Resolutions as directed colimits

A general principle suggests that"anything flat is a directed colimit of countably presentable flats". In this paper, we consider resolutions and coresolutions of modules over a countably coherent ring $R$ (e.g., any coherent ring or any countably Noetherian ring). We show that any $R$-module of flat dimension $n$ is a directed colimit of countably presentable $R$-modules of flat dimension at most $n$, and any flatly coresolved $R$-module is a directed colimit of countably presentable flatly coresolved $R$-modules. If $R$ is a countably coherent ring with a dualizing complex, then any F-totally acyclic complex of flat $R$-modules is a directed colimit of F-totally acyclic complexes of countably presentable flat $R$-modules. The proofs are applications of an even more general category-theoretic principle going back to an unpublished 1977 preprint of Ulmer. Our proof of the assertion that every Gorenstein-flat module over a countably coherent ring is a directed colimit of countably presentable Gorenstein-flat modules uses a different technique, based on results of Saroch and Stovicek. We also discuss totally acyclic complexes of injectives and Gorenstein-injective modules, obtaining various cardinality estimates for the accessibility rank under various assumptions.


Introduction
The classical Govorov-Lazard theorem [16,27] tells that all flat modules are directed colimits of finitely generated projective modules.What about modules of flat dimension n, for a fixed n ≥ 1?
Any module of projective dimension 1, over an arbitrary ring, is a directed colimit of finitely presentable modules of projective dimension at most 1.Indeed, such a module is a cokernel of an injective morphism of free modules, and such a morphism is a directed union of injective morphisms of finitely generated free modules.This fact is mentioned, and a generalization to projective dimension n is discussed, in the paper [5, Section 4]; see specifically [5,Theorem 4.6].
According to [3,Theorem 3.5], any module of flat dimension 1 over a commutative integral domain is a directed colimit of finitely presentable modules of projective dimension at most 1.Nevertheless, over commutative Noetherian local rings already, a module of flat dimension 1 need not be a directed colimit of (finitely generated) modules of projective dimension at most 1 [5,Example 8.5], [21,Theorem B].A further discussion can be found in the book [15,Section 9.2].
In this paper we show that, over any right countably coherent ring, any right module of flat dimension n is a directed colimit of countably presentable modules of flat dimension at most n (see Corollary 5.2).Here a ring R is said to be right countably coherent if every finitely generated right ideal in R is countably presentable as a right R-module.In other words, in the language of [1, Chapter 2], the category of right R-modules of flat dimension ≤ n is ℵ 1 -accessible, and its ℵ 1 -presentable objects are the countably presentable modules of flat dimension ≤ n (i.e., countably presentable as objects of the category of all modules).Why is this interesting?
From our perspective, the significance of flat modules with small cardinalities of generators and relations lies in the fact that such modules have finite projective dimensions.In fact, any countably presentable flat module has projective dimension at most 1 [15,Corollary 2.23].More generally, any flat module with less than ℵ m generators and relations has projective dimension at most m [24, Proposition 5.3] (see also our Corollary 2.4).
In this context, a natural question is whether every module of projective dimension n is a direct summand of a module filtered by modules of projective dimension at most n admitting a resolution by finitely generated projective modules.The answer is negative for n = 1 and commutative Noetherian local rings already [5,Theorem 8.6 or Lemma 9.1], [20,Theorem 3.14], [21,Theorem A].So a module of projective dimension 1 need not be a direct summand of a module filtered by finitely generated modules of projective dimension 1.
On the other hand, over a right countably Noetherian ring, any right module of projective dimension n is filtered by countably generated modules of projective dimension at most n [39, Corollaire II.3.2.5], [2,Proposition 4.1].Here a ring R is called right countably Noetherian if every right ideal in R is countably generated.A far-reaching generalization of this result can be found in [41,Theorem 3.4].
An approach to accessibility based on deconstructibility is possible, but its results may be suboptimal.Let κ be a regular cardinal and S be a set of κ-presentable modules over a ring R. Then it follows from the Hill lemma that every module filtered by S is a κ-directed union of κ-presentable modules filtered by S. Arguing in this way, and using the mentioned result from [41] together with purification considerations [6, Lemma 1 and Proposition 2], one can show that any R-module of flat dimension n is a directed colimit of κ-presentable R-modules of flat dimension at most n, where κ is an uncountable regular cardinal greater than the cardinality of R.This is not as good as the result of our Corollary 5.2, which gives κ = ℵ 1 for modules over countably coherent rings.
The main techniques presented in this paper are category-theoretic in nature, based on a general principle going back to an unpublished 1977 preprint of Ulmer [44] and exemplified by the Pseudopullback Theorem of Raptis and Rosický [7, Proposition 3.1], [38,Theorem 2.2].Another exposition can be found in the recent preprint [34] by the present author.
With these methods, we can and do treat coresolutions on par with resolutions.In particular, one says that an R-module M is flatly coresolved if there exists an exact sequence of R-modules 0 −→ M −→ F 0 −→ F 1 −→ F 2 −→ • • • with flat R-modules F i .We show that, over a countably right coherent ring R, any flatly coresolved right R-module is a directed colimit of countably presentable flatly coresolved right R-modules (see Corollary 6.2).
The latter result brings us closer to the presently popular subject of Gorenstein homological algebra.Approaching it with our category-theoretic methods, we show that, over a right countably coherent ring R with a dualizing complex, any F-totally acyclic complex of flat right R-modules (in the sense of [12,Section 2]) is an ℵ 1 -directed colimit of F-totally acyclic complexes of countably presentable flat modules (see Theorem 8.3).Consequently, any Gorenstein-flat right R-module is a directed colimit of countably presentable Gorenstein-flat modules.Furthermore, any countably presentable Gorenstein-flat right R-module is a direct summand of a (Gorenstein-flat) module admitting an F-totally acyclic two-sided resolution by countably presentable flat R-modules (Corollary 8.5).The same results apply to commutative Noetherian rings with at most countable spectrum (instead of a dualizing complex).
A more powerful approach is based on difficult results of the paper [40], or specifically [40,Theorems 4.9 and 4.11(4)].It allows to prove that, over any right countably coherent ring, any Gorenstein-flat right module is an ℵ 1 -directed colimit of countably presentable Gorenstein-flat modules (Theorem 10.2).
We also consider Gorenstein-injective modules over a left Noetherian ring R with a dualizing complex.Here we prove that, if the cardinality of R is smaller than κ, then any totally acyclic complex of injective left R-modules is a κ-directed colimit of totally acyclic complexes of injective modules of cardinality less than κ (see Theorem 15.1).Consequently, every Gorenstein-injective left R-module is a directed colimit of Gorenstein-injective modules of cardinality less than κ.Furthermore, any Gorenstein-injective left R-module of cardinality less than κ is a direct summand of a (Gorenstein-injective) module admitting a totally acyclic two-sided resolution by injective R-modules of cardinality less than κ (Corollary 15.5).
More generally, let us say that an R-module M is injectively resolved if there exists an exact sequence of R-modules We show that, over a left Noetherian ring R, any injectively resolved left R-module is a directed colimit of injectively resolved R-modules of cardinality not exceeding ℵ 0 plus the cardinality of R.
Let us say a few more words about the proofs.The main arguments in this paper are based on a very general category-theoretic principle going back to [44] and rediscovered in [34].The Pseudopullback Theorem [7, Proposition 3.1], [38,Theorem 2.2] is one of the instances of this principle, and it is essentially sufficient for our purposes.In full generality, the principle involves a regular cardinal κ and a smaller infinite cardinal λ < κ, and the claim is that the class of all κ-accessible categories with directed colimits of λ-indexed chains is stable under many category-theoretic constructions, including the pseudopullbacks, the inserters, and the equifiers.Moreover, there is a good control over the full subcategories of κ-presentable objects in the categories arising under such constructions; this is crucial for applications.
A general principle that anything flat is a directed colimit of countably presentable flats is mentioned in the abstract.This is illustrated by the results of the preprints [34, Sections 10.2, 10.4, and 10.5], [37,, [33,Sections 3,4,10,and 11], and [35,Sections 4,6,and 7].This is also confirmed by the results of the present paper, such as Corollaries 5.2 and 6.2, and Theorem 10.2.
Sections 1-3 contain various kinds of preliminary material and preliminary discussions.In Sections 4-6 we discuss modules with two-sided resolutions by modules from accessible classes and two particular cases, modules of bounded flat dimension and flatly coresolved modules.In the subsequent Sections 7-10 we study F-totally acyclic complexes and Gorenstein-flat modules.In the next Sections 11-12 we construct modules of bounded injective dimension and injectively resolved modules as directed colimits.The final Sections 13-16 are dedicated to accessibility properties of totally acyclic complexes of injectives and Gorenstein-injective modules.brought [40,Theorems 4.9 and 4.11(4)] to my attention, and suggested to include the topic of Gorenstein-injective modules.Lemma 13.1 and Corollary 13.2 are also due to Jan. Thanks are also due to an anonymous referee for careful reading of the manuscript and suggesting a number of small corrections, as well as the reference [24] in connection with Corollary 2.4.The author is supported by the GA ČR project 23-05148S and the Czech Academy of Sciences (RVO 67985840).

Accessible Categories and Acyclic Complexes
We use the book [1] as the main reference source on accessible categories.In particular, we refer to [1, Definition 1.4, Theorem and Corollary 1.5, Definition 1.13(1), and Remark 1.21] for the discussion of κ-directed posets vs. κ-filtered small categories and, accordingly, κ-directed vs. κ-filtered colimits.Here κ denotes a regular cardinal.Let us just mention that a poset Ξ is said to be κ-directed if every subset of cardinality less than κ has an upper bound in Ξ.
The category K is called κ-accessible [1, Definition 2.1] if there is a set of κ-presentable objects S ⊂ K such that every object of K is a κ-directed colimit of objects from S. If this is the case, then the κ-presentable objects of K are precisely all the retracts of the objects from S. A κ-accessible category K is called locally κ-presentable if all colimits exist in K [1, Definition 1.17 and Theorem 1.20].
Given a category K with κ-directed colimits and a class of objects T ⊂ K, we denote by lim − →(κ) T ⊂ K the class of all κ-directed colimits of objects from T. In the case of κ = ℵ 0 , we will simply write lim − → T instead of lim − →(ℵ0) T. The following proposition is essentially well-known.Proposition 1.1.Let κ be a regular cardinal and K be a κ-accessible category.Let T ⊂ K be a set of κ-presentable objects.Then the full subcategory lim − →(κ) T is κ-accessible, and the κ-presentable objects of lim − →(κ) T are precisely all the retracts of objects from T.
T if and only if, for every object S ∈ K <κ , any morphism S −→ K in K factorizes through an object from T.
Proof.In the particular case of finitely accessible additive categories, versions of this result were discussed in [28, Proposition 2.1], [9, Section 4.1], and [26,Proposition 5.11].(The terminology "finitely presented categories" was used in [9,26] for what we call finitely accessible categories.)The key step is to prove the "if" part of the last assertion; then the remaining arguments are easy.Proposition 1.2.Let κ be a regular cardinal, and let (K ξ ) ξ∈Ξ be a family of κ-accessible categories, indexed by a set Ξ of cardinality less than κ.Then the Cartesian product category K = ξ∈Ξ K ξ is also κ-accessible.The κ-presentable objects of K are precisely all the collections of objects (S ξ ∈ K ξ ) ξ∈Ξ such that the object S ξ is κ-presentable in K ξ for every ξ ∈ Ξ.
Proof.This is a corrected version of [1, proof of Proposition 2.67].We refer to [34, Proposition 2.1] for the details.
In the next two theorems, we consider a regular cardinal κ and a smaller infinite cardinal λ < κ (so κ is necessarily uncountable).The cardinal λ is viewed as an ordinal, and directed colimits of λ-indexed chains are considered.Here a λ-indexed Let K 1 , K 2 , and L be three categories, and let Φ 1 : K 1 −→ L and Φ 2 : K 2 −→ L be two functors.The pseudopullback C of the pair of functors Φ 1 and Φ 2 is defined as the category of triples (K 1 , K 2 , θ), where K 1 ∈ K 1 and K 2 ∈ K 2 are two objects, and Theorem 1.3.Let κ be a regular cardinal and λ < κ be a smaller infinite cardinal.Let K 1 , K 2 , and L be three κ-accessible categories where colimits of λ-indexed chains exist.Assume that two functors Φ 1 : K 1 −→ L and Φ 2 : K 2 −→ L preserve κ-directed colimits and colimits of λ-indexed chains, and take κ-presentable objects to κ-presentable objects.Then the pseudopullback C is a κ-accessible category.The κ-presentable objects of C are precisely all the triples (S 1 , S 2 , θ) ∈ C such that the object S 1 is κ-presentable in K 1 and the object S 2 is κ-presentable in K 2 .
Let K and L be two categories, and let Φ 1 , Φ 2 : K ⇒ L be two parallel functors.The isomorpher C of the pair of functors Φ 1 and Φ 2 is defined as the category of pairs (K, θ), where K ∈ K is an object and θ : Φ Theorem 1.4.Let κ be a regular cardinal and λ < κ be a smaller infinite cardinal.Let K and L be two κ-accessible categories where colimits of λ-indexed chains exist.Assume that two functors Φ 1 and Φ 2 : K ⇒ L preserve κ-directed colimits and colimits of λ-indexed chains, and take κ-presentable objects to κ-presentable objects.Then the isomorpher C is a κ-accessible category.The κ-presentable objects of C are precisely all the pairs (S, θ) ∈ C such that the object S is κ-presentable in K.
Let R be an associative ring.We will denote by Mod-R the category of right R-modules and by R-Mod the category of left R-modules.The abelian category Mod-R is locally finitely presentable, hence locally κ-presentable for every regular cardinal κ [1, Remark 1.20].The κ-presentable objects of Mod-R are precisely all the R-modules with less than κ generators and less than κ relations, i. e., in other words, the cokernels of morphisms of free R-modules with less than κ generators.
For any additive category A, we denote by Com(A) the category of (unbounded) complexes in A.
Lemma 1.5.For any associative ring R, the following assertions hold: (a) The abelian category Com(Mod-R) of complexes of right R-modules is locally finitely presentable.Consequently, this category is locally κ-presentable for any regular cardinal κ.
(b) The finitely presentable objects of Com(Mod-R) are precisely all the bounded complexes of finitely presentable R-modules.
(c) For every uncountable regular cardinal κ, the κ-presentable objects of Com(Mod-R) are precisely all the (unbounded) complexes of κ-presentable R-modules.
Proof.Essentially, complexes are modules over a suitable "ring with many objects" (viz., the objects are indexed by the integers n ∈ Z).As usual, the results about modules over rings are applicable to modules over rings with many objects, and provide the assertions of the lemma.Alternatively, part (c) can be deduced by applying an additive version of [17,Theorem 1.2] or [34,Theorem 6.2].
The category of epimorphisms of right R-modules has epimorphisms of right R-modules L −→ M as objects and commutative squares Lemma 1.6.For any ring R and every regular cardinal κ, the category of epimorphisms of right R-modules is κ-accessible.The κ-presentable objects of this category are the epimorphisms of κ-presentable right R-modules.
Proof.This is a particular case of [34,Lemma 10.7].
We will say that an associative ring R is right < κ-coherent if, for every κ-presentable right R-module S, any submodule in S having less than κ generators is κ-presentable.Equivalently, R is right < κ-coherent if and only if every right ideal in R with less than κ generators is κ-presentable as a right R-module, and if and only if every finitely generated right ideal in R is κ-presentable.We will call right < ℵ 1 -coherent rings right countably coherent.
Corollary 1.7.For any regular cardinal κ and any right < κ-coherent ring R, the category of short exact sequences of right R-modules is κ-accessible.The κ-presentable objects of this category are the short exact sequences of κ-presentable right R-modules.
Proof.The first assertion is a restatement of the first assertion of Lemma 1.6 and holds for any ring R; but one needs the < κ-coherence assumption in order to obtain the second assertion of the corollary from the second assertion of the lemma (cf.[34,Corollary 10.13]).
Proposition 1.8.For any uncountable regular cardinal κ and any right < κ-coherent ring R, the category of (unbounded) acyclic complexes of right R-modules is κ-accessible.The κ-presentable objects of this category are the acyclic complexes of κ-presentable right R-modules.
Proof.The argument is similar to [34,proof of Corollary 10.14].Notice that an acyclic complex of modules C • is the same thing as a collection of short exact sequences of modules 0 −→ K n −→ C n −→ M n −→ 0, n ∈ Z, together with an isomorphism of modules M n ≃ K n+1 for every n ∈ Z.This observation allows to construct the category of acyclic complexes of R-modules from the category of short exact sequences of R-modules using Cartesian products and the isomorpher construction.
Specifically, put L = n∈Z Mod-R, and let K be the Cartesian product of the categories of short exact sequences of right R-modules taken over all n ∈ Z.Let Φ 1 : K −→ L be the functor assigning to a family of short exact sequences (0 → K n → L n → M n → 0) n∈Z the family of modules (M n ) n∈Z , and let Φ 2 be the functor assigning to the same family of short exact sequences the family of modules (K n+1 ) n∈Z .Then the isomorpher category C is equivalent to the desired category of acyclic complexes of right R-modules.
The categories K and L are κ-accessible by Corollary 1.7 and Proposition 1.2, and Theorem 1.4 is applicable (for λ = ℵ 0 ).The theorem tells that the category C is κ-accessible, and provides the desired description of its full subcategory of κ-presentable objects.

Modules of Small Presentability Rank as Small Directed Colimits
The aim of this section is to prove the following proposition, which is purported to complement the main results of this paper.Recall that an R-module is said to be κ-presentable in our (category-theoretic) terminology if it is the cokernel of a morphism of free R-modules with less than κ generators.Proposition 2.1.Let R be an associative ring and S be a set of finitely presentable right R-modules.Let C ∈ Mod-R be an ℵ m -presentable R-module belonging to lim − → S ⊂ Mod-R (where m ≥ 0 is an integer).Let D ∈ S ⊥ ≥1 be a right R-module such that Ext i R (S, D) = 0 for all S ∈ S and i > 0. Then Ext i R (C, D) = 0 for all i > m.
Lemma 2.2.Let R be an associative ring and S be a set of finitely presentable R-modules.Let κ be a regular cardinal and C be a κ-presentable R-module belonging to lim − → S. Then C is a direct summand of a directed colimit of modules from S indexed by a directed poset of cardinality less than κ.
Proof.Let T denote the class of all directed colimits of modules from S indexed by directed posets of cardinality less than κ.Then, following [1, proof of Theorem 2.11(iv) ⇒ (i) and Example 2.13(1)] (with λ = ℵ 0 and µ = κ), every module from lim − → S is a κ-directed colimit of modules from T. For a κ-presentable module C ∈ lim − → S, it follows that C is a direct summand of a module from T.
The following lemma can be found in [24, Théorème 4.2].
Lemma 2.3.Let Ξ be a directed poset, (S ξ ) ξ∈Ξ be a Ξ-indexed diagram of right R-modules, and D be a right R-module.Let lim ← − n ξ∈Ξ denote the derived functors of Ξ-indexed limit of abelian groups.Then there is a spectral sequence Proof.Let J • be an injective coresolution of the R-module D, and let B • be the barcomplex of the diagram (S ξ ) ξ∈Ξ .Consider the bicomplex of abelian groups ) for all n ≥ 0. On the other hand, for every q ≥ 0, one has H q (A p,• ) = Ext q R (B p , D).The complex H q (A •,• ) is the cobar-complex computing the derived functor of Ξ-indexed limit lim ← − * ξ∈Ξ Ext q R (S ξ , D), so one has Proof.Let S be the set of all finitely generated projective (or free) R-modules, D be an arbitrary R-module, and apply Proposition 2.1.

Deconstructibility and Directed Colimits
In this section we discuss the deconstructibility-based approach to accessibility.Both the deconstructible classes and the right Ext 1 -orthogonal classes to deconstructible classes are considered.
Let R be an associative ring, F be an R-module, and α be an ordinal.An α-indexed filtration of F is a family of submodules F β ⊂ F , indexed by the ordinals 0 ≤ β ≤ α, satisfying the following conditions: • F 0 = 0 and • one has F β = γ<β F γ for all limit ordinals β ≤ α.An R-module F endowed with an α-indexed filtration (F β ) 0≤β≤α is said to be filtered by the quotient modules F β+1 /F β , 0 ≤ β < α.Given a class of R-modules S ⊂ Mod-R, one says that an R-module F is filtered by S if there exists an ordinal α and an α-indexed filtration on F such that the quotient module F β+1 /F β is isomorphic to a module from S for every 0 ≤ β < α.
The class of all R-modules filtered by S is denoted by Proposition 3.1.Let R be an associative ring, κ be a regular cardinal, and S be a set of κ-presentable R-modules.Then any R-module filtered by S is a κ-directed colimit of κ-presentable R-modules filtered by S. In other words, for any κ-deconstructible class of modules F, all modules from F are κ-directed colimits (in fact, κ-directed unions) of κ-presentable modules from F.
Proof.This is a direct corollary of the Hill lemma [43,Theorem 6], [15,Theorem 7.10], [42,Theorem 2.1].Let F be a module filtered by modules from S. Then the Hill lemma provides a complete lattice of submodules in F such that every subset of cardinality less than κ in F is contained in a κ-presentable submodule of F belonging to this complete lattice, and every module belonging to the lattice is filtered by S. The family of all κ-presentable submodules of F belonging to the lattice is κ-directed by inclusion, and F is the directed union of these submodules.
An R-module is said to be < κ-generated if it has a set of generators of cardinality less than κ.An associative ring R is said to be right < κ-Noetherian if every submodule of a < κ-generated right R-module is < κ-generated, or equivalently, every right ideal in R is < κ-generated.Left < κ-Noetherian rings are defined similarly.The < ℵ 1 -generated modules are called countably generated, and the right < ℵ 1 -Noetherian rings are called right countably Noetherian.Proposition 3.2.Let κ be an uncountable regular cardinal and R be a right < κ-Noetherian associative ring.Then, for every integer m ≥ 0, the class of all right R-modules of projective dimension at most m is κ-deconstructible.
For any set X, we denote by |X| the cardinality of X.The successor cardinal of a cardinal ν is denoted by ν + .We will use the notation ρ = |R| + ℵ 0 for the minimal infinite cardinal greater than or equal to the cardinality of the ring R. Notice that, for any given cardinal ν ≥ ρ, an R-module is ν + -presentable if and only if it has cardinality at most ν.Given two cardinals ν and λ, one denotes by ν <λ the supremum of the cardinals ν µ taken over all the cardinals µ < λ.The following lemma is a generalization of [15,Lemma 10.5]; it can be found in [8, Lemma 4.1].Lemma 3.5.Let R be a ring; put ρ = |R| + ℵ 0 .Let M be an R-module, λ be a regular cardinal, and ν be a cardinal such that ρ ≤ ν and ν <λ = ν.Then for every subset X ⊂ M of cardinality at most ν there exists a submodule N ⊂ M of cardinality at most ν such that X ⊂ N and the following property holds: Every system of less than λ nonhomogeneous R-linear equations in less than λ variables, with parameters from N, has a solution in N provided that it has a solution in M.
Proof.The argument is similar to the one in [15,Lemma 10.5].Notice that one has λ ≤ ν.The submodule N ⊂ M is constructed as the union of an increasing chain of submodules (N i ⊂ M) 0≤i<λ , with the cardinality of N i not exceeding ν for every i.Let N 0 be the submodule spanned by X in M.
For a successor ordinal j = i + 1 < λ, we define N i+1 by adjoining to N one solution of every system of less than λ nonhomogeneous R-linear equations in less than λ variables with parameters from N i that has a solution in M. As such a system of equations has less than λ coefficients from R and less than λ parameters from N i , the cardinality of the set of all such systems of equations is not greater than ν.For a limit ordinal j < λ, we put N j = i<j N i .Now if a system of less than λ nonhomogeneous R-linear equations in less than λ variables has parameters in N = i<λ N i , then all these parameters belong to N i for some i < λ.Hence such system of equations has a solution in N i+1 .
An R-module S is said to be FP 2 [15, Section 5.2] if there exists an exact sequence P 2 −→ P 1 −→ P 0 −→ S −→ 0 with finitely generated projective R-modules P 0 , P 1 , and P 2 .So, over a right coherent ring, all finitely presentable right modules are FP 2 .Similarly, let us say that an R-module S is λ-P 2 if there exists an exact sequence P 2 −→ P 1 −→ P 0 −→ S −→ 0, where P 0 , P 1 , and P 2 are projective R-modules with less than λ generators.Proposition 3.6.Let R be a ring; put ρ = |R| + ℵ 0 .Let λ be a regular cardinal, ν be a cardinal such that ρ ≤ ν and ν <λ = ν, and S be a set of λ-P 2 right R-modules.Then the full subcategory S ⊥ 1 ⊂ Mod-R is closed under λ-directed colimits, and every module M ∈ S ⊥ 1 is a ν + -directed union of the ν + -directed poset of all the ν + -presentable submodules of M belonging to S ⊥ 1 .Consequently, the category S ⊥ 1 is ν + -accessible with directed colimits of λ-indexed chains, and the ν + -presentable objects of S ⊥ 1 are precisely all the ν + -presentable right R-modules belonging to S ⊥ 1 .
Proof.One can easily see that for any λ-P 2 module S there exists an exact sequence P 2 −→ P 1 −→ P 0 −→ S −→ 0 such that P 0 , P 1 , and P 2 are free modules with less than λ generators.Then, for any R-module M, the group Ext 1 R (S, M) can be computed as the middle cohomology group of the complex Hom R (P 0 , M) −→ Hom R (P 1 , M) −→ Hom R (P 2 , M).This proves the first assertion of the proposition.Furthermore, the property of an R-module M to belong to the class {S} ⊥ 1 ⊂ Mod-R is expressed by solvability of certain systems of less than λ nonhomogeneous R-linear equations in less than λ variables with parameters in M. Hence the second assertion of the proposition follows from Lemma 3.5.Notice that an R-module is ν + -presentable if and only if its cardinality does not exceed ν.
Corollary 3.7.Let λ be a regular cardinal and R be a right < λ-Noetherian ring.Let ν be a cardinal such that ρ = |R| + ℵ 0 ≤ ν and ν <λ = ν.Then the full subcategory of injective R-modules is closed under λ-directed colimits in Mod-R.The category of injective right R-modules is ν + -accessible with directed colimits of λ-indexed chains, and the ν + -presentable objects of this category are precisely all the injective R-modules that are ν + -presentable in Mod-R (i.e., injective R-modules of cardinality at most ν).Moreover, every injective right R-module is a ν + -directed union of ν + -presentable injective R-modules.
Proof.Take S to be the set of all cyclic right R-modules R/I (where I ranges over all the right ideals in R), and apply Proposition 3.6.

Two-Sided Resolutions by Accessible Classes
In this section we begin our discussion of accessibility of categories of (co)resolutions based on the techniques described in Section 1.We start with a general abstract formulation before passing to a finite/countable special case.Proposition 4.1.Let κ be a regular cardinal and λ < κ be a smaller infinite cardinal.Let R be a right < κ-coherent ring, and let (T n ) n∈Z be a sequence of classes of κ-presentable right R-modules, T n ⊂ (Mod-R) <κ .Assume that, for every n ∈ Z, the class of R-modules T n is closed under direct summands and the class of R-modules lim − →(κ) T n is closed under colimits of λ-indexed chains in Mod-R.Then the category C of all acyclic complexes of right R-modules T n for every n ∈ Z is κ-accessible.The κ-presentable objects of the category C are all the acyclic complexes of R-modules T • with T n ∈ T n for every n ∈ Z. Consequenly, every acyclic complex of R-modules C • with the terms Proof.This is an application of Theorem 1.3 together with Propositions 1.1, 1.2, and 1.8.Denote by K 1 the category of acyclic complexes of right R-modules from Proposition 1.8, and let K 2 = n∈Z (lim − →(κ) T n ) be the Cartesian product of the full subcategories lim − →(κ) T n ⊂ Mod-R, taken over all the integers n ∈ Z.Let L = n∈Z Mod-R be the Cartesian product of Z copies of the abelian category of right R-modules.Consider the following functors Φ 1 : The functor Φ 1 takes an acyclic complex of right R-modules A • to the collection of modules (A n ) n∈Z .The functor Φ 2 is the Cartesian product of the identity inclusion functors lim − →(κ) T n −→ Mod-R, taken over all n ∈ Z. Then the pseudopullback of the pair of functors Φ 1 and Φ 2 is equivalent to the desired category C.
The category K 1 is κ-accessible by Proposition 1.8.The categories K 2 and L are κ-accessible by Propositions 1.1 and 1.2.Theorem 1.3 is applicable; it tells that the category C is κ-accessible and provides the desired description of the full subcategory of κ-presentable objects in C. Theorem 4.2.Let R be a right countably coherent ring, and let (S n ) n∈Z be a sequence of sets of finitely presentable right R-modules, S n ⊂ (Mod-R) <ℵ 0 .Then the category C of all acyclic complexes of right R-modules C • with C n ∈ lim − → S n for every n ∈ Z is ℵ 1 -accessible.The ℵ 1 -presentable objects of the category C are all the acyclic complexes T • such that T n ∈ lim − → S n and T n is a countably presentable R-module for every n ∈ Z. Consequently, every acyclic complex of R-modules C • with the terms Proof.This is a particular case of Proposition 4.1.Put λ = ℵ 0 , κ = ℵ 1 , and denote by T n ⊂ Mod-R the class of all direct summands of countable directed colimits of modules from S n , for every n ∈ Z. Then lim T n , as per the proof of Lemma 2.2.The class lim − → S n is closed under directed colimits in Mod-R by Proposition 1.1.
Given an acyclic complex of R-modules C • , we denote by Z 0 (C • ) the module of degree 0 cocycles in Proof.Let us emphasize that there is no claim about closedness of the class M under ℵ 1 -directed colimits in this corollary.The assertion of the corollary follows directly from the last assertion of Theorem 4.2.One needs to observe that, in any acyclic complex of countably presentable modules, the modules of cocycles are also countably presentable.
Corollary 4.4.In the context of Corollary 4.3, let M be a countably presentable R-module belonging to the class M. Then M is a direct summand of an R-module N = Z 0 (T • ), where T • is an acyclic complex of countably presentable R-modules with the terms for some acyclic complexes of countably presentable modules T • ξ with the terms T n ξ ∈ lim − → S n .Now if M is countably presentable, then it follows that there exists ξ ∈ Ξ such that M is a direct summand of N ξ .

Modules of Finite Flat Dimension
Recall from the introduction that, already over commutative Noetherian local rings R of Krull dimension 2, a module of flat dimension 1 need not be a directed colimit of modules of projective dimension at most 1 [5, Example 8.5], [21,Theorem B].Any finitely generated R-module of flat dimension ≤ 1 would, of course, have projective dimension ≤ 1 (since finitely presentable flat modules are always projective).So a module of flat dimension 1 need not be a directed colimit of finitely generated flat modules of flat dimension at most 1.
By contrast, in this section we show that any module of flat dimension m is a directed colimit of countably presentable modules of flat dimension ≤ m.

presentable objects of this category are precisely all the exact sequences
Proof.This is a particular case of Theorem 4. Proof.It is easy to see that the full subcategory F m is closed under directed colimits in Mod-R (since the functor Tor preserves directed colimits).In view of Proposition 1.1, it suffices to check the last assertion of the corollary, which follows immediately from Theorem 5.1.Notice that the cokernel of any morphism of countably presentable modules is countably presentable.

Flatly Coresolved Modules
By a flat coresolution we mean an exact sequence , where F n are flat modules for all n ≥ 0. An R-module M is said to be flatly coresolved if there exists an exact sequence of R-modules Theorem 6.1.Let R be a right countably coherent ring.Then the category of flat coresolutions

Consequently, every flat coresolution
Proof.This is another particular case of Theorem 4.2.Take S n to be the set of all finitely generated projective (or free) right R-modules for all n ≥ 0, S −1 to be the set of all finitely presentable right R-modules, and S n = {0} for all n ≤ −2.Proof.This is a corollary of Theorem 6.1 and a particular case of Corollary 4.3.Let us emphasize once again that there is no claim about the class of flatly coresolved modules being closed under ℵ 1 -directed colimits in this corollary.To deduce the assertion of the corollary from Theorem 6.1, one needs to observe that the kernel of any morphism of countably presentable right modules over a right countably coherent ring R is countably presentable.Proof.This is a corollary of Theorem 6.1 and a particular case of Corollary 4.4.

Dualizing Complexes and F-Totally Acyclic Complexes
An acyclic complex of flat right R-modules F • is said to be F-totally acyclic [12, Section 2] if the complex of abelian groups F • ⊗ R J is acyclic for every injective left R-module J.A right R-module M is said to be Gorenstein-flat if there exists an F-totally acyclic complex of flat right R-modules F • such that M ≃ Z 0 (F • ) is its module of cocycles.
The aim of the following four sections, Sections 7-10, is to establish some accessibility properties of the categories of F-totally acyclic complexes and Gorenstein-flat modules, under suitable assumptions.In particular, the present section is purported to prepare ground for the next one.
We recall that a right R-module K is said to be fp-injective (or "absolutely pure") if Ext 1 R (T, K) = 0 for all finitely presentable right R-modules T .The definition of an fp-injective left module is similar.
Given an abelian category A, we denote by D b (A) the bounded derived category of A. A bounded complex of left R-modules K • is said to have finite fp-injective dimension if it is isomorphic, as an object of the derived category of left R-modules D b (R-Mod), to a bounded complex of fp-injective left R-modules.
Let R and S be two associative rings.Assume that the ring S is right coherent.In the following definition we list some properties of a complex of R-S-bimodules D • that are relevant for the purposes of this section.For comparison, see the definition of a dualizing complex in [32, Section 4].
We will say that a bounded complex of R-S-bimodules D • is a right dualizing complex for the rings R and S if it satisfies the following conditions: (i) the terms of the complex D • are fp-injective as right S-modules, and the whole complex D • has finite fp-injective dimension as a complex of left R-modules; (ii) the R-S-bimodules of cohomology of the complex D • are finitely presentable as right S-modules; ) is an isomorphism of graded rings.The following proposition is the main result of this section.It is our version of [25, Lemma 1.7].
Proposition 7.1.Let R be a ring, S be a right coherent ring, and D • be a right dualizing complex of R-S-bimodules.Then an acyclic complex of flat right R-modules F • is F-totally acyclic if and only if the complex of right S-modules F The proof of Proposition 7.1 is based on a sequence of lemmas.Lemma 7.2.Let F • be a complex of flat right R-modules and L • be a bounded acyclic complex of left R-modules.Then the complex of abelian groups Proof.The point is that the complex of abelian groups F n ⊗ R L • is acyclic for every n ∈ Z.The total complex of any bounded acyclic complex of complexes of abelian groups is acyclic.Lemma 7.3.Let F • be a complex of right R-modules such that the complex of abelian groups F • ⊗ R J is acyclic for every injective left R-module J. Then the complex of abelian groups F • ⊗ R K is acyclic for every fp-injective left R-module K.
Proof.This is [29, proof of Lemma 2.8 (1) ⇒ (2)].Let an fp-injective R-module K be a submodule in an injective R-module J; then K is a pure submodule in J (see, e. g., [15, Definition 2.6 and Lemma 2.19]).The point is that, whenever some left R-module M is a pure submodule in a left R-module L, and F • is a complex of right R-modules such that the complex of abelian groups F • ⊗ R L is acyclic, it follows that the complex of abelian groups F • ⊗ R M is acyclic as well.One can prove this by observing that Hom Corollary 7.4.Let F • be an F-totally acyclic complex of flat right R-modules and K • be a bounded complex of left R-modules.Assume that the complex K • has finite fp-injective dimension.Then the complex of abelian groups Proof.Combine Lemmas 7.2 and 7.3.Lemma 7.5.Let S be a ring, H • be an acyclic complex of right S-modules, and G • be a bounded complex of flat left S-modules.Then the complex of abelian groups Proof.The point is that the total complex of any bounded complex of acyclic complexes of abelian groups is acyclic.Lemma 7.6.Let R be a ring, S be a right coherent ring, E be an R-S-bimodule that is fp-injective as a right S-module, and J be an injective left R-module.Then the left S-module Hom R (E, J) is flat.
Proof.This is [32, Lemma 4.1(b)].The point is that the functor N −→ N ⊗ S Hom R (E, J) ≃ Hom R (Hom S (N, E), J) is exact on the abelian category of finitely presentable right S-modules N.
Lemma 7.7.Let R be a ring, S be a right coherent ring, and D • be a right dualizing complex of R-S-bimodules.Then the natural (evaluation) morphism of complexes of left R-modules D • ⊗ S Hom R (D • , J) −→ J is a quasi-isomorphism for every injective left R-module J.
Proof.This lemma only uses conditions (ii-iii) and the first part of condition (i) from the definition of a right dualizing complex above.See [32,  Proof of Proposition 7.1.The "only if" implication is provided by Corollary 7.4.To prove the "if", let J be an injective left R-module.By Lemma 7.7, we have a quasiisomorphism of bounded complexes of left R-modules D • ⊗ S Hom R (D • , J) −→ J.In view of Lemma 7.2, in order to show that the complex F • ⊗ R J is acyclic, it suffices to check that so is the complex F • ⊗ R D • ⊗ S Hom R (D • , J).Now, by assumption, the complex of right S-modules F • ⊗ R D • is acyclic.By Lemma 7.6, Hom R (D • , J) is a bounded complex of flat left S-modules.It remains to refer to Lemma 7.5.
Remark 7.8.The concepts of Gorenstein homological algebra become less complicated over a ring which is itself Gorenstein in a suitable sense.Notice that, for any acyclic complex of flat right R-modules F • and any left R-module G of finite flat dimension, the complex of abelian groups F • ⊗ R G is acyclic.This follows from Lemmas 7.2 and 7.5.Now if every injective left R-module has finite flat dimension, then every acyclic complex of flat right R-modules is F-totally acyclic, and every flatly coresolved right R-module is Gorenstein-flat.Assuming additionally that the ring R is right countably coherent, the results of Theorem 4.2, Theorem 6.1, Corollary 6.2, and Corollary 6.3 become directly applicable (essentially) as properties of F-totally acyclic complexes and Gorenstein-flat modules.

F-Totally Acyclic Complexes as Directed Colimits
We start with a very general result of Šaroch and Št'ovíček.Proof.These assertions are parts of [40,Corollary 4.12].
We will need to assume another condition in addition to the conditions (i-iii) in the definition of a right dualizing complex in Section 7. We will say that a right dualizing complex of R-S-bimodules D • has right countable type if (iv) there exist a right countably coherent ring S ′ and a bounded complex of R-S ′ -bimodules D • whose terms are countably presentable as right S ′ -modules such that the complexes D • and D • are isomorphic as objects of the bounded derived category of left R-modules D b (R-Mod).
Lemma 8.2.Let R and S be two rings, T be a right R-module, and E be an R-S-bimodule.Assume that the right R-module T is κ-presentable and the right S-module E is κ-presentable, for a given regular cardinal κ.Then the right S-module T ⊗ R E is also κ-presentable.
Proof.Represent T as the cokernel of a morphism of free R-modules with less than κ generators, and use the fact that the class of all κ-presentable S-modules is closed under colimits indexed by categories with less than κ morphisms [1, Proposition 1.16].
Theorem 8.3.Let R be a right countably coherent ring, S be a right coherent ring, and D • be a right dualizing complex of R-S-bimodules.Assume that D • is a right dualizing complex of right countable type.Then the category of F-totally acyclic complexes of flat right R-modules is ℵ 1 -accessible.The ℵ 1 -presentable objects of this category are precisely all the F-totally acyclic complexes of countably presentable flat right R-modules.Consequently, every F-totally acyclic complex of flat right R-modules is an ℵ 1 -directed colimit of F-totally acyclic complexes of countably presentable flat right R-modules.
Proof.This is an application of Theorem 1.3 together with Theorem 4.2 and Proposition 7.1.Notice first of all that the full subcategory of F-totally acyclic complexes of flat modules is obviously closed under directed colimits in the ambient abelian category of all complexes of modules.Now let D • be a bounded complex of S ′ -countably presentable R-S ′ -bimodules provided by the definition of a right dualizing complex of right countable type (item (iv)) above.It is clear from Lemma 7.2 that, for any complex of flat right R-modules F • , the complex of right S-modules F Denote by K 1 the category of acyclic complexes of flat right R-modules.Applying Theorem 4.2 for S n being the set of all finitely generated projective (or free) R-modules, for every n ∈ Z, we see that K 1 is an ℵ 1 -accessible category, and obtain a description of its ℵ 1 -presentable objects (cf.Theorem 6.1).
Furthermore, denote by K 2 the category of acyclic complexes of right S ′ -modules.Proposition 1.8 tells that K 2 is an ℵ 1 -accessible category and provides a description of its full subcategory of ℵ 1 -presentable objects.Finally, let L be the abelian category of arbitrary complexes of right S ′ -modules.Lemma 1.5(a,c) tells that L is a locally ℵ 1 -presentable category and describes its full subcategory of ℵ 1 -presentable objects.
Let the functor Φ 1 : K 1 −→ L take any acyclic complex of flat right R-modules F • to the complex of right S ′ -modules the identity inclusion of the category of acyclic complexes of right S ′ -modules into the category of all complexes of right S ′ -modules.Then the pseudopullback C of the two functors Φ 1 and Φ 2 is the category of all acyclic complexes of flat right R-modules F • for which the complex of right S ′ -modules F • ⊗ R D • is acyclic.By Proposition 7.1, and in view of the argument with Lemma 7.2 above, the category C is the desired category of F-totally acyclic complexes of flat right R-modules.
Finally, by Lemma 8.2, the functor Φ 1 takes ℵ 1 -presentable objects to ℵ 1 -presentable objects.All the other assumptions of Theorem 1.3 (for κ = ℵ 1 and λ = ℵ 0 ) are clearly satisfied.Theorem 1.3 tells that C is an ℵ 1 -accessible category, and provides the desired description of its full subcategory of ℵ 1 -presentable objects.
Corollary 8.4.Let R be a right countably coherent ring, S be a right coherent ring, and D • be a right dualizing complex of R-S-bimodules.Assume that D • is a right dualizing complex of right countable type.Then the category of Gorenstein-flat right R-modules GF is ℵ 1 -accessible.The ℵ 1 -presentable objects of GF are precisely all the Gorenstein-flat right R-modules that are countably presentable in Mod-R.So every Gorenstein-flat right R-module is an ℵ 1 -directed colimit of countably presentable Gorenstein-flat right R-modules.
Proof.The full subcategory GF ⊂ Mod-R is closed under directed colimits in Mod-R by Proposition 8.1(a).On the other hand, it is obvious from Theorem 8.3 that any Gorenstein-flat right R-module is an ℵ 1 -directed colimit of Gorenstein-flat right R-modules that are countably presentable in Mod-R.In view of Proposition 1.1, all the assertions of the corollary follow.ξ .Now if M is countably presentable, then it follows that M is a direct summand of N ξ for some ξ ∈ Ξ.

Commutative Noetherian Rings with Countable Spectrum
In this section, we prove results similar to those of Section 8, but under a different set of assumptions.Instead of existence of a dualizing complex, we assume the ring R to be commutative Noetherian with small cardinality of the spectrum.
For any R-module M, we denote by E R (M) an injective envelope of M. For a prime ideal p in a commutative ring R, we denote by R p the local ring (R \ p) −1 R. The following proposition is standard commutative algebra material.Theorem 9.2.Let κ be an uncountable regular cardinal and R be a commutative Noetherian ring with the cardinality of the spectrum | Spec R| < κ.Then the category of F-totally acyclic complexes of flat R-modules is κ-accessible.The κ-presentable objects of this category are precisely all the F-totally acyclic compexes of < κ-generated flat R-modules.Consequently, every F-totally acyclic complex of flat R-modules is a κ-directed colimit of F-totally acyclic complexes of < κ-generated flat R-modules.
Proof.This is an application of Theorem 1.3 together with Proposition 1.2 and Proposition 4.1.The argument is somewhat similar to the proof of Theorem 8.3.
Denote by K 1 the category of acyclic complexes of flat R-modules.Applying Proposition 4.1 for T n being the class of all < κ-generated flat R-modules for every n ∈ Z (cf.[34,Proposition 10.2]) and λ = ℵ 0 , we see that K 1 is a κ-accessible category and obtain a description of its κ-presentable objects.
Furthermore, denote by K 2 the Cartesian product of the categories of acyclic complexes of R p -modules, taken over all the prime ideals p ∈ Spec R. Proposition 1.8 together with Proposition 1.2 tell that K 2 is a κ-accessible category and provide a description of its full subcategory of κ-presentable objects.Finally, denote by L the Cartesian product of the abelian categories of arbitrary complexes of R p -modules, taken over all the spectrum points p ∈ Spec R. Lemma 1.5(a,c) together with Proposition 1.2 tell that L is a locally κ-presentable category and describe its full subcategory of κ-presentable objects.
Let the functor Φ 1 : K 1 −→ L take any acyclic complex of flat R-modules F • to the collection of all complexes of R p -modules F • ⊗ R E R (R/p).Let Φ 2 : K 2 −→ L be the Cartesian product of the identity inclusions of the categories of acyclic complexes of R p -modules into the respective categories of all complexes of R p -modules.Then the pseudopullback C of the two functors Φ 1 and Φ 2 is the category of all F-totally acyclic complexes of flat R-modules (in view of Proposition 9.1(a-b)).
Finally, it follows from Proposition 9.1(c) and Lemma 8.2 (for S = R p ) that the functor Φ 1 takes κ-presentable objects to κ-presentable objects.All the other assumptions of Theorem 1.3 (for the given cardinal κ and λ = ℵ 0 ) are clearly satisfied.Theorem 1.3 tells that C is a κ-accessible category, and provides the desired description of its full subcategory of κ-presentable objects.
Corollary 9.3.Let κ be an uncountable regular cardinal and R be a commutative Noetherian ring with the cardinality of the spectrum | Spec R| < κ.Then the category GF of Gorenstein-flat R-modules is κ-accessible.The κ-presentable objects of GF are precisely all the < κ-generated Gorenstein-flat R-modules.So every Gorenstein-flat R-module is a κ-directed colimit of < κ-generated Gorenstein-flat R-modules.
Proof.Similar to the proof of Corollary 8.4 and based on Theorem 9.2 together with Proposition 8.1(a).
For a far-reaching generalization of Corollary 9.3, see Theorem 10.2 below.Corollary 9.4.Let κ be an uncountable regular cardinal and R be a commutative Noetherian ring with the cardinality of the spectrum | Spec R| < κ.Let M be a < κ-generated Gorenstein-flat R-module.Then M is a direct summand of an R-module N admitting an F-totally acyclic two-sided resolution T • with < κ-generated flat R-modules T n , n ∈ Z.
Proof.Similar to the proof of Corollary 8.5 and based on Theorem 9.2.
In particular, if the cardinality of Spec R is at most countable, then Theorem 9.2 and Corollaries 9.3-9.4 are applicable for κ = ℵ 1 , providing a description of F-totally acyclic complexes of flat R-modules as ℵ 1 -directed colimits of F-totally acyclic complexes of countably generated flat R-modules.

Gorenstein-Flat Modules as Directed Colimits
In this section we use powerful and difficult results of the paper [40] in order to deduce a common generalization of Corollaries 8.4 and 9.3 applicable to all right countably coherent rings R.
Lemma 10.1.Let R be an associative ring, κ be a regular cardinal, and λ < κ be a smaller infinite cardinal.Let S and T be two classes of κ-presentable right R-modules.Assume that the class of R-modules T is closed under direct summands, the class of R-modules lim − →(κ) T is closed under colimits of λ-indexed chains in Mod-R, and the class S contains all the κ-presentable R-modules belonging to Fil(S).Let F ∈ lim − →(κ) T and N ∈ Fil(S) be two R-modules, and let f : F −→ N be a surjective R-module morphism.Then the morphism f is a κ-directed colimit of surjective R-module morphisms t : T −→ S with T ∈ T and S ∈ S.
Proof.This is another application of Theorem 1.3 together with Lemma 1.6 and the Hill lemma.Let K 1 be the category of epimorphisms of right R-modules.Lemma 1.6 tells that the category K 1 is κ-accessible and provides a description of its full subcategory of κ-presentable objects.Let L = Mod-R × Mod-R be the Cartesian square of the abelian category of right R-modules.The category L is locally κ-presentable, and Proposition 1.2 provides a description of its full subcategory of κ-presentable objects.
Finally, let K 2 = F × N be the Cartesian product of the following two categories.The category F is the class of modules F = lim − →(κ) T, viewed as a full subcategory in Mod-R.To construct the category N, we need to recall the Hill lemma [43, Theorem 6], [15, Theorem 7.10], [42,Theorem 2.1].In application to the R-module N with its given filtration by modules from S, the Hill lemma produces a certain complete lattice of submodules of N. The category N is this complete lattice of submodules in N, viewed as a poset, and this poset is viewed as a category.(Notice that the category N is not additive.) As any complete lattice interpreted as a category, the category N has all limits and colimits.In particular, the colimits in N are just the joins in the lattice.Furthermore, in the situation at hand it follows from Hill lemma that N is a locally κ-presentable category, and the κ-presentable objects of N are precisely all the κ-presentable submodules of N belonging to the lattice.On the other hand, the category F is κ-accessible, and T ⊂ F is its full subcategory of κ-presentable objects, by Proposition 1.1.Once again, Proposition 1.2 tells that the category K 2 = F × N is κ-accessible, and describes its full subcategory of κ-presentable objects.
The functor Φ 1 : The pseudopullback category C of the pair of functors Φ 1 and Φ 2 is the category of R-module epimorphisms G −→ M, where G ∈ F and M ⊂ N, M ∈ N. In particular, f : F −→ N is an object of C. Theorem 1.3 tells that the category C is κ-accessible, and provides a description of its full subcategory of κ-presentable category are precisely all the exact sequences T Proof.Recall first of all that the full subcategory of injective R-modules is closed under λ-directed colimits in Mod-R by Corollary 3.7.The same corollary tells that the category of injective right R-modules is ν + -accessible, and describes its full subcategory of ν + -presentable objects as consisting precisely of all the injective R-modules of cardinality at most ν.Now we apply Proposition 4.1 for κ = ν + (notice that λ ≤ ν < κ).Take T n to be the class of all injective right R-modules of cardinality at most ν for all 0 ≤ n ≤ m, T −1 to be the class of all right R-modules of cardinality at most ν, and T n = {0} for all n ≥ m + 1 and n ≤ −2.Remark 11.3.The full subcategory I m ⊂ Mod-R can be also described as I m = S ⊥ 1 , where S denotes the set of all (< λ-generated) m-th syzygy modules of < λ-generated (or just cyclic) right R-modules.This allows to obtain the assertion of Corollary 11.2 directly as a particular case of Proposition 3.6.

Injectively Resolved Modules
By an injective resolution we mean an exact sequence • • • −→ J 2 −→ J 1 −→ J 0 , where J n are injective modules for all n ≥ 0. An R-module M is said to be injectively resolved if there exists an exact sequence of R-modules In this section, as in the previous one, our category-theoretic approach based on Theorems 1.3 and 1.4 via Proposition 4.1 produces results which can be also obtained with the deconstructibility-based approach using a suitable version of Proposition 3.6.
Theorem 12.1.Let λ be a regular cardinal and R be a right < λ-Noetherian ring.Let ν be a cardinal such that ρ = |R| + ℵ 0 ≤ ν and ν <λ = ν.Then the category of injective resolutions Proof.Let us emphasize that there is no claim about the class of injectively resolved modules being closed under µ-directed colimits for any cardinal µ in this corollary.The assertion of the corollary follows immediately from Theorem 12.1.

presentable objects of this category are precisely all the injective resolutions
Corollary 12.3.Let λ be a regular cardinal and R be a right < λ-Noetherian ring.Let ν be a cardinal such that ρ = |R| + ℵ 0 ≤ ν and ν <λ = ν.Let M be an injectively resolved right R-module of cardinality ≤ ν.Then M is a direct summand of an R-module N admitting an injective resolution Proof.This is a corollary of Theorem 12.1 provable similarly to Corollary 4.4.It is clear from the theorem that every injectively resolved right R-module M is the colimit of a ν + -directed diagram of right R-modules (N ξ ) ξ∈Ξ such that N ξ admits an injective resolution , n ≥ 0, of cardinality at most ν for every ξ ∈ Ξ.Now if M has cardinality at most ν, then M is ν + -presentable, and it follows that there exists ξ ∈ Ξ such that M is a direct summand of N ξ .Here I ranges over all the right ideals of R, and the construction of the right Ext 1 -orthogonal class of objects S ⊥ 1 is applied in the abelian category of complexes Com(Mod-R).This assertion is a special case of [14,Proposition 4.6].Therefore, Theorem 12.1 can be also obtained as a particular case of a suitable version of Proposition 3.6 for the category of complexes Com(Mod-R).

Totally Acyclic Complexes of Injectives over a Left Noetherian Ring
An acyclic complex of injective left R-modules I • is said to be totally acyclic [23, Section 5.2] if the complex of abelian groups Hom R (J, I • ) is acyclic for every injective left R-module J.A left R-module M is said to be Gorenstein-injective if there exists a totally acyclic complex of injective left R-modules I • such that M ≃ Z 0 (I • ) is its module of cocycles.The definitions for right modules are similar.
In the following four sections, Sections 13-16, we study accessibility properties of the categories of totally acyclic complexes of injective modules and of Gorensteininjective modules.In the present section, the assumptions are more restrictive than in Section 16, but we obtain a better cardinality estimate.
We start with a lemma and a corollary suggested to the author by Jan Št'ovíček.
Lemma 13.1.For any left Noetherian ring R, there exists a functor assigning to every Gorenstein-injective left R-module M one of its totally acyclic two-sided injective resolutions, i. e., a totally acyclic complex of injective left R-modules I • together with an isomorphism M ≃ Z 0 (I • ).
Proof.First of all we observe that, for any Gorenstein-injective R-module M and an arbitrary injective R-module coresolution 0 • remains exact after applying the functor Hom R (J, −) for any injective R-module J. Furthermore, it is well-known that there exists a functor M −→ (M ֒→ I(M)) assigning to an R-module M its embedding into an injective R-module I(M).One can use a functorial version of the small object argument as in [10, Theorem 2], or simply let I(M) be the direct product of copies of the R-module Hom Z (R, Q/Z) indexed over all the abelian group maps M −→ Q/Z.These observations allow to construct the positive cohomological degree part of the desired functorial two-sided resolution of M. This part of the argument does not use the Noetherianity assumption on R.
The construction of the negative cohomological degree part of the functorial twosided resolution is based on the results of [36,Lemma 2.1 and Example 6.4].By [30, Theorem 2.5], every injective left R-module is a direct sum of indecomposable injective R-modules.It is easy to see that there is only a set of isomorphism classes of indecomposable injectives.Therefore, there exists an injective left R-module J such that all the injective left R-modules are direct summands of direct sums of copies of J. Starting with a Gorenstein-injective left R-module M, we denote by I −1 (M) the direct sum of copies of J indexed over all the R-module maps J −→ M, that is I −1 (M) = J (Hom R (J,M )) .Clearly, the natural R-module map I −1 (M) −→ M is surjective (since M is Gorenstein-injective).By [36, Lemma 2.1(c)] (for A = R-Mod and T = J), the kernel of this map is a Gorenstein-injective left R-module again, and so the inductive process can be continued indefinitely, producing the desired negative part of a totally acyclic two-sided injective resolution in a functorial way.
Corollary 13.2.For any left Noetherian ring R, the class of all Gorenstein-injective left R-modules is closed under ρ + -directed colimits in R-Mod, where ρ = |R| + ℵ 0 .Furthermore, the class of all totally acyclic complexes of injective left R-modules is closed under ρ + -directed colimits in Com(R-Mod) as well.
Proof.Let (M ξ ) ξ∈Ξ be a diagram of Gorenstein-injective left R-modules, indexed by some poset Ξ.By Lemma 13.1, one can construct a Ξ-indexed diagram of totally acyclic two-sided injective resolutions (I • ξ ) ξ∈Ξ for the given diagram of R-modules (M ξ ) ξ∈Ξ .So the first assertion of the corollary follows from the second one.
In order to prove the second assertion, notice first of all that the class of injective left R-modules is closed under directed colimits in R-Mod (e. g., by the left-right opposite version of Corollary 3.7 for λ = ℵ 0 and ν = ρ).Acyclicity of complexes of modules is obviously preserved by directed colimits.
To show that the class of totally acyclic complexes of injective left R-modules is closed under ρ + -directed colimits, notice that in the definition of a totally acyclic complex of injectives it suffices to check the preservation of acyclicity by the functors Hom R (J, −), where J ranges over indecomposable injective left R-modules.Then it remains to observe that each indecomposable injective left R-module has cardinality at most ρ (again by Corollary 3.7 for the same λ and ν).
We refer to the recent preprint [22] for a general discussion of directed colimitclosedness properties of Gorenstein-injective modules.For a version of Corollary 13.2 with more restrictive assumptions and stronger conclusion, see Corollary 15.2 below.Theorem 13.3.Let R be a left Noetherian ring; put ρ = |R| + ℵ 0 .Let ν be an infinite cardinal such that ν ρ = ν.Then the category of totally acyclic complexes of injective left R-modules is ν + -accessible.The ν + -presentable objects of this category are precisely all the totally acyclic complexes of injective left R-modules of cardinality at most ν.Consequently, every totally acyclic complex of injective left R-modules is a ν + -directed colimit of totally acyclic complexes of injective left R-modules of cardinality at most ν.
Proof.This is an application of Theorem 1.3 together with Proposition 4.1.For a comparable or somewhat similar argument, see the proof of Theorem 15.1 below.
Let K 1 denote the category of acyclic complexes of injective left R-modules.By the left-right opposite versions of Corollary 3.7 and Proposition 4.1 for λ = ℵ 0 , κ = ν + , and T n being the set of all injective R-modules of cardinality at most ν, for every n ∈ Z, we know that K 1 is a ν + -accessible category, and have a description of its ν + -accessible objects (cf.Theorem 12.1).
Furthermore, let K 2 denote the Cartesian product of copies of the category of acyclic complexes of abelian groups, taken over a set of representatives of isomorphism classes of indecomposable injective left R-modules J. Notice that every indecomposable injective left R-module is an injective envelope of a cyclic R-module R/I, where I is a left ideal in R; and there is at most ρ of these.Clearly, ρ < ν < ν + .Proposition 1.8 for κ = ν + and the ring of integers Z in the role of R, together with Proposition 1.2, tell that K 2 is a ν + -accessible category and provide a description of its full subcategory of ν + -presentable objects.
Finally, let L be the Cartesian product of copies of the abelian category of arbitrary complexes of abelian groups, taken over the same set of indecomposable injectives J. Lemma 1.5(a,c) together with Proposition 1.2 tell that L is a locally ν + -presentable category and describe its full subcategory of ν + -presentable objects.
Let Φ 1 : K 1 −→ L be the functor taking any acyclic complex of injective left R-modules I • to the collection of complexes of abelian groups Hom R (J, I • ), indexed over the indecomposable injective left R-modules J. Let Φ 2 : K 2 −→ L be the Cartesian product of the identity inclusions of the category of acyclic complexes of abelian groups into the category of all complexes of abelian groups.Then the pseudopullback C of the two functors Φ 1 and Φ 2 is the desired category of totally acyclic complexes of injective left R-modules.
All the other assumptions of Theorem 1.3 are clearly satisfied.Theorem 1.3 tells that C is a ν + -accessible category, and provides the desired description of its full subcategory of ν + -presentable objects.
Corollary 13.4.Let R be a left Noetherian ring; put ρ = |R| + ℵ 0 .Let ν be an infinite cardinal such that ν ρ = ν.Then the category of Gorenstein-injective left R-modules GI is ν + -accessible.The ν + -presentable objects of GI are precisely all the Gorenstein-injective left R-modules of cardinality at most ν.So every Gorensteininjective left R-module is a ν + -directed colimit of Gorenstein-injective left R-modules of cardinality at most ν.
Proof.The full subcategory GI ⊂ R-Mod is closed under ρ + -directed colimits in R-Mod by Corollary 13.2.Since any R-module of cardinality at most ν is ν + -presentable in R-Mod, it follows that any Gorenstein-injective left R-module of cardinality at most ν is ν + -presentable in the category of Gorenstein-injective left R-modules.In view of Proposition 1.1, it remains to check the last assertion of the corollary, which follows immediately from Theorem 13.3.

Totally Acyclic Complexes of Injectives as Directed Colimits
In the present section, the assumptions are somewhat restrictive, as they include existence of a (strong) right dualizing complex; but the cardinality estimates are even better than in Section 13.
In addition to the conditions (i ′ ) and (ii-iii) in the definition of a strong right dualizing complex in Sections 7 and 14, we will need to assume another condition (which should be compared to condition (iv) from Section 8).We will say that a strong right dualizing complex of R-S-bimodules D • has left type < λ if (iv ′ ) D • is isomorphic, as an object of the bounded derived category of left R-modules D b (R-Mod), to a bounded complex of < λ-generated left R-modules D • .Condition (iv ′ ) will be used for left < λ-Noetherian rings R, so the classes of < λ-generated and λ-presentable left R-modules coincide.In particular, in the case of the cardinal λ = ℵ 0 , we will speak about dualizing complexes of left finite type.
Theorem 15.1.Let λ be a regular cardinal and R be a left < λ-Noetherian ring.Let ν be a cardinal such that ρ = |R| + ℵ 0 ≤ ν and ν <λ = ν.Let S be a right coherent ring and D • be a strong right dualizing complex of R-S-bimodules.Assume that D • is a strong right dualizing complex of left type < λ.Then the category of totally acyclic complexes of injective left R-modules is ν + -accessible.The ν + -presentable objects of this category are precisely all the totally acyclic complexes of injective left R-modules of cardinality at most ν.Consequently, every totally acyclic complex of injective left R-modules is a ν + -directed colimit of totally acyclic complexes of injective left R-modules of cardinality at most ν.
Proof.This is an application of Theorem 1.3 together with Proposition 4.1 and Proposition 14.1.The argument bears some similarity to the proof of Theorem 13.3.First of all, let us show that the full subcategory of totally acyclic complexes of injective left R-modules is closed under λ-directed colimits in the ambient abelian category Com(R-Mod).
By Corollary 3.7, the full subcategory of injective left R-modules is closed under λ-directed colimits in R-Mod.It is also clear that acyclicity of complexes is preserved by all directed colimits.Now let I • be an acyclic complex of injective left R-modules, and let D • be a bounded complex of < λ-generated left R-modules provided by the definition of a strong right dualizing complex of left type < λ (item (iv ′ )) above.It is clear from Lemma 14.2 that the complex of left S-modules Hom R (D • , I • ) is acyclic if and only if the complex of abelian groups Hom R ( D • , I • ) is acyclic.By Proposition 14.1, we can conclude that the acyclic complex of injective R-modules I • is totally acyclic if and only if the complex of abelian groups Hom R ( D • , I • ) is acyclic.The latter condition is obviously preserved by λ-directed colimits of the complexes I • .Now denote by K 1 the category of acyclic complexes of injective left R-modules.Applying the left-right opposite version of Proposition 4.1 for κ = ν + and T n being the set of all injective R-modules of cardinality at most ν, for every n ∈ Z, and using Corollary 3.7 again, we see that K 1 is a ν + -accessible category, and obtain a description of its ν + -accessible objects (cf.Theorem 12.1).Notice that λ ≤ ν, so the ring R is left < ν + -coherent.
Furthermore, let K 2 denote the category of acyclic complexes of abelian groups.Proposition 1.8 for κ = ν + and the ring of integers Z in the role of R tells that K 2 is a ν + -accessible category and provides a description of its full subcategory of ν + -presentable objects.Finally, let L be the abelian category of arbitrary complexes of abelian groups.Lemma 1.5(a,c) tells that L is a locally ν + -presentable category and describes its full subcategory of ν + -presentable objects as the category of complexes of abelian groups of cardinality at most ν.
Let the functor Φ 1 : K 1 −→ L take any acyclic complex of injective left R-modules I • to the complex of abelian groups Hom R ( D • , I • ).Let Φ 2 : K 2 −→ L be the identity inclusion of the category of acyclic complexes of abelian groups into the category of all complexes of abelian groups.Then the pseudopullback C of the two functors Φ 1 and Φ 2 is the category of all acyclic complexes of injective left R-modules I • for which the complex of abelian groups Hom R ( D • , I • ) is acyclic.As we have seen in the second paragraph of this proof, the category C is the desired category of totally acyclic complexes of injective left R-modules.
Corollary 15.2.Let R be a left Noetherian ring, S be a right coherent ring, and D • be a right dualizing complex of R-S-bimodules.Assume that D • is a (strong) right dualizing complex of left finite type.Then the class of all Gorenstein-injective left R-modules is closed under directed colimits in R-Mod.Furthermore, the class of all totally acyclic complexes of injective left R-modules is closed under directed colimits in Com(R-Mod) as well.
Proof.In view of the first paragraph of the proof of Corollary 13.2, it suffices to prove the second assertion, which is a particular case of the first two paragraphs of the proof of Theorem 15.1 (for λ = ℵ 0 ).
In the case of a commutative Noetherian ring R with a dualizing complex, the first assertion of Corollary 15.2 can be obtained by combining [22,Theorem 2] with [19,Lemma 2.5(b)].
Corollary 15.3.Let λ be a regular cardinal and R be a left < λ-Noetherian ring.Let ν be a cardinal such that ρ = |R| + ℵ 0 ≤ ν and ν <λ = ν.Let S be a right coherent ring and D • be a strong right dualizing complex of R-S-bimodules.Assume that D • is a strong right dualizing complex of left type < λ.Then every Gorenstein-injective left R-module is a ν + -directed colimit of Gorenstein-injective left R-modules of cardinality at most ν.
is the desired one.Proof of Proposition 2.1.In view of Lemma 2.2, one can assume without loss of generality that C = lim − →ξ∈Ξ S ξ , where Ξ is a directed poset of cardinality smaller than ℵ m and S ξ ∈ S for all ξ ∈ Ξ.Then the spectral sequence from Lemma 2.3 degenerates to a natural isomorphism Ext n R (C, D) ≃ lim ← − n ξ∈Ξ Hom R (S ξ , D) for all n ≥ 0. It remains to recall that the derived functor of Ξ-indexed limit in the category of abelian groups has cohomological dimension at most m [31].The following corollary can be found in [24, Proposition 5.3].It is a partial generalization of [15, Corollary 2.23].Corollary 2.4.Let R be an associative ring and F be an ℵ m -presentable flat R-module.Then the projective dimension of F does not exceed m.

Proposition 3 . 3 .Lemma 3 . 4 .
Let R be an associative ring; put ρ = |R| + ℵ 0 .Then (a) the class of all flat R-modules is ρ + -deconstructible; (b) for every integer m ≥ 0, the class of all R-modules of flat dimension at most m is ρ + -deconstructible.Proof.Part (a) is [6, Lemma 1 and Proposition 2].Part (b) follows from part (a) by virtue of [41, Theorem 3.4].Given a class of R-modules S ⊂ Mod-R, one denotes by S ⊥ 1 ⊂ Mod-R the class of all modules M ∈ Mod-R such that Ext 1 R (S, M) = 0 for all S ∈ S. The following result is known as the Eklof lemma [10, Lemma 1], [15, Lemma 6.2].For any class of modules S ⊂ Mod-R one has S ⊥ 1 = Fil(S) ⊥ 1 .

Corollary 4 . 3 .
Let R be a right countably coherent ring, and let (S n ) n∈Z be a sequence of sets of finitely presentable right R-modules, S n ⊂ (Mod-R) <ℵ 0 .Denote by M the class of all right R-modules M of the form M = Z 0 (C • ), where C • is an acyclic complex of R-modules with the terms C n ∈ lim − → S n .Then every module from M is an ℵ 1 -directed colimit of countably presentable modules from M.

Theorem 5 . 1 .
Let R be a right countably coherent ring and m ≥ 0 be an integer.Then the category of all exact sequences of flat right R-modules 0 2. Take S n to be the set of all finitely generated projective (or free) right R-modules for all −m − 1 ≤ n ≤ −1, S 0 to be the set of all finitely presentable right R-modules, and S n = {0} for all n ≤ −m − 2 and all n ≥ 1.Then lim − → S n is the class of all flat right R-modules for all −m−1 ≤ n ≤ −1, lim − → S 0 = Mod-R, and lim − → S n = {0} for all n ≤ −m − 2 and all n ≥ 1.So the category C from Theorem 4.2 is equivalent to the category of finite flat resolutions we are interested in.Corollary 5.2.Let R be a right countably coherent ring and m ≥ 0 be an integer.Denote by F m ⊂ Mod-R the full subcategory of all right R-modules of flat dimension at most m.Then the category F m is ℵ 1 -accessible.The ℵ 1 -presentable objects of F m are precisely all the countably presentable right R-modules of flat dimension ≤ m (i.e., those modules from F m that are countably presentable in Mod-R).So every right R-module of flat dimension m is an ℵ 1 -directed colimit of countably presentable R-modules of flat dimension at most m.

Corollary 5 . 3 .
Let R be a right countably coherent ring and m ≥ 0 be an integer.Then every right R-module of flat dimension m is an ℵ 1 -directed colimit of countably presentable right R-modules of projective dimension at most m + 1.Proof.This is a corollary of Corollary 5.2.The point is that any countably presentable right R-module of flat dimension ≤ m has projective dimension ≤ m + 1, since any countably presentable flat module has projective dimension at most 1 (by[15,  Corollary 2.23]  or Corollary 2.4 above).Notice that Corollary 5.2 (or more generally, Proposition 4.1) gives a better cardinality estimate for the accessibility rank of the category of R-modules of flat dimension ≤ m than Proposition 3.1 combined with Proposition 3.3(b).
Then lim − → S n is the class of all flat right R-modules for all n ≥ 0, lim − → S −1 = Mod-R, and lim − → S n = 0 for all n ≤ −2.So the category C from Theorem 4.2 is equivalent to the category of flat coresolutions we are interested in.Corollary 6.2.Let R be a right countably coherent ring.Then any flatly coresolved right R-module is an ℵ 1 -directed colimit of countably presentable flatly coresolved right R-modules.

Corollary 6 . 3 .
Let R be a right countably coherent ring and M be a countably presentable flatly coresolved right R-module.Then M is a direct summand of an R-module N admitting a flat coresolution 0 −→ N −→ T 0 −→ T 1 −→ T 2 −→ • • • with countably presentable flat right R-modules T n , n ≥ 0.

Proposition 8 . 1 .
Let R be an associative ring.Then (a) the class of Gorenstein-flat modules is closed under directed colimits in Mod-R; (b) the class of Gorenstein-flat R-modules is ρ + -deconstructible, where ρ = |R|+ℵ 0 .
For a far-reaching generalization of Corollary 8.4, see Theorem 10.2 below.Corollary 8.5.Let R be a right countably coherent ring, S be a right coherent ring, and D • be a right dualizing complex of R-S-bimodules.Assume that D • is a right dualizing complex of right countable type.Let M be a countably presentable Gorenstein-flat right R-module.Then M is a direct summand of an R-module N admitting an F-totally acyclic two-sided resolution T • with countably presentable flat right R-modules T n , n ∈ Z.Proof.The argument is similar to the proof of Corollary 4.4.It is clear from Theorem 8.3 that every Gorenstein-flat right R-module M is the colimit of an ℵ 1 -directed diagram of R-modules (N ξ ) ξ∈Ξ such that N ξ = Z 0 (T • ξ ) for some F-totally acyclic complexes of countably presentable flat right R-modules T •

Proposition 9 . 1 .
Let R be a commutative Noetherian ring.In this setting: (a) All injective R-modules are direct sums of indecomposable injective R-modules.(b) The indecomposable injective R-modules, viewed up to isomorphism, correspond bijectively to prime ideals of R. For every prime ideal p ∈ Spec R, the corre-sponding indecomposable injective R-module is the injective envelope E R (R/p) of the R-module R/p.(c) For every prime ideal p ∈ Spec R, the R-module E R (R/p) is a module over the local ring R p .The R p -module E R (R/p) is (at most) countably generated.Proof.All these results are due to Matlis[30].Part (a) is [30, Theorem 2.5].Part (b) is [30, Proposition 3.1].Part (c) is [30, Theorems 3.6 and 3.11].
Then lim − →(κ)T n is the class of all injective right R-modules for all 0 ≤ n ≤ m, lim − →(κ) T −1 = Mod-R, and lim − →(κ) T n = 0 for all n ≥ m + 1 and n ≤ −2.So the category C from Proposition 4.1 is equivalent to the category of finite injective coresolutions we are interested in.Corollary 11.2.Let λ be a regular cardinal and R be a right < λ-Noetherian ring.Let ν be a cardinal such that ρ = |R| + ℵ 0 ≤ ν and ν <λ = ν.Let m ≥ 0 be an integer.Denote by I m ⊂ Mod-R the full subcategory of all right R-modules of injective dimension at most m.Then the category I m is ν + -accessible.The ν + -presentable objects of I m are precisely all the right R-modules of cardinality ≤ ν and of injective dimension ≤ m.So every right R-module of injective dimension m is a ν + -directed colimit of right R-modules of cardinality at most ν and of injective dimension at most m.Proof.A right R-module K has injective dimension ≤ m if and only if Ext m+1 R (R/I, K) = 0 for all right ideals I ⊂ R.Over a right < λ-Noetherian ring R, the functor Ext * R (M, −) preserves λ-directed colimits for every < λ-generated right R-module M. Consequently, the full subcategory I m is closed under λ-directed colimits in Mod-R.In view of Proposition 1.1, it remains to check the last assertion of the corollary, which follows immediately from Theorem 11.1.
of cardinality at most ν.Proof.Similarly to the proof of Theorem 11.1, the argument in based on Corollary 3.7 and Proposition 4.1 for κ = ν + .Take T n to be the class of all injective right R-modules of cardinality at most ν for all n ≤ −1, T 0 to be the class of all right R-modules of cardinality at most ν, andT n = {0} for all n ≥ 1.Then lim − →(κ) T n is the class of all injective right R-modules for all n ≤ −1, lim − →(κ) T 0 = Mod-R, and lim − →(κ)T n = 0 for all n ≥ 1.So the category C from Proposition 4.1 is equivalent to the category of injective resolutions we are interested in.Corollary 12.2.Let λ be a regular cardinal and R be a right < λ-Noetherian ring.Let ν be a cardinal such that ρ = |R| + ℵ 0 ≤ ν and ν <λ = ν.Then any injectively resolved right R-module is a ν + -directed colimit of injectively resolved right R-modules of cardinality at most ν.