Well Generatedness and Adjoints for Homotopy Categories of N-complexes

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We follow the approach of Saorín and Stovicek [31], where the key notions are that of deconstructible and decomposable classes of modules (see Definition 2.3).
Let B be an additive category and C, a full subcategory of B. We say that C is precovering if every object B of B has a C-precover, i. e., a morphism f : X → B with X ∈ C, such that Hom B (X , f ) is an epimorphism for each X ∈ C. We say that C is coreflective in B if the inclusion functor C → B has a right adjoint. This is equivalent to every object B of B having a coreflection, that is, a C-precover f : X → B with Hom(X , f ) bijective for every X ∈ C.
The definition of well generation commonly used for triangulated categories is based on the notion of small object. Suppose that B has arbitrary direct sums and let κ be an infinite cardinal. An object B of B is κ-small if for any family of objects, {B i | i ∈ I }, and any morphism f : B → i∈I B i , there exists a subset I ⊆ I of cardinality less than κ such that f factors through i∈I B i . The definition of well generated category that we use here is [31,Definition 4.6], which is equivalent in triangulated categories to Neeman's original definition [28] (see the comment before [31,Definition 4.6]). Definition 2.1 Let B be an additive category with arbitrary direct sums and κ an infinite regular cardinal. We say that B is κ-well generated if there exists a set of objects S of B such that: (1) For each non-zero object B of B there exists an object S ∈ S and a non-zero morphism f : S → B. (2) Each object in S is κ-small.
(3) For any morphism in B of the form f : S → i∈I B i with S ∈ S and {B i | i ∈ I } ⊆ B, there exists a morphism f i : S i → B i with S i ∈ S for each i ∈ I , such that f factorizes as

Remark 2.2
Notice that in (2) of [31,Definition 4.6] it is claimed that every object S in S is κ-small relative to the class of all split monomorphisms, i. e., Hom(S, −) preserves direct limits of κ-sequences of split monomorphisms. This condition is equivalent to the notion of κ-smallness used in this paper, since the direct limit of a κ-sequence of split monomorphisms is the direct sum of the objects in the direct system.  [7]. If for any conflation we call f an inflation and g a deflation, the axioms essentially say that the identity morphism of every object is an inflation and a deflation, that inflations and deflations are closed under compositions, that the pushout of any inflation along any morphism is an inflation, and that the pullback of any deflation along any morphism is a deflation. Through this paper we assume that all exact categories satisfy:

Assumption 1 Transfinite compositions of inflations exist and are inflations.
This assumption guarantees that there exist direct sums in A and that direct sums of conflations are conflations [31,Lemma 1.4].
We introduce two types of classes of objects which will play a fundamental role in our results: decomposable and deconstructible classes. Deconstructible classes in exact categories were introduced in [31]. (1) We say that A is G-decomposable if there exists a family of objects of G, {G i | i ∈ I }, such that A ∼ = i∈I G i . We denote by Sum(G) the class consisting of all G-decomposable objects and, if κ is a cardinal, by Sum κ (G) the class consisting of all G-decomposable objects which are isomorphic to a direct sum of less than κ objects belonging to G. (2) We say that F is decomposable if there exists a set of objects S ⊆ F such that F = Sum(S). (3) We say that A is G-filtered if A is the direct limit of a λ-sequence in A, (G α , u αβ | α < β < μ) satisfying that u αβ is an inflation and Coker u αβ ∈ G for each α < β. We denote by Filt E (G) or simply Filt(S) (if the exact structure is understood) the class consisting of all G-filtered objects. When u αβ are inclusions between modules or complexes, we simply denote the filtration by (G α | α < κ).
We will use the following immediate characterization of decomposable and deconstructible classes. Let (A; E) be a Frobenius exact category, that is, an exact category with enough injectives and projectives and such that the classes of all projective and of all injective objects coincide. The stable category associated to (A; E) is the category A whose class of objects is A and whose set of morphisms Hom A (A, B), for each pair of objects A, B ∈ A, is the quotient group Hom A (A, B)/I (A, B), where I (A, B) is the subgroup of all morphisms that factors through an injective object. If A and f is an object and a morphism, respectively, in A, we denote by A and f the corresponding object and morphism in the stable category. The same notation is used for subcategories of A.
The stable category can be turned into a triangulated category in the following way: • For any object A ∈ A, fix a conflation with E A injective, and define the suspension functor as A = C A . • For any morphism f : A → B in A, we can construct the following commutative diagram in which the left-hand square is the pullback of u A and f . The sequence in S, is called a standard triangle. The exact triangles in A are those sequences isomorphic in A to a standard triangle.
Recall that a full subcategory X of a triangulated category T is closed under cones Now we consider N -complexes over the additive category B for N a natural number greater than or equal to 2. An N -complex is a sequence of morphisms in B Given a natural number 0 ≤ r ≤ N and an integer n, we denote by d ) the category whose objects are all unbounded N -complexes (resp. bounded below, bounded above, bounded) with components in B and whose morphisms are all cochain maps.
Given a complex X in C N (B) and an integer n, we denote by τ ≥n X the hard truncation of X , i. e., the complex of C + N (B) given by (τ ≥n X ) i = X i and d i We consider the class S N (B) consisting of all semi-split conflations in C N (B), that is, those kernel-cokernel pairs in C N (B) which are degreewise split exact. We call the inflations (resp. deflations) associated to this exact structure semi-split monomorphisms (resp. semi-split epimorphisms). It is well known that A cochain map f between N -complexes X and Y is called null-homotopic if there exists, for each i ∈ Z, a morphism s i : The homotopy category of unbounded cochain N -complexes is the category K N (B) whose objects are all N -complexes and such that, for each N -complexes X and Y , coincides with the stable category of the Frobenius exact category (C N (B), S N (B)). Following the notation fixed for stable categories, we denote by A, f and X the corresponding object, morphisms and subcategory in K N (B) of an object A, morphism f and subcategory X of B.
All rings in this paper are associative with unit and not necessarily commutative, and all modules are right modules. Given a cardinal number κ, every < κ-generated module trivially is κ-small in the sense of Definition 2.
One of the main technical tools we will use is the deconstruction of a class of complexes. This procedure is based on the Hill Lemma, which we state now for module categories [20, Theorem 4.2.6]. We will not need the more general version for Grothendieck categories [32, Theorem 2.1].

H2) H is closed under arbitrary sums and intersections.
(H3) For any N , P ∈ H with N ≤ P, P N is S-filtered. (H4) For any N ∈ H and X ⊆ M with cardinality smaller than κ, there exists P ∈ H such that N ∪ X ⊆ P and P N is < κ-presented. Now assume that B = Mod−R for some ring R. In this case, since C N (Mod−R) is an abelian category, we have the abelian exact structure which consists of all kernelcokernel pairs which are degreewise exact. Let X ∈ C N (Mod−R) be an N -complex. We denote: Given a class of modules F, we denote by C N (F) the class of all N -complexes X with X n ∈ F and by A N (F) the class of all N -acyclic complexes in C N (F). Let us state some properties of N -acyclic complexes.

Lemma 2.6
Let R be a ring. (2) First, notice that for every s ∈ {1, . . . , N −1} with s ≤ r and n ∈ Z, the equality In particular, the hypotheses of (2) imply that this is true for s = N − r . Now, take x ∈ X n and notice that is trivially closed under unions. The 2-out-of-3 property is proved by diagram chasing and using (1) of this lemma.
As in module categories, we can define κ-presented complexes in C N (Mod−R) for every infinite regular cardinal κ. These complexes are κ-small as the following result shows, which essentially is the extension of [29, Lemma 4.5] to any cardinal κ and N > 2.

Lemma 2.7 Let R be a ring, κ a cardinal with uncountable cofinality and X
Proof (1) Take a family of N -complexes {X i : i ∈ I } and a cochain map f : X → i∈I X i . For each integer n, since X n is < κ-generated, there exists a subset I n of I with cardinality less than κ such that f n factors through i∈I n X n i in Mod−R. Setting J = n∈Z I n , we obtain a subset of I with cardinality less than κ (because κ has uncountable cofinality) such that that f factors through j∈J X j .
As in the case of modules, if X is a deconstructible class of complexes such that X = Filt(S) for a set consisting of < κ-presented complexes for some regular cardinal κ, we say that X is κ-deconstructible.

Coreflective Subcategories of Homotopy Categories of N-complexes
In this section, we prove that certain subcategories of K N (Mod−R) are coreflective.
Since the homotopy category of N -complexes is the stable category of some Frobenius exact category, we first establish a characterization of coreflective subcategories of stable categories using ideas from [9, Section 5], where a similar result is obtained for stable categories of abelian categories. Let (A; E) be a Frobenius exact category. We assume that A has split idempotents, which means that for every object A of A and idempotent endomorphism e of A, there exists an object B and morphisms p : A → B and i : B → A with pi = 1 B and i p = e. This condition implies that retracts and direct summands are the same thing (in general, every direct summand is a retract and the converse is true when the category has split idempotents). In general, A need not have split idempotents, even if A has. However, in some common cases, A does have split idempotents [9,Proposition 5.9].
Let X be an additive subcategory of A (i. e., a full subcategory containing the zero object and closed under finite direct sums). If A and B are objects in A which are isomorphic in A, then A is isomorphic in A to a direct summand of B ⊕ P for some projective object P. In particular, if X contains all projective modules and is closed under direct summands, then A ∈ X provided that B ∈ X .
We now consider some closure properties of additive subcategories, which will allow us to characterize certain coreflective subcategories of the stable category. (1) X is precovering in A if and only if X is precovering in A.
(2) If X is closed under direct summands, then X is closed under cones if and only if X is closed under cokernels of inflations (i. e., every inflation X → Y with X , Y ∈ X has its cokernel in X as well).
Proof (1) Clearly, if X is precovering in A, then X is precovering in A. Conversely, let A be an object in A and take an X -precover of A, u : X → A. Since there are enough projective objects, there exists a deflation v : P → A in A with P projective. Then, it is easy to see that u ⊕ v : X ⊕ P → A is an X -precover of A.
(2) Suppose that X is closed under cones and take a conflation [22, 2.7], this conflation induces a triangle in A, which, by hypothesis, satisfies that Z ∈ X . Conversely, take a standard triangle Since the left-hand square is a pushout, there exists, by [7, Proposition 2.12], a conflation 0 Using that E X ⊕ Y ∈ X and that X is closed under cokernels of conflations, we conclude that Z ∈ X as required. Finally, from this and the fact that X is closed under direct summands, it is easy to see that any exact triangle with its first two terms in X satisfies that the cone is in X as well.
Using this result and [9, Corollary 4.5] we immediately get: (A; E) be a Frobenius exact category with split idempotents such that the stable category A has split idempotents as well. Let X be an additive subcategory of A closed under direct summands and containing all projective objects. The following assertions are equivalent: (1) X is a coreflective subcategory of A and X ⊆ X .
(2) X is precovering in A and closed under under cokernels of inflations, i. e., if is a conflation with X 1 and X 2 belonging to X , then X 3 belongs to X as well.
Applied to the homotopy category of N -complexes, this result gives our criterion for coreflectivity:

Corollary 3.3 Let R be a ring and X , a full subcategory of C N (Mod−R) which is precovering, closed under cokernels of semi-split monomorphisms and that contains all disks. Then X is a coreflective subcategory of K N (Mod−R) that satisfies X ⊆ X .
Proof First, notice that C N (Mod−R) is Grothendieck and, consequently, has split idempotents. Moreover, K N (Mod−R) has split idempotents as a consequence of [9, Proposition 5.9]. Now, since X is precovering it is closed under direct sums and, as it contains all disks, it contains all contractible complexes by [ [8,Theorem 3.3] for different proofs in the category of modules). So, we are going to prove that certain classes of N -complexes are deconstructible. The deconstruction of 2-complexes has been treated in [32] and in [10].
First, we prove that C N (F) is deconstructible when F is, extending [32, Proposition 4.3] to N -complexes.

Theorem 3.4 Let R be a ring, κ, an infinite regular cardinal and F, a κ-deconstructible class of modules. Then C N (F) is κ-deconstructible.
Proof First, notice that every C N (F)-filtered module belongs to C N (F), since F is closed under filtrations. In view of Lemma 2.4, we only have to prove that every complex in C N (F) is filtered by < κ-presented complexes belonging to C N (F). The idea of the proof is essentially the one used in [32,Proposition 4.3].
Let X ∈ C N (F) and fix, using the Hill Lemma 2.5, a family of submodules of X n , H n , associated to some F-filtration of X n by < κ-presented modules. We construct the F-filtration (X α | α < μ) of X recursively on α with the additional properties that X n α ∈ H n and X n α+1 X n α being < κ-presented for each n ∈ Z and α < μ. This implies that X α ∈ C N (Mod−R) by (H3) of 2.5, since F is closed under filtrations, and that X α+1 X α is < κ-presented by Lemma 2.7.
If α = 0, set X α = 0. If α is limit, set X α = γ <α X γ . Notice that X n α ∈ H n by (H2) of Hill Lemma 2.5. If α is successor, say α = γ +1, fix n ∈ Z such that X n γ = X n and take x ∈ X n − X n γ . Set X m α = X m γ if m < n. For m = n, use (H4) of Hill Lemma 2.5 to find X n α ∈ H n such that X n γ +x R ≤ X n α and X n α X n γ is < κ-presented. By (H3) of Hill Lemma and the induction hypothesis, this quotient belongs to F as well. For m ≥ n, if X m α has been already constructed, write X m α = X m γ + W m for some < κ-generated module W m , and apply again (H4) of Hill Lemma to find X m+1 This quotient belongs to F by (H3) of Hill Lemma 2.5, which concludes the construction. Now, we see that A N (F) is deconstructible when F is. We need the following extension of [10,Lemma 4.2] to N -complexes, whose proof is essentially the same but with the obvious modifications. Recall that if κ is an infinite regular cardinal greater than the cardinality of R, a right R-module M is < κ-presented if and only if it is < κ-generated if and only if |M| < κ. We will use this fact freely in the proof of the following result. Lemma 3.5 Let R be a ring, κ, an uncountable regular cardinal with |R| < κ and S, a set of < κ-presented modules. Let X be an N -complex in A N (Filt(S)) and let, for each m ∈ Z, H m be the family of submodules associated to some S-filtration of X m (given by the Hill Lemma 2.5). Let Z 1 and Z 2 be subcomplexes of X satisfying:

Then there exists an N -acyclic subcomplex Y of X such that Z
Proof We denote d m X simply by d m . We are going to construct a family of modules, and, consequently, is < κ-generated by (B). In particular, . Suppose that n > 1 and that we have already constructed H m n−1 for each m ∈ Z. Set H m n = 0 for each m < −n. For m = −n, apply again (H4) of Hill Lemma 2.5 to find H −n n ∈ H −n such that Z −n 1 + Z −n 2 ≤ H −n n and If m > −n, we distinguish between n being odd or even.
n is even. Since finite filtration by < κ-generated modules and is < κ-generated as well). In particular, Using that X is N -acyclic, we can find U ≤ X m a < κgenerated submodule such that T ≤ d m (U . This concludes the construction. Finally, denote Y m = n∈N H m n for each m ∈ Z. These modules define an Nacyclic subcomplex Y of X by (E) and Lemma 2.6. Moreover, it contains Z 1 + Z 2 by (B) and (D), and by (H2) of Hill Lemma 2.5, Y m ∈ H m , since H m n ∈ H m for each m ∈ Z and n ∈ N. Now, Y m Z m 1 has a countable filtration by < κ-presented modules as a consequence of (B) and (E). Using that κ is regular and uncountable, Y m Z m 1 is < κ-presented, and so is Y Z 1 by Lemma 2.7. As a consequence of this result we get: Theorem 3.6 Let R be a ring, κ, an uncountable regular cardinal with |R| < κ and F a κ-deconstructible class of modules. Then A N (F) is κ-deconstructible.
Proof First notice that A N (F) is closed under filtrations as a direct consequence of Lemma 2.6. Now, in order to find a set S ⊆ A N (F) of < κ-presented complexes such that every complex in C N (F) is S-filtered, the argument of the proof of [10,Theorem 4.3] works in this setting using Lemmas 2.6 and 3.5.
Combining Theorems 3.4 and 3.6 with Corollary 3.3 we obtain:

Corollary 3.7 Let R be a ring and F be a deconstructible class of modules. Then K N (F) and A N (F) are coreflective subcategories of K N (Mod−R).
Now we apply this result to the classes of projective, injective, flat, Gorensteinprojective, Gorenstein-injective and Gorenstein-flat modules, respectively, which are known to be deconstructible for some types of rings. In particular, we obtain [18, Theorem 3.8] for non-Noetherian rings, and [2, Lemma 4.3], which extends [30,Theorem 3.2] to N -complexes.
Recall that a ring R is right Gorenstein regular if it has finite right global Gorenstein dimension, and is right -pure injective if R (I ) is pure-injective for every set I , that is, it is injective with respect to every pure monomorphism.

Well Generated N-homotopy Categories
In this section we study when certain subcategories of K N (Mod−R) are well generated. One useful result to study the well generatedness of a stable category is [31,Theorem 4.2(4)]. We give here a more direct proof of this fact: Proof We prove that S is the set of generators of Filt(S). Since direct sums in A are computed in A and then reflected into A, the hypotheses of the theorem imply that S satisfies (2) and (3) of Definition 2.1. Let us prove that S satisfies (1) of Definition 2.1. Take X ∈ Filt(S) an object such that the only morphism f : S → X with S ∈ S is the zero morphism. We prove that X = 0.
If α = 0, then X 0 = 0, so that u 0 = 0. Suppose that we have just proved that u α = 0 for some α < μ, and let us see that u α+1 = 0. Since u α = 0, there exists an injective object E and morphisms v : X α → E and w : E → X such that wv = u α . Since u αα+1 is an inflation, we can extend u αα+1 to a v : X α+1 → E. Now notice that (u α+1 − wv )u αα+1 = 0 which implies that, if p : X α+1 → C is the cokernel of u αα+1 , there exists h : C → X with hp = u α+1 − wv . Using the hypothesis and the fact that C ∈ S, we conclude that hp = 0 and, consequently, that u α+1 = 0.
Going back to our setting of N -complexes, in order to find well generated subcategories of the homotopy category, we need to find deconstructible classes of N -complexes in the exact structure defined by the semi-split short exact sequences. While, as we have seen in the preceding section, deconstructible classes of modules give deconstructible classes of N -complexes in the abelian exact structure, we will see that decomposable classes of modules give deconstructible classes of complexes If F is a decomposable class of modules then, in general, C N (F) is not a decomposable class of complexes. As a consequence of Theorem 3.4, C N (F) is deconstructible. What we prove now is that it is deconstructible in the exact cate- . Given an N -complex X and a C N (F)-filtration (X α | α < λ) of X with the inclusion X α → X β a semi-split monomorphism for each α < β, we obtain, for each n ∈ Z, an F-filtration (X n α | α < λ) of X n such that X n α is a direct summand of X n β for each α < β. If we take a submodule Y n α of X n α+1 such that X n α+1 = X n α ⊕ Y n α , then, reasoning as in [21,Lemma 3.3], X n ∼ = α<λ Y n α , so that X n ∈ F and X ∈ C N (F).
The hypotheses of the theorem imply that we can find an infinite regular cardinal κ and a set of < κ-generated modules S ⊆ F such that F = Sum(S). By Lemma 2.4, the proof will be done if we see that every complex in C N (F) is filtered by complexes belonging to C + N (Sum κ (S)) in the exact structure defined by S N (Mod−R). Take a family {F n i : i ∈ I n } ⊆ S such that X n = i∈I n F n i for each n ∈ Z. We are going to prove that X has a filtration, (X α | α < λ), such that, for each n ∈ Z, there exists an ascending chain of subsets of I n , {I n α : α < λ}, satisfying: (1) For each n ∈ Z: I n = α<λ I n α . (2) For each n ∈ Z and α < λ: Then, (X α | α < λ) will be a C + N (Sum κ (S))-filtration of X in the exact structure defined by S N (Mod−R).
We construct the X α and I α recursively on α.
If α = 0, take X 0 = 0 and I n 0 = ∅ for each n ∈ Z. If α is limit, set X α = γ <α X γ and I n α = γ <α I n γ for each n ∈ Z. Notice that X n α = i∈I n α F n i . Suppose that we have constructed the complex X α and the sets I n α for some ordinal α, and let us construct X α+1 and I n α+1 for each n ∈ Z. If X α = X there is nothing to construct. Otherwise, there exists n ∈ Z such that X n α = X n . Then set I m α+1 = I m α and X m α+1 = X m α for each m < n. In order to construct X n α+1 take a non-zero element x ∈ X n − X n α . We can find a finite set J n α such that X n α + x R ≤ i∈I n α ∪J n α F i α , so that we can take I n α+1 = I n α ∪ J n α and X n α+1 = i∈I n α ∪J n α F i α . Now we can construct X m α+1 and I m α+1 recursively on m for m > n. Assume that we have constructed X m α+1 and I m α+1 for some m ≥ n. Since I m α+1 − I m α has cardinality smaller than κ, F m i is < κ-generated for each i ∈ I m and κ is regular, there exists a < κ-generated submodule W of X m+1 such that d m X (X m α+1 ) = X m+1  (F) is κ-well generated by the set C + N (Sum κ (S)), where S is a set of < κ-presented modules satisfying Sum(S) = F. Proof Let S be a set of < κ-presented modules in F such that F = Sum(S). By Theorem 4.2, C N (F) = Filt S N (Mod−R) (C + N (Sum κ (S))). By Lemma 2.7, every module in C + N (Sum κ (S)) is < κ-small. Then, the result will follow from Theorem 4.1 once we prove that C + N (Sum κ (S)) satisfies (3) of Definition 2.1.
Take a morphism f : X → i∈I F i with X ∈ C + N (Sum κ (S)) and F i ∈ C N (F) for each i ∈ I . Using Theorem 4.2, we can find a regular cardinal μ and a filtration (F iα | α < μ) of F i by complexes in C + N (Sum κ (S)) in the exact structure S N (Mod−R). Denote by p j : i∈I F i → F j the canonical projection for each j ∈ I . Since Im p j f is < κ-generated, we can find α j ≤ μ with α j < κ such that Im p j f ≤ F jα j for each j ∈ J . Since α j < κ, F jα j ∈ C N (Sum κ (S)). Now, the family of morphisms { p j f | j ∈ I } induces a morphism g : X → j∈J F jα j and, clearly, the morphism f factors as X j∈J τ ≥m F jα j i∈I F i g where m is the integer satisfying that X n = 0 for each n < m. This concludes the proof.
As a consequence of this result, we can extend [30, Theorem 1.1] to the homotopy category of N -complexes of projective modules.

Corollary 4.4
Let R be a ring. Then, the homotopy category K N (Proj) is ℵ 1 -well generated by the set C + N (P), where P is the set of all countable presented projective modules.
Proof Follows immediately from the preceding corollary by noting that the class of all projective modules is ℵ 1 -decomposable by Kaplansky's theorem. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.