A new approach to projectivity in the categories of complexes

Recently, several authors have adopted new alternative approaches in the study of some classical notions of modules. Among them, we find the notion of subprojectivity which was introduced to measure in a way the degree of projectivity of modules. The study of subprojectivity has recently been extended to the context of abelian categories, which has brought to light some interesting new aspects. For instance, in the category of complexes, it gives a new way to measure, among other things, the exactness of complexes. In this paper, we prove that the subprojectivity notion provides a new sight of null-homotopic morphisms in the category of complexes. This will be proven through two main results. Moreover, various results which emphasize the importance of subprojectivity in the category of complexes are also given. Namely, we give some applications by characterizing some classical rings and establish various examples that allow us to reflect the scope and limits of our results.


Introduction
In this paper, we will work mainly on an abelian category with enough projectives, although we will find the biggest applications in Sect. 4 on the category of modules over an associative ring with unit. The connection between abelian categories and module categories is well known from Gabriel's theorem: an abelian category is equivalent to a module category if and only if it is cocomplete and has a finite projective generator (see, for instance, [1, page 211]).
Throughout the paper, A will denote an abelian category with enough projectives and R will denote an associative (non-necessarily commutative) ring with a unit element 1 R ∈ R . The category of left R-modules will be denoted by R−Mod . Modules are, unless otherwise explicitly stated, left R-modules.
The notion of subprojectivity was introduced in [2] as a new treatment in the analysis of the projectivity of a module. However, the study of the subprojectivity goes beyond that goal and, indeed, provides, among other things, a new and interesting perspective on some other known notions. Flatness has also been studied recently with a similar approach in [3], by defining general (sub)domains and then studying their particularities related to flatness. However, the results presented in [3] are quite different from those we give in this paper, mainly due to significant differences between the classes of flat and projective modules.
An alternative perspective on the projectivity of an object of an abelian category A with enough projectives was investigated in [4], where, in addition, it was shown that subprojectivity can be used to measure characteristics different from the projectivity and that subprojectivity domains may not be restricted to a single object. On the contrary, the subprojectivity domains of a whole class of objects can be computed, giving rise to very interesting characterizations. For instance, the subprojectivity domain of the whole class of DG-projective complexes is very useful to measure the exactness of complexes (see [4,Proposition 2.5]).
Recall that, given two objects M and N of A , M is said to be N-subprojective if for every epimorphism g ∶ B → N and every morphism f ∶ M → N , there exists a morphism h ∶ M → B such that gh = f , or equivalently, if every morphism M → N factors through a projective object (see [4,Proposition 2.7]). The subprojectivity domain of any object M, denoted −1 A (M) , is defined as the class of all objects N such that M is N-subprojective, and the subprojectivity domain of a whole class ℭ of A , −1 A (ℭ) , is defined as the class of objects N such that every C of ℭ is N-subprojective.
In this paper, we go deeper in the investigation of subprojectivity in the category of complexes of A which has enough projectives since A is supposed to have enough projectives. In this sense, when studying subprojectivity of complexes, it is observed that the concept of subprojectivity is relatively closely linked to that of null-homotopy of morphisms. Therefore, what we intend in the two main results of this paper (Theorems 1 and 2) is to deepen the understanding of this relationship. Namely, in Theorem 1, we prove that if is the homotopy category of C(A) ) for every short exact sequence of complexes 0 → K → P → N → 0 with P projective. The proof of this theorem is based on a new characterization of the subprojectivity of an object in any abelian category with enough projectives in terms of the splitting of some particular short exact sequences (Proposition 1).
The second main result of the paper (Theorem 2) assures that for any two complexes M and N with N n+1 ∈ −1 A (M n ) for every n ∈ ℤ , the conditions N ∈ (P[n]) for every n ∈ ℤ , where P is a projective generator of A (Corollary 1). Motivated by this result, we asked whether subprojectivity can measure the exactness of a complex N at each N i . In fact, we prove that, for any complex N and any n ∈ ℤ , N ∈ . This result allows us to answer two interesting questions. Namely, we provide an example showing that the subprojectivity domains are not closed under kernel of epimorphisms (see Example 2). And, we give an example showing that the equivalence of Theorem 1 mentioned above does not hold in general if we replace the condition "P is projective" with P ∈ −1 C(A) (M) (see Remark 1 and Example 3). The necessity and the importance of the conditions given in the main Theorems 1 and 2 are deeply discussed in Propositions 2 and 5, respectively, and Example 1. It is worth noting that semisimple categories (in the sense that every object is projective) are also characterized in terms of subprojectivity. In fact, this was a consequence of the study of the condition " N n+1 ∈ −1 A (M n ) for every n ∈ ℤ " assumed in Theorem 2. Namely, we prove that the category A must be semisimple when this condition implies the condition N ∈ Finally, Section 4 is devoted to some applications. Namely, we give, as consequences of Theorem 1, some new characterizations of some classical rings. In Proposition 10, we characterize left hereditary rings in terms of subprojectivity as those rings for which every subcomplex of a DG-projective complex is DG-projective. Furthermore, we do it without the condition "Every exact complex of projective modules is projective" needed in [5,Proposition 2.3].
Following the same context, subprojectivity also makes it possible to characterize rings of weak global dimension at most 1, and using subprojectivity domains we prove that these rings are the ones over which subcomplexes of DG-flat complexes are always also DG-flat (Proposition 11). As a consequence, left semi-hereditary rings are also characterized in terms of subprojectivity (Corollary 2).

Preliminaries
In this section, we fix some notations from [6] and recall some definitions and basic results that will be used throughout this article. By a complex X of objects of A , we mean a sequence of objects and morphisms such that d n d n+1 = 0 for all n ∈ ℤ . If Im d n+1 = ker d n for all n ∈ ℤ , then we say that X is exact, and given an object M of A , X is said to be Hom A (M, −)-exact if the complex of abelian groups Hom A (M, N) is exact. We denote by X n ∶ X n → Im d n the canonical epimorphism and by X n ∶ Ker(d n−1 ) → X n−1 the canonical monomorphism. The n th boundary (respectively, cycle, homology) of a complex X is defined as ), and it is denoted by B n (X) (respectively, Z n (X) , H n (X)).
Throughout the paper, we use the following particular kind of complexes:

Disc complex.
Given an object M, we denote by M the complex with all terms 0 except M in the degrees 1 and 0. Sphere complex. Also, for an object M, we denote by M the complex with all terms 0 except M in the degree 0.

Shift complex.
Let X be a complex with differential d X and fix an integer n. We denote by X[n] the complex consisting of X i−n in degree i with differential Now, by a morphism of complexes f ∶ X → Y , we mean a family of morphisms f n ∶ X n → Y n such that d Y n f n = f n−1 d X n for all n ∈ ℤ . The category of complexes of A will be denoted by C(A) . In particular, the category of complexes of modules over the ring R will be denoted by C(R).
A morphism of complexes f ∶ X → Y is said to be null-homotopic if, for all n ∈ ℤ , there exist morphisms s n ∶ X n → Y n+1 such that for any n we have f n = d Y n+1 s n + s n−1 d X n , and then we say that f is null-homotopic by s. For a complex X, id X is null-homotopic if and only if X is of the form ⊕ n∈ℤ M n [n] for some family of objects M n . A complex of this special type is called contractible.
Two morphisms of complexes f and g are homotopic, f ∼ g in symbols, if f − g is null-homotopic. The relation f ∼ g is an equivalence relation. The homotopy category K(A) is defined as the one having the same objects as C(A) , and which morphisms are homotopy equivalence classes of morphisms in C(A).
For complexes X and Y, we let Hom • (X, Y) denote the complex of abelian groups with

Subprojectivity and null-homotopy
As mentioned in the introduction, subprojectivity of complexes is closely related to null-homotopy of morphisms of complexes and kernels of epimorphisms. The aim of this section is to deepen the understanding of this relationship.
We start with a new characterization of subprojectivity in terms of splitting short exact sequences which will be considered somehow as the subprojectivity analogue of the classical characterization of projectivity. We fix the following notation: the pullback of two morphisms g ∶ C → B and f ∶ A → B will be denoted by (D, g � , f � ).

Proposition 1 Let
A be an abelian category with enough projectives. If M and N are two objects of A, the following conditions are equivalent.
, there exists a morphism h ∶ M → K such that the following diagram commutes Then, by the universal property of pullbacks, there exists a morphism k ∶ M → D such that g � k = id M . Hence g ′ splits, as desired.
2. ⇒ 3. This is clear since the category A is supposed to have enough projectives.
3. ⇒ 4. This is clear since every projective object belongs to The following two lemmas will be useful in the proof of Theorem 1. N) , then there exist two morphisms ∶ P → N and ∶ M → P such that P is projective and f = (see [4,Proposition 2.7]). Now, id P is null-homotopic since P is contractible; thus, the composition id P is null-homotopic. Therefore,

Lemma 1 For two complexes M and N with
Proof Let ∶ X → A n and ∶ X → C n be two morphisms of A such that f n = g n and consider the two morphisms of complexes ∶ X[n − 1] → A and ∶ X[n − 1] → C induced by and , respectively. It is straightforward to verify that f = g , so there exists a unique morphism of complexes h ∶ X[n − 1] → D such that g � h = and f � h = . Then, g � n h n = and f � n h n = . The unicity of h n ∶ X → D n comes from the unicity of h. ◻ Now, we give the first main result of the paper.

Theorem 1
Let M and N be two complexes such that N n ∈ −1 A (M n ) for every n ∈ ℤ . Then, the following statements are equivalent.
2. For every short exact sequence 0 → K → P → N → 0 with P projective, the equation There exists a short exact sequence 0 → K → P → N → 0 with P projective such that n ∈ ℤ , respectively. Hence, the third column is also exact. Now, applying the Snake Lemma to the following commutative diagram with exact rows and columns we get the exact sequence . This is not true in general. Indeed, we can always consider, over a non-semisimple ring R, two modules X and Y with Y ∉ theless, the answer to the question would be positive if we assume, furthermore, that N belongs to  Another natural question at this point is whether the inverse implication of Proposition 2 is true or not. Namely, given two complexes M and N, is the condition " N n ∈ −1 A (M n ) for every n ∈ ℤ ", sufficient to assure that N ∈ −1 C(A) (M) ? Again, this is not true in general since, for instance, for exact complexes of modules it only holds over left hereditary rings (see Proposition 10).
We have studied so far the relation between subprojectivity and null-homotopic morphisms involving kernels of epimorphisms. We will now see that this relation can also be described without considering such kernels (Theorem 2).
We start by characterizing when a contractible complex holds in the subprojectivity domain of another complex. We need the following lemma.

Lemma 4 Let M be a complex, N be an object of
The following result characterizes subprojectivity in terms of factorization of morphisms through contractible complexes and through complexes in subprojectivity domains.

Proposition 4
Let M and N be two complexes. The following conditions are equivalent.  3. ⇒ 2. Apply Proposition 3. ◻ Notice that conditions 1. and 2. of Proposition 4 are equivalent in any abelian category with enough projectives.

Every morphism M → N factors through a complex of
Lemma 5 Let f ∶ X → Y be a null-homotopic morphism of complexes by a morphism s. If every morphism s n ∶ X n → Y n+1 of A factors through an object L n+1 , then f ∶ X → Y factors through the contractible complex ⊕ n∈ℤ L n+1 [n] .
Proof Suppose that for any n there exist two morphisms n ∶ X n → L n+1 and n ∶ L n+1 → Y n+1 such that s n = n n . Then, we have the situation For every n ∈ ℤ , let p 1 n+1 ∶ L n+1 ⊕ L n → L n+1 and p 2 n ∶ L n+1 ⊕ L n → L n be the canonical projections, and k 1 n+1 ∶ L n+1 → L n+1 ⊕ L n and k 2 n ∶ L n → L n+1 ⊕ L n be the canonical injections. Now, call Z the complex ⊕ n∈ℤ L n+1 [n] and consider, for every n ∈ ℤ , the two morphisms of A h n ∶ L n+1 ⊕ L n → Y n given by h n = d Y n+1 n p 1 n+1 + n−1 p 2 n , and g n ∶ X n → L n+1 ⊕ L n given by g n = ( n , n−1 d X n ) . We claim that both h ∶ Z → Y and g ∶ X → Z are morphisms of complexes.
For any n ∈ ℤ , we have d Y n h n = d Y n (d Y n+1 n p 1 n+1 + n−1 p 2 n ) = d Y n n−1 p 2 n , and h n−1 d Z n = (d Y n n−1 p 1 n + n−2 p 2 n−1 )k 1 n p 2 n = d Y n n−1 P 1 n k 1 n p 2 n = d Y n n−1 p 2 n , so h is a morphism of complexes, and for any n ∈ ℤ we have so g is also a morphism of complexes.
Now we see that f = hg since for any n ∈ ℤ we have Therefore, f ∶ X → Y factors through the contractible complex Z = ⊕ n∈ℤ L n+1 [n] . In the following result, we prove that this condition suffices for exact complexes if and only if A is semisimple.

Proposition 5
The following conditions are equivalent.

1.
A is semisimple. (M n ) for every n ∈ ℤ . This is not true in general. For instance, in the category of R-modules, if we take any non-projective module X and choose any other module Y out of the subprojectivity domain of X (such modules exist over any non-semisimple ring). Then, the complex Y [2] belongs to

For every object M of
However, if we add the condition " N ∈  ◻ From now on we will assume in this section that A has a projective generator P. If we let M = P in Proposition 8, then the condition "N is Hom A (P, −)-exact" means that N is exact ( since P preserves and reflects exactness by its definition). This leads to the following characterization of exact complexes in terms of subprojectivity. Corollary 1 Let P be a projective generator of A and N be a complex. The following assertions are equivalent.
There is now a natural question which comes to mind after Corollary 1: we have described, for the projective generator P, how the subprojectivity domain of the set of complexes {P[n], n ∈ ℤ} is, so, what about the subprojectivity domain of each of the complexes P[n] ? Can we describe them as well?
Given a complex N, we know, by Theorem 2, that N ∈ Now, with Proposition 9 in hand, it is easy to see that subprojectivity domains are not closed under kernels of epimorphisms in general.

Example 2 Consider the short exact sequence of complexes
where P is the projective generator of A . It is clear by Proposition 9 that P [1] and P both hold in −1 C(A) (P) , but P does not. Therefore, the subprojectivity domain of P is not closed under kernels of epimorphisms.
Moreover, Proposition 9 helps us to answer a question raised in Remark 1. Precisely, it is understood by the equivalence (1 ⇔ 4) in Theorem 1 that the second assertion remains equivalent to the first assertion even if we replace the condition "Q is projective" with Q ∈ −1 C(A) (M) . However, this fact does not hold true. Namely, the following example shows that if we replace "Q is projective" with Q ∈

Applications
Recall that a the ring R is said to be left hereditary if any left R-submodule of a projective left R-module is projective. Recall also that a complex P is said to be DG-projective if its components are projective and Hom • (P, E) is exact for every exact complex E. In [5,Proposition 2.3] it is proved that, under certain conditions, a ring is left hereditary if and only if every subcomplex of a DG-projective complex is DG-projective. Among these conditions, the authors included: "Every exact complex of projective modules is projective". In this section, using the properties of subprojectivity domains, we will show that the latter equivalence holds without the mentioned assumption.