COVERS INDUCED BY EXT PAUL C. EKLOF AND JAN TRLIFAJ Abstract. We prove a generalization of the Flat Cover Conjecture by showing for any ring R that (1) each (right R-) module has a Ker Ext(−, C)-cover, for any class of pure-injective modules C, and that (2) each module has a Ker Tor(−, B)-cover, for any class of left R-modules B. For Dedekind domains, we describe Ker Ext(−, C) explicitly for any class of cotorsion modules C; in particular, we prove that (1) holds, and that Ker Ext(−, C) is a cotilting torsion-free class. For right hereditary rings, we prove the consistency of the existence of special Ker Ext(−, G)-precovers for any set of modules G. 1. Introduction A classical result of Eckmann-Schopf says that if I is the class of all injective (right R-) modules, then each module has an I-envelope. Bass proved that if P is the class of all projective modules, then each module has a P-cover iff R is a right perfect ring. Bass’ result is often interpreted as a lack of duality for modules over non-right perfect rings. Call a module M a dual module provided that there are a ring S, an R, Sbimodule N , and an injective cogenerator, Q, of Mod-S such that M ∼ = HomS (N, Q) (as right R-modules). There are two important instances of dual modules: if S = Z and Q = Q/Z, then the dual module is called the character module of the left R-module N ; if R is a k-algebra over a field k and S = Q = k, then any module M which is finite dimensional over k is a dual module. A well-known result (cf. [15]) says that the dual module M is injective iff N is a flat left R-module. So a natural candidate for dualizing the Eckmann-Schopf result (to arbitrary rings) is obtained by replacing P by F, the class of all flat modules. This led Edgar Enochs [11] to formulate the Flat Cover Conjecture (FCC): “every module over every ring has an F-cover”. Only recently the conjecture has been proved, independently by Enochs and El Bashir [4]. Enochs’ proof proceeds by showing that the hypothesis of Corollary 11 of the authors’ [10] is true for any ring R. The heart of his argument is a proof that there is a cardinal κ (depending only on R) such that every flat R-module A is the union of an increasing continuous sequence (Aα | α < σ) of pure submodules (for some σ depending on A) such that for all α + 1 < σ, card(Aα+1 /Aα ) ≤ κ and Aα+1 /Aα is 1991 Mathematics Subject Classification. Primary: 16E30, 18G25; Secondary: 03C60, 03E45, 13F05, 16D40, 16E60, 18G15, 20K40. Key words and phrases. Precover, cover, right approximation, Ext, Tor, pure-injective module, Dedekind domain, Axiom of Constructibility, Flat Cover Conjecture. Research partially supported by grants NSF DMS 98-03126, GAČR 201/97/1162, and CEZ: J13/98:113200007. 1 2 PAUL C. EKLOF AND JAN TRLIFAJ flat. The hypothesis of [10, Cor. 11] then follows as in the proof of Corollary 10 below. This motivates the following definition Definition 1. For any right R-module A and any cardinal κ, a κ-refinement of A (of length σ) is an increasing sequence (Aα | α ≤ σ) of pure submodules of A such that A0 = 0, Aσ = A, Aα = ∪β<α Aβ for all limit ordinals α ≤ σ, and card(Aα+1 /Aα ) ≤ κ for all α + 1 ≤ σ. In homological terms, the FCC says that every module has an A-cover, where A is the kernel of the contravariant Ext functor Ext(−, C) and C is the class of all dual modules (or, respectively, A is the kernel of the covariant Tor functor Tor(−, B) and B is the class of all left R-modules. Precise definitions are given in the next section.) Using κ-refinements, we will generalize the FCC by replacing C by any class of pure-injective modules (resp., replacing B by any class of left R-modules. See Corollaries 10 and 11). In Theorem 16, we prove that C can be any class of cotorsion modules when R is a Dedekind domain; in that case, we also give a full description of the kernel. Assuming Gödel’s Axiom of Constructibility (V = L), we prove existence of special Ker Ext(−, G)-precovers for any set of modules G provided that R is a right hereditary ring (Theorem 14). 2. Preliminaries For a ring R, denote by Mod-R the category of all right R-modules. We will use “module” to mean right R-module. Also, Hom, Ext and Tor will stand for HomR , Ext1R and TorR 1 , respectively. Definition 2. Let C ⊆ Mod-R and let B be a class of left R-modules. We define ⊥ C = Ker Ext(−, C) = {D | Ext(D, C) = 0 for all C ∈ C}, and similarly C ⊥ = Ker Ext(C, −) = {D | Ext(C, D) = 0 for all C ∈ C}, Ker Tor(−, B) = {A | Tor(A, B) = 0 for all B ∈ B}. For a module C, we will write ⊥ C instead of ⊥ {C}. We start by recalling a lemma relating κ-refinements to the vanishing of Ext: Lemma 3. Let C be a module. Suppose that A = Aµ is the union of a continuous chain of submodules, A = ∪α<µ Aα such that A0 ∈ ⊥ C and for all α + 1 < µ, Aα+1 /Aα ∈ ⊥ C. Then A ∈ ⊥ C. proof. Well-known (see [7, Thm. 1.2], [12, Lemma IV.2.1], or [10, Lemma 1]). ✷ We will often use the following notions and facts concerning precovers and covers: COVERS INDUCED BY EXT 3 Definition 4. Let A ⊆ Mod-R and M ∈ Mod-R. A homomorphism φ ∈ Hom(A, M ) with A ∈ A is called an A-precover of M if the induced map Hom(A′ , A) → Hom(A′ , M ) is surjective for all A′ ∈ A. An A-precover φ ∈ Hom(A, M ) is an A-cover provided that each ψ ∈ Hom(A, A) satisfying φ = φψ is an automorphism of A.1 A precover φ is called special provided that Ker(φ) ∈ A⊥ and φ is surjective. A (special) A-preenvelope and an A-envelope are defined dually; see [22, §1.2]. If φ : A → M is surjective, A ∈ A and Ker(φ) ∈ A⊥ , then φ is a special A-precover of M (see [22, 2.1.3]). Furthermore, by [22, 2.1.1 and 2.2.12], we have: Theorem 5. Let A ⊆ Mod-R be a class containing all projective modules and closed under direct limits and extensions. Assume that a module M has an Aprecover. Then M has an A-cover, the A-cover is special, and it is uniquely determined up to isomorphism. A submodule A of a module B is a pure submodule (A ⊆∗ B, for short) if for each finitely presented module F , the functor Hom(F, −) preserves exactness of the sequence 0 → A → B → B/A → 0. (See, for example, [17, pp. 53f] or [14, pp. 94ff].) We will need the following properties of pure submodules: Lemma 6. Let R be a ring and let κ ≥ card(R) + ℵ0 . (i) Let M be a module and X be a subset of M with card(X) ≤ κ. Then there is a pure submodule N ⊆∗ M such that X ⊆ N and card(N ) ≤ κ. (ii) Assume C ⊆∗ A and B/C ⊆∗ A/C. Then B ⊆∗ A. (iii) Assume A0 ⊆ · · · ⊆ Aα ⊆ Aα+1 ⊆ . . . is a chain of pure submodules of M . Then ∪α Aα is a pure submodule of M . proof. Well-known (see [14, Theorem 6.4]). ✷ For convenience we state a consequence of [10, Theorem 10] in the terminology of this paper. Theorem 7. If C is a class of modules such that (⊥ C)⊥ = Q⊥ for some module Q, then every module has a special ⊥ C-precover. proof. This follows from [10, Theorem 10] because (⊥ C, (⊥ C)⊥ ) is a cotorsion theory and to say that it has enough projectives is to say that every module has a special ⊥ C-precover. ✷ 3. Covers induced by Ext and Tor Modules that are injective with respect to pure embeddings are called pureinjective [14, §7]. For example, any dual module is pure injective. Theorem 8. Let R be a ring and C be a class of pure-injective modules. Let κ = card(R) + ℵ0 . Then the following conditions are equivalent for any module A: (i) A ∈ ⊥ C; (ii) there is a cardinal λ such that A has a κ-refinement (Aα | α ≤ λ) with Aα+1 /Aα ∈ ⊥ C for all α < λ. 1 Here, we follow the terminology of Enochs and Xu [22]. The corresponding terminology of Auslander, Reiten and Smalø (e.g. in [2]) is that of a right approximation and a minimal right approximation. 4 PAUL C. EKLOF AND JAN TRLIFAJ proof. (i) implies (ii): Let κ = card(R) + ℵ0 . If card(A) ≤ κ, we can let λ = 1, A0 = 0, and A1 = A. So we can assume that card(A) > κ. Let λ = card(A). Then A∼ = F/K where F = R(λ) is a free module. We enumerate the elements of F in a λ-sequence: F = {xα | α < λ}. By induction on α, we will define a sequence (Aα | α ≤ λ) so that for all α ≤ λ, Aα is pure in A and belongs to ⊥ C. Since each C ∈ C is pure-injective, it will follow from the long exact sequence induced by 0 → Aα → Aα+1 → Aα+1 /Aα → 0 ⊥ that Aα+1 /Aα ∈ C for all α < λ. Aα will be constructed so that it equals (R(Iα ) + K)/K for some Iα ⊆ λ such that R(Iα ) ∩K is pure in K. Let A0 = 0. Assume Aβ has been defined for all β < σ. Suppose first that σ = α + 1. By induction on n < ω we will define an increasing chain F0 ⊆ F1 ⊆ . . . and then put Aα+1 = ∪n<ω (Fn + K)/K. We require that card(Fn+1 /Fn ) ≤ κ for all n < ω, and furthermore: for n odd, (Fn + K)/K is pure in F/K; for n even, Fn = R(Jn ) for some Jn ⊇ Jn−2 ⊇ · · · ⊇ J0 and Fn ⊇ Kn′ where Fn−1 ∩ K ⊆ Kn′ ⊆∗ K. First, put F0 = R(Iα ) and let J0 = Iα and K0′ = R(Iα ) ∩ K. Assume Fn−1 has been constructed and n is odd. By part (i) of Lemma 6 there is a pure submodule (Fn + K)/(Fn−2 + K) ⊆∗ F/(Fn−2 + K) of cardinality ≤ κ containing (xα R + Fn−1 + K)/(Fn−2 + K). Moreover, we can choose Fn so that card(Fn /Fn−1 ) ≤ κ. By part (ii) of Lemma 6, (Fn + K)/K is pure in F/K. Assume n > 0 is even. We first define Kn′ : by part (i) of Lemma 6, we find a pure ′ ′ ′ . of cardinality ≤ κ containing (Fn−1 ∩ K)/Kn−2 ⊆∗ K/Kn−2 submodule Kn′ /Kn−2 ′ This is possible since Kn−2 ⊇ Fn−3 ∩ K and (Fn−1 ∩ K)/(Fn−3 ∩ K) embeds in Fn−1 /Fn−3 , so it has cardinality ≤ κ. By part (ii) of Lemma 6, we have Kn′ ⊆∗ K. We can choose Jn ⊆ λ such that card(Jn − Jn−2 ) ≤ κ and Fn−1 + Kn′ ⊆ R(Jn ) = Fn . This is possible since card((Fn−1 + Kn′ )/Fn−2 ) ≤ κ; indeed, we have the exact sequence 0 → Fn−1 /Fn−2 → (Fn−1 + Kn′ )/Fn−2 → (Fn−1 + Kn′ )/Fn−1 → 0 and (Fn−1 + Kn′ )/Fn−1 ∼ = Kn′ /(Fn−1 ∩ K) has cardinality ≤ κ because it is a ′ . homomorphic image of Kn′ /Kn−2 Now, define Aα+1 = ∪n<ω (Fn + K)/K and Iα+1 = ∪n<ω J2n . By part (iii) of Lemma 6, Aα+1 ⊆∗ A. Clearly, card(Aα+1 /Aα ) ≤ κ. We have Aα+1 ∼ = F ′ /K ′ , where F ′ = ∪n<ω F2n and K ′ = F ′ ∩ K. Also, F ′ = (Iα+1 ) ′ is free, and K ′ = ∪n<ω K2n is pure in K by construction and part (iii) of R Lemma 6. Let C ∈ C. In order to prove that Ext(Aα+1 , C) = 0, we have to extend any f ∈ Hom(K ′ , C) to an element of Hom(F ′ , C). First, f extends to K, since K ′ ⊆∗ K and C is pure-injective. By the assumption (i), we can extend further to F , and then restrict to F ′ . Finally, if σ ≤ λ is a limit ordinal, let Aσ = ∪β<σ Aβ ; that Aσ has the desired properties follows from Lemma 3 and part (iii) of Lemma 6. (ii) implies (i): This is clear by Lemma 3. ✷ Lemma 9. If A ⊆ Mod-R is equal to ⊥ C for a class C of pure-injective modules, then every module M which has an A-precover has an A-cover. proof. This follows from Theorem 5 and the following observation of AngeleriMantese-Tonolo-Trlifaj: Assume P is a pure-injective module. Then ⊥ P is closed COVERS INDUCED BY EXT 5 under homomorphic images of pure epimorphisms. The canonical map of a direct sum onto a direct limit is well-known to be a pure epimorphism (cf. [21, 33.9(2)]). So ⊥ P is closed under direct limits. ✷ Corollary 10. Let R be a ring and C be a class of pure-injective modules. Then every module has a ⊥ C-cover. proof. Let κ = card(R)+ℵ0 . Denote by H the direct sum of a representative set of the class {A | card(A) ≤ κ & Ext(A, C) = 0}. Clearly, (⊥ C)⊥ ⊆ H ⊥ . Conversely, take D ∈ H ⊥ . Let A ∈ ⊥ C; by Theorem 8, A has a κ-refinement (Aα | α ≤ λ). By choice of H, Ext(Aα+1 /Aα , D) = 0 for all α < λ and hence, by Lemma 3, Ext(A, D) = 0. So D ∈ (⊥ C)⊥ . This proves that (⊥ C)⊥ = H ⊥ . By Theorem 7, every module has a special ⊥ C-precover. An application of Lemma 9 finishes the proof. ✷ If C is the class of all pure-injective modules then ⊥ C is the class of all flat modules, so Corollary 10 implies the FCC. However, in general, ⊥ C will be larger than the class of flat modules. Theorem 8 and Corollary 10 remain true for any notion of “pure” that satisfies properties (i) – (iii) in Lemma 6. For example, this happens for the RD-purity [12, II.§3]; hence we get analogous results for the particular case when C is a class of RD-injective modules. There is an analogue of Theorem 8 for the bifunctor Tor: Corollary 11. Let R be a ring and B be any class of left R-modules. Let κ = card(R) + ℵ0 . The following conditions are equivalent for any module A: (i) A ∈ Ker Tor(−, B), (ii) there is a cardinal λ such that A has a κ-refinement (Aα | α ≤ λ) such that Aα+1 /Aα ∈ Ker Tor(−, B) for all α < λ. proof. For each B ∈ B, let C(B) = HomZ (B, Q/Z) be the character module of B. Put C = {C(B) | B ∈ N }. Then C is a class of pure-injective modules. By [6, Proposition VI.5.1], ⊥ C = Ker Tor(−, B), so the assertion follows from Theorem 8. ✷ Theorem 12. Let R be a ring. (i) Let B be a class of left R-modules. Then every module has a Ker Tor(−, B)cover. (ii) Let D be a class of dual modules. Then every module has a ⊥ D-cover. proof. (i) As above, we have A = Ker Tor(−, B) = ⊥ C where C is a class of pure-injective modules. Then every module has an A-cover by Corollary 10. (ii) Since any dual module is pure-injective, every module has a ⊥ D-cover by Corollary 10. ✷ Taking B to be the class of all left R-modules, we obtain the FCC again, this time as a consequence of Theorem 12(i). Corollary 13. (i) Let k be a field and R be a k-algebra. Let M be a class of k-finite dimensional modules. Then every module has a ⊥ M-cover. (ii) Assume that R is a right pure-semisimple ring. Let M be any class of modules. Then every module has a ⊥ M-cover. 6 PAUL C. EKLOF AND JAN TRLIFAJ proof. (i) Since any finite dimensional module is dual (in the k-vector space duality), the assertion follows from Theorem 12(ii). (ii) Since every R-module is pure-injective (see [14, Theorem 8.4]) this follows from Corollary 10. ✷ 4. Hereditary rings By [10, Theorem 10], every module has a special M ⊥ -preenvelope, for any module M . When M is pure-injective, Theorem 8 (for the class C = {M }) yields the dual assertion that every module has a special ⊥ M -precover. It is an open problem (even for R = Z) whether for every M (or even for M = Z), every module has a special ⊥ M -precover. However, we can prove a consistency result in the case when R is a right hereditary ring: Theorem 14. Assume V = L. Let R be a right hereditary ring and G be a set of Q modules. Let κ = G∈G card(G) + card(R) + ℵ0 . (i) Let A be a module of cardinality ρ > κ such that A ∈ ⊥ G. Then there is a κ-refinement (Aα | α ≤ ρ) such that Aα+1 /Aα ∈ ⊥ G for all α < ρ. (ii) Every module has a special ⊥ G-precover. Q proof. Replacing G by G∈G G, we can, without loss of generality, assume that G = {G} for a single module G. Part (i) is then a consequence of Theorem 5.5(2), page 50, of [8], which is proved there for the ring Z, but which has the same proof for any hereditary ring R. The sequence (Aν | ν ≤ cf(ρ)) given there has quotients Aν+1 /Aν which are of cardinality < ρ, but by induction on ρ ≥ κ+ , we can refine this sequence by inserting between Aν and Aν+1 , whenever card(Aν+1 /Aν ) > κ, a sequence (Aν,τ | τ ≤ ρν ) such that ρν = card(Aν+1 /Aν ), Aν,0 = Aν , Aν,ρν = Aν+1 and for all τ < ρν , card(Aν,τ +1 /Aν,τ ) ≤ κ. Moreover, one can check that each member of the refined sequence (Aν,τ | ν ≤ cf(ρ), τ ≤ ρν ) has fewer than ρ predecessors, and hence the whole sequence has length ρ. (See also [3, Thm. 3.1] where the result is proved for G = {R}.) Part (ii) follows as in Corollary 10. ✷ Remark 15. (1) The proof does not extend to proper classes of modules because of the dependence of κ on the cardinality of G. This contrasts with Theorem 8, where κ = card(R) + ℵ0 does not depend on C. Also, Theorem 14 (ii) cannot be improved to claim the existence of ⊥ G-covers. Indeed, if R is right hereditary, but not right perfect, and G = {F } where F is the free module of rank 2card(R) , then (under V = L) ⊥ G is the class of all projective modules, cf. [20, Theorem 3.13(ii)]. Since R is not right perfect, there exist modules without ⊥ G-covers, by the classical result of Bass. (2) In order to be able to conclude that every module has a special ⊥ G-precover, it is not necessary that the length of the refined sequence be a cardinal, ρ = card(A), rather than just an ordinal. We do not know if it is provable in ZFC (say for R = Z) that for every G there is a κ such that every A satisfying Ext(A, G) = 0 has a κ-refinement (of some length σ) whose factors (i.e. Aα+1 /Aα ) are in ⊥ G. (3) For the case of G = Z = R, in any model of ZFC in which there are non-free Whitehead groups, there exists A ∈ ⊥ Z such that there is no ℵ0 -refinement of A whose factors are in ⊥ Z: take A to be a non-free Whitehead group and use the fact COVERS INDUCED BY EXT 7 that countable Whitehead groups are free. Furthermore, for any explicitly given cardinal κ (e.g. κ is ℵ586 or ℵω1 +ω3+29 ), there is no theorem of ZFC which says that every A ∈ ⊥ Z has a κ-refinement whose factors are in ⊥ Z; this is because there is a model of ZFC in which there are non-free Whitehead groups but every Whitehead group of size ≤ κ is free (see [9, 2.8]). (4) There is a model of ZFC + GCH such that for any non-cotorsion Z-module G, and any κ, there is an A such that Ext(A, G) = 0 but there is no κ-refinement of A of length = card(A) whose factors are in ⊥ G. We use a model of the uniformization principle designated UP in [20, p. 1526]. As in there, or in [19], given κ, we can construct a Z-module A of some cardinality λ > κ such that Ext(A, G) = 0 and A has a λ-filtration ∪ν<λ A′ν such that for a stationary set of ν, A′ν+1 /A′ν ∼ = Q. A standard argument then shows that for any κ-refinement (Aα | α ≤ λ), there is an α < β < λ such that Q is a submodule of Aβ /Aα , and hence Ext(Aβ /Aα , G) 6= 0 since G is not cotorsion. In contrast to Remark 15(4) we have the following theorem for cotorsion modules over Dedekind domains. Recall that a module C is cotorsion if Ext(F, C) = 0 for every flat module F (cf. [22, p. 52]). For example, any pure-injective module is cotorsion. If R is a Dedekind domain, then C is cotorsion iff Ext(Q(R), C) = 0 where Q(R) is the quotient field of R. For a module M , denote by Cog(M ) the class of all modules cogenerated by M (i.e., the class of all submodules of products of copies of M ). A module M is cotilting if ⊥ M = Cog(M ). Cog(M ) is then a torsion-free class, called the cotilting torsion-free class, cf. [5, §1]. Theorem 16. Let R be a Dedekind domain and Spec(R) be the spectrum of R. Let C be a class of cotorsion modules. (i) There is a set SC ⊆ Spec(R) such that ⊥ C = {A ∈ Mod-R | ∀P ∈ SC : R/P * A}. In fact, SC = {P ∈ Spec(R) | ∃C ∈ C : R/P ∈ / ⊥ C}. (ii) There is a class P of pure-injective modules such that ⊥ C = ⊥ P. This is a consequence Q of any one of the following facts for an arbitrary cotorsion module C: (a) ⊥ C = ⊥ {R̂P | P ∈ SC } where R̂P = Hom(E(R/P ), E(R/P )); (b) ⊥ C = ⊥ PE(C) where PE(C) is the pure-injective envelope of C; (c) ⊥ C = ⊥ F where F is the flat cover of C; moreover, F is pure-injective. (iii) ⊥ C is a cotilting torsion-free class and every module has a ⊥ C-cover. proof. (i) Let A be a module. Denote by T (A) the torsion part of A. Since every element of C is cotorsion and A/T (A) is flat, we have A ∈ ⊥ C iff T (A) ∈ ⊥ C. We also have Soc(E(T (A)) E T (A) E E(T (A)) and Soc(E(T (A)) = Soc(T (A)) ∼ = ⊕06=P ∈Spec(R) (R/P )(αP ) for some cardinals αP . By Matlis’ theory [16] (see also [17, Theorem 18.4]) we have E(R/P ) = ∪n<ω P −n (R/P ), so E(R/P ) has an (infinite) composition series with factors isomorphic to R/P , for every 0 6= P ∈ Spec(R). By Lemma 3 we get that T (A) ∈ ⊥ C iff Soc(T (A)) ∈ ⊥ C iff R/P ∈ ⊥ C for all 0 6= P ∈ Spec(R) such that R/P is a submodule of A. Note that R/P ∈ ⊥ C iff P ∈ / SC . It follows that A ∈ ⊥ C iff R/P is not a submodule of A for all P ∈ SC . (ii) (a) Let 0 6= P ∈ Spec(R). By part (i), ⊥ R̂P = {A | ∀Q ∈ SR̂P : R/Q * A}, / ⊥ R̂P }. where SR̂P = {Q ∈ Spec(R) | R/Q ∈ 8 PAUL C. EKLOF AND JAN TRLIFAJ By Matlis’ theory, if q ∈ R \ P , then q. is an automorphism of E(R/P ), and hence of R̂P . Since R̂P = Hom(E(R/P ), ⊕Q∈Spec(R) E(R/Q)), R̂P is pure-injective and flat, but not injective. Since q. is a monomorphism of E(R̂P ) we infer that the torsion module MP = E(R̂P )/R̂P is q-torsion-free. We also have Ext(R/Q, R̂P ) = Hom(R/Q, MP ) for all 0 6= Q ∈ Spec(R). It follows that Ext(R/Q, R̂P ) = 0 for all Q 6= P . Since Soc(MP ) 6= 0 and Soc(MP ) is a direct sum of copies of R/P , we get Ext(R/P, R̂P ) 6= 0. Q This proves that SR̂P = {P }. If J = {R̂P | P ∈ SC }, then J is pure-injective and SJ = SC , so ⊥ C = ⊥ J by part (i). (b) By part (i) it suffices to show that for all P in Spec(R), R/P ∈ ⊥ C if and only if R/P ∈ ⊥ PE(C). But C is elementarily equivalent to PE(C) ([18]; see also [14, Thm 7.51]). Once we show that there is a first-order sentence θP in the language of R-modules such that for any module M , Ext(R/P, M ) = 0 if and only if M |= θP , we are done. Now P is generated by two elements, say p1 , p2 , and is finitely P2 presented; say the relations are generated by { i=1 rij pi = 0 | j = 1, . . . , m}. Also, Ext(R/P, M ) = 0 if and only if every homomorphism from P to M extends to a homormorphism from R to M . Therefore Ext(R/P, M ) = 0 if and only if M |= ∀x1 ∀x2 [( 2 m X ^ rij xi = 0) ⇒ (∃y(∧2i=1 pi y = xi ))]. j=1 i=1 (c) Since C is cotorsion, F is flat and cotorsion, hence pure-injective [22, Lemma 3.2.3]. For each P ∈ Spec(R), denote by RP the localization of R at P , by PP the (unique) maximal ideal Q of RP , and by k(P ) the residue field RP /PP . By [22, Theorem 4.1.15], F ∼ = P ∈Spec(R) TP , where TP is the completion of a free RP module of rank πP in the PP -adic topology. The cardinals πP (P ∈ Spec(R)), are uniquely determined by C, and called the 0-th dual Bass numbers of C, [22, §5.2]. Xu’s formula for computing dual Bass numbers [22, Theorem 5.2.2] gives πP = dimk(P ) k(P )⊗RP Hom(RP , C). In particular, πP = 0 iff k(P )⊗RP Hom(RP , C) = 0 iff Im(νP ⊗RP 1) = Hom(RP , C), where νP is the embedding of PP into RP . The latter is equivalent to PP .Hom(RP , C) = Hom(RP , C). Since RP is a noetherian valuation domain, the ideal PP is principal, PP = s.RP for some s ∈ PP . So πP = 0 iff s.Hom(RP , C) = Hom(RP , C). On the other hand, if 0 6= P ∈ Spec(R), then R/P ∼ = k(P ) as R-modules, so R/P ∈ ⊥ C iff Hom(νP , C) is surjective. The latter is equivalent to s.Hom(RP , C) = Hom(RP , C), and hence to πP = 0. It follows that SC = {P ∈ Spec(R) | P 6= 0 & πP 6= 0}. By [22, Lemma 4.1.5], TP ∼ = Hom(E(R/P ), E(R/P )(πP ) ), so q. is an automorphism of TP for each q ∈ R \ P , and as in part (a), we get STP = {P } whenever P 6= 0 and πP 6= 0. Since ST0 = ∅, we infer that SF = SC , so ⊥ C = ⊥ F by part (i). (iii) By part (ii) and Corollary 10, every module has a special ⊥ C-cover. Since ⊥ C is closed under submodules and products, [1, Theorem 2.5] gives that ⊥ C is a cotilting torsion-free class. ✷ In [13, §2], cotilting torsion-free classes of abelian groups were characterized. We have the following for modules over Dedekind domains: Corollary 17. Let R be a Dedekind domain and T be a class of modules. Then the following conditions are equivalent: COVERS INDUCED BY EXT 9 (i) T is a cotilting torsion-free class such that T is closed under direct limits. (ii) There is a set of non-zero prime ideals, P, such that T = {A ∈ Mod-R | ∀P ∈ P : R/P * A}. proof. (i) implies (ii): We have T = ⊥ C for a cotilting module C. Since T is closed under direct limits and contains all projective modules, C is cotorsion. By part (i) of Theorem 16, we can take P = SC . (ii) implies (i): By the proof of part (ii) of Theorem 16, we have SR̂P = {P } Q for each non-zero prime ideal P . So T = ⊥ {R̂P | P ∈ P}, and T is a cotilting torsion-free class closed under direct limits by (the proof of) part (iii). ✷ 5. Open Problems (1) Characterize the rings R such that for each M ∈ Mod-R, every module has a special ⊥ M -precover. By Theorem 14, this is the case for any right hereditary ring R assuming Gödel’s Axiom of Constructibility (V = L). Also, this is true in ZFC in the case when R is right pure-semisimple, by Corollary 13(ii). (2) Denote by W the class of all Whitehead groups, [9]. Does every abelian group have a special W-precover (in ZFC)? This is a particular case of (1) for R = M = Z. Under V = L, every Whitehead group is free, so the answer is positive. (3) Can Theorem 16 be extended to wider classes of rings (such as Prüfer domains or commutative Noetherian rings of finite Krull dimension)? In particular, for which rings is it the case that for every class C of cotorsion modules, every module has a special ⊥ C-precover? References [1] L. Angeleri Hügel, A. Tonolo and J. Trlifaj, Tilting preenvelopes and cotilting precovers, Algebras and Representation Theory 3(2000), to appear. [2] M. Auslander and S. Smalø, Preprojective modules over artin algebras, J. Algebra 66 (1980), 61-122. [3] T. Becker, L. Fuchs, and S. Shelah, Whitehead modules over domains, Forum Math. 1 (1989), 53–68. [4] L. Bican, R. 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Xu, Flat Covers of Modules, Lecture Notes in Mathematics No. 1634, Springer (1996). (Eklof) Department of Mathematics, UCI, Irvine, CA 92697-3875 E-mail address: peklof@math.uci.edu (Trlifaj) Katedra algebry MFF UK, Sokolovská 83, 186 75 Prague 8, Czech Republic E-mail address: trlifaj@karlin.mff.cuni.cz