Flat Mittag-Leffler modules over countable rings

We show that over any ring, the double Ext-orthogonal class to all flat Mittag-Leffler modules contains all countable direct limits of flat Mittag-Leffler modules. If the ring is countable, then the double orthogonal class consists precisely of all flat modules and we deduce, using a recent result of \v{S}aroch and Trlifaj, that the class of flat Mittag-Leffler modules is not precovering in Mod-R unless R is right perfect.


Introduction
The notion of a Mittag-Leffler module was introduced by Raynaud and Gruson [10], who used the concept to prove a conjecture due to Grothendieck that the projectivity of infinitely generated modules over commutative rings is a local property. This is a crucial step for defining and working with infinitely generated vector bundles, as considered by Drinfeld in [3], where we also refer for more explanation.
The main step behind this geometrically motivated result is a completely general characterization of projective modules over any (in general noncommutative) ring R. Namely, one can show (consult [3]) that an R-module is projective if and only if M satisfies the following three conditions: (1) M is flat, (2) M is Mittag-Leffler, (3) M is a direct sum of countably generated modules. As mentioned by Drinfeld, the proof of projectivity of a given module might be non-constructive even in very simple cases, because it requires the Axiom of Choice. This applies for instance to the ring Q of rational numbers and the Q-module R of real numbers. The main trouble there is condition (3). Thus, one might consider replacing projective modules by flat Mittag-Leffler modules (these are called "projective modules with a human face" in a preliminary version of [3]).
However, a surprising result in [5, §5] indicates that if one is interested in homological algebra, this might not be a good idea at all. Namely, the class of flat Mittag-Leffler abelian groups does not provide for precovers (sometimes also called right approximations). In the present paper, we use recent results due toŠaroch and Trlifaj [11] to show this is a much more general phenomenon and applies to many geometrically interesting examples. Namely, we prove in Theorem 6 that the class of flat Mittag-Leffler R-modules over a countable ring R is precovering if and only if R is a right perfect ring. Note that in that case the classes of projective modules, flat Mittag-Leffler modules and flat modules coincide, so the flat Mittag-Leffler precovers are just the projective ones.

Preliminaries
In this paper, R will always be an associative, not necessarily commutative ring with a unit. If not specified otherwise, a module will stand for a right R-module. We will denote by D the class of all modules which are flat and satisfy the Mittag-Leffler condition in the sense of [10,8]: A crucial closure property of the class D has been obtained in [8]: Let R be a ring and (F i , u ji : F i → F j ) be a direct system of modules from D indexed by (I, ≤). Assume that for each increasing chain (i n | n < ω) in I, the module lim − →n<ω F in belongs to D.
Let us look more closely at countable chains of modules and their limits. Recall that given a sequence of morphisms we have a short exact sequence such that ϕ is defined by ϕι n = ι n − ι n+1 u n , where ι n : F n → F m are the canonical inclusions. Note the following simple fact: . . of morphisms as above and a number n 0 < ω, the middle term of ( * ) decomposes as where m < n 0 and x m ∈ F m . It follows easily that one can uniquely express each y = (y n ) ∈ n<ω F n as y = z + w, where z ∈ ϕ m<n 0 F m and w ∈ m≥n 0 F m . Namely, we take z = (y 0 , . . . , y n 0 −1 , y ′ n 0 , 0, 0, . . . ) with −y ′ n 0 = u n 0 −1 (y n 0 −1 ) + u n 0 −1 u n 0 −2 (y n 0 −2 ) + · · · + u n 0 −1 u n 0 −2 . . . u 1 u 0 (y 0 ) and w = y − z ∈ m≥n 0 F m .
We will also need a few simple results concerning infinite combinatorics, starting with a well known lemma.
The next lemma deals with a construction of a large family of "almost disjoint" maps f : ω → λ. The result is well known in the literature and it has many different proofs. We refer for instance to [2,Lemma 2.3] or [4, Proposition II.5.5].
Lemma 5. Let λ be an infinite cardinal. Then there is a subset J ⊆ λ ω of cardinality λ ℵ 0 such that for any pair of distinct maps f, g : ω → λ of J, the set formed by the x ∈ ω on which the values f (x) and g(x) coincide is a finite initial segment of ω.
Proof. Consider the tree T of the finite sequences of elements of λ, i.e.

Main result
Now we are in a position to state our main result, which is inspired by [8, §5]. It will be proved by using a cardinal argument similar to the one in [2, Proposition 2.5]. Note that the result sharpens [11,Theorem 2.9] by removing the additional set-theoretical assumption of Singular Cardinal Hypothesis, and also [8, Corollaries 7.6 and 7.7] by removing the assumption that D is closed under products.
Regarding the notation and terminology, given a class C ⊆ Mod-R, we Recall that a module is called cotorsion if it cannot be non-trivially extended by a flat module.
We recall also the notion of a precover, or sometimes called right approximation. If X is any class of modules and M ∈ Mod-R, a homomorphism f : X → M is called an X -precover of M if X ∈ X and for every homomorphism f ′ ∈ Hom R (X ′ , M ) with X ′ ∈ X there exists a homomorphism g : X ′ → X such that f ′ = f g. The class X is called precovering if each M ∈ Mod-R admits an X -precover. Theorem 6. Let R be a ring and D be the class of all flat Mittag-Leffler right R-modules. Given any countable chain of morphisms such that F n ∈ D for all n < ω, we have lim − → F n ∈ ⊥ (D ⊥ ). If, moreover, R is a countable ring, then the following hold: (1) D ⊥ is precisely the class of all cotorsion modules.
(2) D is a precovering class in Mod-R if and only if R is right perfect.
Proof. Assume we have a countable direct system (F n , u n ) as above, put F = lim − → F n , and fix a module C ∈ D ⊥ . We must prove that Ext 1 R (F, C) = 0. Let us fix an infinite cardinal λ, depending on C, such that we have λ ≥ |Hom R (F n , C)| for each n < ω and λ ℵ 0 = 2 λ ; we can do this using Lemma 4. Applying Lemma 5, we find a subset J ⊆ λ ω of cardinality 2 λ such that the values of each pair f, g : ω → λ of distinct elements of J coincide only on a finite initial segment of ω. We claim that there is a short exact sequence of the form such that E ∈ D and |Hom R (P, C)| ≤ 2 λ . Let us construct such a sequence. First, denote for each α < λ by F n,α a copy of F n , and by P the direct sum F n,α taken over all pairs (n, α) such that n < ω and α = f (n) for some f ∈ J. Note that P is a summand in Next, we will construct E. Given f ∈ J, let be the split inclusion which sends each F n to F n,f (n) . Using the short exact sequence ( * ) from page 2, we can extend P by F via the following pushout diagram: Now, we can put these extensions for all f ∈ J together. Namely, let σ : P (J) → P be the summing map and consider the pushout diagram: For each g ∈ J, the composition of the canonical inclusion ν g : E g → f ∈J E f with the morphism π yields a monomorphism E g → E. In fact, if y ∈ E g is such that πν g (y) = 0, then ρ(ν g (y)) = 0, hence the exact sequence ε g gives that y is in the image of P and the composition of the canonical embedding µ g : P → P (J) with the morphism σ is a monomorphism. From now on we shall without loss of generality view these monomorphisms E g → E as inclusions.
To prove the existence of ( †), it suffices to show that E ∈ D in ε. To this end, denote for any subset S ⊆ J by M S the module Then the family (M S | S ⊆ J & |S| ≤ ℵ 0 ) with obvious inclusions forms a direct system and we claim that its union is the whole of E. Indeed, it is straightforward to check, using diagram (∆) and the construction of the embeddings E g ⊆ E, that E = P + f ∈J Im ϑ f . Further, the left hand square of diagram (∆) is a pull-back, which implies P ∩ Im ϑ f = Im ι f and Thus, E = f ∈J Im ϑ f and the claim is proved.
Moreover, the union of any chain M S 0 ⊆ M S 1 ⊆ M S 2 ⊆ . . . from the direct system belongs to the direct system again. Therefore, if we prove that M S ∈ D for each countable S ⊆ J, it will follow from Proposition 2 that E ∈ D. Our task is then reduced to prove the following lemma: Lemma 7. With the notation as above, the following hold: (1) Given S ⊆ T ⊆ J with S and T finite and such that |T | = |S|+1, the inclusion M S ⊆ M T splits and there is n 0 < ω such that M T /M S ∼ = m≥n 0 F m . (2) Given a countable subset S ⊆ J, the module M S is isomorphic to a countable direct sum with each summand isomorphic to some F n , n < ω. In particular, M S ∈ D.
Proof. Let us focus on (1) since (2) is an immediate consequence. Denote by f : ω → λ the single element of T \ S, and let n 0 < ω be the smallest number such that f (n 0 ) = g(n 0 ) for each g ∈ S.
We claim that the following are satisfied by the construction: The second equality holds simply because ϑ f • ϕ = ι f by diagram (∆). For the first, note that M S ∩Im ϑ f as a submodule of E, is contained in P . Since P ∩ Im ϑ f = Im ι f , we have by the construction of P . This proves the claim. Invoking Lemma 3, we further deduce that In particular, the inclusion M S ∩ Im ϑ f ⊆ Im ϑ f splits and so does the inclusion M S ⊆ M S + Im ϑ f = M T . Moreover, we have the isomorphisms which finishes the proof of the lemma.
To finish the proof of Theorem 6, suppose R is a countable ring. Since each F ∈ D is flat, D ⊥ contains all cotorsion modules. On the other hand, if C is not cotorsion, there is a countable flat module F such that Ext 1 R (F, C) = 0; see for instance [6, Theorems 4.1.1 and 3.2.9]. By the first part of Theorem 6, we know that F ∈ ⊥ (D ⊥ ), so C ∈ D ⊥ . Hence D ⊥ consists precisely of cotorsion modules.
The fact that D is not precovering unless R is right perfect (and D is then the class of projective modules) follows directly from [11,Theorem 2.10]. This finishes the proof of Theorem 6.
Remark 8. The proof of Theorem 6 is to some extent constructive. Namely, if R is a countable ring and C is a module which is not cotorsion, the theorem gives us a recipe how to construct E ∈ D such that Ext 1 R (E, C) = 0, and it allows us to estimate the size of E based on the size of C. Note that if R is non-perfect, the size of E must grow with the size of C. This is because for any set S ⊆ D, we have ⊥ (S ⊥ ) ⊆ D Flat-R by [