A construction of relatively pure submodules

In [BEE01] the flat cover conjecture for module categories has been solved positively, giving two independent approaches to prove the existence of flat covers. The first one due to Enochs uses certain properties of cotorsion theories. It seems to be better understood than the second one and has been used to generalize the existence theorems for flat covers to other categories; see for instance [A + 01]. The second method due to Bican and El Bashir is based on a complex set theoretic argument, where relatively pure submodules are constructed. This line of thought has been generalized to Grothendieck categories in [El 06, Thm. 2.1], where a different proof has been given. To our knowledge these argument have not been used very widely, although there are several applications as [Kra10] and [HJ08] show. The reluctance to use these arguments seems to be related to the fact that the proof given in [BEE01] is short and difficult to understand. We try to remedy this fact by giving a detailed explanation of the proof and several corollaries. The main theorem 4.2 is a slight generalizations of the classical theorem by Bican and El Bashir. As our goal is to popularize the approach of Bican and El Bashir we show how to deduce this result from the classical theorem. In addition an interesting application will be discussed in section 5. However, the main results were already known (i.e. they appeared as [BEE01, Theorem 5], resp. [CPT10, Theorem 2.1]). Nevertheless, the author thinks that detailed proofs are indispensable to understand these important theorems. Hence our aim is to remedy the lack of such proofs for these important results.

We fix an arbitrary small additive category C and let S be a skeleton for this category, i.e. a set of representatives of the objects Ob C. 2.2 Definition (projectively generated purities) A be a class of modules in Mod C. We consider the class σ A of all short exact sequences in Mod C, for which all A ∈ A are projective, i.e. for every η ∈ σ A the induced sequence Hom Mod C (A, η) is exact in Ab for all A ∈ A. The class σ A will be called a purity projectively generated by A.

taken to an exact sequence in Ab by the functor Hom Mod C (A, -) for every A ∈ A.

We remind the reader that the representable functors Hom(•, A), A ∈ C form a generating set of projective objects for Mod C. Thus Mod C has enough projective objects. The next Lemma is preparatory in nature and will come in handy to prove the theorem of Bican and El Bashir:

Let A be some set of modules in Mod C and N M be in Mod C. Fix for every A ∈ A a short exact sequence ρ A : 0 → K ι → P → A → 0 with P some projective object. The submodule N is σ A -pure, if and only if for every pair of morphisms s : K → N and s : P → M inducing a commutative diagram

there is a homomorphism r : P → N , such that rι = s.

) is an exact sequence in Ab and we derive a morphism k : A → M , such that πk = s . The required morphism exists by the homotopy Lemma (cf. [JL94, Lemma B1]. Conversely let N be module with the above property and we wish to show that Hom Mod C (A, η) is exact. Since the functor is left exact it suffices to show that Hom(A, π) is an epimorphism. To this end let f : A → Coker ι be an arbitrary morphism. Now P is projective, which implies that there is a morphism k : P → M with πk = f a and since N is the kernel of π, there is another morphism k : K → N such that ιk = ka. The module N has the above property with respect to the pair of morphisms k , k, there is r : P → N such that ra = k . Again by the Homotopy Lemma, there is a morphism w : A → M such that Hom(A, π)(w) = f proving our claim.

3 Revisiting the approach of Bican and El Bashir to flat covers

Throughout this section we are working in a module category Mod R over some (unital) ring R. We may think of R as a category with a single object and therefore Mod R is in a canonical way a functor category over a small additive category with one object.

Viewing Mod R as a special case of Mod C we may consider the objects defined in the last section in the setting of a module category. It is then easy to see that the notion of projectively generated purity coincides with the definition of Bican and El Bashirs paper (cf. [BEE01]). We may now formulate the main Theorem due to Bican and El Bashir:

3.1 Theorem Let R be an arbitrary (unital) ring and σ a purity projectively generated by some set of modules A. For every cardinal λ there is a cardinal κ, such that for every module M in Mod R and L M with |M | ≥ κ and |M/L| ≤ λ the submodule L contains another non trivial submodule, which is σ-pure in M .

Proof of Theorem 3.1. In order to avoid problems later on, we may assume that the set A is non-empty and contains at least one non trivial R-module, i.e. A = {0}. For every A ∈ A fix a short exact sequence 0 → U A → V A → A → 0 with a projective module V A . Choose a set of elements X A which generates U A and a set Y A generating V A as an R-module. As a technical requirement

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