Kaplansky classes of complexes

In this paper, we introduce and study Kaplansky classes of complexes. We give some results by which one can construct many Kaplansky classes of complexes. We also give some relations between Kaplansky classes of complexes and cotorsion pairs.


Theorem 1.1 Let ℵ ≥ |R| be a cardinal number. If F is a strongly ℵ-Kaplansky class of R-modules, then the following statements hold.
(1) F is an ℵ-Kaplansky class of complexes.
(2) If F is closed under direct limits, then E ∩ #F is an ℵ-Kaplansky class of complexes.
We also study some relations between Kaplansky classes of complexes and cotrosion pairs by showing the following result (see Theorem 3.10). In particular, as an application of the above results, we prove the following result (see Example 4.9).

Theorem 1.3 Let R be a commutative Noetherian ring and C a semidualizing R-module. Then (
and (E ∩ #A C , (E ∩ #A C ) ⊥ ) are perfect cotorsion pairs, where A C is the Auslander class with respect to C.
This paper consists of four sections. In Sect. 2, we will recall some notions which are necessary for our proofs of the main results of this paper. In Sect. 3, we will study Kaplansky classes of complexes. In particular, we will prove Theorem 1.1 and Theorem 1.2 as above. In Sect. 4, we will consider some consequences of the results in Sect. 3. In particular, Theorem 1.3 will be proved in this section.

Preliminaries
Throughout this paper, R denotes an associative ring with identity and all modules are unitary left R-modules unless otherwise stated. The letter ℵ always denotes an infinite cardinal number.

2.1
Let A be an abelian category with enough projectives and injectives. Given a class F of objects of A. Following [8], a morphism φ : An F-precover and an F-cover of X are defined dually. It is immediate that envelopes and covers, if they exist, are unique up to isomorphism.

2.2
Given a class F of objects of A, write F ⊥ = {C ∈ Ob(A)| Ext 1 (F, C) = 0 for all F ∈ F} and ⊥ F = {C ∈ Ob(A)| Ext 1 (C, F) = 0 for all F ∈ F}. A pair (F, G) of classes of objects of A is called a cotorsion pair (or cotorsion theory) [12] if F ⊥ = G and ⊥ G = F. Two simple examples of cotorsion pairs in the category of R-modules are (Proj, R-Mod) and (R-Mod, Inj), where Proj (resp., Inj) is the class of projective (resp., injective) R-modules. A cotorsion pair (F, G) is said to be cogenerated by a set X of objects of A if X ⊥ = G. A cotorsion pair (F, G) is said to be perfect if every object of A has an F-cover and a G-envelope.

A (cochain) complex · · ·
/ / X −1 by (X, δ X ) or simply X. The nth boundary (resp., cycle, homology) of X is defined as Imδ n−1 (resp., Kerδ n , Kerδ n /Imδ n−1 ) and it is denoted by B n (X ) (resp., Z n (X ), H n (X )). A complex X is called exact if Z n (X ) = B n (X ) (or equivalently, H n (X ) = 0) for any n ∈ Z. We let E denote the class of exact complexes. C will denote the abelian category of complexes of R-modules. This category has enough projectives and injectives (see, e.g., [15,Proposition 3.2]). If X and Y are both complexes of R-modules, then by a map (or morphism) f : X / / Y of complexes we mean a sequence of R-homomorphisms f n : X n / / Y n such that f n+1 δ n X = δ n Y f n for each n ∈ Z, and f is denoted by { f n } n∈Z . Following [10], Hom R (X, Y ) denotes the set of maps of complexes from X to Y and Ext i R (X, Y ) (i ≥ 1) are right derived functors of Hom (these extension functors are not the same as those defined in [3]).
Let F be a class of R-modules. A complex X is called a #-F complex [20] if all terms X i are in F for i ∈ Z, and a complex X is called an F-complex [15] if X is exact and all cycle modules Z i (X ) are in F for i ∈ Z. Then F-complexes are #-F complexes whenever F is closed under extensions. The classes of #-F complexes and F-complexes will be denoted by #F and F, respectively.

2.4
Given a class F of R-modules, we say that F is an ℵ-Kaplansky class if, for every F ∈ F and every x ∈ F, there exists an S ∈ F with x ∈ S ⊆ F and |S| ≤ ℵ and F/S ∈ F. Also F is called a Kaplansky class if it is an ℵ-Kaplansky class for some cardinal number ℵ.
Similarly, we give the following definition.

2.5
Given a class F of complexes, we say that F is an ℵ-Kaplansky class if, for every F ∈ F and every x ∈ F k (k ∈ Z arbitrary), there exists a subcomplex T of F such that x ∈ T k , T, F/T ∈ F and |T | ≤ ℵ. Also F is called a Kaplansky class if it is an ℵ-Kaplansky class for some cardinal number ℵ.

Kaplansky classes of complexes
In this section, we study Kaplansky classes of complexes defined in (2.5). Doing so will lead us to the notion of strongly Kaplansky classes of modules, which we will need in order to prove Theorems 3.3 and 3.5.
Definition 3.1 Let F be a class of R-modules. We say that F is a strongly ℵ-Kaplansky class if, for every M ∈ F and every K ≤ M with |K | ≤ ℵ, there exists an R-module S such that K ≤ S ≤ M and |S| ≤ ℵ, and S, M/S ∈ F. Also F is called a strongly Kaplansky class if F is a strongly ℵ-Kaplansky class for some cardinal ℵ. Let ℵ ≥ |R| be a cardinal number. If F is an strongly ℵ-Kaplansky class of R-modules, then F is an ℵ-Kaplansky class. In the following, we show that if F is a strongly ℵ-Kaplansky class of R-modules then F is an ℵ-Kaplansky class of complexes. However, we do not know whether F (strongly) ℵ-Kaplansky class implies that F is (strongly) ℵ-Kaplansky.
be any complex in F. Without loss of generality, we may assume that x ∈ F 0 . Then δ 0 (Rx) ≤ Kerδ 1 ∈ F and |δ 0 (Rx)| ≤ ℵ, and so there exists an R-module S 1 such that δ 0 (Rx) ≤ S 1 ≤ Kerδ 1 and |S 1 | ≤ ℵ, and then there exists an R-module S 0 such that Ker(δ 0 | A 0 ) ≤ S 0 ≤ Kerδ 0 and |S 0 | ≤ ℵ, and S 0 , (Kerδ 0 )/S 0 ∈ F. Then we get an exact sequence By continuing in this way, this time for Ker(δ 0 | A 0 +S 0 ), we can get an exact sequence Now, by the construction above, one can check easily that T is a subcomplex of F such that x ∈ T 0 , T, F/T ∈ F and |T | ≤ ℵ. This implies that F is an ℵ-Kaplansky class of complexes.
With Remark 3.2(2), we get the following corollary.

Theorem 3.5 Let ℵ ≥ |R| be a cardinal. If F is a strongly ℵ-Kaplansky class of R-modules closed under direct limits, then E ∩ #F is an ℵ-Kaplansky class of complexes.
Proof Let F be a strongly ℵ-Kaplansky class of R-modules. Then, for every M ∈ F and every K ≤ M with |K | ≤ ℵ, there exists an R-module S such that K ≤ S ≤ M, |S| ≤ ℵ and S, M/S ∈ F. Now let be any complex in E ∩ #F. Without loss of generality, we may assume that x ∈ F 0 . Note that Rx ≤ F 0 ∈ F and |Rx| ≤ ℵ, then there exists an R-module S 0 1 such that |S 0 ). Continue this process, we get an exact sequence ). Continue this process, we get an exact sequence of R-modules such that |(F2) i | ≤ ℵ for any i ∈ Z. Obviously, F1 ≤ F2 ≤ F. Note that A 0 2 ≤ F 0 ∈ F and |A 0 2 | ≤ ℵ, then there is an R-module S 0 3 such that |S 0 3 | ≤ ℵ, A 0 2 ≤ S 0 3 ≤ F 0 and S 0 3 , F 0 /S 0 3 ∈ F. Using the similar argument as above we get an exact sequence If we continue this "Zig-Zag" procedure, we can get exact sequences Fm for all m ∈ N such that Fi ≤ F j for i ≤ j, x ∈ (Fm) 0 and |Fm| ≤ ℵ for all m ∈ N (since |(Fm) i | ≤ ℵ for any i ∈ Z). Furthermore, for each i ∈ Z, there are infinitely many m with (Fm) i ∈ F.
Let T = lim − → Fm. Then T ≤ F is exact such that x ∈ T 0 , T i ∈ F and |T i | ≤ ℵ 0 · ℵ = ℵ for any i ∈ Z, and so |T | ≤ ℵ. Finally, Then, for each i ∈ Z, (F/T ) i ∈ F since there are infinitely many m such that (F/Fm) i ∈ F by construction. Thus F/T ∈ E ∩ #F.  In the following, we consider some relations between Kaplansky classes and cotorsion pairs. Recall that a continuous chain of subcomplexes of a given complex C is a set of subcomplexes of C, {C α | α < λ} (for some ordinal number λ), such that C α is a subcomplex of C β for all α ≤ β < λ, and that C γ = α<γ C α whenever γ < λ is a limit ordinal.
The next lemma was originally stated and proved for the category of modules (see [6] or [7]). In fact, it is also true in the category of complexes or more general categories (without modifying the proofs given in [6] or [7]) (see, for example, [12, Proposition 3.1.1]).

Lemma 3.8 Let X and Y be complexes of R-modules. If X is the direct union of a continuous chain {X α | α < λ} of subcomplexes for an ordinal number
The following lemma can be proved using a similar method as proved in [11,Theorem 2.8].

Lemma 3.9 Let F be a Kaplansky class of complexes closed under extensions and well ordered direct limits.
Then the pair (F, F ⊥ ) is cogenerated by a set. Furthermore, every complex has an F ⊥ -envelope.
The "module version" of the next result was given in [11,Theorem 2.9].

Applications
In this section, we consider some consequences of the results in Sect. 3.