THE PHANTOM COVER OF A MODULE

IVO HERZOG

Abstract. A morphism f : M → N of left R-modules is a phantom morphism if for any morphism g : A → M, with A finitely presented, the composition fg factors through a projective module. Equivalently, Tor_{1}(X, f) = 0 for every right R-module X. It is proved that every R-module possesses a phantom cover, whose kernel is pure injective.

If \underline{mod}-R is the category of finitely presented right R-modules modulo projectives, then the association M ↦ Tor_{1}(−, M) is a functor from the category of left R-modules to that of the flat functors on \underline{mod}-R. The phantom cover is used to characterize when this functor is faithful or full. It is faithful if and only if the flat cover of every module has a pure injective kernel; this is equivalent to the flat cover being the phantom cover. The question of fullness is only reasonable when the functor is restricted to the subcategory of cotorsion modules. This restriction is full if and only if every phantom cover of a cotorsion module is pure injective.

The notion of a phantom morphism, which has its origins in homotopy theory [21], was introduced into the setting of a triangulated category [22] by Christensen and Strickland [12]. If R is a Quasi-Frobenius ring, as in the case of a finite group ring k[G], then the stable category R-\underline{Mod} has the structure of a triangulated category [10]. Gnacadja [15] calls a morphism f : M → N of k[G]-modules a phantom morphism if the induced morphism \underline{f} : \underline{M} → \underline{N} of stable modules is a phantom in R-\underline{Mod}. Equivalently, every restriction of f to a finitely generated submodule of M factors through a projective module. In this article, we consider the general setting of modules over an associative ring R with identity. A morphism f : M → N of left R-modules is called a phantom morphism if for any morphism g : A → M, with A finitely presented, the composition fg : A → M → N factors through a projective module. Equivalently, the induced morphism Tor_{1}(X, f) : Tor_{1}(X, M) → Tor_{1}(X, N) of abelian groups is 0 for every right R-module X.

This paper is a continuation of our work [17, 18] on pure injective modules. The main result is

Theorem 7. Given a left R-module N, there exists a short exact sequence

[figure: 0; K; k; Ph(N); c; N; 0,]

where the morphism c : Ph(N) → N is the phantom cover of N in R-Mod and K is a pure injective R-module.

The theorem indicates that the notion of a phantom morphism may also be useful in the study of representations of Lie algebras, infinite groups and finite-dimensional algebras that are not necessarily Quasi-Frobenius.

The connection between phantom morphisms and purity was already noticed by Christensen and Strickland (see also [11]), who proved [12, Prop 4.14] the existence of phantom precovers in the category of homotopy spectra. They also described the ideal of phantom morphisms in terms of the first derived functor lim_{←}^{1} of the projective limit functor [16]. For modules over a finite group ring k[G], a similar description was found by Gnacadja [15]. The relationship between phantom morphisms in k[G]-Mod and purity was further investigated by Benson and Gnacadja [7]. Their Proposition 4.2.2 states that a morphism f : M → N of k[G]-modules is a phantom morphism if and only if the induced morphism \underline{f} : \underline{M} → \underline{N} of stable modules corresponds to a pure exact sequence under the natural isomorphism

2000 Mathematics Subject Classification. 16B50, 16E05, 16E30, 18G15.

Key words and phrases. phantom map, phantom cover, stable category, flat functor, flat cover, Brown representability.

Partially supported by NSF Grant DMS05-01207.

\underline{Hom}_{k[G]}(\underline{M}, \underline{N}) ≅ Ext_{k[G]}^{1}(M, Ω(N)).

For a compactly generated triangulated category T , a theory of purity has been developed independently by Beligiannis [4, 5] and H. Krause [20]. They define a morphism c : M → N to be a phantom if for every morphism f : C → M, with C a compact object in T , the composition cf : C → M → N is 0. In their terminology, a triangle

[figure: K; k; M; c; N; e; Σ(K)]

is pure if the morphism c : M → N is phantom. By [22, Lemma 1.1.10], this is equivalent to the morphism k : K → M being a pure epimorphism: for every compact object C, the morphism of abelian groups Hom(C, e) : Hom(C, K) → Hom(C, M) is an epimorphism. In analogy with the classical case of pure exact sequences, this is equivalent to e : N → Σ(K) being a pure monomorphism. If the category T satisfies the Brown Representability Condition, then for every object N ∈ T , there is a triangle ([20, Thm D] and [4, §8]) as above where c : M → N is a phantom cover. Equivalently, the morphism e : N → Σ(K) is the pure injective envelope of N. In that case, the object K is, like its suspension Σ(K), pure injective.

In the category R-Mod of left R-modules, the kernel k : K → P h(N) is a monomorphism, but it is close to being a pure epimorphism, in the sense that if we take any epimorphism g : Z → Ph(N) from a flat module Z, then the coproduct of morphisms k ⨿ g : K ⊕ Z → Ph(N) is a pure epimorphism (Proposition 3).

Because K is pure injective, Theorem 21 of [18] implies that the element of Ext^{1}(N, K) represented by the phantom cover of N corresponds to a morphism Δ : (−, N) → Ext^{1}(−, K) in the category ((R-mod)op, Ab) of contravariant additive functors F : (R-mod)op → Ab from the category of finitely presented left R-modules to the category Ab of abelian groups. If R-\underline{mod} is the category of finitely presented left R-modules, modulo morphisms that factor through a projective, then the natural projection morphism from R-mod to R-\underline{mod} induces an exact embedding of functor categories

((R-\underline{mod})op, Ab) ⊆ ((R-mod)op, Ab).

This inclusion identifies ((R-\underline{mod})op, Ab) with a torsion-free class of ((R-mod)op, Ab). If the morphism Δ is factored

[figure: (−, N); φ_{N}; (−, \underline{N}); η; Ext^{1}(−, K)]

as a composition of an epimorphism φ_{N} followed by a monomorphism η, then

φ_{N} : (−, N) → (−, \underline{N}) := \underline{Hom}(−, \underline{N})

is the torsion-free quotient morphism of the object (−, N), and η : (−, \underline{N}) → Ext^{1}(−, K) is the injective envelope of (−, \underline{N}) in ((R-\underline{mod})op, Ab). Since K is the unique [18, Thm 23] injective-free pure injective module for which Ext^{1}(−, K) is the injective envelope of (−, \underline{N}), this permits one to recover the kernel K, without direct reference to the phantom cover of N.

Bican, El Bashir and Enochs [9] proved that for every module N, there exists a short exact sequence

[figure: 0; Ω^{♭}(N); FC(N); b; N; 0,]

where b : FC(N) → N is the flat cover of N. The flat cover of N is obviously a phantom morphism and a phantom morphism f : M → N is called trivial if it factors through a flat module. This is equivalent to f factoring through the flat cover of N, which may therefore be regarded as the ”trivial” phantom cover of N. The flat cover resembles the phantom cover in that its kernel Ω^{♭}(N) is a cotorsion module. Recall [27] that a module C is cotorsion if Ext^{1}(Z, C) = 0 for every flat module Z, and that cotorsion modules represent a homologically defined generalization of pure injective modules. If R is a Quasi-Frobenius ring this resemblance is not detectable, because every flat R-module is projective, and so every R-module is cotorsion.

Let R-\underline{Mod} be the category of left R-modules, modulo morphisms that factor through a flat module. If R is a Quasi-Frobenius ring, then R-\underline{Mod} is the usual stable category modulo projectives (resp., injectives). The general relationship between phantom morphisms and purity is given by

Theorem 9. Given left R-modules M and N, there is a natural embedding

\underline{Hom}(\underline{M}, \underline{N}) \overline{}Ext^{1}(M, Ω^{♭}(N))

with the property that a morphism f : M → N is phantom if and only if the image of \underline{f} is pure exact.

In a compactly generated triangulated category satisfying the Brown Representability Condition, an object N admits no nontrivial phantom morphism f : M → N if and only if it is pure injective [20, Lemma 1.4]. In the category R-Mod, such objects N are characterized in

Theorem 10. A left R-module N admits no nontrivial phantom morphism f : M → N if and only if Ω^{♭}(N) is pure injective. Equivalently, the flat cover b : FC(N) → N is the phantom cover of N.

The main idea behind the proof of Theorem 10 is that any phantom epimorphism f : M → N (e.g., the flat cover of N) with a pure injective kernel is a phantom precover (Proposition 5). If R is a right coherent ring and N is a pure injective left module over R, then [27, Lemma 3.2.4] Ω^{♭}(N) is pure injective. It follows (Corollary 11) that there are no nontrivial phantom morphisms f : M → N in that case.

The functor M ↦ (−, \underline{M}) from the category R-Mod of all left R-modules to ((R-\underline{mod})op, Ab) induces a functor on the category R-\underline{Mod},

\underline{Υ} : R-\underline{Mod} → Flat((R-\underline{mod})op, Ab).

Theorem 10 may be used to characterized those rings R for which the functor \underline{Υ} is faithful. Indeed, a global version of Theorem 10 is given by Corollary 12, which implies that this functor \underline{Υ} is faithful if and only if Ω^{♭}(N) is pure injective, for every left R-module N.

The existence of phantom covers is also useful in the study of when the functor \underline{Υ} is full.

Proposition 14 Let N be an R-module. The morphism of abelian groups

\underline{Hom}(\underline{M}, \underline{N}) → [(−, \underline{M}), (−, \underline{N})]

induced by the functor \underline{Υ} is an epimorphism for every R-module M if and only if the torsion-free quotient morphism φ_{N} : (−, N) → (−, \underline{N}) is a flat precover in the category ((R-mod)op, Ab).

If the morphism φ_{N} : (−, N) → \underline{Hom}(−, \underline{N}) is a flat precover, then the module N may be decomposed N = Z ⊕M where Z is flat and M is a flat-free (having no nonzero flat pure submodules) cotorsion module. The question of when the functor \underline{Υ} is full is therefore only reasonable when we consider the restriction of \underline{Υ},

\underline{Ψ} : R-\underline{Cotor} → Flat((R-\underline{mod})op, Ab)

to the subcategory of R-\underline{Mod} represented by cotorsion modules.

Theorem 18 The following conditions are equivalent for a ring R :

(1) the functor \underline{Ψ} is full;

(2) for every cotorsion module N, the morphism φ_{N} : (−, N) → (−, \underline{N}) is a flat precover in ((R-mod)op, Ab);

(3) for every cotorsion module N, the kernel of the morphism φ_{N} : (−, N) → (−, \underline{N}) is cotorsion;

(4) for every cotorsion module N, the phantom cover Ph(N) is pure injective;

(5) every cotorsion module N admits a direct decomposition N = Z ⊕ M, where Z is a pure injective flat module and M is flat-free, and Ω^{♭}(N) has pure injective dimension at most 1.

The last condition of Theorem 18 resembles a condition considered by Beligiannis and H. Krause [6] that is equivalent to the local fullness of the functor that brings a module M over a finite group ring k[G] to the associated graded module Ĥ^{∗}(G, M) over the Tate cohomology ring Ĥ^{∗}(G, k). Rings satisfying the condition that every cotorsion left R-module be a direct sum N = Z ⊕ M with Z a flat pure injective and M a flat-free cotorsion module have been studied by Rothmaler [24]. They are the rings over which every left flat cotorsion module is pure injective (Proposition 17).

An example of Benson and Gnacadja [8] shows that not every flat object of the category (R-\underline{mod})op, Ab) is of the form \underline{Hom}(−, \underline{M}) for some R-module M. Nevertheless, we prove (Theorem 29) that every object in (R-\underline{mod})op, Ab) admits a \underline{Hom}-precover. This is applied to prove

Corollary 32 Let R be a ring with the property that every left cotorsion module is pure injective. Then the functor \underline{Ψ} : R-\underline{Cotor} → Flat((R-\underline{mod})op, Ab) is an equivalence of categories and the category (R-\underline{mod})op, Ab) is pure semisimple (cf. [13]).

Rings all of whose left cotorsion modules are pure injective have been studied by Xu [27, Thm 3.5.1], who characterized them as the rings for which the class of left pure injective modules is closed under extension.

Throughout the article, R will denote an associative ring with identity. The unadorned term Rmodule will refer to a unital left R-module. The category of (right) R-modules is denoted by R-Mod (Mod-R); the subcategory of finitely presented (right) R-modules by R-mod (mod-R). If M and N are R-modules, we use the abbreviations (M, N) := Hom_{R}(M, N) and (\underline{M}, \underline{N}) := \underline{Hom}(\underline{M}, \underline{N}).

1. Preliminaries

Let ((R-mod)op, Ab) denote the category of contravariant additive functors F : (R-mod)op → Ab from the category of finitely presented left R-modules to the category Ab of abelian groups. The morphisms in this category are the natural transformations η : F → G between functors; the morphism groups will be denoted by [F, G]. An object F of ((R-mod)op, Ab) is representable if there is finitely presented module A such that F ≅ (−, A). Yoneda’s Lemma [25, Prop 7.3] asserts that the association A ↦ (−, A), is an equivalence between R-mod and the representable objects of ((R-mod)op, Ab). It implies that every representable object is a finitely generated projective object of ((R-mod)op, Ab), and that the class of representable functors forms a skeletally small generating set of ((R-mod)op, Ab). Because idempotents split in R-mod, every finitely generated projective object of ((R-mod)op, Ab) is representable.

An object F ∈ ((R-mod)op, Ab) is flat (cf. [13]) if it is isomorphic to a direct limit of finitely generated projective functors. The subcategory of ((R-mod)op, Ab) of flat functors is denoted by Flat((R-mod)op, Ab). Because every finitely generated projective functor is representable, every flat functor F is a direct limit of the form

F ≅ lim_{→} (−, A_{i}).

The equivalence between R-mod and the category of representable functors permits us to consider the corresponding direct limit M = lim_{→} A_{i} in R-Mod. A well-known characterization [25, Prop. V.3.4] of finitely presented modules implies that

F ≅ lim_{→}(−, A_{i}) ≅ (−, lim_{→} A_{i}) ≅ (−, M).

Proposition 1. [13, Thm 1.4] The functor Υ : R-Mod → ((R-mod)op, Ab), given by M ↦ (−, M), is a fully faithful left exact functor. It yields an equivalence between the category R-Mod of Rmodules and the category Flat((R-mod)op, Ab).

Given R-modules M and N, let Flat(M, N) ⊆ (M, N) be the subgroup of morphisms f : M → N that factor through a flat R-module. The category R-\underline{Mod} is defined to be the category whose objects \underline{M} are represented by R-modules M, with morphism groups given by

\underline{Hom}(\underline{M}, \underline{N}) = (M, N)/Flat(M, N).

The category of (projectively) stable finitely presented left R-modules is the subcategory R-\underline{mod} ⊆ R-\underline{Mod} of classes \underline{A}, where A is a finitely presented R-module. If A is finitely presented, then every morphism f : A → Z to a flat module Z factors through a finitely generated projective module. So if M is an arbitrary R-module, then subgroup Flat(A, M) = proj(A, M) is the group of morphisms f : A → M that factor through a finitely generated projective module. Thus the definition of R-\underline{mod} given here coincides with the standard definition, where morphisms are defined to be the morphisms of R-modules modulo those that factor through a projective module.

The functor category ((R-\underline{mod})op, Ab) has as its objects the contravariant functors F : (R-\underline{mod})op → Ab. Mimicking the situation above, we associate to every R-module M the functor (−, \underline{M}) := \underline{Hom}(−, \underline{M}) ∈ ((R-\underline{mod})op, Ab). The association M ↦ (−, \underline{M}) is functorial, and if M is flat, then clearly (−, \underline{M}) = 0. Consequently, if a morphism f : M → N belongs to Flat(M, N), then (−, \underline{f}) = 0. The functor M → (−, \underline{M}) thus induces a functor

\underline{Υ} : R-\underline{Mod} → ((R-\underline{mod})op, Ab),

defined by \underline{M} ↦ (−, \underline{M}). If A is a finitely presented R-module, then, by Yoneda’s Lemma, the representable functor (−, \underline{A}) is a finitely generated projective object of ((R-\underline{mod})op, Ab), and the skeletally small class of all such functors is a generating set for the category ((R-\underline{mod})op, Ab). However, all that may be inferred in this situation is that every finitely generated projective object of ((R-\underline{mod})op, Ab) is a coproduct factor of a representable object.

The stable category \underline{mod}-R of finitely presented right R-modules is defined similarly, and the Auslander-Bridger Transpose [2], Tr : \underline{mod}-R → (R-\underline{mod})op, is an equivalence that induces an equivalence of functor categories Tr_{∗} : ((R-\underline{mod})op, Ab) → (\underline{mod}-R, Ab). By [1, Prop. I.3.2], there is an isomorphism of functors,

(1)  Tr_{∗}[(−, \underline{M})] ≅ Tor_{1}^{R}(−, M),

natural in M. The functor \underline{M} ↦ Tor_{1}(−, M) from R-\underline{Mod} to (\underline{mod}-R, Ab) is the composition Tr_{∗} ∘ \underline{Υ}.

As above, a functor F ∈ ((R-\underline{mod})op, Ab) is flat if it is isomorphic to a direct limit of finitely generated projective objects, and the subcategory of flat objects of ((R-\underline{mod})op, Ab) is denoted by Flat((R-\underline{mod})op, Ab). Since every finitely generated projective object is a direct limit of representable objects, a flat functor on R-\underline{mod} may be represented as a direct limit of representable objects. In view of the isomorphism (1), every object Tor_{1}(−, A), with A finitely presented is a projective object in (\underline{mod}-R, Ab). If an R-module M is represented as a direct limit M = lim_{→} A_{i} of finitely presented module A_{i}, then, because the functor Tor_{1}(X_{R}, −) : R-Mod → Ab commutes with direct limits, we get that

Tor_{1}(−, M) ≅ lim_{→} Tor_{1}(−, A_{i})

is a flat object of (\underline{mod}-R, Ab), and hence that (−, \underline{M}) is a flat object of ((R-\underline{mod})op, Ab). The functor \underline{Υ} therefore takes its values in the subcategory of flat functors,

\underline{Υ} : R-\underline{Mod} → Flat((R-\underline{mod})op, Ab).

An example of Benson and Gnacadja [8] shows that not every object of Flat((R-\underline{mod})op, Ab) need lie in the image of \underline{Υ}.

The natural projection π : R-mod → R-\underline{mod}, defined by A ↦ \underline{A} on objects and f ↦ \underline{f} on morphisms, induces a fully faithful exact functor

π_{∗} : ((R-\underline{mod})op, Ab) → ((R-mod)op, Ab),

given by π_{∗}(G) = π∘G. It permits us to identify the category ((R-\underline{mod})op, Ab) with the subcategory of ((R-mod)op, Ab) whose objects are the functors that vanish on R. Similarly, the projection functor from mod-R to \underline{mod}-R induces a fully faithful exact functor from (\underline{mod}-R, Ab) to (mod-R, Ab) that allows us to identify the category (\underline{mod}-R, Ab) with its image in (mod-R, Ab), the subcategory of covariant functors that vanish on the right R-module R_{R}. The equivalence Tr_{∗} induced by the Auslander-Bridger Transpose affords us the option of choosing an ambient context for the category ((R-\underline{mod})op, Ab). When it is viewed as a subcategory of (mod-R, Ab), the theory of injective objects (cf. [16]) in (mod-R, Ab) may be applied when taking injective coresolutions. As a subcategory of ((R-mod)op, Ab), it may be analyzed using the theory of flat resolutions developed in [18].

2. Purity

Recall that a short exact sequence in R-Mod

[figure: 0; M; f; N; g; K; 0]

is pure exact if the corresponding sequence in (mod-R, Ab),

[figure: 0; − ⊗_{R} M; − ⊗ f; − ⊗_{R} N; − ⊗ g; − ⊗_{R} K; 0]

is exact. In that case, the morphism f : M → N is called a pure monomorphism. The module M is pure injective if every pure monomorphism f : M → N is a split monomorphism. For example, if the module K in the short exact sequence is flat, then Tor_{1}(−, K) = 0 and the long exact sequence indicates that this short exact sequence is pure exact. A module M is cotorsion if Ext^{1}(K, M) = 0 for every flat module K. We conclude that every pure injective module is cotorsion. A cotorsion functor C in the category ((R-mod)op, Ab) is defined similarly, if

Ext^{1}[(−, M), C] = 0,

for every R-module M.

Proposition 2. [18, Thm 4] A flat functor (−, M) is a cotorsion object of ((R-mod)op, Ab) if and only if the R-module M is pure injective. If e : M → PE(M) is the pure injective envelope of M, then the induced morphism (−, e) : (−, M) → (−, PE(M)) of flat functors is the cotorsion envelope of (−, M).

Various proofs of the existence of pure injective envelopes have been given by Kiełpiǹski [19], Warfield [26] and Ziegler [28].

Pure exact sequences may also be characterized as the short exact sequences

[figure: 0; M; f; N; g; K; 0]

for which the corresponding sequence in (mod-R, Ab),

[figure: 0; (−, M); (−, f); (−, N); (−, g); (−, K); 0]

is exact. The morphism g : N → K is called a pure epimorphism and the module K is pure projective if every pure epimorphism g : N → K is a split epimorphism. For example, if A is a finitely presented module, then the representable functor (−, A) is projective, so that A is a pure projective module. Because the representable functors (−, A) form a generating class of functors, we conclude that a module is pure projective if and only if it is isomorphic to a direct summand of a direct sum of finitely presented modules [26]; equivalently, if and only if the flat functor (−, M) is projective in ((R-mod)op, Ab).

Proposition 3. (cf. [18, Prop 15]) An epimorphism g : N → K of R-modules is a pure epimorphism if and only if the induced morphism Tor_{1}(−, g) : Tor_{1}(−, N) → Tor_{1}(−, K) is an epimorphism in (\underline{mod}-R, Ab).

Proof. Let f : M → N be the kernel of g. The short exact sequence

[figure: 0; M; f; N; g; K; 0]

induces a long exact sequence of functors in the category (mod-R, Ab). The first few terms are given by

[figure: Δ; − ⊗_{R} M; − ⊗ f; − ⊗_{R} N; − ⊗ g; − ⊗_{R} K; 0; ⋯; Tor_{1}(−, N); Tor_{1}(−, g); Tor_{1}(−, K); Δ]

Then Tor_{1}(−, g) is an epimorphism if and only if Δ = 0 if and only if − ⊗ f is a monomorphism if and only if g is a pure epimorphism. □

3. Phantom morphisms

As a subcategory of ((R-mod)op, Ab), the category ((R-\underline{mod})op, Ab) is closed under subobjects, extensions, and direct products. It is therefore a torsion-free class. The corresponding torsion class is Gen(−, R), the subcategory of objects generated by (−, R) (cf. [18, §2.1]). A functor F belongs to Gen(−, R) provided there is a free module R^{(α)} for which there exists an epimorphism η : (−, R^{(α)}) ≅ (−, R)^{(α)} → F. The torsion subobject of a functor G ∈ ((R-mod)op, Ab) is denoted by t_{R}(G), and the torsion-free quotient object by \underline{G} = G/t_{R}(G). The functor G ↦ \underline{G} is the left adjoint of the exact functor π_{∗} : ((R-\underline{mod})op, Ab) → ((R-mod)op, Ab). It is therefore right exact and preserves direct limits. Furthermore, if G = (−, M), then \underline{G} = (−, \underline{M}), so that it also preserves representable and flat objects [18, Prop 14]. Given an R-module M, let us denote the torsion-free quotient morphism by

φ_{M} : (−, M) → (−, \underline{M}).

Any morphism μ : (−, M) → G, with G ∈ ((R-\underline{mod})op, Ab) factors uniquely through φ_{M},

(−, M)_{φ_{M}}[figure] μ (−, \underline{M}) \underline{μ} ; \overline{}G.

Given a morphism f : M → N of R-modules, the naturality of the torsion-free quotient object yields a commutative diagram

[figure: (−, M); (−, f); (−, N); φ_{M}; φ_{N}; (−, \underline{M}); (−, \underline{f}); (−, \underline{N}).]

A morphism f : M → N is a phantom morphism if the induced morphism

(−, \underline{f}) : (−, \underline{M}) → (−, \underline{N})

is 0. In other words, for every morphism g : A → M, with A finitely presented, the composition fg factors through a finitely generated projective module. The equivalence Tr_{∗} induced by the Auslander-Bridger Transpose shows that this is equivalent to the equation

Tor_{1}(−, f) = Tr_{∗}[(−, \underline{f})] = 0

in the functor category (\underline{mod}-R, Ab).

Let us construct a flat resolution in the category ((R-mod)op, Ab) of the functor (−, \underline{N}). The kernel of the torsion-free quotient morphism φ_{N} is given by the torsion subobject t_{R}(−, N). This is described by the short exact sequence

[figure: 0; t_{R}(−, N); (−, N); φ_{N}; (−, \underline{N}); 0.]

To continue a flat resolution of (−, \underline{N}) beginning with the morphism φ_{N}, we need to characterize the epimorphisms η : (−, M) → t_{R}(−, N).

Proposition 4. A morphism η : (−, M) → t_{R}(−, N) is an epimorphism if and only if the composition with the inclusion t_{R}(−, N) ⊆ (−, N) is given by (−, f) : (−, M) → (−, N), where f : M → N is a phantom epimorphism.

Proof. The commutative square above shows that f : M → N is a phantom if and only if Im (−, f) ⊆ t_{R}(−, N). On the other hand, f is an epimorphism if and only if t_{R}(−, N) ⊆ Im (−, f). Thus f is a phantom epimorphism if and only if t_{R}(−, N) = Im (−, f) = Im η. □

The flat resolution may be completed by letting K = Ker f. One obtains

[figure: 0; (−, K); (−, k); (−, M); (−, f); (−, N); φ_{N}; (−, \underline{N}); 0.]

If this exact sequence of functors is evaluated at the R-module R, the result is a short exact sequence

[figure: 0; K; k; M; f; N; 0,]

where f : M → N is a phantom morphism. A flat resolution of t_{R}(−, N) is given by

[figure: 0; (−, K); (−, k); (−, M); (−, f); t_{R}(−, N); 0.]

A phantom morphism c : M → N is a phantom precover of N if for any phantom morphism g : M′ → N, there is a morphism h : M′ → N such that the diagram

[figure: M′; h; g; M; c; N]

commutes. A flat precover is defined similarly, with the term ”phantom morphism” replaced by ”morphism with flat domain.” If an epimorphism g : Z → N from a flat R-module Z has a cotorsion kernel, then it is a flat precover [27, Prop 2.1.3]. The analogous result for phantom precovers is given by

Proposition 5. Let f : M → N be a phantom epimorphism. If Ker f is pure injective, then f : M → N is a phantom precover.

Proof. Let K = Ker f. We wish to complete the diagram

[figure: (−, M′); (−, h); (−, g); 0; (−, K); (−, M); (−, f); t_{R}(−, N); 0.]

As K is pure injective, the functor (−, K) is cotorsion (Proposition 2). The functor (−, M′) is flat, so there is a morphism h : M′ → M as indicated. □

Just as a flat cover is called special [27, Prop 2.1.3] when its kernel is cotorsion, we will call a phantom precover c : M → N special if the kernel of c is pure injective. Given a module N, a special phantom precover may be easily produced.

Proposition 6. Let g : Z → N be a phantom epimorphism. In the morphism of short exact sequences

[figure: 0; K; k; Z; f; N; 0; e; 0; PE(K); h; M; c; N; 0]

gotten by pushout along the pure injective envelope e : K → PE(K), the morphism c : M → N is a special phantom precover.

Proof. Let us consider the pushout in the category ((R-mod)op, Ab)

[figure: 0; (−, K); (−, k); (−, Z); t_{R}(−, N); 0; (−, e); γ; 0; (−, PE(K)); G; t_{R}(−, N); 0,]

where e : K → PE(K) is the pure injective envelope of K. Since e is a pure monomorphism, the cokernel of (−, e) is the flat functor (−, L), where L = PE(K)/K. The pushout property ensures that the cokernel of γ is isomorphic to (−, L) and is therefore flat. By [18, Prop 3], the functor G = (−, M) is itself flat. Evaluating the commutative diagram at the module R, yields the diagram in the statement of the proposition. Since the bottom row of the diagram in ((R-mod)op, Ab) is a flat resolution of t_{R}(−, N), the morhism c : M → N obtained by evaluation at R is a phantom morphism. □

A phantom epimorphism as in the hypothesis of Proposition 6 is found by taking any epimorphism g : Z → N where Z is flat. For example, if Z = P (N) is projective, then the kernel of g : P (N) → N is a first syzygy Ω(N) of N. The construction above provides a special phantom precover of N whose kernel is the pure injective envelope of Ω(N). If R = Λ is an artin algebra and N is finitely presented, Auslander and Reiten [3] use the short exact sequence

[figure: 0; Ω(N); P (N); N; 0]

to induce a resolution of (−, \underline{N}) by finitely generated projective objects in ((R-mod)op, Ab). A phantom precover c : M → N is a phantom cover if every endomorphism h of M over N,

[figure: M; h; c; M; c; N,]

is an isomorphism. This implies that the domain of a phantom cover of N is unique up to isomorphism over N. We will denote the phantom cover of N by c : Ph(N) → N. By [27, Prop 2.2.7], the flat cover of a module is special. The analogous result for phantom covers is included in

Theorem 7. Every R-module N has a phantom cover c : Ph(N) → N. The phantom cover is a special precover.

Proof. By [14, Thm 2.7] or [18, Thm 11], the torsion subobject t_{R}(−, N) of (−, N) has a flat cover η : (−, M) → t_{R}(−, N) in the category ((R-mod)op, Ab). The kernel C of η is a cotorsion object in ((R-mod)op, Ab),

[figure: 0; C; (−, M); η; t_{R}(−, N); 0.]

Composing the flat cover η with the inclusion t_{R}(−, N) ⊆ (−, N), it is seen that the kernel of η is isomorphic to the kernel of the composition (−, c) : (−, M) → (−, N). Thus C = (−, K), where K = Ker c. By Proposition 2, the module K is pure injective, and c : M → N is a special phantom precover.

To see that it is a cover, let h : M → M be an endomorphism over N, c = ch. Since we have that t_{R}(−, N) = Im (−, c), a commutative diagram

[figure: (−, M); (−, h); η; (−, M); η; t_{R}(−, N)]

arises in the category ((R-mod)op, Ab). Since η : (−, M) → t_{R}(−, N) is the flat cover, the endomorphism (−, h) : (−, M) → (−, M) is necessarily an isomorphism. Thus h is an automorphism of M, and M ≅ Ph(N). □

A standard argument [27, Thm. 1.2.7] shows that every phantom precover contains the phantom cover as a direct summand.

Theorem 23 of [18] asserts that if R-\overline{Pinj} denotes the category of injective-free (containing no nonzero injective submodules) pure injective modules, then there is an equivalence of categories

(2)  R-\overline{Pinj} ≅ Inj((R-\underline{mod})op, Ab),

given by K ↦ Ext^{1}(−, K), where Inj((R-\underline{mod})op, Ab) denotes the subcategory of the injective objects of ((R-\underline{mod})op, Ab). The kernel K of the phantom cover c : Ph(N) → N certainly contains no nonzero injective submodules, for the kernel of a cover cannot contain any of its direct summands.

If K is a pure injective module, then [18, Thm 21] states that for any R-module N, there is a natural isomorphism

(3)  Ext^{1}(N, K) ≅ [(−, N), Ext^{1}(−, K)].

If Δ ∈ Ext^{1}(N, K) is represented by the short exact sequence

[figure: 0; K; M; N; 0,]

then the corresponding morphism Δ : (−, N) → Ext^{1}(−, K) is found to be the first connecting morphism in the induced long exact sequence,

[figure: 0; (−, K); (−, M); (−, N); Δ; Ext^{1}(−, K); · · · .]

Theorem 8. Let N be an R-module and K the kernel of the phantom cover c : Ph(N) → N. If Δ ∈ Ext^{1}(N, K) is represented by the short exact sequence

[figure: 0; K; k; Ph(N); c; N; 0,]

then the corresponding morphism Δ : (−, N) → Ext^{1}(−, K) is the composition

[figure: (−, N); φ_{N}; (−, \underline{N}); η; Ext^{1}(−, K),]

where η : (−, \underline{N}) → Ext^{1}(−, K) is the injective envelope of (−, \underline{N}) in ((R-\underline{mod})op, Ab).

Proof. Since Δ is the first connecting morphism in the long exact sequence, it is immediate that (−, \underline{N}) is the image of Δ. Because K is pure injective, the functor Ext^{1}(−, K) is an injective object of ((R-\underline{mod})op, Ab), so the inclusion Im Δ ⊆ Ext^{1}(−, K) yields a coproduct decomposition

Ext^{1}(−, K) = E[Im Δ] ⨿ I,

where E[Im Δ] is the injective envelope of Im Δ in ((R-\underline{mod})op, Ab). By [18, Thm 23], there is direct sum decomposition K = K_{1} ⊕ K_{2} such that E[Im Δ] ≅ Ext^{1}(−, K_{1}) and I ≅ Ext^{1}(−, K_{2}). Now Ker Ext^{1}(−, k) = Im Δ, so the restriction k_{2} of k to K_{2} induces a split monomorphism Ext^{1}(−, k_{2}) : Ext^{1}(−, K_{2}) → Ext^{1}(−, Ph(M)). Since K_{2} is pure injective, [18, Thm 16] implies that the retraction of Ext^{1}(−, k_{2}) is of the form Ext^{1}(−, g), where g : Ph(M) → K_{2}. As K_{2} is injective-free, Theorem 23 of [18] implies that the composition gk_{2} : K_{2} → K_{2} is an isomorphism, and hence that k_{2} : K_{2} → Ph(N) is a split monomorphism. But the kernel of a phantom cover contains none of its nonzero summands. Thus K_{2} = 0, and the result is proved. □

4. Fidelity

In this section, we use the phantom cover to characterize the situation when the functor \underline{Υ} : R-\underline{Mod} → Flat((R-\underline{mod})op, Ab) given by \underline{M} ↦ (−, \underline{M}) is faithful. A phantom morphism f : M → N is trivial if it factors through a flat module. Equivalently, the induced morphism \underline{f} : \underline{M} → \underline{N} is 0 in R-\underline{Mod}. Let b : FC(N) → N be the flat cover of N [9], and denote the kernel of b by Ω^{♭}(N). Thus f : M → N is a trivial phantom morphism if and only if there exists a factorization as indicated in the commutative diagram

[figure: M; f; ξ : 0; Ω^{♭}(N); FC(N); b; N; 0.]

The following generalizes the relationship between phantom morphisms and purity observed by Benson and Gnacadja [7, Prop 4.2.2] in the case of finite group rings.

Theorem 9. Let N be an R-module. The morphism

\underline{Hom}(\underline{M}, \underline{N}) \overline{}Ext^{1}(M, Ω^{♭}(N))

defined by \underline{f} : \underline{M} → \underline{N} ↦ Ext^{1}(f, Ω^{♭}(N))(ξ) is a monomorphism natural in M. A morphism f : M → N is a phantom if and only if the image of \underline{f} is pure exact.

Proof. The morphism associates to a representative f : M → N the short exact sequence obtained by pushout of the flat cover ξ along f. The morphism is a monomorphism, because the obtained short exact sequence splits if and only if f factors through the flat cover of N. Consider the morphism of long exact sequences induced by f,

[figure: 0; (−, Ω^{♭}(N)); (−, Y); (−, M); Δ′; Ext^{1}(−, Ω^{♭}(N)); (−, f); 0; (−, Ω^{♭}(N)); (−, FC(N)); (−, b); (−, N); Δ; Ext^{1}(−, Ω^{♭}(N)).]

Because b : FC(N) → N, is a phantom epimorphism, the image of (−, b) is t_{R}(−, N), by Proposition 4. Thus f is a phantom morphism if and only if the composition Δ′ = Δ(−, f) = 0 if and only if the short exact sequence obtained by pushout along f : M → N is pure exact. □

The kernel Ω^{♭}(N) of the flat cover of N is a cotorsion module [27, Prop 2.27].

Theorem 10. The following conditions are equivalent for an R-module N :

(1) every phantom morphism g : M → N is trivial;

(2) the phantom cover of N is flat;

(3) the phantom cover of N is the flat cover of N;

(4) the module Ω^{♭}(N) is pure injective.

Proof. (1) ⇔ (2). If the phantom cover of N is trivial, it factors through the flat cover of N. The flat cover of N is thus a phantom precover of N. The phantom cover, being a direct summand of any phantom precover, is therefore flat. If the phantom cover of N is flat, then any phantom morphism f : M → N factors through the flat module Ph(N).

(2) ⇔ (3). If the phantom cover of N is flat, then it factors through the flat cover. Thus the flat cover and phantom cover factor through each other. By the uniqueness property of covers, they must be isomorphic.

(3) ⇔ (4). If Ω^{♭}(N) is pure injective, then by Proposition 5, the flat cover of N is a special phantom precover. The flat and phantom covers of N then factor through each other, and are therefore isomorphic. □

Corollary 11. If R is right coherent, and N is a pure injective module, then every phantom morphism f : M → N is trivial.

Proof. By [27, Lemma 3.2.4], Ω^{♭}(N) is pure injective. □

The situation described by Theorem 10 coincides with the flat cover of t_{R}(−, N) itself being a torsion object. Indeed, if Z is a flat module, then any epimorphism g : N → Z is a pure epimorphism. If we take an epimorphism g : R^{(α)} → Z from a free module, then the morphism (−, g) induced in ((R-mod)op, Ab) is an epimorphism to (−, Z) from the object (−, R^{(α)}) ≅ (−, R)^{(α)}. The flat functor (−, Z) therefore belongs to the torsion class Gen(−, R). Conversely, it is not difficult to show that if a flat functor (−, N) is torsion, then N must be a flat R-module.

Let g : N → K be a morphism with the property that Tor_{1}(−, g) is an epimorphism. Consider a morphism h : Z → K from a flat module Z such that the coproduct morphism g ⨿ h : N ⊕ Z → K is an epimorphism. The induced morphism

Tor_{1}(−, g ⨿ h) : Tor_{1}(−, N ⊕ Z) → Tor_{1}(−, K)

is an epimorphism, because Tor_{1}(−, N ⊕ Z) = Tor_{1}(−, N) and Tor_{1}(−, g ⨿ h) = Tor_{1}(−, g). Proposition 3 implies that g ⨿ h is a pure epimorphism. Call a morphism \underline{g} : \underline{N} → \underline{K} in R-\underline{Mod} a Tor-epimorphism if the morphism Tor_{1}(−, g) is an epimorphism in (\underline{mod}-R, Ab). For example, in the short exact sequence

[figure: 0; K; k; Ph(N); c; N; 0,]

where c : Ph(N) → N is the phantom cover of N, we have that Tor_{1}(−, k) is an epimorphism in (\underline{mod}-R, Ab). This is because Tor_{1}(−, c) = 0 and the long exact sequence. Thus the induced morphism \underline{k} : \underline{K} → \underline{Ph(N)} is a Tor-epimorphism.

The following is a global version of Theorem 10.

Corollary 12. The following are equivalent for a ring R :

(1) the functor \underline{Υ} : R-\underline{Mod} → Flat((R-\underline{mod})op, Ab) is faithful;

(2) the phantom cover of every R-module is flat;

(3) the phantom cover of every R-module is its flat cover;

(4) for every R-module N, Ω^{♭}(N) is pure injective;

(5) every Tor-epimorphism \underline{g} : \underline{N} → \underline{K} is an epimorphism in R-\underline{Mod}.

Proof. The equivalence of the first four conditions is immediate from Theorem 10. So assume that \underline{Υ} is faithful and that \underline{g} : \underline{N} → \underline{K} is a Tor-epimorphism. If \underline{h} : \underline{K} → \underline{L} is such that \underline{h}\underline{g} = 0, then Tor_{1}(−, h) ∘ Tor_{1}(−, g) = 0. As \underline{g} is a Tor-epimorphism, Tor_{1}(−, h) = \underline{Υ}(\underline{h}) = 0. As \underline{Υ} is faithful, \underline{h} = 0.

Conversely, suppose that every Tor-epimorphism is an epimorphism in R-\underline{Mod}. Given an Rmodule N, the foregoing discussion shows that the kernel k : K → Ph(N) of the phantom cover induces a Tor-epimorphism \underline{k} : \underline{K} → \underline{Ph(N)} in R-\underline{Mod}. As \underline{c}\underline{k} = 0, our assumption implies that \underline{c} : \underline{Ph(N)} → \underline{N} is 0. Thus the phantom cover of N factors through a flat module, and must itself be flat. □

5. Fullness

If M is a pure projective module, then any morphism \underline{η} : (−, \underline{M}) → (−, \underline{N}) is of the form (−, \underline{f}), where f : M → N is a morphism of R-modules. Indeed, the diagram

[figure: (−, M); η; (−, N); φ_{M}; φ_{N}; (−, \underline{M}); \underline{η}; (−, \underline{N})]

may be completed as indicated, because (−, M) is projective and φ_{N} : (−, N) → (−, \underline{N}) is an epimorphism. As η : (−, M) → (−, N) is of the form η = (−, f) for some R-morphism f : M → N, it follows that \underline{η} = (−, \underline{f}).

Proposition 13. If M is a pure projective module, then the morphism of abelian groups

(\underline{M}, \underline{N}) → [(−, \underline{M}), (−, \underline{N})]

induced by the functor \underline{Υ} : R-\underline{Mod} → ((R-\underline{mod})op, Ab) is an epimorphism.

In this section of the paper, we consider conditions on N, the module in the second variable, that are sufficient for the morphism (\underline{M}, \underline{N}) → [(−, \underline{M}), (−, \underline{N})] to be onto.

Proposition 14. Let N be an R-module. The morphism of abelian groups

(\underline{M}, \underline{N}) → [(−, \underline{M}), (−, \underline{N})]

is an epimorphism for every R-module M if and only if the torsion-free quotient morphism φ_{N} : (−, N) → (−, \underline{N}) is a flat precover.

Proof. Consider the diagram

[figure: (−, M); φ_{M}; (−, \underline{M}); (−, f); η; (−, N); φ_{N}; (−, \underline{N}).]

If φ_{N} : (−, N) → (−, \underline{N}) is a flat precover, it may be completed to a commutative diagram as indicated, and thus η = (−, \underline{f}).

Conversely, if (\underline{M}, \underline{N}) → [(−, \underline{M}), (−, \underline{N})] is onto and α : (−, M) → (−, \underline{N}) is a morphism from a flat functor, then the induced morphism \underline{α} : (−, \underline{M}) → (−, \underline{N}) is of the form (−, \underline{f}), for some f : M → N. But then the diagram

[figure: (−, M); (−, f); α; (−, N); φ_{N}; (−, \underline{N})]

commutes. □

Proposition 14 suggests we distinguish the modules M for which the torsion-free quotient morphism φ_{M} : (−, M) → (−, \underline{M}) is a flat precover.

Proposition 15. If φ_{M} : (−, M) → (−, \underline{M}) is a flat precover, then any maximal flat pure submodule Z of M is a direct summand.

Proof. Let Z ⊆ M be a maximal flat pure submodule. Such a module exists, by Zorn’s Lemma. The top row of the commutative diagram

[figure: 0; (−, Z); (−, M); (−, q); (−, M/Z); 0; φ_{Z}; φ_{M}; φ_{M/Z}; (−, \underline{Z}); (−, \underline{M}); (−, \underline{q}); (−, \underline{M/Z}); 0,]

is exact, by purity. The bottom row is exact because the torsion-free quotient morphism F ↦ \underline{F} is right exact. Because Z is flat, (−, \underline{Z}) = 0 and so (−, \underline{q}) is an isomorphism. By hypothesis, the morphism (−, \underline{q})^{−1}φ_{M/Z} : (−, M/Z) → (−, \underline{M}) is a flat precover, and therefore so is φ_{M/Z} : (−, M/Z) → (−, \underline{M/Z}). If we can show that φ_{M/Z} is a flat cover, then (−, \underline{q})^{−1}φ_{M/Z} would be a flat cover, and the morphism (−, q) : (−, M) → (−, M/Z) would be a split epimorphism. It would follow therefore that Z is a direct summand of M.

We have that φ_{M/Z} : (−, M/Z) → (−, \underline{M/Z}) is a flat precover. By [27, Thm 1.2.7] a flat precover may be decomposed as (−, M/Z) = (−, Z′)⨿(−, N), where (−, Z′) ⊆ Ker φ_{M/Z} and the restriction of φ_{M/Z} to (−, N) is the flat cover of (−, \underline{M/Z}). But the restriction of φ_{M} to (−, Z′) is just φ_{Z′} = 0, which implies that Z′ is flat. By the choice of Z, M/Z has no nonzero flat pure submodules, let alone nonzero flat summands. Thus Z′ = 0 and φ_{M/Z} is indeed a flat cover. □

Let us call a module M flat-free if there are no nonzero flat pure submodules Z ⊆ M.

Proposition 16. If G ∈ ((R-\underline{mod})op, Ab) and η : (−, M) → G is a flat cover, then M is a flat-free cotorsion module.

Proof. The module M is cotorsion by [18, Thm 12], so suppose that Z ⊆ M is a flat pure submodule. Then the short exact sequence

[figure: 0; Z; M; q; M/Z; 0]

is pure exact and so the top row in the commutative diagram below is exact. Since Z is a flat module, the functor (−, Z) is torsion and the restriction of φ_{M} to (−, Z) is 0,

[figure: 0; (−, Z); (−, M); (−, q); (−, M/Z); 0; 0; φ_{M}; α; G.]

But φ_{M} is the flat cover, so the morphism (−, q) : (−, M) → (−, M/Z) must have a section. It is therefore a monomorphism, and hence Z = 0. □

The preceding propositions suggest that the fullness condition on the functor

\underline{Υ} : R-\underline{Mod} → Flat((R-\underline{mod})op, Ab)

is reasonable only if the domain is restricted to the subcategory R-\underline{Cotor} ⊆ R-\underline{Mod} of classes \underline{N} with N cotorsion. The cotorsion envelope CE(Z) of a flat R-module Z is itself a flat module, so if a morphism f : M → C, with C a cotorsion module, factors through a flat module Z, then it necessarily factors through the flat cotorsion module CE(Z). The category R-\underline{Cotor} may therefore be defined as the category of cotorsion modules modulo morphisms that factor through a flat cotorsion module.

Let us denote by \underline{Ψ} the restriction of \underline{Υ} to R-\underline{Cotor},

\underline{Ψ} : R-\underline{Cotor} → Flat((R-\underline{mod})op, Ab).

The image of the functor \underline{Ψ} is the same as the image of \underline{Υ}. To see this, recall that the cotorsion envelope m : M → CE(M) = C of any R-module M has a flat cokernel Z = C/M ([27, Thm 3.4.2]). The short exact sequence

[figure: 0; M; m; C; Z; 0]

induces a long exact sequence of functors in (mod-R, Ab), part of which is given by

[figure: Tor_{2}(−, Z); Tor_{1}(−, M); Tor_{1}(−, m); Tor_{1}(−, C); Tor_{1}(−, Z),]

and lies in the subcategory (\underline{mod}-R, Ab). As Z is flat, Tor_{2}(−, Z) = Tor_{1}(−, Z) = 0, so the morphism Tor_{1}(−, m) : Tor_{1}(−, M) → Tor_{1}(−, C) induced by the cotorsion envelope is an isomorphism of functors. Applying the equivalence Tr_{∗} induced by Auslander-Bridger Transpose, we obtain that

\underline{Υ}(M) = (−, \underline{M}) ≅ (−, \underline{C}) = \underline{Ψ}(C).

The foregoing comments, together with Proposition 15 indicate the relevance of rings R satisfying

Condition R. Every maximal flat pure submodule Z of a cotorsion module M is a direct summand.

Rings satisfying Condition R have been studied by Rothmaler [24] and have several nice characterizations. Condition (2) of the following proposition implies that if R is right coherent, then it satisfies Condition R.

Proposition 17. [24] The following are equivalent for a ring R :

(1) Condition R;

(2) the pure injective envelope of every flat module Z is flat;

(3) the cotorsion envelope of every flat module Z is its pure injective envelope;

(4) every flat cotorsion module is pure injective;

(5) every cotorsion module is a direct sum C = Z ⊕ N where N is flat-free and Z is a pure injective flat module.

Proof. (1) ⇒ (2). Consider the pure injective envelope of a flat module Z ⊆ PE(Z). By Zorn’s Lemma, there is a maximal flat pure submodule of PE(Z) containing Z. Since it is a direct summand of PE(Z), it must be the pure injective envelope.

(2) ⇔ (3). If the pure injective envelope Z ⊆ PE(Z) is flat, then so is the cokernel PE(Z)/Z. A monomorphism from Z into a cotorsion module, such as PE(Z), with a flat cokernel, must be a cotorsion preenvelope [27, Prop 2.1.4]. The cotorsion envelope is therefore a direct summand of PE(Z), and is thus pure injective. For the converse, recall that the cotorsion envelope of a flat module is itself flat [27, Thm 3.4.2].

(3) ⇒ (1). Let C be a cotorsion module and Z ⊆ C a maximal flat pure submodule of C. The inclusion morphism factors through the cotorsion envelope z : Z → CE(Z),

[figure: CE(Z); z; ι′; Z; ι; C.]

Since the cotorsion envelope of Z is the pure injective envelope, it is a pure essential extension. As the restriction of ι′ : CE(Z) → C to Z is a pure monomorphism, the morphism ι′ must itself be a pure monomorphism. Since CE(Z) is also flat, our choice of the module Z yields the equation Z = CE(Z) = PE(Z), which implies that Z is a direct summand of C.

(3) ⇔ (4) is immediate, as are (5) ⇒ (4) and [(1) ∧ (4)] ⇒ (5). □

If N is a cotorsion module, then, because both the flat cover FC(N) and the phantom cover Ph(N) have cotorsion kernels, they must themselves be cotorsion. Condition R implies that the flat cover FC(N) is pure injective.

Recall from [27, Prop 2.1.3] that if the kernel of a morphism α : (−, M) → G is a cotorsion object of ((R-mod)op, Ab), then α is a flat precover of G. Such a flat precover is called special.

Theorem 18. The following conditions are equivalent for a ring R :

(1) the functor \underline{Ψ} : R-\underline{Cotor} → Flat((R-\underline{mod})op, Ab) is full;

(2) for every cotorsion module M, the morphism φ_{M} : (−, M) → (−, \underline{M}) is a flat precover in ((R-mod)op, Ab);

(3) for every cotorsion module M, the morphism φ_{M} : (−, M) → (−, \underline{M}) is a special flat precover;

(4) for every cotorsion module M, the phantom cover Ph(M) is pure injective;

(5) every cotorsion module M admits a direct sum decomposition M = Z ⊕ N, where N is flat-free and Z is a flat pure injective module, and Ω^{♭}(M) has pure injective dimension at most 1.

Proof. (1) ⇔ (2). To prove (1) ⇒ (2), consider the flat cover α : (−, N) → (−, \underline{M}). By Proposition 16, the module N is cotorsion. The hypothesis ensures that the induced morphism \underline{α} : (−, \underline{N}) → (−, \underline{M}) is of the form \underline{α} = (−, \underline{f}) for some morphism f : N → M. We may therefore complete the diagram

[figure: (−, N); φ_{N}; (−, \underline{N}); (−, f); \underline{α}; (−, M); φ_{M}; (−, \underline{M})]

as indicated to obtain a commutative diagram. But then φ_{M} : (−, M) → (−, \underline{M}) is a flat precover. The converse follows from the proof of Proposition 14.

(2) ⇒ (3). By Propositions 15 and 17, the ring satisfies Condition R. If M is a cotorsion module and φ_{M} : (−, M) → (−, \underline{M}) is a flat precover, we already noted once how (−, M) may be decomposed [27, Thm 1.2.7] as (−, M) = (−, Z) ⨿ (−, N), where (−, Z) ⊆ Ker φ_{M} and the restriction of φ_{M} to (−, N) is the flat cover of (−, \underline{M}). The restriction of φ_{M} to (−, Z) is just φ_{Z} = 0, which implies that Z is a flat cotorsion module. By Condition R, Z is pure injective, and so (−, Z) is cotorsion. The kernel of φ_{M} is therefore the coproduct of (−, Z) and the kernel of φ_{M} restricted to (−, N), which is also cotorsion, because this restriction is a flat cover.

(3) ⇔ (4). Let k : K → Ph(M) be the kernel of the phantom cover of M. A flat resolution of t_{R}(−, M) is given by

[figure: 0; (−, K); k; (−, Ph(M)); t_{R}(−, M); 0.]

Now φ_{M} is a special flat precover if and only if t_{R}(−, M) is cotorsion. Because K is pure injective, the flat functor (−, K) is cotorsion. Thus t_{R}(−, M) is cotorsion if and only if (−, Ph(M)) is cotorsion if and only if Ph(M) is pure injective. [(2) ∧ (4)] ⇒ (5). If Z is a flat module, then its phantom cover is given by 1_{Z} : Z → Z, so if Z is cotorsion, then it must be pure injective. Thus Condition (4) implies Condition R, and we have that every cotorsion module M admits a decomposition M = Z ⊕ N, where Z is a flat pure injective module and N is flat-free. Now consider the short exact sequence obtained from the flat cover of M, and take the pushout along the pure injective envelope e : Ω^{♭}(M) → E of the kernel,

[figure: 0; 0; 0; Ω^{♭}(M); k; FC(M); b; M; 0; e; 0; E; X; c′; M; 0; L; L; 0; 0.]

Since FC(M) is pure injective, the kernel k : Ω^{♭}(M) → FC(M) factors through the pure injective envelope E as indicated. This induces the splitting of the middle column and yields the direct sum decomposition X = FC(M) ⊕ L, which shows that the module Ω^{♭}(M) has pure injective dimension at most 1 if and only if the pushout X is pure injective. But the middle row induces a flat resolution of the torsion subobject t_{R}(−, M) given by

[figure: 0; (−, E); (−, X); (−, c′); t_{R}(−, M); 0.]

By Condition (2), the torsion subobject is cotorsion. Since E is pure injective, the functor (−, E) is also cotorsion. It follows that (−, X) is cotorsion, and that X is pure injective.

(5) ⇒ (4). Let M be a cotorsion module and consider the pushout diagram above. The first conjuct of Condition (5) is equivalent to Condition R, so that FC(M) is pure injective, and hence the same splitting of the middle column is affected. The pushout X = FC(M) ⊕ L is pure injective, because L = E/Ω^{♭}(M) and Ω^{♭}(M) has pure injective dimension at most 1. By Proposition 6, the morphism c′ : X → M is a phantom precover. The phantom cover Ph(M) is therefore a direct summand of X, which implies that it is also pure injective. □

By Proposition 17, the first conjuct of Condition (5) is equivalent to Condition R. In this form of Condition R, Condition (5) resembles a condition considered by Beligiannis and H. Krause [6] that is equivalent to the local fullness of the functor that brings a module M over a finite group ring k[G] to the associated graded module Ĥ^{∗}(G, M) over the Tate cohomology ring Ĥ^{∗}(G, k). Note also that the converse of Condition (4) holds in general: if the phantom cover Ph(M) of a module M is pure injective, then M must be cotorsion.

Corollary 19. If R is a left perfect, right coherent countable ring (for example, a countable Quasi-Frobenius ring). Then the functor

\underline{Υ} : R-\underline{Mod} → Flat((R-\underline{mod})op, Ab),

given by \underline{M} → (−, \underline{M}) is full.

Proof. Since R is left perfect, R-\underline{Cotor} = R-\underline{Mod} is the stable category modulo projectives. As R is right coherent, it satisfies Condition R. Because it is countable, every module has pure injective dimension at most 1 [16]. □

Corollary 20. Let R be a ring with the property that the functor

\underline{Ψ} : R-\underline{Cotor} → Flat((R-\underline{mod})op, Ab)

is faithful. Then \underline{Ψ} is full if and only if R satisfies Condition R.

Proof. If \underline{Ψ} is full, then Condition (5) of Theorem 18 implies Condition R. Conversely, if \underline{Ψ} is faithful and M is cotorsion, then the phantom cover Ph(M) is flat, by Theorem 10. Because the phantom cover of a cotorsion module is itself cotorsion, Condition R implies that it is pure injective, so that Condition (4) of Theorem 18 is satisfied. □

Recall from [18] that a flat resolution of an object in ((R-mod)op, Ab) is called special if every morphism in the resolution has a cotorsion kernel.

Corollary 21. The following are equivalent for a ring R :

(1) the functor \underline{Ψ} : R-\underline{Cotor} → Flat((R-\underline{mod})op, Ab) is fully faithful;

(2) the phantom cover of every cotorsion module is flat and pure injective;

(3) for every cotorsion module C, the flat cover of C induces a special flat resolution of (−, \underline{C}) in ((R-mod)op, Ab).

Proof. The equivalence of Conditions (1) and (2) follows from Theorems 10 and 18. Consider the short exact sequence associated to the flat cover of C,

[figure: 0; Ω^{♭}(C); k; FC(C); g; C; 0.]

The induced flat resolution of (−, \underline{C}) in ((R-mod)op, Ab) is given by

[figure: 0; (−, Ω^{♭}(C)); (−, k); (−, FC(C)); (−, g); (−, C); D_{0}; (−, \underline{C}); 0.]

It is special if and only if both (−, Ω^{♭}(C)) and (−, FC(C)) are flat cotorsion objects, which is equivalent to both Ω^{♭}(C) and FC(C) being pure injective. By Conditions (4) of Theorem 10 and Theorem 18, respectively, these are equivalent to the faithfulness and fullness of \underline{Ψ}, respectively. □

6. Examples

If R is a right coherent ring and N a pure injective left R-module, then the flat cover g : FC(N) → N and its kernel Ω^{♭}(N) are pure injective [27, Lemma 3.2.4]. By Theorem 10, the flat cover of N is its phantom cover and there are no nontrivial phantom morphisms f : M → N. Since the phantom cover is pure injective, it implies that the restriction of \underline{Υ} : R-\underline{Mod} → ((R-\underline{mod})op, Ab) to the subcategory R-\underline{Pinj} of classes represented by pure injective R-modules is full and faithful. Because R is right coherent, the pure injective envelope of a flat module is flat, so that R-\underline{Pinj} may be defined as the category of pure injective modules modulo morphisms that factor through a flat pure injective module.

Definition. A ring R will be called (left) Xu if it satisfies any of the following equivalent [27, Thm 3.5.1] conditions:

(1) every left cotorsion module over R is pure injective; (2) the class of pure injective left R-modules is closed under extension;

(3) for every left R-module M, the cokernel PE(M)/M of the pure injective envelope of M is

flat. Every left pure semisimple ring is left Xu, because every module is pure injective. Every von Neumann regular ring is Xu, because every module is flat. Over a Xu ring, Ω^{♭}(N) is pure injective, for every R-module N. By Theorem 10, every flat cover is the phantom cover, and the functor \underline{Υ} : R-\underline{Mod} → ((R-\underline{mod})op, Ab) is faithful. Since the phantom cover of a cotorsion module is cotorsion, every phantom cover of a cotorsion module is flat and pure injective. The functor \underline{Ψ} : R-\underline{Cotor} → ((R-\underline{mod})op, Ab) is therefore full and faithful.

Proposition 22. The ring R is Xu if and only if every morphism η : (−, M) → G with G ∈ ((R-\underline{mod})op, Ab) and M pure injective is a special flat precover.

Proof. Let α : (−, N) → Ker η be the flat cover of the kernel of η. The composition of α with the inclusion Ker η ⊆ (−, M) is given by (−, g) : (−, N) → (−, M), where g : N → M. If K = Ker g, then the functor (−, K) is the kernel of (−, g). It is therefore cotorsion and K is a pure injective module. We have a short exact sequence

[figure: 0; K; N; g; M; 0]

of R-modules with K and M pure injective, and a short exact sequence

[figure: 0; (−, K); (−, N); (−, g); Ker η; 0]

in ((R-mod)op, Ab). As (−, K) is cotorsion, Ker η is cotorsion if and only if (−, N) is cotorsion if and only if N is pure injective. □

Clearly, every Xu ring satisfies

Condition X. Given a short exact sequence

[figure: 0; M; f; N; g; K; 0]

with M and N pure injective, the module K is also pure injective.

Proposition 23. Suppose that the ring R satisfies Condition X. Then R is Xu if and only if the functor \underline{Ψ} : R-\underline{Cotor} → ((R-\underline{mod})op, Ab) is full.

Proof. We noted already that the functor \underline{Ψ} is full if R is Xu. To prove the converse, let M be a cotorsion module, and consider its phantom cover. If \underline{Ψ} is full, then Condition (4) of Theorem 18 implies that Ph(M) is pure injective. As the kernel of the phantom cover is pure injective, Condition X implies that M is pure injective. □

Xu [27, Thm 3.5.1] proved that if a right coherent ring R satisfies Condition X, then it is Xu.

Theorem 24. The following are equivalent for a ring R :

(1) the ring R satisfies Condition X;

(2) every object G ∈ ((R-\underline{mod})op, Ab) is cotorsion in ((R-mod)op, Ab);

(3) every nonzero object H ∈ (\underline{mod}-R, Ab) has injective dimension 2 in (mod-R, Ab).

Proof. (1) ⇔ (2). Every monomorphism f : M → N of pure injective modules yields a special flat resolution of an object G ∈ ((R-\underline{mod})op, Ab),

[figure: 0; (−, M); (−, f); (−, N); (−, g); (−, K); G; 0.]

Conversely, every object G ∈ ((R-\underline{mod})op, Ab) has a special flat resolution as above [18, Thm 11], with f : M → N a monomorphism of pure injective modules. Then the image Im (−, g) is cotorsion and we have the short exact sequence

[figure: 0; Im (−, g); (−, K); G; 0,]

which shows that G is cotorsion if and only if (−, K) is cotorsion if and only if K is pure injective.

(1) ⇔ (3). By [16], an object of the category (mod-R, Ab) is injective if and only if it is of the form − ⊗_{R} M, where M is a pure injective R-module. Every monomorphism f : M → N of pure injective modules thus induces an injective copresentation of an object H ∈ (\underline{mod}-R, Ab), that is, an object for which H(R_{R}) = 0,

[figure: 0; H; − ⊗ M; − ⊗ f; − ⊗ N.]

Conversely, the minimal injective copresentation of an object H ∈ (\underline{mod}-R, Ab) has this form. The tensor functor is right exact, so we obtain a resolution

[figure: 0; H; − ⊗ M; − ⊗ f; − ⊗ N; − ⊗ K; 0,]

where K = Coker f. Now the injective dimension of H is 2 if and only if − ⊗ K is injective if and only if K is pure injective. □

Corollary 25. If the ring R satisfies Condition X, then every functor of the form (−, \underline{M}) is projective in ((R-\underline{mod})op, Ab).

Proof. Consider the diagram

[figure: (−, M); α; φ_{M}; 0; G; F; η; (−, \underline{M}); 0,]

where the bottom row is an exact sequence in ((R-\underline{mod})op, Ab). Since R satisfies Condition X, Theorem 24 implies that the object G is cotorsion in ((R-mod)op, Ab), whence the morphism α : (−, M) → F. The induced morphism \underline{α} : (−, \underline{M}) → F is a section of η. □

For a contravariant cotorsion functor, the question of its injective dimension may be answered in general.

Theorem 26. Every cotorsion object of ((R-mod)op, Ab) has injective dimension at most 2.

Proof. Let G be a cotorsion object in ((R-mod)op, Ab). We will show that

Ext^{3}(F, G) = Ext^{1}(Ω^{2}(F), G) = 0,

for any object F ∈ ((R-mod)op, Ab). It suffices to prove that the second syzygy Ω^{2}(F) is flat. But consider a projective presentation

[figure: (−, M_{1}); (−, d_{1}); (−, M_{0}); π; F; 0]

of F in ((R-mod)op, Ab), with M_{0} and M_{1} pure projective modules. If M_{2} is the kernel of d_{1}, then Ω^{2}(F) ≅ (−, M_{2}) is flat. □

7. \underline{Hom}-Precovers

Benson and Gnacadja [8] found a group ring R = k[G] with the property that there exists a flat object in the category ((R-\underline{mod})op, Ab) that is not of the form (−, \underline{M}). Nevertheless, it is shown in this section that the class of flat objects of the form (−, \underline{M}) = \underline{Hom}(−, \underline{M}) has some appealing features.

Proposition 27. Every projective object P in ((R-\underline{mod})op, Ab) is of the form (−, \underline{M}) for some R-module M.

Proof. Let η : (−, M) → P be an epimorphism in ((R-mod)op, Ab) from a projective object (−, M). This morphism factors according to the commutative diagram

[figure: (−, M); φ_{M}; η; (−, \underline{M}); \underline{η}; P.]

As P is projective in ((R-\underline{mod})op, Ab), the morphism \underline{η} : (−, \underline{M}) → P has a section \underline{σ} : P → (−, \underline{M}), \underline{η}\underline{σ} = 1_{P} . The composition \underline{φ} = \underline{σ}\underline{η} is an idempotent endomorphism of (−, \underline{M}) with the property that P = Im \underline{φ}. By Proposition 13, there is an endomorphism f : M → M such that (−, \underline{f}) = \underline{φ}. Consider the directed system M_{n}, indexed by ℕ such that M_{n} = M for every n, with structural morphisms given by f : M_{n} → M_{n+1}. The direct limit is a module N with the property that (−, \underline{N}) is the colimit of the the directed system indexed by ℕ, where every object is (−, \underline{M}) and the morphism between consecutive objects is given by the idempotent endomorphism \underline{φ}. Thus (−, \underline{N}) ≅ \underline{φ}[(−, \underline{M})] = P. □

If the ring R satisfies Condition X, then Corollary 25 and Proposition 27 imply that the projective objects of ((R-\underline{mod})op, Ab) are precisely those of the form (−, \underline{M}) for some R-module M.

A flat object F ∈ ((R-\underline{mod})op, Ab) is countably presented if there is an exact sequence

[figure: ⨿_{j∈J}Q_{j}; ⨿_{i∈I}P_{i}; F; 0,]

where the P_{i} and Q_{j} are finitely generated projective objects, and the index sets I and J are both countable. By [23], every such flat module is a direct limit of finitely generated objects indexed by the partial order ℕ of natural numbers. A standard argument shows that F is then the colimit of a directed system of finitely generated representable objects (−, A_{n}), A_{n} ∈ R-mod, also indexed by ℕ. Such a system clearly lifts to a system A_{n}, in R-mod, whose direct limit M has the property that F ≅ (−, \underline{M}). This proves

Proposition 28. Every countably presented flat object of ((R-mod)op, Ab) is of the form (−, \underline{M}) for some R-module M.

Let η : (−, M) → G be the flat cover in ((R-mod)op, Ab) of an object G ∈ ((R-\underline{mod})op, Ab). This flat cover factors through the torsion-free quotient object (−, \underline{M}) as

[figure: (−, M); φ_{M}; η; (−, \underline{M}); \underline{η}; G.]

Let us show that the morphism \underline{η} : (−, \underline{M}) → G is a \underline{Hom}-precover of G in ((R-\underline{mod})op, Ab), in the sense that any morphism \underline{α} : (−, \underline{N}) → G from an object in the image of \underline{Υ} may be factored through \underline{η} as indicated in the commutative diagram

[figure: (−, \underline{N}); \underline{β}; \underline{α}; (−, \underline{M}); \underline{η}; G.]

Indeed, the torsion-free quotient morphism φ_{N} : (−, N) → (−, \underline{N}), composed with \underline{α} : (−, \underline{N}) → G yields a morphism α : (−, N) → G from the flat object (−, N). By the definition of a flat cover, the morphism α factors through η : (−, M) → G via the morphism (−, f) : (−, N) → (−, M) as indicated in the commutative diagram

[figure: (−, N); φ_{N}; (−, \underline{N}); \underline{α}; G; (−, f); \underline{β}; (−, M); φ_{M}; (−, \underline{M}); \underline{η}; G.]

The induced morphism \underline{β} = (−, \underline{f}) yields the desired factorization. We have proved

Theorem 29. Let G ∈ ((R-\underline{mod})op, Ab). If η : (−, M) → G is the flat cover of G in ((R-mod)op, Ab), then the induced morphism \underline{η} : (−, \underline{M}) → G on the torsion-free quotient object is a \underline{Hom}-precover of G in ((R-\underline{mod})op, Ab).

Corollary 30. If η : (−, M) → (−, \underline{N}) is the flat cover of (−, \underline{N}) in ((R-mod)op, Ab), then (−, \underline{N}) is a coproduct factor of (−, \underline{M}).

Proof. By Theorem 29, the induced morphism \underline{η} : (−, \underline{M}) → (−, \underline{N}) is a split epimorphism. □

A \underline{Hom}-precover \underline{η} : (−, \underline{M}) → G is a \underline{Hom}-cover if any endomorphism \underline{α} of (−, \underline{M}) over G,

[figure: (−, \underline{M}); \underline{α}; \underline{η}; (−, \underline{M}); \underline{η}; G.]

is an automorphism of (−, \underline{M}).

Theorem 31. Let M be an R-module with the property that the torsion-free quotient morphism φ_{M} : (−, M) → (−, \underline{M}) is a flat cover in ((R-mod)op, Ab). Given an object G ∈ ((R-\underline{mod})op, Ab) a morphism η : (−, M) → G is the flat cover of G in ((R-mod)op, Ab) if and only if the induced morphism \underline{η} : (−, \underline{M}) → G is the \underline{Hom}-cover of G in ((R-\underline{mod})op, Ab).

Proof. Suppose that η : (−, M) → G is the flat cover in ((R-mod)op, Ab). By Proposition 29, the induced morphism \underline{η} : (−, \underline{M}) → G is a \underline{Hom}-precover. To see that it is a \underline{Hom}-cover, let \underline{α} : (−, \underline{M}) → (−, \underline{M}) be an endomorphism of (−, \underline{M}) over G. The following commutative diagram arises,

[figure: (−, M); φ_{M}; (−, \underline{M}); \underline{η}; G; (−, f); \underline{α}; (−, M); φ_{M}; (−, \underline{M}); \underline{η}; G.]

The endomorphism (−, f) : (−, M) → (−, M) exists, because φ_{M} : (−, M) → (−, \underline{M}) is a flat precover. Since η : (−, M) → G is the flat cover of G, the morphism f : M → M is necessarily an automorphism. But then so is \underline{α} = (−, \underline{f}).

Suppose now that the induced morphism \underline{η} : (−, \underline{M}) → G is a \underline{Hom}-precover and that the morphism φ_{M} : (−, M) → (−, \underline{M}) is a flat precover in ((R-mod)op, Ab). Let us show that the composition η : (−, M) → G is a flat precover in ((R-mod)op, Ab). If α = \underline{α}φ_{N} : (−, N) → G is a morphism from a flat object, it factors through (−, \underline{M}) according to the commutative diagram

[figure: (−, N); φ_{N}; (−, \underline{N}); \underline{α}; G; (−, f); \underline{β}; (−, M); φ_{M}; (−, \underline{M}); \underline{η}; G;]

the morphism (−, f) : (−, N) → (−, M) exists because the morphisms \underline{η} and φ_{M} are a \underline{Hom}-precover and flat precover, respectively.

If \underline{η} : (−, \underline{M}) → G is a \underline{Hom}-cover and φ_{M} : (−, M) → (−, \underline{M}) is a flat cover in ((R-mod)op, Ab), then it is readily verified that any endomorphism (−, f) of (−, M) is necessarily an automorphism.

□

The category ((R-\underline{mod})op, Ab) is said to be pure semisimple [13] provided that every flat object is projective. Equivalently, every object in ((R-\underline{mod})op, Ab) has a projective cover.

Corollary 32. If R is a Xu ring, then the category ((R-\underline{mod})op, Ab) is pure semisimple, and the functor \underline{Ψ} : R-\underline{Cotor} → Flat((R-\underline{mod})op, Ab) is an equivalence.

Proof. Let F ∈ ((R-\underline{mod})op, Ab) and consider the flat cover η : (−, M) → F of F in ((R-mod)op, Ab). By Proposition 16, M is a flat-free cotorsion module. As R is Xu, M is pure injective. By Proposition 22, the morphism φ_{M} : (−, M) → (−, \underline{M}) is a flat precover. But M is flat-free, so has no nonzero flat summands. It follows that the kernel of φ_{M} contains no direct summands of (−, M) and hence that φ_{M} is the flat cover of (−, \underline{M}). By Theorem 31, the morphism \underline{η} : (−, \underline{M}) → F induced by the flat cover is the \underline{Hom}-cover of F. Since R is Xu, Corollary 25 and Proposition 27 imply that an object of ((R-\underline{mod})op, Ab) is projective if and only if it is of the form (−, \underline{M}). Thus the morphism \underline{η} : (−, \underline{M}) → F is the projective cover of F in ((R-\underline{mod})op, Ab).

The foregoing implies that the functor \underline{Ψ} : R-\underline{Cotor} → Flat((R-\underline{mod})op, Ab) is dense. We noted that it is full and faithful just before Proposition 22.

□

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