariants here are sfli R, the supremum of the flat lengths (dimensions) of injective modules, and its analogue sfli R op for the opposite ring R op .Using in an essential way results in [26], it was proved by Christensen, Estrada and Thompson in [11] that the following conditions are equivalent: (i) Gwgl.dimR < ∞, (ii) Gfd R M < ∞ for any module M, (iii) the invariants sfli R and sfli R op are finite.If these conditions are satisfied, then Gwgl.dimR = sfli R = sfli R op .III. Cotorsion pairs and Hovey triples.Let A be an exact additive category, in the sense of Quillen [8], and consider a full subcategory B ⊆ A. A morphism f : B −→ A in A is called a B-precover of the object A ∈ A if: (i) B ∈ B and (ii) the induced map f * : Hom A (B ′ , B) −→ Hom A (B ′ , A) is surjective for any B ′ ∈ B. The reader is referred to [20] for a thorough and systematic study of precovers.
The Ext 1 -pairing induces an orthogonality relation between subclasses of A. If B ⊆ A, then we define the left orthogonal ⊥ B of B as the class consisting of those objects X ∈ A, which are such that Ext A (C, D) = 0 for all i > 0 and all objects C ∈ C and D ∈ D. The cotorsion pair is complete if for any object A ∈ A there exist short exact sequences (conflations), usually called approximation sequences A Hovey triple on A is a triple (C, W, F ) of subclasses of A, which are such that the pairs (C, W ∩ F ) and (C ∩ W, F ) are complete cotorsion pairs and the class W is closed under direct summands and satisfies the 2-out-of-3 property for short exact sequences (conflations) in A. The fundamental work of Gillespie [18], which is based on work of Hovey [23], gives a bijection between Hovey triples on a (weakly) idempotent complete exact category A and certain, socalled exact, Quillen model structures on A; cf.[18,Theorem 3.3].In the context of Gillespie's bijection, it is proved in [18,Proposition 5.2] that for an exact model structure on A that has its associated complete cotorsion pairs hereditary, the class C ∩F is a Frobenius exact category with projective-injective objects equal to C ∩ W ∩ F .Then, a result of Happel [21] implies that the associated stable category, which is C ∩ F modulo its projective-injective objects, is triangulated.The upshot of this connection is that the (Quillen) homotopy category of an exact model structure is triangulated equivalent to the stable category of the Frobenius exact category C ∩ F ; cf.[18, Proposition 4.4 and Corollary 4.8].

Modules of finite PGF-dimension
In this section, we define the notion of PGF-dimension for a module and show that the resulting class PGF(R) of modules of finite PGF-dimension has many standard closure properties.
We recall that the class PGF(R) is projectively resolving and closed under direct sums and direct summands.The following result is a formal consequence of these properties of PGF(R); cf.[1, Lemma 3.12].Lemma 2.1.Let M be an R-module, n a non-negative integer and

Proposition 2.2. The following conditions are equivalent for an R-module M and a nonnegative integer n:
(i) There exists an exact sequence of modules (ii) For any exact sequence of modules Proof.The implication (i)→(ii) is a consequence of Lemma 2.1, whereas the implication (ii)→ (i) follows by considering a truncated projective resolution of M.
If the equivalent conditions in Proposition 2.2 are satisfied, then we say that the module M has a PGF-resolution of length n and write PGF-dim R M ≤ n.In the case where PGF-dim R M ≤ n and M has no PGF-resolution of length < n, we say that M has PGF-dimension equal to n and write PGF-dim R M = n.Finally, we say that M has infinite PGF-dimension and write PGF-dim R M = ∞, if M has no PGF-resolution of finite length.We now consider the class PGF(R) of all modules of finite PGF-dimension and describe certain closure properties of that class.
It remains to show that we also have sup To that end, assume that PGF-dim R M = n < ∞ and consider for any i an exact sequence and our assumption on the PGF-dimension of M imply that i K i is a PGF-module.Since PGF(R) is closed under direct summands, it follows that K i is a PGF-module for all i.Then, M i has a PGF-resolution of length n and hence PGF-dim R M i ≤ n for all i, as needed.
In particular, the class PGF(R) has the 2-out-of-3 property: if two out of the three modules that appear in a short exact sequence have finite PGF-dimension, then so does the third.Proof.(i) Assume that max{PGF-dim R M ′ , PGF-dim R M ′′ } = n and consider two projective resolutions P ′ * −→ M ′ −→ 0 and P ′′ * −→ M ′′ −→ 0 of M ′ and M ′′ respectively.Then, we may construct by the standard step-by-step process a projective resolution P * −→ M −→ 0 of M, such that P i = P ′ i ⊕ P ′′ i and the corresponding syzygy module Ω i M is an extension of Ω i M ′′ by Ω i M ′ for all i.Since both M ′ and M ′′ have PGF-dimension ≤ n, the modules Ω n M ′ and Ω n M ′′ are both PGF-modules.Then, the short exact sequence and the closure of PGF(R) under extensions show that Ω n M is a PGF-module as well.Then, the exact sequence (ii) We can prove this assertion by using the same argument as the one used in order to prove assertion (i) above, by invoking the closure of PGF(R) under kernels of epimorphisms.
(iii) We fix a short exact sequence where P is a projective module, and consider the pullback of the short exact sequence given in the statement of the Proposition along p Since P is projective, the horizontal short exact sequence in the middle of the diagram splits and hence X ≃ P ⊕ M ′ .We now invoke Proposition 2.3 and conclude that PGF-dim R X = PGF-dim R M ′ .Then, the vertical short exact sequence in the middle of the diagram and assertion (ii) above show that Since we may splice any PGF-resolution of K of length PGF-dim R K with the short exact sequence (1) and obtain a PGF-resolution of As a consequence of the equality PGF(R) ∩ Flat(R) = Proj(R), we obtain the following result on the relation between the projective dimension and the PGF-dimension of flat modules and, analogously, the relation between the projective dimension and the flat dimension of PGF-modules.
In order to prove the reverse inequality, it suffices to assume that PGF-dim R M = n < ∞.Then, the truncation of a projective resolution of M provides us with an exact sequence where P 0 , . . ., P n−1 are projective modules and K ∈ PGF(R).Since M is flat, it follows that K is also flat and hence K ∈ PGF(R) ∩ Flat(R) = Proj(R).We conclude that M admits a projective resolution of length n and hence pd R M ≤ n = PGF-dim R M, as needed.
(ii) Since projective modules are flat, we always have fd R M ≤ pd R M. In order to prove the reverse inequality, it suffices to assume that fd R M = n < ∞.Then, the truncation of a projective resolution of M provides us with an exact sequence where P 0 , . . ., P n−1 are projective modules and K is flat.Since M is a PGF-module and the class PGF(R) is projectively resolving, it follows that K is also a PGF-module.Then, K ∈ PGF(R) ∩ Flat(R) = Proj(R) and hence M admits a projective resolution of length n, i.e. pd R M ≤ n = fd R M.
Remark 2.6.If we denote by Proj(R) and Flat(R) the classes of modules of finite projective dimension and finite flat dimension respectively, then PGF(R) ∩ Flat(R) = Proj(R).Indeed, the inclusion Proj(R) ⊆ PGF(R) ∩ Flat(R) is clear, since any projective resolution of finite length is both a PGF-resolution and a flat resolution of finite length.Conversely, if M is a module contained in PGF(R) ∩ Flat(R), then the n-th syzygy module Ω n M in a projective resolution of M is a flat and PGF-module for n ≫ 0. Since PGF(R) ∩ Flat(R) = Proj(R), it follows that Ω n M is projective for n ≫ 0 and hence M ∈ Proj(R).

Approximation sequences
In this section, we show that the finiteness of PGF-dimension can be detected by the existence of suitable approximation sequences, in analogy with the case of the finiteness of Gorenstein projective dimension.
The next result is akin to [22, Theorem 2.10].
Proposition 3.1.Let M be a module with PGF-dim R M = n.Then, there exists a short exact sequence Proof.The result is clear if n = 0 and hence we may assume that n ≥ 1.Since PGF-dim R M = n, there exists an exact sequence where P 0 , . . ., P n−1 are projective modules and N ∈ PGF(R).Then, there exists another exact sequence 0 , where Q 0 , . . ., Q −n+1 are projective modules and G ∈ PGF(R).Since all kernels of the latter exact sequence are PGF-modules as well, it follows from [26, Corollary 4.5] that the exact sequence remains exact after applying the functor Hom R ( , P ) for any projective module P .We conclude that there exists a morphism of complexes The unlabelled vertical arrows induce a quasi-isomorphism between the corresponding complexes and hence we may consider the associated mapping cone, which is an acyclic complex Note that G⊕P 0 is a PGF-module and the module K = ker π has projective dimension ≤ n−1.
In fact, our assumption that PGF-dim R M = n implies that the inequality pd , where the latter inclusion is a consequence of the inclusion PGF(R) ⊆ GProj(R), we conclude that π is indeed a PGF(R)-precover of M.
Corollary 3.2.If M is a module with PGF-dim R M ≤ 1, then the following conditions are equivalent: R (M, P ) = 0 for any projective module P .Proof.The implication (i)→(ii) follows from [26,Theorem 4.4], whereas the implication (ii)→(iii) is obvious.In order to prove that (iii)→(i), we use Proposition 3.1 and note that the hypothesis PGF-dim R M ≤ 1 implies the existence of a short exact sequence where P is projective and G ∈ PGF(R).By our assumption, the group Ext 1 R (M, P ) is trivial and hence the exact sequence splits.It follows that M is a direct summand of G. Since the class PGF(R) is closed under direct summands, we conclude that M ∈ PGF(R) as well.The case where n = 0 is trivial.Assume that n > 0 and let K be the kernel of the map G 0 −→ M, so that there is a short exact sequence Since G 0 ∈ PGF(R), the group Ext i R (G 0 , P ) is trivial and hence Ext i R (K, P ) = Ext i+1 R (M, P ) is also trivial for all i > 0 and all projective modules P .The module K admits a PGF-resolution of length n − 1 and our induction hypothesis implies that K ∈ PGF(R).Therefore, it follows that PGF-dim R M ≤ 1.Since Ext 1 R (M, P ) = 0 for any projective module P , we finish the proof by invoking Corollary 3.2.
In view of Proposition 2.4(iii), the existence of a short exact sequence as in the statement of Proposition 3.1 is equivalent to the finiteness of the PGF-dimension of M. In fact, we may complement this assertion and prove the following result.Here, condition (iii) is analogous to [10, Lemma 2.17] (see also [25,Lemma 1.9]) and conditions (iv) and (v) are inspired by the Remark following [26,Theorem 4.11].
Theorem 3.4.The following conditions are equivalent for a module M and a non-negative integer n: (ii) There exists a short exact sequence where G is a PGF-module and pd R K = n − 1.If n = 0, this is understood to mean K = 0.If n = 1, we also require that the exact sequence be non-split.(iii) There exists a short exact sequence where G is a PGF-module and pd R K = n.
(iv) There exists a projective module P , such that the module which remains exact after applying the functor Hom R ( , Q) for any module where G is a PGF-module and pd R K = n.
(v) There exists a PGF-module P , such that the module where G is a PGF-module and pd R K = n.If n = 1, we also require that the exact sequence remain exact after applying the functor Hom R ( , Q) for any projective module Q.

Proof. (i)→(ii):
The existence of the short exact sequence follows from Proposition 3.1.If n = 1, then the exact sequence cannot split.(Indeed, if the short exact sequence were split, then M would be a direct summand of the PGF-module G and hence M would be itself a PGF-module; this is absurd, since PGF-dim R M = 1.) (ii)→(iii): Consider a short exact sequence as in (ii).Since G ∈ PGF(R), there exists a short exact sequence 0 where P is a projective module and G ′ ∈ PGF(R).By considering the pushout of that short exact sequence along the given epimorphism G −→ M, we obtain a commutative diagram with exact rows and columns We claim that the rightmost vertical exact sequence is of the required type.Indeed, if n = 0, then K = 0 and hence K ′ = P is a projective module.If n = 1, then K is projective and the monomorphism K −→ P is not split.(Indeed, if that monomorphism were split, then the monomorphism K −→ G would be split as well, contradicting our assumption.)It follows that the module K ′ = coker (K −→ P ) is not projective and hence pd R (K, ) = 0; it follows that pd R K ′ = n.(iii)→(iv): Consider a short exact sequence as in (iii) and let 0 −→ K ′ −→ P −→ K −→ 0 be a short exact sequence, where P is a projective module and pd R K ′ = n − 1. (If n = 0, then K is projective and we choose P = K and K ′ = 0.) By considering the pullback of that short exact sequence along the given monomorphism M −→ K, we obtain a commutative diagram with exact rows and columns 0 0 Since the class PGF(R) is projectively resolving, the horizontal short exact sequence in the middle shows that G ′ is a PGF-module.Then, the definition of the pullback and the surjectivity of the the map P −→ K imply that there is a short exact sequence In order to show that this short exact sequence has the required additional property, we note that for any module Q ∈ PGF(R) ⊥ the two horizontal short exact sequences in the diagram above induce a commutative diagram of abelian groups with exact rows It follows readily that there is an induced sequence of abelian groups (iv)→(v): This is immediate, since projective modules are contained in both classes PGF(R) and PGF(R) ⊥ .
(v)→(i): Consider an exact sequence as in (v) and note that Proposition 2.3 implies that PGF-dim R M ′ = PGF-dim R M. Therefore, it suffices to prove that PGF-dim R M ′ = n.Since G is a PGF-module and PGF-dim R K ≤ pd R K = n, we may invoke Proposition 2.4(i) and conclude that PGF-dim R M ′ ≤ n.It remains to show that the latter inequality cannot be strict.Indeed, let us assume that n ≥ 1 and PGF-dim R M ′ ≤ n − 1.
If n = 1, then M ′ is a PGF-module and hence PGF-dim R K ≤ 1.Since the short exact sequence is assumed to remain exact after applying the functor Hom R ( , Q) for any projective module Q and M ′ ∈ PGF(R) ⊆ ⊥ Proj(R), it follows that the abelian group Ext1 R (K, Q) is trivial for any projective module Q.Then, Corollary 3.2 implies that K ∈ PGF(R); in particular, K ∈ GProj(R).As shown in [22,Proposition 2.27], any Gorenstein projective module of finite projective dimension is necessarily projective.We therefore conclude that the module K is projective. 1 This is absurd, since pd R K = 1.
We now consider the case where n > 1.Since the PGF-module G is Gorenstein projective, the functor Ext n−1 R (G, ) vanishes on projective modules.Since PGF-dim R M ′ ≤ n − 1 and PGF(R) ⊆ GProj(R), we also have Gpd R M ′ ≤ n − 1.Therefore, [22,Theorem 2.20] implies that the functor Ext n R (M ′ , ) vanishes on projective modules as well.It follows that the functor Ext n R (K, ) vanishes on projective modules.This contradicts our assumption that pd R K = n; indeed, if we consider a projective resolution P * −→ K −→ 0 of length n, then the monomorphism P n −→ P n−1 is not split and hence Ext n R (K, P n ) = 0.
Remarks 3.5.(i) In the case where n = 1, it is necessary to impose some restrictions on the short exact sequences appearing in Theorem 3.4(ii),(v).Indeed, if P is any non-zero projective module and M ∈ PGF(R), then the (split) short exact sequence is of the type appearing in Theorem 3.4(ii), but PGF-dim R M = 0 = 1.On the other hand, if K is a module with pd R K = 1, then a projective resolution of K provides an exact sequence of the type appearing in Theorem 3.4(v), but PGF-dim R P 0 = 0 = 1.
(ii) It is clear from the proof of Theorem 3.4 that the analogues of conditions (iv) and (v) therein for Gorenstein projective modules are equivalent to the analogues of conditions (i), (ii) and (iii) for such modules, thereby complementing the characterizations of the finiteness of the Gorenstein projective dimension given in [22,Theorem 2.10] and [10,Lemma 2.17].
The next result is a characterization of modules of finite PGF-dimension, that parallels the characterization of modules of finite Gorenstein projective dimension in [22,Theorem 2.20].
Proposition 3.6.The following conditions are equivalent for a module M of finite PGFdimension and a non-negative integer n: and fix a flat module F .Since the functors Ext j R ( , F ) vanish on the class of PGF-modules for all j > 0 (cf.[26, Corollary 4.5]), we may deduce the desired vanishing by dimension shifting. (ii)→(i Since the modules M, G 0 , . . ., G n−1 are of finite PGF-dimension, an iterated application of Proposition 2.4(ii) shows that the module K has finite PGF-dimension as well.On the other hand, our hypothesis and the dimension shifting argument employed in the proof of the implication (i)→(ii) above show that the functors Ext i R (K, ) vanish on flat modules for all i > 0. Invoking Corollary 3.3, we conclude that K ∈ PGF(R), as needed.
The implication (ii)→(iii) follows by induction on the flat dimension of the module F , whereas the implication (iii)→(ii) is immediate.
Finally, the implications (i)↔(ii)'↔(iii)' that involve projective modules can be proved by using exactly the same arguments as those used above for the implications that involve flat modules.
An immediate consequence of the characterization above is that the PGF-dimension is a refinement of the ordinary projective dimension, whereas the Gorenstein projective dimension is a refinement of the PGF-dimension.
) vanish for all i > n and Ext i R (M, P ) = 0 for a suitable projective module P .Since PGF-dim R M ≤ n, the equality PGF-dim R M = n follows from Proposition 3.6.
(ii Since PGF(R) ⊆ GProj(R), it follows from [22,Theorem 2.20] that Ext 1 R (M, P ) = 0 whenever M ∈ PGF(R) and P ∈ Proj(R); this is precisely the assertion of Proposition 3.6(iii)' in the case where n = 0 therein.In fact, this vanishing provides a characterization of PGF-modules and modules of finite projective dimension, if we restrict to modules of finite PGF-dimension.Proposition 3.8.Let N be a module of finite PGF-dimension.Then: Proof.(i) As we noted above, the Ext-group is trivial if N ∈ PGF(R).Conversely, assume that N is a module of finite PGF-dimension contained in ⊥ Proj(R).Proposition 3.1 implies the existence of a short exact sequence where G ∈ PGF(R) and K ∈ Proj(R).In view of our assumption on N, this sequence splits and hence N is a direct summand of the PGF-module G. Since the class PGF(R) is closed under direct summands, we conclude that N is a PGF-module.
(ii) As we noted above, the Ext-group is trivial if N ∈ Proj(R).Conversely, assume that N is module of finite PGF-dimension contained in PGF(R) ⊥ .Then, Theorem 3.4(iii) implies the existence of a short exact sequence where G ∈ PGF(R) and K ∈ Proj(R).In view of our assumption on N, this sequence splits and hence N is a direct summand of K.Then, pd R N ≤ pd R K < ∞ and hence N ∈ Proj(R), as needed.
We now examine the special case of Gorenstein flat modules and show that the values of their PGF-dimension are controlled by the values of the projective dimension of flat modules.We let splf R be the supremum of the projective lengths (dimensions) of flat modules.Proposition 3.9.We have an equality sup{PGF-dim R M : In order to prove the reverse inequality, it suffices to assume that splf R < ∞, so that any flat module has finite projective dimension.If M is any Gorenstein flat module, then [26,Theorem 4.11] implies that there exists a short exact sequence where F is flat and G ∈ PGF(R).Since F has finite projective dimension, Theorem 3.4(iii) implies that PGF-dim R M = pd R F ≤ splf R. We conclude that s ≤ splf R, as needed.
Considering the projective dimension of direct sums of flat modules, it is easily seen that Flat(R) ⊆ Proj(R) if and only if splf R < ∞.In the same way, we may consider the PGFdimension of direct sums of Gorenstein flat modules (cf.Proposition 2.3) and conclude that GFlat(R) ⊆ PGF(R) if and only if s < ∞.Therefore, the final statement in the Proposition follows from the equality s = splf R.
We may complement the characterization of the finiteness of PGF-dimension given in Theorem 3.4, in the case of a Gorenstein flat module M, by requiring that the module K that appears in assertions (ii), (iii), (iv) and (v) therein be also flat.To that end, we note that any Gorenstein flat module of finite projective dimension is necessarily flat.Indeed, such a module must also have finite flat dimension and its flatness follows then from [6, §2]; see also [15,Remark 1.5].
Proposition 3.10.The following conditions are equivalent for a Gorenstein flat module M and a non-negative integer n: (ii) There exists a short exact sequence where G is a PGF-module and K is a flat module with pd R K = n − 1.If n = 0, this is understood to mean K = 0.If n = 1, we also require that the exact sequence be non-split.
(iii) There exists a short exact sequence where G is a PGF-module and K is a flat module with pd R K = n.
(iv) There exists a projective module P , such that the module M ′ = M ⊕P fits into an exact sequence 0 −→ G −→ M ′ −→ K −→ 0, which remains exact after applying the functor Hom R ( , Q) for any module Q ∈ PGF(R) ⊥ , where G is a PGF-module and K is a flat module with pd R K = n.
(v) There exists a PGF-module P , such that the module where G is a PGF-module and K is a flat module with pd R K = n.If n = 1, we also require that the exact sequence remain exact after applying the functor Hom R ( , Q) for any projective module Q. Proof.We proceed as in the proof of Theorem 3.4, showing that (i)→(ii)→(iii)→(iv)→(v)→(i).Since M is Gorenstein flat, PGF(R) ⊆ GFlat(R) and the class of Gorenstein flat modules is projectively resolving (cf.[26,Corollary 4.12]), the module K appearing in (ii) and (iii) is a Gorenstein flat module of finite projective dimension; as noted above, this forces K to be flat.We also note that the argument in the proof of the implication (iii)→(iv) in Theorem 3.4 provides a short exact sequence as in (iv) with K being the same module K that appears in (iii).

Hovey triples on PGF(R) and GFlat(R)
We shall now relate the results obtained in the previous section to the theory of exact model structures and describe a hereditary Hovey triple in the exact category PGF(R) of modules of finite PGF-dimension, which is such that the homotopy category of the associated exact model structure is equivalent as a triangulated category to the stable category of PGF-modules.We shall also describe the stable category of PGF-modules, up to triangulated equivalence, as the homotopy category of the exact model structure associated with a similar Hovey triple in the exact category GFlat(R) of Gorenstein flat modules.
It is easily seen that PGF(R) is an exact Frobenius category with projective-injective objects given by the projective modules.The proof of the latter claim is essentially identical to the proof of the corresponding claim for the class of Gorenstein projective modules, which can be found for instance in [13,Proposition 2.2].
The category PGF(R) of modules of finite PGF-dimension is an extension closed subcategory of the abelian category of all modules (cf.Proposition 2.4(i)), which is also closed under direct summands (cf.Proposition 2.3).Therefore, PGF(R) is an idempotent complete exact additive category [8].The following result is an analogue of [13,Theorem 3.7].The idea is that in order to realize the stable category of PGF-modules as the homotopy category of a Quillen model structure, it suffices to work on the subcategory PGF(R) of modules of finite PGF-dimension.We note that the class Proj(R) of modules of finite projective dimension is closed under direct summands and has the 2-out-of-3 property for short exact sequences.
Theorem 4.1.The triple PGF(R), Proj(R), PGF(R) is a hereditary Hovey triple in the idempotent complete exact category PGF(R).The homotopy category of the associated exact model structure is equivalent, as a triangulated category, to the stable category of PGF-modules.Proof.We need to prove that the pairs PGF(R), Proj(R) ∩ PGF(R) and PGF(R) ∩ Proj(R), PGF(R) are complete and hereditary cotorsion pairs in the exact category PGF(R).Since any PGFmodule is Gorenstein projective, we conclude that where the latter equality follows from [22,Proposition 2.27].On the other hand, projective modules are contained in both classes PGF(R) and Proj(R) and hence PGF(R) ∩ Proj(R) = Proj(R). 2Thus, the two pairs displayed above become PGF(R), Proj(R) and Proj(R), PGF(R) .
We begin by considering the pair PGF(R), Proj(R) and note that Proposition 3.8 states precisely that this is indeed a cotorsion pair in PGF(R).Theorem 3.4 provides the approximations referring to completeness, whereas Proposition 3.6(iii)', applied to the case where n = 0, shows that the cotorsion pair is hereditary.
We now consider the pair Proj(R), PGF(R) and note that PGF(R) is obviously the right orthogonal of Proj(R) within PGF(R).In order to prove that Proj(R) is the left orthogonal of PGF(R) within PGF(R), we let M be a module of finite PGF-dimension which is also contained in ⊥ PGF(R) and consider a short exact sequence 0 −→ M ′ −→ P −→ M −→ 0, where P is projective.Then, Proposition 2.4(ii) implies that M ′ has also finite PGF-dimension and hence Ext 1 R (M, M ′ ) = 0.In particular, the exact sequence above splits.It follows that M is a direct summand of P and hence M is projective.The cotorsion pair Proj(R), PGF(R) in PGF(R) is hereditary (since all higher Ext's with a projective first argument vanish) and complete (since the class PGF(R) is projectively resolving). The where G is a PGF-module and F is flat; cf.[26,Theorem 4.11(2)].In view of our assumption on N, this short sequence splits and hence N is a direct summand of the PGF-module G. Since the class PGF(R) is closed under direct summands, we conclude that N ∈ PGF(R).
(ii) As we noted above, the Ext-group is trivial if N is flat.Conversely, assume that N is a Gorenstein flat module contained in PGF(R) ⊥ .Then, there exists a short exact sequence where G is a PGF-module and F is flat; cf.[26,Theorem 4.11(4)].In view of our assumption on N, this short sequence splits and hence N is a direct summand of the flat module F .Therefore, N is flat.
We note that the class Flat(R) of flat modules is closed under direct summands and has the 2-out-of-3 property within the class of Gorenstein flat modules.Of course, Flat(R) is closed under extensions and kernels of epimorphisms.Moreover, if the cokernel of a monomorphism between flat modules is Gorenstein flat, then that cokernel is necessarily flat. 3The proof of the following result is very similar to the proof of Theorem 4.1.
We begin by considering the pair (PGF(R), Flat(R)) and note that Proposition 4.2 states precisely that this is indeed a cotorsion pair in the exact category GFlat(R).Completeness of the cotorsion pair follows from the exact sequences in [26,Theorem 4.11(2),( 4)], whereas Proposition 3.6(ii), applied to the case where n = 0, shows that the cotorsion pair is hereditary.
We now consider the pair (Proj(R), GFlat(R)) and note that GFlat(R) is obviously the right orthogonal of Proj(R) within GFlat(R).In order to prove that Proj(R) is the left orthogonal of GFlat(R) within GFlat(R), we let M be a Gorenstein flat module which is also contained in ⊥ GFlat(R) and consider a short exact sequence 0 −→ M ′ −→ P −→ M −→ 0, where P is projective.Since the class GFlat(R) is projectively resolving (cf.[26, Corollary 4.12]), we deduce that M ′ is also Gorenstein flat.Therefore, Ext 1 R (M, M ′ ) = 0 and the exact sequence above splits.It follows that M is a direct summand of P and hence M is projective.The cotorsion pair (Proj(R), GFlat(R)) in GFlat(R) is hereditary (since all higher Ext's with a projective first argument vanish) and complete (since the class of Gorenstein flat modules is projectively resolving). The

The finiteness of the PGF global dimension
In this section, we characterize those rings over which all modules have finite PGF-dimension, in terms of classical homological invariants.As a consequence of this description, we generalize a result by Jensen [24] (on commutative Noetherian rings) and another result by Gedrich and Gruenberg [17] (on group rings of groups over a commutative Noetherian coefficient ring).We define the (left) PGF global dimension PGF-gl.dimR of the ring R, by letting PGF-gl.dimR = sup{PGF-dim R M : M a left R-module}.
(ii)→(iii): Since PGF(R) is contained in both classes GProj(R) and GFlat(R), our hypothesis implies that any module M has both finite Gorenstein projective dimension and finite Gorenstein flat dimension.(In fact, both Gpd R M and Gfd R M are bounded by PGF-dim R M < ∞.)Then, assertion (iii) follows from the characterization of the finiteness of the Gorenstein global dimension and the Gorenstein weak global dimension of R; cf.§1.II.
(iii)→(iv): This is straightforward.(iv)→(ii): Assume that spli R = n < ∞ and fix a module M.Then, the construction by Gedrich and Gruenberg in [17, §4] provides us with an acyclic complex of projective modules shows that PGF-dim R M ≤ n < ∞, as needed.The final claim in the statement of the Theorem is an immediate consequence of Corollary 3.7(ii), which implies that PGF-gl.dimR = Ggl.dimR, if PGF-gl.dimR is finite.
As an immediate consequence of the equivalence between assertions (iii) and (iv) in Theorem 5.1 above, we obtain the following result.
Corollary 5.2.Let R be a ring, such that both invariants spli R and sfli R op are finite.Then, silp R = spli R.
We may obtain a left-right symmetric assertion, as follows.
Proposition 5.3.Let R be a ring, such that both invariants spli R and spli R op are finite.Then, we have silp R = spli R and silp R op = spli R op .Proof.Since projective (left or right) modules are flat, we have sfli R ≤ spli R < ∞ and sfli R op ≤ spli R op < ∞.
Then, the result follows by applying Corollary 5.2 for the ring R and its opposite R op .Corollary 5.4.If R is a ring which is isomorphic with its opposite R op , then silp R ≤ spli R with equality if spli R < ∞.Proof.The inequality is obvious if spli R = ∞ and hence it suffices to consider the case where spli R < ∞.Then, spli R op = spli R is also finite and we may invoke Proposition 5.3.
We recall that a ring R is called left (resp.right) ℵ 0 -Noetherian if any left (resp.right) ideal of R is countably generated.For example, countable rings and countably generated algebras over fields are both left and right ℵ 0 -Noetherian.Remarks 5.5.(i) Let k be a commutative ring, G a group and R = kG the associated group algebra.Then, R is isomorphic with its opposite R op and hence Corollary 5.4 implies that silp R ≤ spli R. In the special case where the coefficient ring k is Noetherian of finite selfinjective dimension, this inequality was proved by Gedrich and Gruenberg in [17, Theorem 2.4], using the Hopficity of the group algebra R.
(ii) Let k be a commutative ℵ 0 -Noetherian ring, G a group and R = kG the associated group algebra.Then, we may invoke [14, Proposition 4.3] and conclude that the inequality in

1 A
(X, B) = 0 for all B ∈ B. Analogously, the right orthogonal B ⊥ of B is the class consisting of those objects Y ∈ A, which are such that Ext 1 A (B, Y ) = 0 for all B ∈ B. If C, D are two subclasses of A, then the pair (C, D) is a cotorsion pair in A (cf. [16]) if C = ⊥ D and C ⊥ = D.The cotorsion pair is called hereditary if Ext i
0 for all i > n and any flat module F .(ii)' Ext i R (M, P ) = 0 for all i > n and any projective module P .(iii) Ext i R (M, F ) = 0 for all i > n and any module F of finite flat dimension.(iii)' Ext i R (M, P ) = 0 for all i > n and any module P of finite projective dimension.Proof.(i)→(ii): We consider a PGF-resolution of length n 0 follows from Proposition 3.6 and [22, Theorem 2.20].

( 3 )
• • • −→ P n+1 −→ P n −→ Q n−1 −→ Q n−2 −→ • • • ,which coincides in degrees ≥ n with a projective resolution• • • −→ P n+1 −→ P n −→ P n−1 −→ • • • −→ P 0 −→ M −→ 0 of M.Since the acyclic complex (3) consists of projective (and hence flat) modules, it remains acyclic by applying the functor L ⊗ R for any right module L of finite flat dimension; this follows easily by induction on the flat dimension of L. In particular, our assumption about the finiteness of sfli R op implies that the complex (3) remains acyclic by applying the functor I ⊗ R for any injective right module I. Therefore, the module K = coker (P n+1 −→ P n ) is a PGF-module.Then, the exact sequence0 −→ K −→ P n−1 −→ • • • −→ P 0 −→ M −→ 0 [26, of the statement follows from [18, Proposition 4.4 and Corollary 4.8].The category GFlat(R) of Gorenstein flat modules is also closed under extensions and direct summands; this follows from[26, Corollary 4.12].Hence, GFlat(R) is an idempotent complete exact category as well.As shown in[26, Theorem 4.4], the group Ext 1 R (M, F ) is trivial whenever M is a PGF-module and F is flat.This vanishing actually provides a characterization of PGF-modules and flat modules, if we restrict to Gorenstein flat modules.Proposition 4.2.Let N be a Gorenstein flat module.Then: (i) N ∈ PGF(R) if and only if Ext 1 R (N, F ) = 0 for any flat module F .(ii) N is flat if and only if Ext 1R (M, N) = 0 for any M ∈ PGF(R).Proof.(i) As we noted above, the Ext-group is trivial if N is a PGF-module.Conversely, assume that N is a Gorenstein flat module contained in ⊥ Flat(R).Then, there exists a short exact sequence 0 final statement follows from [18, Proposition 4.4 and Corollary 4.8].
[19,rk 4.4.Another model for the stable category of PGF-modules can be obtained from the Hovey triple (PGF(R), PGF(R) ⊥ , R-Mod) on the category R-Mod of all modules; cf.[26, Theorem 4.9] and[19, Proposition 37].A possible advantage of the Hovey triples presented in this section is that the classes of modules that are involved herein admit a more manageable description.