{'fraction_non_alphanumeric': 0.055967425959359274, 'fraction_numerical': 0.015019398455806092, 'mean_word_length': 3.242137852370865, 'pattern_counts': {'":': 0, '<': 9, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Degreewise n-projective and n-flat model structures on chain complexes

In the paper"Cotorsion Pairs in C(R-Mod)", the authors construct an abelian model structure on the category of chain complexes Ch(R), where the class of cofibrant objects is given by the class of degreewise projective chain complexes. Using a generalization of a well known theorem by I. Kaplansky, we generalize the method used in that paper in order to obtain, for each integer n>1, a new abelian model structure on Ch(R), where the class of cofibrant objects is the class of chain complexes whose terms have projective dimension smaller or equal than n (dw-n-projective complexes), provided the ring R is noetherian. We also present another method, based on the paper"Covers and Envelopes in Grothendieck Categories: Flat Covers of Complexes with Applications", to construct this model structure. This method also works to construct an abelian model structure whose cofibrant objects are the dw-n-flat complexes.


Introduction
A cotorsion pair in an abelian category C is a pair (A, B), where A and B are classes of objects of C such that they are orthogonal to each other with respect to the Ext functor. A model category is a bicomplete category with three classes of morphisms, called cofibrations, fibrations and weak equivalences, satisfying certain conditions. It turns out to be that these two notions have a deep connection. As far as the author knows, the first person who described this connection was M. Hovey in the paper Cotorsion pairs, model category structures and representation theory, where he proved that any two compatible and complete cotorsion pairs (A, B ∩ E) and (A ∩ E, B), in a bicomplete abelian category C, give rise to a unique abelian model structure on C where A is the class of cofibrant objects, B is the class of fibrant objects, and E is the class of trivial objects.
From this point there has been an increasing interest in constructing new model structures, specially on Ch(R−Mod). One of the most influential researchers in this matter has been J. Gillespie, who has provided several results that allows us to induce cotorsion pairs in the category Ch(C) of chain complexes over an abelian category C, from a certain cotorsion pair in C. One of those results states that given a cotorsion pair (A, B) in an abelian category C with enough projective and injective objects, there exist two cotorsion pairs in Ch(C) given by (dw A, (dw A) ⊥ ) and (ex A, (ex A) ⊥ ), where dw A is the class of chain complexes X such that X m ∈ A for every m ∈ Z, and ex A = dw A∩E where E is the class of exact complexes. As an example, if P 0 denotes the class of projective modules in the category R−Mod of left R-modules, then the cotorsion pair (P 0 , R−Mod) induces two cotorsion pairs (dw P 0 , (dw P 0 ) ⊥ ) and (ex P 0 , (ex P 0 ) ⊥ ). In [4], the authors prove that these pairs are compatible and complete, with the help of a theorem by I. Kaplansky, namely that every projective module can be written as a direct sum of countably generated projective modules. Then, using [10, Theorem 2.2], they get a new abelian model structure on Ch(R−Mod) where the class of cofibrant objects is the class dw P 0 , which we shall call the class of degreewise projective complexes. We shall refer to this model structure as the dw-projective model structure.
In [2] it is proven that if P n denotes the class of left R-modules with projective dimension at most n, then (P n , P ⊥ n ) is a complete and hereditary cotorsion pair. It follows we have two induced cotorsion pairs (dw P n , (dw P n ) ⊥ ) and (ex P n , (ex P n ) ⊥ ) in Ch(R−Mod). Our goal is to prove that these two cotorsion pairs are complete for every n > 0, in order two obtain a new abelian model structure on Ch(R−Mod) such that dw P n is the class of cofibrant objects. This paper is organized as follows. First, we recall some definitions and introduce the notation we shall use. Then, we shall give a "generalization" of the Kaplansky Theorem in Mod-R provided that R is left noetherian. Specifically, we shall prove that every module of projective dimension ≤ n has a P ℵ0 n -filtration, where P ℵ0 n is the set of all modules M for which there exists an exact sequence where P k is a countably generated projective module, for every 0 ≤ k ≤ n. Using this result, we shall prove that (dw P n , (dw P n ) ⊥ ) and (ex P n , (ex P n ) ⊥ ) are complete cotorsion pairs. Then, we shall give another method to prove the previous result. The interesting thing of this other method, based on arguments appearing in the proof of [1,Proposition 4.1], is that it can be applied to show that (dw F n , (dw F n ) ⊥ ) and (ex F n , (ex F n ) ⊥ ) are complete cotorsion pairs, where F n denotes the class of left Rmodules having flat dimension at most n. At the end of the paper, we shall give some comments concerning the dw-n-projective and dw-n-flat model structures.

Preliminaries
This section is devoted to recall some notions and to introduce part of the notation we shall use throughout the paper. From now on, we work in the category R−Mod of left R-modules, and the category Ch(R−Mod) of chain complexes over R−Mod.
Given a chain complex X = (X m ) m∈Z with boundary maps ∂ X m : Then we can define the quotient complex X/Y as the complex whose components are given by (X/Y ) m = X m /Y m and whose boundary maps ∂ Let C be either R−Mod or Ch(R−Mod). Let A and B be two classes of objects in C. The pair (A, B) is called a cotorsion pair in C if the following conditions are satisfied: (2) B = A ⊥ := {X ∈ Ob(C) / Ext 1 (A, X) = 0 for every A ∈ A}.
A cotorsion pair (A, B) in C is said to be complete if: (a) (A, B) has enough projectives: for every object X there exist objects A ∈ A and B ∈ B, and a short exact sequence (b) (A, B) has enough injectives: for every object X there exist objects A ′ ∈ A and B ′ ∈ B, and a short exact sequence (4) The following example is probably the most important cotorsion pair we shall consider in this paper, the pair (P n , P ⊥ n ), where P n is the class of modules which have projective dimension ≤ n. Recall that a module M has projective dimension ≤ n is there exists an exact sequence such that P k is a projective module, for every 0 ≤ k ≤ n. Such a sequence is called a projective resolution of M of length n. We shall refer to the modules in P n as n-projective modules. In [2], the authors proved that (P n , P ⊥ n ) is a cotorsion pair cogenerated by the set of all n-projective modules whose cardinality is less or equal than a given infinite cardinal κ with κ ≥ Card(R).
where F k is a flat module, for every 0 ≤ k ≤ n. This sequence is called a flat resolution of length n. In [9,Theorem 4.1.3], it is proven that (F n , F ⊥ n ) is a complete cotorsion pair. Here we shall give a easier proof of this fact. We shall say that f has the left lifting property with respect to g (or that g has the right lifting property with respect to f ) if for every pair of maps A model category is a bicomplete category C equipped with three classes of maps named cofibrations, fibrations and weak equivalences, satisfying the following properties: (1) 3 for 2: If f and g are maps of C such that g • f is defined and two of f , g and g • f are weak equivalences, then so is the third.
(2) If f and g are maps of C such that f is a retract of g and g is a weak equivalence, cofibration, or fibration, then so is f .
Define a map to be a trivial cofibration if it is both a weak equivalence and a cofibration. Similarly, define a map to be a trivial fibration if it is both a weak equivalence and a fibration. (4) Every map f can be factored as f = α • β = γ • δ, where α (resp. δ) is a cofibration (resp. fibration), and γ (resp. β) is a trivial cofibration (resp. trivial fibration).

Degreewise n-projective complexes
We begin this section with the notion of a filtration. Let C be either R−Mod or Ch(R−Mod). Given an object X ∈ C, by a filtration of X indexed by an ordinal λ we shall mean a family (X α : α < λ) of subobjects of X such that: (1) X = α<λ X α .
If S is some class of objects in Ch(R−Mod), we say that a filtration (X α : α < λ) of X is a S-filtration if for each α + 1 < λ we have that X 0 and X α+1 /X α are isomorphic to an element of S.
The construction of the model structure given in [4] is based on a theorem by I.

Kaplansky, namely:
Theorem 3.1 (Kaplansky's Theorem). If P is a projective module then P is a direct sum of countable generated projective modules.
So when one thinks of a possible generalization of the dw-projective model structure for n-projective modules, a good question would be if it is possible to generalize the Kaplansky's Theorem for such modules. Let M ∈ P n be an n-projective module: By Kaplansky's Theorem we can write P k = i∈I k P i k , where P i k is a countably generated projective module, for every i ∈ I k and every 0 ≤ k ≤ n. Then we can rewrite the previous resolution as From now on we shall write any projective resolution of length n by using such direct sum decompositions. We shall denote by P ℵ0 n the set of all modules M having a projective resolution as above, where I k is a countable set for each 0 ≤ k ≤ n.
For any class of modules A, we denote by dw A (resp. ex A) the class of (resp. exact) chain complexes such that each term belongs to A. We shall prove that (dw P n , (dw P n ) ⊥ ) is a cotorsion pair cogenerated by the set dw P ℵ0 n . We shall name dw P n the class of dw-n-projective chain complexes. The fact that (dw P n , (dw P n ) ⊥ ) is a cotorsion pair in Ch(R−Mod) is a consequence of the following result (which is proven by its author for any abelian category): We shall prove that every dw-n-projective complex has a dw P ℵ0 n -filtration. Then the completeness of (dw P n , (dw P n ) ⊥ ) shall be a consequence of Theorem 2.1 and the following result: In order to construct dw P ℵ0 n -filtrations of dw-n-projective complexes, we need the following generalization of the Kaplansky's Theorem: Proof. Let M ∈ P n and let be a projective resolution of M . We shall construct a P ℵ0 n -filtration (M α : α < λ) of M , with N ⊆ M 1 , by using transfinite induction. For α = 0 set M 0 = 0. Now we construct M 1 . Let G be a countable set of generators of N . Since f 0 is surjective, for every g ∈ G we can choose y g ∈ i∈I0 P i 0 such that g = f 0 (y g ). Consider the set Y = {y g : g ∈ G}. Since Y is a countable subset of i∈I0 P i 0 , we have that Y is a countably generated submodule of P 0 . Choose a countable subset I 1,0 then f (b) = 0 and by exactness of the above sequence there exists Then Y ′ is a countably generated submodule of i∈I1 P i 1 . Hence there exists a countable subset I 1,0 Use the same argument to find a countable subset I 1,0 2 ⊆ I 2 such that Repeat the same argument until find a count- Repeat this process until find a countable subset I 1,0 0 ⊆ I 1,1 0 ⊆ I 0 satisfying f 1 What we have been doing so far is called the zig-zag procedure. Keep repeating this procedure infinitely many times, and set I 1 for every 0 ≤ k ≤ n. By construction, we get the following exact sequence We take the quotient of the resolution of M by the resolution of M ′ , and get the following commutative diagram: where the third row is an exact sequence since the class of exact complexes is closed under taking cokernels. Then we have a projective resolution of length n for M/M 1 .
Repeat the same procedure above for M/M 1 , by choosing in order to get an exact sequence where the first and third rows are exact sequences, and then so is the second since the class of exact complexes is closed under extensions. We have that M 1 and M 2 are n-projective modules such that M 1 ∈ P ℵ0 n , M 2 /M 1 ∈ P ℵ0 n . Now suppose that there is an ordinal β such that: (1) M α is an n-projective module, for every α < β.
If β is a limit ordinal, then set M β = α<β M α . Otherwise there exists an ordinal α < β such that β = α + 1. In this case, construct M α+1 ∈ P n from M α as we constructed M 2 from M 1 , such that M α+1 /M α ∈ P ℵ0 n . By transfinite induction, we obtain a P ℵ0 n -filtration (M α : α < λ) of M , for some ordinal λ, such that M 1 ⊇ N and M 1 ∈ P ℵ0 n .
From now on, R shall be a noetherian ring. Now we are ready to prove the main result of this section.
Theorem 3.2. Every chain complex X ∈ dw P n has a dw P ℵ0 n -filtration.
Proof. Let X ∈ dwP n and write For each k one has a projective resolution of X k of length n: We shall construct a dw P ℵ0 n -filtration of X by using transfinite induction. For α = 0 set X 0 = 0. For α = 1, choose m ∈ Z. Let S be a countably generated submodule of X m . By the previous lemma, there exists a submodule P ℵ0 n ∋ X 1 m ⊆ X m such that S ⊆ X 1 m . Note that X 1 m is also countably generated. Then ∂ m (X 1 m ) is a countably generated submodule of X m−1 , and so there exists P ℵ0 Repeat the same procedure infinitely many times in order to obtain a subcomplex of X such that X 1 k ∈ P ℵ0 n for every k ∈ Z (we are setting X 1 k = 0 for every k > m). Hence X 1 ∈ dw P ℵ0 n . Note from the proof of the previous lemma that the quotient X/X 1 is in dw P n . We have where for every k ≤ m one has the following projective resolutions of length n for X 1 k and X k /X 1 k : Apply the same procedure above to the complex X/X 1 , in order to get a subcomplex of X/X 1 , such that for each k ∈ Z one has the following projective resolution of length where each I 2 j − I 1 j ⊆ I j is countable. Now consider the complex As we did in the proof of the previous lemma, we have that is an exact sequence. So X 2 k ∈ P n for every k ∈ Z, and hence X 2 ∈ dw P n , with n . The rest of the proof follows by transfinite induction, as in the end of the proof of the previous lemma.

Exact degreewise n-projective complexes
Consider the class of exact dw-n-projective complexes ex P n = dw P n ∩ E, where E denotes the class of exact complexes. The goal of this section is to prove that (ex P n , (ex P n ) ⊥ ) is a complete cotorsion pair. This pair is a cotorsion pair by the following result by Gillespie: Hom(R, Q/Z) ∈ P ⊥ n , we have that exP n , (exP n ) ⊥ is a cotorsion pair.
Given a module M ∈ P n , consider a projective resolution of M of length n: We shall say that a projective resolution is a nice subresolution of ( * ) if I ′ k ⊆ I k for every 0 ≤ k ≤ n and N ⊆ M .
From now on, fix an infinite cardinal κ such that κ ≥ Card(R). We shall say that a set S is small if Card(S) ≤ κ. We shall also say that a chain complex X = (X m ) m∈Z We shall prove that every exact dw-n-projective complex has a ex P ≤κ n -filtration.
of ( * ) such that N ⊆ N ′ . Moreover, if N has an small and nice subresolution of M , then ( * * * ) can be constructed in such a way that it contains the given resolution of N .
Proof. Since f 0 is surjective, for every x ∈ N choose y x ∈ i∈I0 P i 0 such that Repeat the same argument until find a small subset I 0 n ⊆ I n such that f n i∈I 0 n is a small submodule of i∈In−1 P i n−1 , we can choose a small subset From this point just use the zig-zag procedure in order to get small subsets I ′ k = j≥0 I j k ⊆ I k and an exact sequence Now suppose that N has a small and nice subresolution of ( * ). Take the quotient of ( * ) by this resolution and get Repeat the argument above using this sequence and the small submodule z + N , where z ∈ N . Then we get a projective subresolution of the previous one: where each set I ′ k − I N k is a small set. As we did in the proof of Lemma 3.2, we have that is small and nice subresolution of ( * ), and that contains the resolution of N as a nice subresolution.
Lemma 4.2. Let X ∈ dw P n and let Y be a bounded above subcomplex of X such that Proof. We are given the following commutative diagram Since X m is an n-projective module, we have a projective resolution By the previous lemma, there exists a submodule Y ′ m of X m containing Y m , along with a small and nice subresolution has a small and nice subresolution of a fixed resolution of X m−1 . Repeat this process infinitely many times in order to obtain a complex Theorem 4.1. Every X ∈ ex P n has a ex P ≤κ n -filtration.
Proof. Let X ∈ ex P n . We construct a ex P ≤κ n -filtration of X using transfinite induction. For α = 0 set X 0 = 0. For the case α = 1 let m ∈ Z be arbitrary and let T 1 ⊆ X m be a small submodule of X m . Then there exists a small submodule Y 1 m of X m such that T 1 ⊆ Y 1 m and that Y 1 m has a small and nice projective subresolution of a given resolution of and that Y 1 m−1 has a small and nice projective subresolution of a given resolution of X m−1 . Keep repeating this argument infinitely many times. We obtain a complex which is a subcomplex of X and Y 1 ∈ dw P ≤κ n . Note that Y 1 is not necessarily exact. We shall construct a complex X 1 from Y 1 such that X 1 ⊆ X and X 1 ∈ ex P ≤κ n . The rest of this proof uses an argument similar to the one used in [4,Theorem 4.6]. Fix any p ∈ Z. Then Card(Y 1 p ) ≤ κ and so Card(Z p Y 1 ) ≤ κ. Since X is exact and for every j > p + 1, and that each C j with j ≤ p has a small and nice projective subresolution of a given resolution of X j . Since Y 1 +C is a bounded above subcomplex of X, there exists a small subcomplex Y 2 of X such that Y 1 + C ⊆ Y 2 and that each Y 2 j has a small and nice projective subresolution of a given resolution of X j . Note ). Construct Y 3 from Y 2 as we just constructed Y 2 from Y 1 , and so on, making sure to use the same p ∈ Z at each step. Set X 1 = ∞ j=1 Y j ⊆ X. Note that X 1 is exact at p. Repeat this argument to get exactness at any level.
So we may assume that X 1 is an exact complex. Every X 1 k has a small and nice projective subresolution of the given resolution of X k . For every j one has a projective subresolution of the form where I 1 l (k) ⊆ I 2 l (k) ⊆ · · · for every 0 ≤ l ≤ n, by Lemma 4.1. If we take the union of all of the previous sequences, then we obtain the following exact sequence: where j≥1 I j l (k) ⊆ I l (k) for every 0 ≤ l ≤ n. Therefore, X 1 ∈ ex P ≤κ n . Now consider the quotient complex Note that each X k /X 1 k is n-projective and that X/X 1 is exact. We apply the same procedure above to the complex X/X 1 in order to get a complex X 2 /X 1 ⊆ X/X 1 such that X 2 /X 1 ∈ ex P ≤κ n . Note that X 2 is an exact complex since the class of exact complexes is closed under extensions, and so X 2 ∈ ex P n . The rest of the proof follows by using transfinite induction.
It follows by Proposition 3.2 and the Eklof and Trlifaj Theorem that (ex P n , (ex P n ) ⊥ ) is a complete cotorsion pair cogenerated by ex P ≤κ n .

The staircase zig-zag argument
In this section, we present another method to prove that the induced cotorsion pairs (dw P n , (dw P n ) ⊥ ) and (ex P n , (ex P n ) ⊥ ) are complete. This method also applies for the cotorsion pairs (dw F n , (dw F n ) ⊥ ) and (ex F n , (ex F n ) ⊥ ). From now on, let A denote either the class of projective modules or the class of flat modules, and let A n denote the class of all modules M having an A-resolution of length n: As we mentioned before, in [9] it is proven that (F n , F ⊥ n ) is a complete cotorsion pair. We give a simpler proof of this fact.
of (1) such that S ′ k is a small and pure submodule of F k , for every 0 ≤ k ≤ n, and such that N ⊆ N ′ . Moreover, if N has a subresolution of (1) where S k is a small and pure submodule of F k , for every 0 ≤ k ≤ n, then the above resolution of N ′ can be constructed in such a way that it contains the resolution of N .
Proof. First, note that for every flat module F and for every small submodule N ⊆ F , there exists a small and pure submodule S ⊆ F such that N ⊆ S (for a proof of this, see [6,Lemma 5.3.12]). For every x ∈ N there exists y x ∈ F 0 such that x = f 0 (y x ). Let Y be the set {y x : x ∈ N and f 0 (y x ) = x} and consider the submodule Y ⊆ F 0 . Since Y is small, there exists a small pure submodule S 0 (1) ⊆ F 0 such that Y ⊆ S 0 (1).
Note that f 0 (S 0 (1)) ⊇ N . Now consider Ker f 0 | S0(1) and let A be a set of preimages (1) . It is easy to see that A is a small submodule of F 1 , so we can embed it into a small pure submodule S 1 (1) ⊆ F 1 .
Hence we have f 1 (S 1 (1)) ⊇ Ker f 0 | S0 (1) . Now consider Ker f 1 | S1(1) and repeat the same process above in order to find a small pure submodule S 2 (1) ⊆ F 2 such that f 2 (S 2 (1)) ⊇ Ker f 1 | S1 (1) . Keep doing this until find a small pure submodule (2). Repeat this process until find a small pure submodule S 0 (2) ⊆ F 0 such that f 1 (S 1 (2)) ⊆ S 0 (2). If we now consider Ker f 0 | S0(2) ⊆ F 0 , we repeat the same argument above to find a small pure submodule S 1 (3) ⊆ F 1 such that (2) . Keep repeating this zig-zag procedure infinitely many times and set S k = i≥1 S k (i), for every 0 ≤ k ≤ n. Note that each S k is a pure submodule of F k . By construction, we get an exact complex (2) where Q = CoKer(f 1 | S1 ) ⊆ M . If we take the quotient of (1) by (2), we get an exact complex Since each S k is a pure submodule of F k , we know that S k and F k /S k are flat modules.
Therefore, Q is a small n-flat submodule with N ⊆ Q such that M/Q is also n-flat.
The rest of the proof follows as in Lemma 4.1. Theorem 5.1. Let X ∈ ex A n and let x ∈ X (i.e. x ∈ X m for some m ∈ Z). Then there exists a small complex Y ∈ ex A ≤κ n such that x ∈ Y and X/Y ∈ ex A n .
The following proof is based on an argument given in [1,Proposition 4.1], where the authors prove that (dw F 0 , (dw F 0 ) ⊥ ) and (ex F 0 , (ex F 0 ) ⊥ ) are complete cotorsion pairs. We shall call this argument the staircase zig-zag.
Proof of Theorem 5.1. Assume without loss of generality that x ∈ X 0 . Consider the submodule x ⊆ X 0 . Since X 0 ∈ A and x is small, we can embed x into a submodule A ≤κ ∋ Y 1 0 ⊆ X 0 such that X 0 /Y 1 0 ∈ A. Since X is exact, we can construct a small and exact subcomplex Now construct a small and exact subcomplex Note that it is possible to construct L 2 containing L 1 . Now embed L 2 0 into a sub- Again, construct a small and exact subcomplex containing L 2 . Now let Y 4 1 ∈ A ≤κ be a submodule of X 1 containing L 3 1 such that X 1 /Y 4 1 ∈ A, and construct an exact and small complex Construct an exact and small subcomplex containing L 4 . In a similar way, construct small and exact complexes such that Y 6 −1 ∈ A is a small submodule of X −1 containing Y 6 −1 + ∂ 0 (Y 5 0 ), and Y 7 −2 ∈ A is a small submodule of X −2 containing ∂ −1 (Y 6 −1 ). We have the following commutative diagram of subcomplexes of X: where the k + 1-th complex can be constructed in such a way that it contains the k-th complex. Note that the submodules S k i appear according to the following pattern: It is clear that Y is an exact complex. We check that Y is also a dw A n . For example, consider It is clear that Y 0 is small. At this point, we split the proof in two cases: (1) A = P 0 : Consider a projective resolution of X 0 of length n, say where each direct sum is a direct sum of countably generated projective modules.
By Lemma 4.1, we can construct Y 1 0 containing x with a subresolution where each I 1 k is a small subset of I k . Note that the quotient of (1) by (2) yields a projective resolution of X 0 /Y 1 0 of length n, so X 0 /Y 1 0 ∈ P n . Using Lemma 4.1 again, we can construct a subresolution containing (2) such that X 0 /Y 3 0 ∈ P n . We keep applying Lemma 4.1 to get an ascending chain of subresolutions of (1): . . . Now we take the union of this ascending chain and get an exact complex Since each j I j k is a small subset of I k , we have that the previous sequence is a P ≤κ 0 -subresolution of (1). Note also that the quotient of (1) by (4) yields a projective resolution of X 0 /Y 0 of length n. Then Y 0 ∈ P ≤κ n . In a similar way, we can show that Y m ∈ P ≤κ n and X m /Y m ∈ P n , for every m ∈ Z. Hence, Y ∈ ex P ≤κ n . Note also that the quotient of exact complexes is exact, so we also have X/Y ∈ ex P ≤κ n .
By Lemma 5.1, we can construct a subresolution where x ⊆ Y 1 0 , and each S 1 k is a small and pure submodule of F k . As we did in the previous case, applying Lemma 5.1 infinitely many times, we can get an ascending chain of subresolutions 0 −→ S 1 n −→ · · · −→ S 1 Taking the union of these subresolutions yields an exact sequence where each j S j k is a small and pure submodule of F k , and so it is flat and the quotient F k / j S j k is also flat. Then we have that Y 0 ∈ F ≤κ n and X 0 /Y 0 ∈ F n (take the quotient of (1') by (2')). In a similar way, we can show that Y m ∈ F ≤κ n and X m /Y m ∈ F n , for every m ∈ Z. It follows Y ∈ ex F ≤κ n and X/Y ∈ ex F n .
In both cases, we have constructed a subcomplex Y ⊆ X containing x such that Y ∈ ex A ≤κ n and X/Y ∈ ex A n .
Remark 5.2. Note that using the staircase zig-zag method, there is no need to assume that R is right noetherian.
6 dw-n-projective and dw-n-flat model structures In this section we obtain two model structures on Ch(R−Mod), one from the two complete cotorsion pairs (dw P n , (dw P n ) ⊥ ) and (ex P n , (ex P n ) ⊥ ), and the other one from (dw F n , (dw F n ) ⊥ ) and (ex F n , (ex F n ) ⊥ ).
Recall that a subcategory D of an abelian category C is said to be thick if the following two conditions are satisfied: (a) D is closed under retracts, i.e., given a sequence with g • f = id D ′ and D ∈ D, then D ′ ∈ B. For example, the class E of exact complexes is thick.
The following theorem by Hovey describes how to get an abelian model structure from two complete cotorsion pairs. in [8]. We shall use the following result from [4] to show that (dw P n , (dw P n ) ⊥ ) and (ex P n , (ex P n ) ⊥ ) are compatible cotorsion pairs. In the previous lemma, put (C, D ′ ) = (dw A n , (dw A n ) ⊥ ) and (U, V) = ( ⊥ E, E). In [4, Lemma 5.1] it is proven that ( ⊥ E, E) is cogenerated by a set, so it is complete. In [7], it is proven that ⊥ E is the class of dg-projective complexes, where a complex X is dg-projective if X m ∈ P 0 for every m ∈ Z and every map X −→ Y is nullhomotopic whenever Y is exact. Hence, it is clear that ⊥ E ⊆ dw A n , i.e. U ⊆ C. Setting