INJECTIVE FUNCTORS ON THE STABLE CATEGORY

IVO HERZOG

Abstract. Given an associative ring R, denote by ((R-\underline{mod})op, Ab) the category whose objects are the contravariant functors G : (R-\underline{mod})op → Ab from the stable category R-\underline{mod} of finitely presented left R-modules to the category Ab of abelian groups. It is proved that the subcategory of injective objects of ((R-\underline{mod})op, Ab) is equivalent to the category of pureinjective left R-modules, modulo morphisms that factor through an injective. This yields a description of the contravariant Gabriel spectrum of R, the set of indecomposable injective objects of the functor category ((R-mod)op, Ab). The points are in bijective correspondence with the set of pure-injective indecomposable left R-modules, which correspond to the points of the covariant Gabriel spectrum of R. It is proved that both Gabriel spectra of R may be partitioned into an open and closed set such that this canonical bijection restricts to a homeomorphism on each.

These results follow from a characterization of minimal flat resolutions in the category ((R-mod)op, Ab). Using the characterization, we are able to describe the relationship between the minimal flat resolution of a functor on the stable category and its minimal injective copresentation in ((R-\underline{mod})op, Ab). This relationship is entirely analogous to results obtained by Auslander and Reiten for finitely presented functors in the case when R is an artin algebra.

Let R be an associative ring with identity and denote by R-mod the category of finitely presented left R-modules. Given a left R-module M, there is an associated contravariant functor (−, M) := Hom_{R}(−, M) : (R-mod)op → Ab with values in the category Ab of abelian groups. This rule M ↦ (−, M) constitutes a full and faithful functor from the category R-Mod of all left R-modules into the Grothendieck category ((R-mod)op, Ab) of contravariant functors F : (R-mod)op → Ab. M. Auslander proposed to study the representation theory of R in terms of the ambient category ((R-mod)op, Ab). In his work on coherent functors [1], he showed that if the category R-mod is abelian, that is, if the ring R is left coherent, then the subcategory fp((R-mod)op, Ab) of finitely presented contravariant functors is itself an abelian category.

The category R-Mod may also be embedded into the Grothendieck category (mod-R, Ab) of covariant functors G : mod-R → Ab on the category mod-R of finitely presented right R-modules. This is accomplished by associating to the left R-module M the functor −⊗_{R}M. The category of covariant functors has several advantages over the category of contravariant functors. Auslander [2, §III.2] showed that the subcategory fp(mod-R, Ab) is abelian without any hypotheses on the ring R; Gruson and Jensen [18] characterized the injective objects

2000 Mathematics Subject Classification. 16B50, 16E30, 18A25, 18G05, 18G10, 18G15. Key words and phrases. stable category, functor category, flat cover, injective functor, Gabriel spectrum,

triangulated category.

Partially supported by NSF Grant DMS05-01207.

of (mod-R, Ab) as the functors isomorphic to some − ⊗_{R} M, where M is a pure-injective R-module; and Auslander [4] and Gruson and Jensen [18] noticed independently that there exists a duality between the respective subcategories fp(mod-R, Ab) and fp(R-mod, Ab) of finitely presented functors.

Perhaps the most profound application of the functorial perspective was obtained in the representation theory of an artin algebra R = Λ. Auslander and Reiten [6] analyzed the finer aspects of the category fp((Λ-mod)op, Ab), and then developed a theory of almost split sequences [7]. In this setting, the category of contravariant functors enjoys all the advantages of the category of covariant functors, because the duality D : (Λ-mod)op → mod-Λ (cf. [2, §III.1]) induces an equivalence D_{∗} : (mod-Λ, Ab) → ((Λ-mod)op, Ab) of categories. The objective of this article is to develop a theory for the category ((R-mod)op, Ab) of contravariant functors for a general ring R that is analogous to the theory developed by Auslander and Reiten in [6] for the category fp((Λ-mod)op, Ab) of finitely presented contravariant functors when the ring is an artin algebra Λ.

For example, a fundamental property of a finitely presented contravariant functor F : (Λ-mod)op → Ab is that it possesses a minimal projective resolution, which is of the form

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

where M, N and K are finitely presented left Λ-modules. This may be generalized to the setting where R is a Krull-Schmidt ring [20], but does not hold in general. The analogous property of a general contravariant functor G in ((R-mod)op, Ab) is that it possesses a minimal flat resolution. The flat objects of ((R-mod)op, Ab) have been characterized by Crawley-Boevey [14, Thm. 1.4] as those functors isomorphic to (−, M) for some left Rmodule M. The existence of flat covers in ((R-mod)op, Ab) follows from a result of Enochs, Estrada, Garcia Rozas, and Oyonarte [15, Thm. 2.7]. In the first chapter (Theorem 10) of this article, we characterize minimal flat resolutions in ((R-mod)op, Ab) as the flat resolutions having the form

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

where M and N are pure-injective left R-modules (possibly 0) and f : M → N and g : N → K are nowhere pure morphisms (see §1.4 for the definition). We also indicate how pure-injective envelopes may be used to construct a minimal flat resolution of G starting with a projective presentation. This yields a ”constructive” proof of the existence of flat covers in the category ((R-mod)op, Ab) independent of the (more general) result due to Enochs, et. al. The characterization of minimal flat resolutions is an easy consequence of the following theorem, which is a special case of an observation made in [22]. Recall that a functor F ∈ ((R-mod)op, Ab) is cotorsion if Ext^{1}[(−, N), F] = 0 for every flat object (−, N) of ((R-mod)op, Ab).

Theorem 4. A flat functor (−, M) ∈ ((R-mod)op, Ab) is cotorsion if and only if M is a pure-injective module.

The (projectively) stable category R-\underline{mod} is the category whose objects are those of R-mod, and whose morphisms are the morphisms of R-mod modulo morphisms that factor through a projective module. As in the case of R-mod, the preadditive category R-\underline{mod} may be fully and faithfully embedded, via the functor \underline{M} ↦ \underline{Hom}(−, \underline{M}), into the Grothendieck category ((R-\underline{mod})op, Ab) of contravariant additive functors G : (R-\underline{mod})op → Ab. Auslander and Reiten [6] proved that if R = Λ is an artin algebra and M ∈ Λ-mod, then the functor Ext^{1}(−, M) is a finitely presented object of ((Λ-\underline{mod})op, Ab); they characterized these functors as the injective objects of fp((Λ-\underline{mod})op, Ab). The main result of this paper is the following analogue in the functor category ((R-\underline{mod})op, Ab).

Theorem 23. The functor \overline{M} ↦ Ext^{1}(−, M) is an equivalence between the category R-\overline{Pinj} of pure-injective left R-modules, modulo morphisms that factor through an injective module, and the subcategory of injective objects of ((R-\underline{mod})op, Ab).

Gruson and Jensen [18] proved that if M is an R-module and m : M → PE(M) is the pure-injective envelope of M, then the induced morphism −⊗m : −⊗_{R} M → −⊗_{R} PE(M) is the injective envelope in the category (mod-R, Ab) of the object − ⊗_{R} M. Theorem 31 shows that the induced morphism Ext^{1}(−, m) : Ext^{1}(−, M) → Ext^{1}(−, PE(M)) is the injective envelope of Ext^{1}(−, M) in ((R-\underline{mod})op, Ab).

For an artin algebra Λ, Auslander and Reiten [6, Prop. 4.6] showed that if an object G in fp((Λ-\underline{mod})op, Ab) has the minimal projective resolution

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

in fp((Λ-mod)op, Ab), then the minimal injective copresentation of G in fp((Λ-\underline{mod})op, Ab) is given by

[figure: 0; G; Ext^{1}(−, M); Ext^{1}(−, f); Ext^{1}(−, N).]

This minimal injective copresentation is obtained from the next two terms of the long exact sequence of functors associated to the extension in Ext^{1}(K, M) that induces the minimal projective resolution of G. A similar result (Corollary 33) is proved in this article; it asserts the same property for an object of the category ((R-\underline{mod})op, Ab) for a general ring R, with the minimal projective resolution replaced by a minimal flat resolution. This result has a kind of converse (Corollary 34), which shows how a minimal injective copresentation in ((R-\underline{mod})op, Ab) of an object G may be used to obtain the minimal flat resolution of G in ((R-mod)op, Ab).

The projection functor π : R-mod → R-\underline{mod} induces a full and faithful functor

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

that permits one to identify the category ((R-\underline{mod})op, Ab) with the subcategory of ((R-mod)op, Ab) of functors G that vanish on the regular representation _{R}R, G(R) = 0. The subcategory ((R-\underline{mod})op, Ab) is a torsion class of ((R-mod)op, Ab). Similarly, the category (\underline{mod}-R, Ab) may be viewed as the torsion class of (mod-R, Ab) consisting of functors that vanish on R_{R}. The injective objects of (\underline{mod}-R, Ab) have the form t(− ⊗_{R} M), where M is a pure-injective R-module and t(−) is the torsion subfunctor. We use the dual of an argument found in [7] to prove the following.

Theorem 29. The Auslander-Bridger Transpose Tr : \underline{mod}-R → (R-\underline{mod})op induced an equivalence of categories Tr_{∗} : ((R-\underline{mod})op, Ab) → (\underline{mod}-R, Ab) with the property that

Tr_{∗}[Ext^{1}(−, M)] ≅ t(− ⊗ M),

for every R-module M.

The Gabriel spectrum of ((R-mod)op, Ab) is the space Sp((R-mod)op, Ab) whose points are the indecomposable injective objects of ((R-mod)op, Ab), up to isomorphism. Theorem 23 may be used to give a complete list, without repetition, of the points of the Gabriel spectrum Sp((R-mod)op, Ab). They fall into 2 classes:

With torsion: E[Ext^{1}(−, M)], the injective envelope of Ext^{1}(−, M), where M is an

indecomposable pure-injective left R-module that is not injective; Torsion-free: Hom(−, E), where E is an indecomposable injective left R-module.

This classification of the indecomposable injective objects of ((R-mod)op, Ab) yields a canonical bijective correspondence Ξ_{R} between Sp(mod-R, Ab) and Sp((R-mod)op, Ab) given by

(1)  Ξ_{R} : − ⊗_{R} U ↦ \{ \begin{matrix}E[Ext^{1}(−, U)], & if U is not injective; \\ (−, U), & if U is injective. \end{matrix}

Because the subcategory (\underline{mod}-R, Ab) ⊆ (mod-R, Ab) is a hereditary torsion class, it induces a partition of the Gabriel spectrum

Sp(mod-R, Ab) = Sp(\underline{mod}-R, Ab) ∪^{·} Sp(R-Mod)

into an open subset and a closed subset. The hereditary torsion class ((R-\underline{mod})op, Ab) of ((R-\underline{mod})op, Ab) induces a similar partition of Sp((R-\underline{mod})op, Ab). The canonical bijective correspondence Ξ_{R} respects these partitions (Theorem 36) and restricts to a homeomorphism between the open subsets and closed subsets, respectively. If the ring R is left coherent, then the categories ((R-mod)op, Ab) and (mod-R, Ab) are locally coherent Grothendieck categories, and the correspondence Ξ_{R} induces a similar relationship between the covariant and contravariant Ziegler spectra of R. If R is left semihereditary, then Proposition 51 shows that Ξ_{R} is a homeomorphism between Sp(mod-R, Ab) and Sp((R-mod)op, Ab).

Let R-\overline{Mod} be the category whose objects are the left R-modules, and whose morphisms are the morphisms of R-Mod modulo those that factor through an fp-injective module. If R is a Quasi-Frobenius ring, then the category R-\overline{Mod} is the category of stable modules and may be equipped with the structure of a triangulated category [11, 33, 34]. A theory of purity has been developed for triangulated categories by Beligiannis [8, 9] and H. Krause [24]. In the fourth chapter, Theorem 23 is used to introduce notions in R-\overline{Mod} that generalize those of pure-monomorphism and pure-injective object in the Quasi-Frobenius case. It is shown that every object \overline{M} of R-\overline{Mod} admits an R-\overline{Pinj}-envelope, where R-\overline{Pinj} ⊆ R-\overline{Mod} is the subcategory of classes with a pure-injective representative. Considered as a morphism, the R-\overline{Pinj} envelope \overline{m} : \overline{M} → PE(\overline{M}) is an Ext-monomorphism, that is, Ext^{1}(−, m) is a monomorphism in ((R-\underline{mod})op, Ab). If R is Quasi-Frobenius, the objects of R-\overline{Pinj} are the pure-injective objects of R-\overline{Mod}, considered as a triangulated category, and the Extmonomorphisms are the pure-monomorphisms of R-\overline{Mod}.

If R-Mod is embedded into the category ((R-mod)op, Ab) via the rule M ↦ (−, M), then its objects may be characterized as the flat objects of ((R-mod)op, Ab). Similarly, the objects of (mod-R, Ab) that are isomorphic to −⊗_{R}M, for some R-module M, may be homologically characterized as the fp-injective objects of (mod-R, Ab). In the fifth chapter of the article, we show (Corollary 44) that if the ring R is left coherent, then the objects Ext^{1}(−, M) are fp-injective in the category (mod-R, Ab). This leads to a characterization of the left coherent rings R for which the category ((R-\underline{mod})op, Ab) is locally noetherian.

Theorem 48 The following conditions are equivalent for a left coherent ring R : (1) for every module M, the quotient PE(M)/M is fp-injective;

(2) for every module M, the object Ext^{1}(−, M) is injective in ((R-\underline{mod})op, Ab);

(3) the category ((R-\underline{mod})op, Ab) is locally noetherian.

The fifth chapter also contains a brief treatment of Garavaglia’s notion [29] of elementary Krull dimension. It is shown (Theorem 47) that if a left R-module M has finite elementary Krull dimension n, then its pure-injective dimension is bounded by n.

Throughout the article, R will denote an associative ring with identity and J(R) the Jacobson radical of R. The unadorned term R-module will refer to a unital left R-module. The category of R-modules is denoted by R-Mod; the category of abelian groups by Ab. The category of finitely presented (resp., right) R-modules is denoted by R-mod (resp., mod-R). Given R-modules M and N, we use the abbreviation (M, N) := Hom_{R}(M, N); the unqualified tensor product ⊗ will always refer to the tensor ⊗_{R} over R. The category of contravariant additive functors F : (R-mod)op → Ab is denoted by ((R-mod)op, Ab); the category of covariant functors by (mod-R, Ab). If F and G are objects of ((R-mod)op, Ab), then the set of morphisms [F, G] consists of the natural transformations from F to G. The notation F : A → B for a functor between categories A and B is reserved for covariant functors. Thus a contravariant functor from A to B is indicated by G : Aop → B.

1. Minimal flat resolutions

In this section, we will characterize minimal flat resolutions in ((R-mod)op, Ab) and show how a projective presentation of an object F ∈ ((R-mod)op, Ab) may be used to construct a minimal flat resolution of F.

1.1 Flat functors.

A functor F ∈ ((R-mod)op, Ab) is called representable if it is isomorphic to a functor of the form Hom(−, A), where A is a finitely presented module.

Yoneda’s Lemma. (cf. [31, Prop. IV.7.3]) Let A be a finitely presented R-module and F ∈ ((R-mod)op, Ab). There is an isomorphism Θ_{(A,F)} : [(−, A), F] → F (A) natural in both F and A.

A sequence of functors F →^{μ} G →^{ν} H is exact in ((R-mod)op, Ab) if for every finitely presented R-module A, the corresponding sequence of abelian groups

[figure: F (A); μ(A); G(A); ν(A); H(A)]

is exact. By Yoneda’s Lemma, every representable object of ((R-mod)op, Ab) is projective. Indeed, because the representable functors form a generating set [31, Cor. 7.5], an object G ∈ ((R-mod)op, Ab) is projective if and only if it is isomorphic to a coproduct factor of a coproduct of representable functors. A finite coproduct of representable functors is representable and because idempotents split in R-mod, a functor in ((R-mod)op, Ab) is representable if and only if it is a finitely generated projective object of ((R-mod)op, Ab).

Proposition 1. ([31, Cor. 7.4]) The Yoneda functor Υ : R-mod → ((R-mod)op, Ab), given by A ↦ (−, A), is a full and faithful left exact functor. It is an equivalence between the category R-mod of finitely presented R-modules and the category proj((R-mod)op, Ab) of finitely generated projective objects of ((R-mod)op, Ab).

As in the case of exactness, a direct limit lim_{→} F_{i} of functors in ((R-mod)op, Ab) is calculated objectwise. Thus if A ∈ R-mod, then

(lim_{→} F_{i})(A) := lim_{→} F_{i}(A),

where the latter direct limit is taken in the category Ab of abelian groups. An object G ∈ ((R-mod)op, Ab) is flat if it is isomorphic to a direct limit of finitely generated projective functors,

G ≅ lim_{→} (−, A_{i}).

The (full) subcategory of ((R-mod)op, Ab) of flat functors is denoted by Flat((R-mod)op, Ab). Because the Yoneda functor is an equivalence between R-mod and the category of representable functors, we may consider the corresponding direct limit M = lim_{→} A_{i} in R-Mod. A well-known characterization [31, Prop. V.3.4] of finitely presented modules implies that

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

Proposition 2. [14, Thm. 1.4] The functor Υ : R-Mod → ((R-mod)op, Ab), given by M ↦ (−, M), is a full and faithful left exact functor. It yields an equivalence between the category R-Mod of R-modules and the category Flat((R-mod)op, Ab).

Proposition 2 may be used to see that the category Flat((R-mod)op, Ab) is closed under direct limits, direct products and coproduct factors in ((R-mod)op, Ab). Every projective object of ((R-mod)op, Ab) is flat.

In general, the category R-mod is not abelian. However, it has cokernels, so we may call a contravariant functor F : (R-mod)op → Ab left exact if it takes cokernels to kernels. Since every flat functor in ((R-mod)op, Ab) is isomorphic to a functor of the form (−, M), it is left exact. Let us note that, conversely, if F : (R-mod)op → Ab is left exact, then F ≅ (−, F (R)) is flat. First, there is an obvious isomorphism α(R) : F (R) → (R, F (R)) of abelian groups; it induces for every finitely generated free module R^{n} an isomorphism α(R^{n}) : F (R^{n}) → (R, F (R^{n})). Given A ∈ R-mod, consider a free presentation

[figure: R^{m}; R^{n}; A; 0]

and apply the two functors F and (−, F (R)). An isomorphism α(A) : F (A) → (A, F (R)) is then induced by left exactness,

[figure: 0; F (A); F (R^{n}); F (R^{m}); α(A); ≅; ≅; 0; (A, F (R)); (R^{n}, F (F)); (R^{m}, F (R)).]

This observation together with a routine diagram chase may be used to show the following.

Proposition 3. If the sequence in ((R-mod)op, Ab)

[figure: 0; (−, M); F; (−, K); 0]

is exact, then F is flat.

Let n be a whole number. An object F ∈ ((R-mod)op, Ab) is said to be of flat global dimension at most n if there is a flat resolution

[figure: ⋯; (−, M_{2}); (−, M_{1}); (−, M_{0}); D_{0}; F; 0]

with M_{n+1} = 0. Consider a projective presentation

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

in ((R-mod)op, Ab) of the functor F. By Proposition 2, there is an R-morphism d_{1} : M_{1} → M_{0} such that η = (−, d_{1}). Let d_{2} : M_{2} → M_{1} be the kernel of d_{1}. Because the functor Υ is left exact, the sequence

[figure: 0; (−, M_{2}); (−, d_{2}); (−, M_{1}); (−, d_{1}); (−, M_{0}); D_{0}; F; 0]

yields an exact sequence in ((R-mod)op, Ab), which is a flat resolution of F of length at most 2. Thus every object in the category ((R-mod)op, Ab) has flat dimension at most 2.

Let F ∈ ((R-mod)op, Ab). A morphism φ : (−, M) → F from a flat functor is a flat precover if, given a morphism ψ : (−, N) → F whose domain is a flat object (−, N), there is an R-morphism f : N → M such that the diagram

[figure: (−, N); (−, f); ψ; (−, M); φ; F]

commutes. A flat precover φ : (−, M) → F is a flat cover if any R-endomorphism g : M → M for which the diagram

[figure: (−, M); (−, g); φ; (−, M); φ; F]

commutes is an automorphism of M.

1.2 Cotorsion functors.

A functor F ∈ ((R-mod)op, Ab) is cotorsion if Ext^{1}[(−, M), F] = 0 for every flat object (−, M) of ((R-mod)op, Ab). Any undefended claim that we make about cotorsion objects in ((R-mod)op, Ab) has a proof similar to the corresponding fact about cotorsion modules, and may be found in [35]. If F is cotorsion, then Ext^{i}[(−, M), F] = 0 for all i ≥ 1. This may be verified with the help of the argument above, which shows that in a projective resolution of a flat module (−, M), all the syzygies are flat. Consequently, if F is a cotorsion object and

[figure: 0; F; G; H; 0]

is a short exact sequence in ((R-mod)op, Ab), then G is cotorsion if and only if H is cotorsion. If F is cotorsion in the short exact sequence

[figure: 0; F; (−, M_{0}); D_{0}; G; 0,]

then D_{0} : (−, M_{0}) → G is a flat precover of G. Dually, if G ∈ ((R-mod)op, Ab) is arbitrary and there is a cotorsion object F such that the short exact sequence

[figure: 0; G; η; F; (−, M); 0]

is exact, then the morphism η : G → F is a cotorsion preenvelope; a preenvelope is defined in a manner dual to that of a precover [35, §1.2].

In order to characterize flat cotorsion functors, let us recall the notion of a pure-injective module. A short exact sequence of R-modules

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

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

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

is exact. In that case, the morphism f : M → N is called a pure-monomorphism. A submodule M ⊆ N is called a pure submodule if the inclusion morphism ι : M → N is a puremonomorphism. The morphism g in a pure-exact sequence is called a pure-epimorphism. Thus a morphism g is a pure-epimorphism if and only if for every finitely presented module A, the morphism (A, g) : (A, N) → (A, K) of abelian groups is an epimorphism.

A module M is pure-injective if every pure-monomorphism f : M → N is a split monomorphism. Every module M admits a pure-injective envelope m : M → PE(M), which is unique up to isomorphism over M. Dually, a module K is called pure-projective if every pure-epimorphism is a split epimorphism. It is clear from the definition that every finitely presented module is pure-projective.

On several occasions, we will need the fact that the endomorphism ring S = End_{R}M of a pure-injective module M is an exchange ring [26, §1.3]. This follows from the characterization [27, Thm. 2.1] of exchange rings due to Nicholson and is verified in [19, Lemma 2]. It implies [27, Prop. 1.9] that an endomorphism f : M → M belongs to the Jacobson radical J(S) if and only if the principal left ideal Sf contains no nonzero idempotent elements.

Theorem 4. A flat functor (−, M) ∈ ((R-mod)op, Ab) is cotorsion if and only if M is a pure-injective module. If M is an R-module and m : M → PE(M) is the pure-injective envelope, then (−, m) : (−, M) → (−, PE(M)) is the cotorsion envelope of the flat functor (−, M).

Proof. Suppose that (−, M) is a cotorsion object, and consider the pure-injective envelope m : M → PE(M). The short exact sequence

[figure: 0; M; m; PE(M); PE(M)/M; 0]

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

[figure: 0; (−, M); (−, PE(M)); (−, PE(M)/M); 0]

is exact. As (−, PE(M)/M) is flat, the sequence splits and M = PE(M) is pure-injective. For the converse, suppose that M is a pure-injective module and consider an extension

[figure: 0; (−, M); μ; G; ν; (−, Z); 0]

of (−, M) by a flat functor (−, Z). By Proposition 3, G is isomorphic to the flat functor (−, G(R)). Replacing G by (−, G(R)), we get that μ = (−, f) and ν = (−, g), where

[figure: 0; M; f; G(R); g; Z; 0]

is a pure-exact sequence. As M is pure-injective, the sequence splits.

To prove the second statement, let M be a left R-module. The morphism of functors (−, m) : (−, M) → (−, PE(M)) is a monomorphism into a cotorsion functor (−, PE(M)) whose cokernel is the flat object (−, PE(M)/M). Thus the morphism (−, m) : (−, M) → (−, PE(M)) is a cotorsion preenvelope. To verify that it is a cotorsion envelope, it suffices to check that (−, PE(M)) contains no proper direct summands containing (−, M). But any such summand would have to be of the form (−, M′) where M′ is a proper direct summand of PE(M) containing M. It follows from the definition of the pure-injective envelope that no such summand exists. □

1.3 Special resolutions.

A flat precover is called special [16, p. 153] if it has a cotorsion kernel. Given an object F in ((R-mod)op, Ab), a flat resolution

[figure: 0; (−, M_{2}); (−, d_{2}); (−, M_{1}); (−, d_{1}); (−, M_{0}); D_{0}; F; 0]

of F will be called special if both morphisms D_{0} : (−, M_{0}) → F and (−, d_{1}) : (−, M_{1}) → Im(−, d_{1}) are special flat precovers.

Proposition 5. The flat resolution of F given above is special if and only if M_{1} and M_{2} are pure-injective.

Proof. The kernel of (−, d_{1}) is cotorsion if and only if M_{2} is pure-injective. Assuming that M_{2} is pure-injective, then the Ker D_{0} = Im(−, d_{1}) is cotorsion if and only if (−, M_{1}) is cotorsion if and only if M_{1} is pure-injective. □

Suppose that G has flat dimension at most 1. There is a flat resolution of the form

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

Evaluating the resolution at _{R}R shows that f : M → N is a monomorphism. Let g : N → K be the cokernel of f; one obtains an exact sequence of functors

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

Because G is the cokernel of (−, f), it may be embedded into the flat functor (−, K).

Conversely, suppose that there is a monomorphism ι : G ⊆ (−, K) of G into a flat functor, and consider an epimorphism η : (−, N) → G from a projective functor. The composition is of the form (−, g) : (−, N) → (−, K). If f : M → N is the kernel of g, one obtains a flat resolution of G as above. Thus the flat dimension of G is at most 1. We have proved the following.

Proposition 6. An object G ∈ ((R-mod)op, Ab) is of flat dimension at most 1 if and only if there is a monomorphism ι : G → (−, K) into a flat object.

The flat resolution of G above may be used to produce a special flat resolution, if we take the pushout with the cotorsion envelope (−, m) : (−, M) → (−, PE(M)). One obtains the following commutative diagram with exact rows and columns,

[figure: 0; 0; 0; (−, M); (−, f); (−, N); π; G; 0; 0; (−, PE(M)); F; G; 0; (−, Z); (−, Z); 0; 0,]

where Z = PE(M)/M. By Proposition 3, the functor F ≅ (−, N′) is flat. Because (−, PE(M)) is cotorsion, the flat resolution of G given in the second row is special.

A special resolution of length 1 may be used to produce a cotorsion preenvelope of G. Let n : N → PE(N) be the pure-injective envelope and consider the commutative diagram with exact rows,

[figure: 0; (−, M); (−, f); (−, N); π; G; 0; (−, n); η; 0; (−, M); (−, nf); (−, PE(N)); G′; 0.]

Both M and PE(N) are pure-injective, so that G′ is cotorsion. The morphism η : G → G′ is clearly a monomorphism and because the right square is a pullback-pushout diagram, Coker η = Coker (−, n) ≅ (−, PE(N)/N) is flat. Thus the morphism η : G → G′ is a cotorsion preenvelope. The next proposition may also be deduced from [15, Thm. 2.7].

Proposition 7. Every object F ∈ ((R-mod)op, Ab) has a special flat resolution.

Proof. Consider an epimorphism η : (−, K′) → F from a projective object. The kernel G = Ker η is of flat dimension at most 1. By the foregoing observations, there is a cotorsion preenvelope ι : G → G′ with a flat cokernel. The pushout construction yields a morphism of short exact sequences,

[figure: 0; G; β; (−, K′); η; F; 0; ι; α; 0; G′; β′; H; F; 0,]

where the morphism α : (−, K′) → H is a monomorphism whose cokernel is isomorphic to the flat functor Coker ι. By Proposition 3, H = (−, K) is flat. As G′ is of flat dimension at most 1, it has a special flat resolution of length at most 1 and, because G′ is cotorsion, this special flat resolution consists of flat cotorsion objects. This provides a special flat resolution of F. □

1.4 Nowhere pure morphisms.

A morphism f : M → N will be called nowhere pure if there is no nonzero pure submodule M′ ⊆ M such that the restriction of f to M′ is a pure-monomorphism.

Lemma 8. A composition of morphisms f = gh : M →^{h} N →^{g} K is nowhere pure, if one of f and g is nowhere pure.

Proof. Suppose that f is not nowhere pure. Then there is a pure submodule M′ ⊆ M such that the restriction of f to M′ is a pure-monomorphism. The restriction of h to M′ must also be a pure-monomorphism. It follows that the image h(M′) ⊆ N is a pure submodule and that the restriction of g to h(M′) is also a pure-monomorphism. □

If M is a pure injective module and f : M → N is an R-morphism, then one may apply Zorn’s Lemma to obtain a pure submodule M′ ⊆ M maximal with respect to the property that the restriction of f to M′ is a pure-monomorphism. Since the pure-injective envelope PE(M′) is a pure-essential extension of M′, the restriction of f to PE(M′) is also a pure-monomorphism. It follows from the maximality of M′ that M′ = PE(M′) is a direct summand of M,

M = M′ ⊕ M′′,

where the restriction of f to M′ is a pure-monomorphism and its restriction to M′′ is nowhere pure.

Proposition 9. Let M be a pure-injective module and S = End_{R}M. A morphism f : M → N is a nowhere pure morphism if and only if for every morphism g : N → M, the composition gf ∈ J(S). Equivalently, the morphism f : M → N is not nowhere pure if and only if there is a morphism g : N → M such that gf = e is a nonzero idempotent element of S. In particular, an endomorphism f : M → M is nowhere pure if and only if f ∈ J(S).

Proof. The foregoing comments indicate that a morphism f : M → N from a pure-injective module M is a nowhere pure morphism if and only if there is no nonzero direct summand M′ of M such that the restriction of f to M′ is a pure-monomorphism. Equivalently, for every morphism g : N → M, the left ideal Sgf contains no idempotent elements. Since the endomorphism ring S = End_{R}M is an exchange ring, this is equivalent to gf ∈ J(S). □

A flat resolution of F

[figure: 0; (−, M_{2}); (−, d_{2}); (−, M_{1}); (−, d_{1}); (−, M_{0}); D_{0}; F; 0]

is called minimal if the morphisms D_{0} : (−, M_{0}) → F and (−, d_{1}) : (−, M_{1}) → Im(−, d_{1}) are flat covers. It follows from the definition of a flat cover that a minimal flat resolution {(−, M_{i})}_{i} of F admits a morphism from any other flat resolution of F, and that an endomorphism {(−, f_{i})}_{i} of {(−, M_{i})}_{i} that induces the identity 1_{F} on F is necessarily an automorphism. Thus any two minimal flat resolutions of F are isomorphic over F, and every special flat resolution of F contains a minimal one as a direct summand.

Theorem 10. Let F ∈ ((R-mod)op, Ab). A special resolution of F

[figure: 0; (−, M_{2}); (−, d_{2}); (−, M_{1}); (−, d_{1}); (−, M_{0}); D_{0}; F; 0]

is minimal if and only if d_{1} and d_{2} are nowhere pure.

Proof. Suppose that d_{2} is not nowhere pure. As M_{2} is pure-injective, d_{2} : M_{2} → M_{1} is a puremonomorphism on some nonzero direct summand of M_{2}. The morphism (−, d_{1}) then contains a nonzero summand in its kernel. It cannot be the flat cover of its image, contradicting minimality. That d_{1} is nowhere pure is proved similarly.

Suppose now that d_{1} and d_{2} are both nowhere pure. Since the given flat resolution is special, it suffices to show that any endomorphism

[figure: 0; (−, M_{2}); (−, d_{2}); (−, M_{1}); (−, d_{1}); (−, M_{0}); D_{0}; F; 0; (−, f_{2}); (−, f_{1}); (−, f_{0}); 0; (−, M_{2}); (−, d_{2}); (−, M_{1}); (−, d_{1}); (−, M_{0}); D_{0}; F; 0]

is an automorphism. Because the resolution is special, the endomorphism {(−, f_{i})}_{i} is homotopic to the identity morphism; the homotopy is given by R-morphisms s_{i} : M_{i} → M_{i+1} satisfying the equations

f_{2} − 1_{M_{2}} = s_{1}d_{2}, f_{1} − 1_{M_{1}} = d_{2}s_{1} + s_{0}d_{1}.

The morphisms d_{1} and d_{2} are nowhere pure so Lemma 8 implies that s_{1}d_{2}, d_{2}s_{1}, and s_{0}d_{1} are also nowhere pure. From Proposition 9, we get that f_{2} − 1_{M_{2}} ∈ J(End_{R}M_{2}) and f_{1} − 1_{M_{1}} ∈ J(End_{R}M_{1}). Thus f_{1} and f_{2} are isomorphism, and, because (−, f_{1}) and (−, f_{2}) are isomorphisms, so is (−, f_{0}). □

Let us use Theorem 10 and the knowledge that a minimal flat resolution must be a summand of any special flat resolution to find a minimal flat resolution. By Proposition 7, every functor F ∈ ((R-mod)op, Ab) has a special resolution

[figure: 0; (−, M_{2}); (−, d_{2}); (−, M_{1}); (−, d_{1}); (−, M_{0}); D_{0}; F; 0.]

There is a direct sum decomposition M_{2} = M′_{2} ⊕ M′′_{2}, where the restriction of d_{2} to M′_{2} is a pure-monomorphism and the restriction to M′′_{2} is nowhere pure. Then d_{2}(M′_{2}) is a direct summand of M_{1}, and we may consider the given special flat resolution modulo the resolution of 0 given by

[figure: 0; (−, M′_{2}); (−, d_{2}); (−, d_{2}(M′_{2})); 0; 0; 0.]

The quotient is also a special resolution of F with the further property that the morphism d_{2} is now nowhere pure. Similarly, there is a direct sum decomposition M_{1} = M′_{1} ⊕ M′′_{1} such that the restriction of d_{1} to M′_{1} is a pure-monomorphism and the restriction to M′′_{1} is nowhere pure. Then d_{1}(M′_{1}) is a direct summand of M_{0} and we may factor out by the resolution

[figure: 0; 0; (−, M′_{1}); (−, d_{1}); (−, d_{1}(M′_{1})); 0; 0]

of 0 to obtain a special resolution of F in which both d_{1} and d_{2} are nowhere pure. These considerations complete our proof of [15, Thm. 2.7] in the special case of ((R-mod)op, Ab).

Theorem 11. Every object F ∈ ((R-mod)op, Ab) has a minimal flat resolution.

2. Functors on the stable category

An R-module Z is flat if every epimorphism g : Y → Z is a pure-epimorphism. As in the case of functors, flat modules may be characterized (cf. [14, Thm. 1.4]) as direct limits of finitely generated projective modules. It follows that every morphism f : A → Z from a finitely presented module A to a flat module Z factors through a finitely generated projective module.

Given R-modules M and N, let Flat(M, N) ⊆ Hom_{R}(M, N) be the subgroup of morphisms that factor through a flat module. The category R-\underline{Mod} is the additive category whose objects are of the form \underline{M}, where M is an R-module, and whose morphism groups are given by

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

The (projectively) stable category of finitely presented R-modules is the full subcategory R-\underline{mod} ⊆ R-\underline{Mod} of classes \underline{M} represented by finitely presented modules, modulo morphisms that factor through a flat module. This is equivalent to the standard definition [5], in which the morphism groups are defined to be the quotient groups Hom(A, B)/proj(A, B), where proj(A, B) is the subgroup of morphisms that factor through a finitely generated projective.

2.1 The torsion-free class ((R-\underline{mod})op, Ab).

The projection functor π : R-mod → R-\underline{mod} is defined by π(A) = \underline{A} on objects, and takes a morphism f : A → B to its class modulo Flat(A, B) = proj(A, B). Every contravariant functor G : (R-\underline{mod})op → Ab induces a contravariant functor F = G ∘ π : (R-mod)op → Ab with the property that F (R) = 0. Conversely, if F : (R-mod)op → Ab is a contravariant functor on R-mod such that F (R) = 0, then F (P) = 0 for every finitely generated projective P, and hence F (f) = 0 for every f ∈ proj(A, B). The functor F thus induces a functor G : (R-\underline{mod})op → Ab with the property that F = G ∘ π. We may therefore identify the category ((R-\underline{mod})op, Ab) of contravariant functors on R-\underline{mod} with the full subcategory of ((R-mod)op, Ab) whose objects are the contravariant functors that vanish on the R-module R. This subcategory ((R-\underline{mod})op, Ab) is a Grothendieck category in its own right, and a sequence of morphisms in ((R-\underline{mod})op, Ab) is exact if and only if it is such in ((R-mod)op, Ab).

Because the objects of ((R-\underline{mod})op, Ab) are the contravariant functors on R-mod that vanish on R, they may be characterized as the objects F of ((R-mod)op, Ab) with the property that for any (some) flat presentation

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

the morphism d_{1} : M_{1} → M_{0} is an epimorphism.

Theorem 12. If F ∈ ((R-\underline{mod})op, Ab) and D_{0} : (−, M_{0}) → F is the flat cover of F in ((R-mod)op, Ab), then the module M_{0} is cotorsion.

Proof. The flat cover is part of a minimal flat resolution of F, necessarily of length 2,

[figure: 0; (−, M_{2}); (−, d_{2}); (−, M_{1}); (−, d_{1}); (−, M_{0}); D_{0}; F; 0.]

Evaluating the resolution at R yields a short exact sequence

[figure: 0; M_{2}; d_{2}; M_{1}; d_{1}; M_{0}; 0.]

Since M_{2} and M_{1} are pure-injective (Proposition 5), they are both cotorsion. The module M_{0} is therefore also cotorsion. □

The subcategory ((R-\underline{mod})op, Ab) of ((R-mod)op, Ab) is closed under subobjects, products and extensions. It is therefore a torsion-free class. Denote by Gen(−, R) the subcategory of ((R-mod)op, Ab) consisting of all (−, R)-generated objects. The objects of Gen(−, R) are the quotient objects of coproducts of copies of (−, R). Because (−, R) is a projective object, Gen(−, R) is closed under extensions. It is therefore a torsion class in ((R-mod)op, Ab). Denote by t_{R}(F) ⊆ F the torsion subobject associated to F by the torsion class Gen(−, R).

Proposition 13. If the subcategory ((R-\underline{mod})op, Ab) ⊆ ((R-mod)op, Ab) is regarded as the torsion-free class of a torsion theory, then Gen(−, R) is the corresponding torsion class.

Proof. Clearly, the object (−, R) is torsion with respect to ((R-\underline{mod})op, Ab). It follows that every object of Gen(−, R) is also torsion with respect to ((R-\underline{mod})op, Ab). To prove the reverse inclusion let F ∈ ((R-mod)op, Ab) be an object that is torsion with respect to ((R-\underline{mod})op, Ab). There are no nonzero morphisms from (−, R) to the quotient object F/t_{R}(F), so it must be that F/t_{R}(F) ∈ ((R-\underline{mod})op, Ab) and so F = t_{R}(F). □

The rule F ↦ \underline{F} := F/t_{R}(F) yields a functor from from ((R-mod)op, Ab) to ((R-\underline{mod})op, Ab). For example, if M is an R-module and F = (−, M), then \underline{F} ≅ \underline{Hom}(−, \underline{M}). Indeed, an epimorphism g : R^{(α)} → M from a free module induces a flat presentation

[figure: (−, R)^{(α)}; (−, g); (−, M); \underline{Hom}(−, \underline{M}); 0.]

The sequence is exact, because if a morphism f : A → M with finitely presented domain A factors through a flat object, f = hk, where k : A → Z and Z is flat, then k factors through a finitely generated projective module, and hence so does f. But then f must factor through g : R^{(α)} → M. Now \underline{Hom}(−, \underline{M}) is torsion-free and the image of (−, g) is torsion. Thus Im(−, g) = t_{R}(F) and \underline{F} = \underline{Hom}(−, \underline{M}).

If F ∈ ((R-mod)op, Ab) and G ∈ ((R-\underline{mod})op, Ab), then, as with any torsion theory, there is an isomorphism of abelian groups

[\underline{F} , G] ≅ [F, G],

natural in both F and G. In other words, the functor F ↦ \underline{F} is the left adjoint of the inclusion functor from ((R-\underline{mod})op, Ab) to ((R-mod)op, Ab). By [31, Prop. 9.4], it is right exact and preserves direct limits.

Proposition 14. The functor F ↦ \underline{F} from ((R-mod)op, Ab) to ((R-\underline{mod})op, Ab) preserves representable objects, finitely presented objects and flat objects.

Proof. If A is a finitely presented module and F = (−, A), then \underline{F} is the representable object \underline{Hom}(−, \underline{A}) of ((R-\underline{mod})op, Ab). If F ∈ ((R-mod)op, Ab) is finitely presented, consider a presentation of F by representable objects

[figure: (−, A); (−, f); (−, B); F; 0.]

Because the functor F ↦ \underline{F} is right exact, one obtains a presentation of \underline{F} in ((R-\underline{mod})op, Ab) by representable objects

[figure: \underline{Hom}(−, \underline{A}); (−, \underline{f}); \underline{Hom}(−, \underline{B}); \underline{F}; 0.]

If M is an arbitrary R-module, it is isomorphic to a direct limit M = lim_{→} A_{i} of finitely presented modules A_{i}. The functor \underline{F} preserves direct limits, so if we apply it to both sides of the equation (−, M) = lim_{→}(−, A_{i}), we obtain that

\underline{Hom}(−, \underline{M}) ≅ lim_{→} \underline{Hom}(−, \underline{A}_{i})

is flat. □

Benson and Gnacadja [10] give an example of a group ring R = k[G] where not every flat object of ((R-\underline{mod})op, Ab) is of the form \underline{Hom}(−, \underline{M}).

2.2 The functor Ext^{1}(−, M).

Let M be an R-module. Because Ext^{1}(R, M) = 0, the contravariant functor Ext^{1}(−, M) is a functor on the stable category R-\underline{mod}. A morphism f : M → N in R-Mod induces a morphism Ext^{1}(−, f) : Ext^{1}(−, M) → Ext^{1}(−, N) in the category (R-mod)op, Ab).

Proposition 15. Let f : M → N be a monomorphism. Then f is a pure-monomorphism if and only if the induced morphism Ext^{1}(−, f) : Ext^{1}(−, M) → Ext^{1}(−, N) is a monomorphism in the category ((R-mod)op, Ab).

Proof. The short exact sequence 0 → M →^{f} N →^{g} K → 0 induces a long exact sequence of functors

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

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

A module X is absolutely pure if every monomorphism f : X → Y is a pure-monomorphism. Absolutely pure modules may be characterized as the modules X for which the pure-injective envelope x : X → PE(X) is the injective envelope. Equivalently, Ext^{1}(A, X) = 0 for every finitely presented left R-module A; it is for this reason that absolutely pure modules are also called fp-injective. We may express this by the equation

Ext^{1}(−, X) = 0

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

Let e : M → E be the injective envelope of a left R-module M; the cokernel is denoted by Ω^{−1}(M) := E/M. The short exact sequence

[figure: 0; M; e; E; p; Ω^{−1}(M); 0]

induces a long exact sequence of functors. Because Ext^{1}(−, E) = 0, the first few terms of this long exact sequence are given by

[figure: 0; (−, M); (−, e); (−, E); p; (−, Ω^{−1}(M)); D; Ext^{1}(−, M); 0,]

which is a flat resolution of Ext^{1}(−, M) in ((R-mod)op, Ab). If M is pure-injective, the flat resolution is special, by Proposition 5.

Given R-modules M and N denote by Abs(M, N) ⊆ Hom_{R}(M, N) the subgroup of morphisms f : M → N that factor through an fp-injective module. The preadditive category R-\overline{Mod} is defined to have the objects \overline{M}, where M is a left R-module, and the morphism groups given by

\overline{Hom}(\overline{M}, \overline{N}) : Hom_{R}(M, N)/Abs(M, N).

Because Ext^{1}(−, X) = 0 for every fp-injective module X, the functor M ↦ Ext^{1}(−, M) from R-Mod to ((R-\underline{mod})op, Ab) induces a functor

Ext^{1}(−,?) : R-\overline{Mod} → ((R-\underline{mod})op, Ab)

given by \overline{M} ↦ Ext^{1}(−, M).

Theorem 16. Let M be a left R-module and N a pure-injective left R-module. The induced morphism of abelian groups

Ext^{1}(−,?) : \overline{Hom}(\overline{M}, \overline{N}) → [Ext^{1}(−, M), Ext^{1}(−, N)],

defined by \overline{f} ↦ Ext^{1}(−, f) is an isomorphism.

Proof. To prove that the morphism is onto, let η : Ext^{1}(−, M) → Ext^{1}(−, N) be a morphism in ((R-mod)op, Ab) and consider the respective flat resolutions of Ext^{1}(−, M) and Ext^{1}(−, N).

[figure: 0; (−, M); (−, E(M)); (−, Ω^{−1}(M)); Ext^{1}(−, M); 0; (−, f); (−, g); (−, k); η; 0; (−, N); (−, E(N)); (−, Ω^{−1}(N)); Ext^{1}(−, N); 0]

Because N is pure-injective, the flat resolution of Ext^{1}(−, N) is special. This gives rise to a morphism of flat resolutions, unique up to homotopy, indicated by the dotted arrows. Evaluating the commutative diagram at the module R yields a morphism of short exact sequences,

[figure: 0; M; e_{M}; E(M); p_{M}; Ω^{−1}(M); 0; f; g; k; 0; N; e_{N}; E(N); p_{N}; Ω^{−1}(N); 0.]

Since E(M) and E(N) are injective, the induced morphism of long exact sequences of functors implies that η = Ext^{1}(−, f).

To demonstrate that the morphism of abelian groups is injective, suppose that a morphism of flat resolutions as above induces the morphism η = Ext^{1}(−, f) = 0. It must then be homotopic to the zero morphism, which implies that f : M → N factors through e_{M} : M → E(M). □

Given R-modules M and N, the theorem helps us make sense of what it means to be a morphism α : Ext^{1}(−, M) → Ext^{1}(−, N) in the category (R-\underline{mod})op, Ab). If n : N → PE(N) is the pure-injective envelope, then, according to the theorem, there is an R-morphism f : M → PE(N) that makes the diagram

[figure: Ext^{1}(−, PE(N)); Ext^{1}(−, f); Ext^{1}(−, n); Ext^{1}(−, M); α; Ext^{1}(−, N)]

commute. Given a finitely presented module A and an extension ζ ∈ Ext^{1}(A, M), the extension Ext^{1}(A, f)(ζ) gotten by pushout along f : M → PE(N) arises from the extension α(A)(ζ) ∈ Ext^{1}(A, N) by pushout along the pure-injective envelope of N. Since the morphism Ext^{1}(−, n) is a monomorphism, this property characterizes α(A)(ζ).

2.3 Injective-free pure-injective modules.

The proof of Theorem 16 shows that if a morphism f : M → N, with N pure-injective, factors through an fp-injective module, then it factors through an injective module. This is not surprising, because if f = gh where g : K → N and K is fp-injective, then g factors through the pure-injective envelope k : K → PE(K), which is injective. Thus if M and N are pure-injective, then

Abs(M, N) = Inj(M, N),

where Inj(M, N) ⊆ Hom_{R}(M, N) is the subgroup of morphisms that factor through an injective module. Let R-\overline{Pinj} ⊆ R-\overline{Mod} be the subcategory of classes of the form \overline{M} where M is pure-injective.

Call a pure-injective module M injective-free if it contains no nonzero injective summands.

Proposition 17. Let M be a pure-injective left R-module. There is a direct sum decomposition

M = Mif ⊕ E,

where E is injective and Mif is injective-free.

Proof. By Zorn’s Lemma, there is a maximal fp-injective submodule E ⊆ M. As M is pureinjective, the pure-injective envelope PE(E) is an injective summand of M. By the choice of E, E = PE(E) is therefore injective. Thus

M = Mif ⊕ E,

where Mif is a pure-injective module with no nonzero fp-injective submodules. □

Let M be an injective-free pure-injective module. Then the injective envelope e : M → E is nowhere pure. The cokernel p : E → Ω^{−1}(M) of the injective envelope is also nowhere pure, because it has an essential kernel. By Theorem 10, the flat resolution of Ext^{1}(−, M) given by

[figure: 0; (−, M); (−, e); (−, E); (−, p); (−, Ω^{−1}(M)); D_{0}; Ext^{1}(−, M); 0]

is minimal. If N is any pure-injective R-module, we may decompose it according to Proposition 17 as N = Nif ⊕ E_{N}, where Nif is an injective-free pure-injective, and E_{N} is injective. Then Ext^{1}(−, N) = Ext^{1}(−, Nif), and a minimal flat resolution of Ext^{1}(−, N) is obtained by taking a flat resolution of Ext^{1}(−, Nif) as above.

Lemma 18. Let _{R}M be an injective-free pure-injective module and S = End_{R}M. Then Inj(M, M) ⊆ J(S).

Proof. The endomorphism ring S of a pure-injective module is an exchange ring, so if Inj(M, M) ⊈ J(End_{R}M), then there is a nonzero idempotent e ∈ Inj(M, M). Write e = fg where g : M → E and f : E → M with E injective. Then e = (ef)(ge) = 1_{eM}, where ge : eM → E and fe : E → eM. But then eM is isomorphic to a nonzero summand of E, contradicting the assumption that M is injective-free. □

Lemma 19. Let f : M → N be an isomorphism between injective-free pure-injective modules such that f : \overline{M} → \overline{N} is an isomorphism in R-\overline{Pinj}. Then f is an isomorphism.

Proof. Let g : N → M be a morphism such that \overline{f}\overline{g} = 1\overline{N} and \overline{g}\overline{f} = 1\overline{M} in R-\overline{Pinj}. By Lemma 18, 1_{N} − fg ∈ J(End_{R}N) and 1_{M} − gf ∈ J(End_{R}M). Thus fg ∈ End_{R}N and gf ∈ End_{R}M are both units, and so f is both injective and surjective. □

Let M and N be pure-injective R-modules and consider the respective direct sun decompositions, M = Mif ⊕ E_{M} and N = Nif ⊕ E_{N} given by Proposition 17. If f : M → N is an isomorphism, it may be represented with respect to these decompositions by a 2 × 2 matrix

f = ( \begin{matrix}f_{11} & f_{12} \\ f_{21} & f_{22} \end{matrix} ) ,

where f_{11} : Mif → Nif, f_{12} : E_{M} → Nif, f_{21} : Mif → E_{N}, and f_{22} : E_{M} → E_{N}. In R-\overline{Pinj}, \overline{M} ≅ \overline{M}if and \overline{N} ≅ \overline{N}if and \overline{f} = \overline{f}_{11} is an isomorphism. By Lemma 19, f_{11} : Mif → Nif is an isomorphism. This justifies calling any direct summand of the form Mif the injective-free part of M.

Two pure-injective modules M and N are said to be injectively stably isomorphic if \overline{M} and \overline{N} are isomorphic in R-\overline{Pinj}. Lemma 19 implies that M and N are injectively stably isomorphic if and only if they have isomorphic injective-free parts. Thus M = Mif ⊕ E_{M} and N = Nif ⊕ E_{N} with Mif ≅ Nif. If E = E_{M}^{ω} ⊕ E_{N}^{ω}, then, as in Eilenberg’s trick, we have that

M ⊕ E ≅ Mif ⊕ E ≅ Nif ⊕ E ≅ N ⊕ E.

On the other hand, if there is an injective module E, such that M ⊕ E ≅ N ⊕ E, then M and N are unmistakably injectively stably isomorphic.

2.4 Injective objects.

Every functor F ∈ ((R-\underline{mod})op, Ab) admits a monomorphism into a functor of the form Ext^{1}(−, M). Indeed, a flat resolution of F of length 2 may instead be continued as a long exact sequence of functors

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

This is possible because g : N → K is an epimorphism. As F is isomorphic to the image of the connecting morphism Δ, it may be embedded into Ext^{1}(−, M).

Proposition 20. Every injective object in ((R-\underline{mod})op, Ab) is isomorphic to an object of the form Ext^{1}(−, M), where M is a pure-injective module.

Proof. We may embed F into a functor of the form Ext^{1}(−, M), which in turn admits an embedding into Ext^{1}(−, PE(M)). Thus we may assume that F is isomorphic to a coproduct factor of Ext^{1}(−, M), where M is a pure-injective R-module. As F is injective, there is an idempotent endomorphism η of Ext^{1}(−, M) whose image is F. By Theorem 16, η = Ext^{1}(−, f) for some f ∈ End_{R}M which is idempotent modulo the two-sided ideal Inj(M, M). Since End_{R}M is an exchange ring, idempotents lift modulo any ideal [27, Cor. 1.3], so that there is an idempotent e ∈ End_{R}M such that η = Ext^{1}(−, e). It follows that F ≅ Ext^{1}(−, eM). □

Given R-modules M and N, Yoneda’s Lemma in R-Mod asserts the existence of an isomorphism of abelian groups,

Θ_{N} : [(−, M), Ext^{1}(−, N)]_{R-Mod} ≅ Ext^{1}(M, N),

natural in M, where [(−, M), Ext^{1}(−, N)]_{R-Mod} denotes the abelian group of natural transformations from (−, M) to Ext^{1}(−, N) considered as functors on the category R-Mod of all R-modules. Denote by

Δ_{N}(M) : Ext^{1}(M, N) → [(−, M), Ext^{1}(−, N)]

the morphism obtained by composing Θ_{N}^{−1} with the morphism

[(−, M), Ext^{1}(−, N)]_{R-Mod} → [(−, M), Ext^{1}(−, N)]

obtained by restriction to R-mod.

Theorem 21. If _{R}N is pure-injective, then for every module M, the morphism

Δ_{N}(M) : Ext^{1}(M, N) → [(−, M), Ext^{1}(−, N)]

is an isomorphism.

Proof. M. Auslander [3, Prop. I.10.1] proved that a module N is pure-injective if and only if for every directed system {C_{i}}_{i} of modules, the natural morphism

Ext^{1}(lim_{→} C_{i}, N) → lim_{←} Ext^{1}(C_{i}, N)

is an isomorphism. Write M = lim_{→} C_{i} as a direct limit of finitely presented modules C_{i}. Using Yoneda’s Lemma, we have

Ext^{1}(M, N) = Ext^{1}(lim_{→} C_{i}, N)
≅ lim_{←} Ext^{1}(C_{i}, N)
≅ lim_{←}[(−, C_{i}), Ext^{1}(−, N)]
≅ [lim_{→}(−, C_{i}), Ext^{1}(−, N)]
≅ [(−, M), Ext^{1}(−, N)].
□

The main consequence of Theorem 21 is an analogue of [6, Prop. 3.2].

Proposition 22. If _{R}N is pure-injective, then the functor Ext_{R}^{1}(−, N) is an injective object of ((R-\underline{mod})op, Ab).

Proof. We will show that if F ∈ ((R-\underline{mod})op, Ab), then Ext^{1}[F, Ext^{1}(−, N)] = 0 in the ambient category ((R-mod)op, Ab). Since the subcategory ((R-\underline{mod})op, Ab) is closed under extensions, the same equation will hold there. Consider a resolution of F,

[figure: 0; (−, M_{2}); (−, d_{2}); (−, M_{1}); (−, d_{1}); (−, M_{0}); D_{0}; F; 0,]

where (−, M_{0}) and (−, M_{1}) are projective. Applying the functor [?, Ext_{R}^{1}(−, N)] to this flat resolution yields the commutative diagram

[figure: [(−, M_{0}), Ext^{1}(−, N)]; [(−, M_{1}), Ext^{1}(−, N)]; [(−, M_{2}), Ext^{1}(−, N)]; Δ_{N}(M_{0}); Δ_{N}(M_{1}); Δ_{N}(M_{2}); Ext^{1}(M_{0}, N); Ext^{1}(d_{1}, N); Ext^{1}(M_{1}, N); Ext^{1}(d_{1}, N); Ext^{1}(M_{2}, N).]

Since F (R) = 0, the bottom row is exact. As Δ_{N} is an isomorphism, the top row is also exact. □

These findings may be summarized as follows.

Theorem 23. The functor Ext^{1}(−,?) : R-\overline{Pinj} → ((R-\underline{mod})op, Ab), given by \overline{N} ↦ Ext^{1}(−, N), yields an equivalence of categories between R-\overline{Pinj} and the subcategory of injective objects of ((R-\underline{mod})op, Ab).

2.5 The torsion class ((R-\underline{mod})op, Ab).

The category ((R-\underline{mod})op, Ab) is a subcategory of ((R-mod)op, Ab) closed under subobjects, quotient objects, coproducts and extensions. It therefore constitutes a hereditary torsion class. If F ∈ ((R-mod)op, Ab), we will denote by t(F) the torsion subobject of F corresponding to the torsion class ((R-\underline{mod})op, Ab).

Proposition 24. Let I ∈ ((R-mod)op, Ab) be an injective object. There is an injective-free pure-injective module M (unique up to isomorphism) such that

I = E(Ext^{1}(−, M)) ⨿ I_{0},

where E(Ext^{1}(−, M)) denotes the injective envelope of Ext^{1}(−, M) in ((R-mod)op, Ab) and I_{0} is a torsion-free injective object. The decomposition is unique up to isomorphism.

Proof. The torsion subobject t(I) of I has no essential extensions in the category ((R-\underline{mod})op, Ab). It is therefore injective as an object of ((R-\underline{mod})op, Ab). By Proposition 20, there is a pure-injective module M, which we may assume to be injective-free, such that t(I) ≅ Ext^{1}(−, M). Then I = E(Ext^{1}(−, M)) ⨿ I_{0}, where I_{0} is a torsion-free injective object of ((R-mod)op, Ab). If another such decomposition I = E(Ext^{1}(−, M′)) ∐ I′_{0} is given, then Ext^{1}(−, M′) ≅ t(I) ≅ Ext^{1}(−, M). Since M and M′ are both injective-free pure-injective modules, Theorem 23 implies that M ≅ M′. The identity morphism 1_{I} induces an isomorphism from E(Ext^{1}(−, M)) to E(Ext^{1}(−, M′)). By [31, Lemma V.5.3], the torsion-free parts I_{0} and I′_{0} are also isomorphic. □

If M is an R-module, then the flat object (−, M) is torsion-free. For, suppose that G belongs to ((R-\underline{mod})op, Ab). There is a projective presentation

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

where f : N → K is an epimorphism. Applying the functor [?, (−, M)] yields an exact sequence of abelian groups

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

Since f is an epimorphism, (f, M) : (K, M) → (N, K) is a monomorphism of groups. Thus [G, (−, M)] = 0, and (−, M) is torsion-free.

Proposition 25. If (T , F) is the torsion theory whose torsion class is given by T = ((R-\underline{mod})op, Ab), then the class F of torsion-free objects consists of the objects of flat dimension at most 1.

Proof. Recall that a functor F ∈ ((R-mod)op, Ab) is of flat dimension at most 1 if and only if it is isomorphic to a subfunctor of a flat functor. Thus every functor of flat dimension at most 1 is torsion-free. Conversely, suppose that G is a torsion-free object with projective presentation

[figure: (−, N); (−, f); (−, K); G; 0; (−, g); μ; (−, L)]

If g : K → L denotes the cokernel of f : N → K, then there is unique morphism μ : G → (−, L) as indicated by the dotted arrow. Evaluating the diagram at the module R, shows that μ(R) is a monomorphism, and hence that Ker μ belongs to ((R-\underline{mod})op, Ab). But G is torsion-free, so it must be that Ker μ = 0, and hence that G admits an embedding into a flat object. □

Theorem 26. Let I ∈ ((R-mod)op, Ab) be an injective object. There is an injective-free pure-injective module M and an injective module E such that

I = E(Ext^{1}(−, M)) ⨿ (−, E).

If I = E(Ext^{1}(−, M′)) ⨿ (−, E′) is another such decomposition, then M′ ≅ M and E′ ≅ E.

Proof. By Proposition 24, it suffices to prove that a torsion-free injective object I of ((R-mod)op, Ab) must be of the form I ≅ (−, E), where E is an injective module. By Proposition 25, there is an embedding of I into a flat functor (−, L). The embedding is necessarily a split monomorphism, so that there is a direct summand E of L, such that I ≅ (−, E). Obviously, E must be an injective R-module. □

The proof of Theorem 26 indicates that the torsion theory whose torsion class is T = ((R-\underline{mod})op, Ab) is cogenerated by the injective objects of the form (−, E), as E ranges over the class of injective R-modules.

3. The category of covariant functors.

The category of covariant additive functors F : mod-R → Ab is denoted by (mod-R, Ab). Like the category ((R-mod)op, Ab), it is a locally finitely presented Grothendieck category, and, as in the case of ((R-mod)op, Ab), there is a full and faithful functor from R-Mod to (mod-R, Ab). It is given by the rule M ↦ − ⊗_{R} M, which is a right exact functor, in contrast to the full and faithful functor M ↦ (−, M) into the category of contravariant functors, which is left exact. It is because of our preference for left R-modules and these natural embeddings of R-Mod into ((R-mod)op, Ab) and (mod-R, Ab), respectively, that we choose R-mod as our domain for the contravariant functors and mod-R as the appropriate domain for the category of covariant functors.

3.1 The Auslander-Bridger Transpose.

The stable category \underline{mod}-R of the category of finitely presented right R-modules is defined as for the category R-mod. The functor category (\underline{mod}-R, Ab) may be identified with the subcategory of (mod-R, Ab) of functors F for which F (R_{R}) = 0. The Auslander-Bridger Transpose [5] is an equivalence of categories Tr : \underline{mod}-R → (R-\underline{mod})op. It is defined as follows: let A_{R} be a finitely presented right R-module with a presentation

(2)  R_{R}^{n} ⟶^{f} R_{R}^{m} ⟶^{p} A_{R} → 0

by free modules of finite rank. Applying the left exact contravariant functor (−)^{∗} = (−, R_{R}) yields an exact sequence

(3)  0 ⟶ A^{∗} ⟶^{p^{∗}} R^{m} ⟶^{f^{∗}} R^{n} ⟶ Tr(A) ⟶ 0

of left R-modules, where Tr(A) is the cokernel of f^{∗}. The object Tr(A) is not well-defined, but its class in R-\underline{mod} is. The inverse of Tr is defined similarly and also denoted by Tr : (R-\underline{mod})op → \underline{mod}-R. The Auslander-Bridger Transpose induces an equivalence of functor categories Tr_{∗} : ((R-\underline{mod})op, Ab) → (\underline{mod}-R, Ab), given by Tr_{∗} : F ↦ F ∘ Tr. The category of contravariant functors on the stable category R-\underline{mod} is therefore equivalent to the category of covariant functors on the stable category \underline{mod}-R = Rop-\underline{mod}.

Just as the contravariant functors of the form (−, M) are characterized as the flat objects of ((R-mod)op, Ab), the covariant functors of the form − ⊗_{R} M may be intrinsically characterized as the fp-injective objects of (mod-R, Ab) (cf. [13, Lemma 1.4]). Recall that an object F is fp-injective if Ext^{1}(A, F) = 0 for every finitely presented object A ∈ (mod-R, Ab). The relationship between the functors − ⊗ M and (−, M) is clarified by the following. The argument is dual to the proof of [7, Prop. 2.2].

Proposition 27. For every R-module M, there is an exact sequence in (mod-R, Ab),

[figure: 0; Tr_{∗}[Ext^{1}(−, M)]; − ⊗_{R} M; α_{M}; ((−)^{∗}, M)]

natural in M.

Proof. Let A_{R} be a finitely presented right R-module with the free presentation (2) given above. Applying the right exact functor − ⊗_{R} M to this free presentation, and the left exact functor (−, M) to the exact sequence (3), yields the following commutative diagram.

[figure: R^{n} ⊗_{R} M; f ⊗ M; R^{m} ⊗_{R} M; p ⊗ M; A ⊗_{R} M; 0; ≅; ≅; α_{M}(A); 0; (Tr(A), M); (R^{n}, M); (f^{∗}, M); (R^{m}, M); (p^{∗}, M); (A^{∗}, M).]

The first two vertical arrows are isomorphisms as indicated. The top row is exact, so that A ⊗_{R} M is the cokernel of f ⊗ M. The morphism α_{M}(A) : A ⊗_{R} M → (A^{∗}, M) is induced by the universal property of the cokernel and the fact that (p^{∗}, M)(f^{∗}, M) = 0. It is functorial in A and because

Ext^{1}(Tr(A), M) = Ker (p^{∗}, M)/Im(f^{∗}, M),

Ker α_{M} is naturally isomorphic to Ext^{1}(Tr(−), M) = Tr_{∗}[Ext^{1}(−, M)]. □

It may be shown that a short exact sequence

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

is pure-exact provided the corresponding sequence

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

is exact in (mod-R, Ab). Indeed, if f : M → N is a monomorphism, then the naturality of Proposition 27 yields a morphism of exact sequences as follows,

[figure: 0; Tr_{∗}[Ext^{1}(−, M)]; − ⊗ M; α_{M}; ((−)^{∗}, M); Tr_{∗}[Ext^{1}(−, f)]; − ⊗ f; ((−)^{∗}, f); 0; Tr_{∗}[Ext^{1}(−, N)]; − ⊗ N; α_{M}; ((−)^{∗}, N).]

Because ((−)^{∗}, f) : ((−)^{∗}, M) → ((−)^{∗}, N) is a monomorphism, we see that − ⊗ f is a monomorphism if and only if Tr_{∗}[Ext^{1}(−, f)] is. By Proposition 15, this is equivalent to f being a pure-monomorphism.

If M is pure-injective, then the functor − ⊗_{R} M is an injective object of (mod-R, Ab). Indeed, the injective objects of (mod-R, Ab) have been characterized by Gruson and Jensen [18] as the functors isomorphic to those of the form −⊗_{R} M, where M is a pure-injective module. Given F ∈ (mod-R, Ab), there is an injective copresentation

[figure: 0; F; − ⊗_{R} M; η; − ⊗_{R} N.]

Because the embedding M ↦ − ⊗_{R} M is full, there is a morphism f : M → N of R-modules such that η = − ⊗ f. If K = Coker f is the cokernel of f, then the right exactness of the tensor functor permits one to complete the copresentation to a coresolution of F by fp-injective objects as follows,

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

As in the contravariant case, the subcategory (\underline{mod}-R, Ab) is a hereditary torsion class of (mod-R, Ab). If F ∈ (mod-R, Ab), we will denote the (\underline{mod}-R, Ab)-torsion subobject of F also by t(F). Let us note that if E is an injective R-module, then − ⊗_{R} E is a torsionfree object of (mod-R, Ab). For, suppose that F ∈ (\underline{mod}-R, Ab) and consider an injective copresentation of F,

[figure: 0; F; − ⊗_{R} M_{0}; − ⊗ f; − ⊗_{R} M_{1}.]

Because F (R) = 0, the morphism f : M_{0} → M_{1} of pure-injective modules is a monomorphism. Applying the functor [?, − ⊗_{R} E] yields an exact sequence of abelian groups. The embedding M ↦ − ⊗_{R} M is full, so the sequence is isomorphic to

[figure: (M_{1}, E); (f, E); (M_{0}, E); [F, − ⊗ E]; 0.]

Since f is a monomorphism and E is injective, the morphism (f, E) : (M_{1}, E) → (M_{0}, E) of abelian groups is an epimorphism. By the exactness of the sequence, [F, − ⊗ E] = 0. These considerations may be used to determine the torsion subobject of the functor − ⊗_{R} M.

Proposition 28. Let e : M → E be the injective envelope of M. The torsion subobject of − ⊗ M is part of the exact sequence

[figure: 0; t(− ⊗ M); − ⊗ M; − ⊗ e; − ⊗ E.]

Proof. Obvious, because the kernel of − ⊗ e is torsion, while its image is a subobject of − ⊗ E, and is thus torsion-free. □

By Theorem 23, the injective objects of ((R-\underline{mod})op, Ab) are of the form Ext^{1}(−, M), where M is pure-injective. Because the injective objects of (mod-R, Ab) are of the form − ⊗_{R} M, where M is injective, the injective objects of (\underline{mod}-R, Ab) are the objects of the form t(−⊗_{R}M), M injective. This is because these are precisely the objects with no essential extensions in (\underline{mod}-R, Ab). It is therefore no surprise that if M is pure-injective, then the injective object Ext^{1}(−, M) of ((R-\underline{mod})op, Ab) is associated by Tr_{∗} to the injective object t(− ⊗ M) of (\underline{mod}-R, Ab). More generally, one has the following. The argument is dual to the proof of [7, Prop. 2.2].

Theorem 29. Given an R-module M, there is an isomorphism

Tr_{∗}[Ext^{1}(−, M)] ≅ t(− ⊗ M),

natural in M.

Proof. Let e : M → E be the pure-injective envelope of M. We obtain the following commutative diagram

[figure: 0; t(− ⊗ M); ι; 0; 0; Tr_{∗}[Ext^{1}(−, M)]; ι_{M}; − ⊗_{R} M; α_{M}; ((−)^{∗}, M); Tr_{∗}[Ext^{1}(−, e)]; 0; − ⊗_{R} e; ((−)^{∗}, e); 0; Tr_{∗}[Ext^{1}(−, E)]; ι_{E}; − ⊗_{R} E; α_{E}; ((−)^{∗}, E)]

where the rows and column are exact sequences; the morphism α_{M} : − ⊗ M → ((−)^{∗}, M) is given by Proposition 27. The composition (− ⊗ e)ι_{M} = 0, because it factors through Tr_{∗}[Ext^{1}(−, E)] = 0. Also, the injective envelope e : M → E is a monomorphism, so that ((−)^{∗}, e) : ((−)^{∗}, M) → ((−)^{∗}, E) is a monomorphism of functors. But ((−)^{∗}, e)α_{M}ι = α_{E}(− ⊗ e)ι = 0 which implies that α_{M}ι = 0. The morphisms between Tr_{∗}[Ext^{1}(−, M)] and t(− ⊗_{R} M) are then induced by the universal property of a kernel. They are both monomorphisms, so that Tr_{∗}[Ext^{1}(−, M)] and t(− ⊗ M) represent the same subobject of − ⊗ M. Naturality may be verified using a routine argument. □

By Theorem 29, an injective object − ⊗ M is torsion-free if and only if M is injective. Just as the hereditary torsion class ((R-\underline{mod})op, Ab) of ((R-mod)op, Ab) was cogenerated by the injective objects of the form (−, E), where E is an injective R-module, this shows that the hereditary torsion class (\underline{mod}-R, Ab) of (mod-R, Ab) is cogenerated by the injective objects of the form − ⊗_{R} E, where E is an injective R-module. Theorem 29 may also be used to give an immediate proof of Propositions 20 and 22. Naturality may be used to give an alternative proof of Theorem 23.

Notice that if E is an injective R-module, then a morphism f : E → N is nowhere pure if and only if the kernel of f is essential in E.

Proposition 30. Let M be a pure-injective module. A morphism f : M → N is nowhere pure if and only if Ker (− ⊗ f) ⊆ − ⊗ M is an essential subfunctor.

Proof. The functor M ↦ − ⊗ M is full, so that the endomorphism rings S = End_{R}M and End(− ⊗ M) may be identified. Since M is pure-injective, the object − ⊗ M is injective in the category (mod-R, Ab). An endomorphism h : M → M thus belongs to the Jacobson radical of S if and only if Ker (− ⊗ h) is essential in − ⊗ M. The proposition then follows from Proposition 9. □

Let us describe how the equivalence Tr_{∗} : ((\underline{mod}-R, Ab) → ((R-\underline{mod})op, Ab) acts on a general object G ∈ ((\underline{mod}-R, Ab). Consider a copresentation of G by fp-injective objects in (mod-R, Ab),

[figure: 0; G; − ⊗ M; − ⊗ f; − ⊗ N.]

Because the torsion class ((\underline{mod}-R, Ab) is hereditary, the torsion functor t(−) is left exact. Since t(G) = G, applying the functor t yields the exact sequence

[figure: 0; G; t(− ⊗ M); t(− ⊗ f); t(− ⊗ N).]

Applying the equivalence Tr_{∗} yields an exact sequence

[figure: 0; Tr_{∗}(G); Ext^{1}(−, M); Ext^{1}(−, f); Ext^{1}(−, N)]

in the category ((R-\underline{mod})op, Ab). Since G belongs to (\underline{mod}-R, Ab), the morphism f : M → N is a monomorphism. Let g : N → K be the cokernel of g. It follows that Tr_{∗}(G) is the object of ((R-\underline{mod})op, Ab) given by the flat resolution

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

If the copresentation of G is a minimal injective copresentation in (mod-R, Ab), then the fp-injective coresolution

[figure: 0; G; − ⊗ M; − ⊗ f; − ⊗ N; − ⊗ g; − ⊗ K; 0]

of G has the property that M and N are pure-injective and the kernels of both − ⊗ f and − ⊗ g are essential in − ⊗ M and − ⊗ N, respectively. By Proposition 30, the corresponding flat resolution of Tr_{∗}(G) is minimal.

3.2 Injective envelopes.

Recall from [18] that if m : M → PE(M) is the pure-injective envelope of M, then the morphism − ⊗ m : − ⊗ M → − ⊗ PE(M) is the injective envelope of − ⊗ M in the category (mod-R, Ab).

Theorem 31. Let m : M → PE(M) be the pure-injective envelope of M. Then Ext^{1}(−, m) : Ext^{1}(−, M) → Ext^{1}(−, PE(M)) is the injective envelope of Ext^{1}(−, M) in ((R-\underline{mod})op, Ab).

Proof. The pure-injective envelope m : M → PE(M) is a pure-monomorphism, so the induced morphism −⊗m : (−⊗M) → (−⊗PE(M)) is a monomorphism of covariant functors. Since the torsion theory whose torsion class is (\underline{mod}-R, Ab) is hereditary, the restriction to the torsion subobjects is also a monomorphism

t(− ⊗ m) : t(− ⊗ M) → t(− ⊗ PE(M)).

Its image is essential, because − ⊗ M is essential in − ⊗ PE(M). By Theorem 29, this is just the morphism Ext^{1}(−, m) : Ext^{1}(−, M) → Ext^{1}(−, PE(M)). □

The next result will be used to relate the minimal flat resolution in ((R-mod)op, Ab) of an object G ∈ ((R-\underline{mod})op, Ab) and its minimal injective copresentation in ((R-\underline{mod})op, Ab).

Theorem 32. Let f : M → N be a morphism of R-modules with M pure-injective. The kernel of the induced morphism Ext^{1}(−, f) : Ext^{1}(−, M) → Ext^{1}(−, N) is essential in Ext^{1}(−, M) if and only if the restriction of f to the injective-free part of M is nowhere pure.

Proof. Let fif : Mif → N be the restriction of f to the injective-free part of M. Because f and fif induce the same morphism on Ext^{1}(−, Mif) = Ext^{1}(−, M), we may assume without loss of generality that M = Mif is injective-free.

Suppose that f : M → N is nowhere pure. If the kernel of Ext^{1}(−, f) is not essential in Ext^{1}(−, M), then there is a morphism η : Ext^{1}(−, N) → Ext^{1}(−, M) such that the composition ηExt^{1}(−, f) is a nonzero idempotent endomorphism of Ext^{1}(−, M). By Theorem 16, there is a morphism g : N → M such that η = Ext^{1}(−, g). Thus gf is idempotent modulo Abs(M, M). Since S = End_{R}M is an exchange ring, idempotents lift modulo any two-sided ideal [27, Cor. 1.3], and there is a nonzero idempotent e ∈ S such that gf − e ∈ Abs(M, M) ⊆ J(S), by Lemma 18. But then gf ∉ J(S), which contradicts Proposition 9.

If f : M → N is not nowhere pure, then there is a morphism g : N → M such that gf = e ∈ End_{R}M is a nonzero idempotent. The image of Ext^{1}(−, g)Ext^{1}(−, f) = Ext^{1}(−, e) is Ext^{1}(−, eM), which is a nonzero summand of Ext^{1}(−, M), because eM is not injective. The kernel of Ext^{1}(−, f) is then contained in the kernel of Ext^{1}(−, e), which is not essential in Ext^{1}(−, M). □

The following is an analogue of a result of Auslander and Reiten [6, Prop. 4.6].

Corollary 33. Let F ∈ ((R-\underline{mod})op, Ab) and suppose that a minimal flat resolution of F in ((R-mod)op, Ab) is given by

[figure: 0; (−, M_{2}); (−, d_{2}); (−, M_{1}); (−, d_{1}); (−, M_{0}); D_{0}; F; 0.]

Then the minimal injective copresentation of F in ((R-\underline{mod})op, Ab) is given by

[figure: 0; F; Ext^{1}(−, M_{2}); Ext^{1}(−, d_{2}); Ext^{1}(−, M_{1}).]

Proof. Evaluate the minimal flat resolution of F at R to obtain the short exact sequence

[figure: 0; M_{2}; d_{2}; M_{1}; d_{1}; M_{0}; 0.]

By Proposition 5 and Theorem 10, M_{2} and M_{1} are pure-injective and d_{2} and d_{1} are nowhere pure. The beginning of the associated long exact sequence of functors is given by

0  \overline{}(−, M_{2}) (−, d_{2})_{\overline{}}(−, M_{1}) (−, d_{1})_{\overline{}}(−, M_{0}) Δ_{0}_{\overline{}}Ext^{1}(−, M_{2}) Ext^{1}(−, d_{2})_{\overline{}}.

The functor F is the image of the first connecting morphism Δ_{0}. By exactness, it is the kernel of Ext^{1}(−, d_{2}), which is essential, by Theorem 32, since M_{1} is injective-free. Thus Ext^{1}(−, M_{2}) is the injective envelope of F. Since d_{1} is nowhere pure on M_{1}, its restriction to the injective-free part of M_{1} is also nowhere pure. Theorem 32 then implies that the kernel of Ext^{1}(−, d_{1}) is essential in Ext^{1}(−, M_{1}). By exactness, Ext^{1}(−, M_{1}) is the injective envelope of the image of Ext^{1}(−, d_{2}). Thus the injective copresentation is minimal. □

Next, we prove a converse to Corollary 33. It is a consequence of Theorem 32, whose proof depends on the observation that if f : M → K and g : N → K are nowhere pure maps, with M and N pure-injective, then so is the coproduct f ⨿ g : M ⊕ N → K. This is because the kernel of − ⊗ (f ⨿ g) contains Ker (− ⊗ f) ⨿ Ker (− ⊗ g), which, by Proposition 30, is essential in (− ⊗ M) ⨿ (− ⊗ N) = − ⊗ (M ⊕ N).

Corollary 34. Suppose that the minimal injective resolution of F ∈ ((R-\underline{mod})op, Ab) is given by

[figure: 0; F; Ext^{1}(−, M); Ext^{1}(−, f); Ext^{1}(−, N),]

where f : M → N and M and N are injective-free pure-injective modules. Let g′ : Ker f → E be the injective envelope of Ker f, and g : M → E some extension of g′ to M. The minimal flat resolution of F in ((R-mod)op, Ab) is induced by the short exact sequence

[figure: 0; M; (f, g); N ⊕ E; h; K; 0]

Proof. Considering the long exact sequence of functors induced by the short exact sequence shows that the first few terms give a flat resolution of F in ((R-mod)op, Ab). The modules M and N ⊕ E are both pure-injective, so by Proposition 5, the flat resolution is special. By Theorem 10, it suffices to prove that (f, g) and h are both nowhere pure. To see that (f, g) is nowhere pure, suppose that there existed a morphism s : N ⊕ E such that sf + sg = s(f, g) ∉ J(End_{R}M). By Theorem 32, f : M → N is nowhere pure, so that sf ∈ J(End_{R}M); it must be that sg ∉ J(End_{R}M). Then g : M → E is not nowhere pure and we could find an s′ : E → M such that e = s′g ∈ End_{R}M is a nonzero idempotent. Then eM is a summand of M isomorphic to a summand of E, contradicting the assumption that M was injective-free.

To see that h : N ⊕ E → K is nowhere pure, it suffices to show that the restrictions of h to N and E, respectively, are nowhere pure. The restriction of h to the injective-free pure-injective module N is nowhere pure, because the kernel of Ext^{1}(−, h) is equal to the image of Ext^{1}(−, f), and is therefore is essential in Ext^{1}(−, N). The restriction of h to E is also nowhere pure, because it has an essential kernel. This follows from the fact that Ker f considered as an essential submodule of E is contained in the image of (f, g), which is equal to the kernel of h. □

3.3 The Gabriel spectrum.

Let C be a Grothendieck category. The Gabriel spectrum of C, denoted by Sp(C), is the set whose points are the indecomposable injective objects E - up to isomorphism - of the category C. That this collection indeed forms a set follows from the fact that every indecomposable injective object is the injective envelope of a quotient object of the generator; up to isomorphism, there are only set-many such quotient objects. The Gabriel spectrum is topologized by defining a subset O to be open if there is a hereditary torsion class T ⊆ C such that

O = O(T ) := {E ∈ Sp(C) : E is not T -torsion-free}.

We will call Sp((R-mod)op, Ab) the contravariant Gabriel spectrum of the ring R and and Sp(mod-R, Ab) the covariant Gabriel spectrum of R. A description of the points of the covariant Gabriel spectrum of R follows from the characterization [18] of the injective objects of (mod-R, Ab) as those of the form − ⊗_{R} M, where M is a pure-injective R-module. The points of Sp(mod-R, Ab) are therefore the isomorphism types of the functors − ⊗_{R} U, where U is an indecomposable pure-injective R-module. A similar description of the points of the contravariant Gabriel spectrum of R follows from Theorem 26.

Corollary 35. The following is a list, complete and without repetition, of the injective indecomposable objects of ((R-mod)op, Ab) :

(1) E(Ext^{1}(−, M)), the injective envelope of Ext^{1}(−, M), where M is a non-injective indecomposable pure-injective R-module;

(2) (−, E), where E is an indecomposable injective R-module.

The corollary implies that the points of Sp(mod-R, Ab) and Sp((R-mod)op, Ab) may be placed in bijective correspondence, according to the rule

(4)  Ξ_{R} : − ⊗_{R} U ↦ \{ \begin{matrix}E[Ext^{1}(−, U)], & if U is not injective; \\ (−, U), & if U is injective. \end{matrix}

A hereditary torsion class T ⊆ C is itself a Grothendieck category and so has a Gabriel spectrum associated to it. To every indecomposable injective I ∈ Sp(T ) in the category T , we may associate its injective envelope E_{C}(I) in C, an indecomposable object, because I is uniform in C. The association I ↦ E_{C}(I) yields a continuous function Sp(E_{C}) : Sp(T ) → Sp(C) that consitutes a homeomorphism from Sp(T ) to its image, the open subset O(T ).

For example, the functor category ((R-\underline{mod})op, Ab) is a hereditary torsion class in ((R-mod)op, Ab). The Gabriel spectrum Sp((R-\underline{mod})op, Ab) of the contraviant functors on the stable category is therefore homeomorphic to the open subset O((R-\underline{mod})op, Ab) of the contravariant Gabriel spectrum of R. Similarly, the functor category (\underline{mod}-R, Ab) is a hereditary torsion class in (mod-R, Ab), and so the Gabriel spectrum Sp(\underline{mod}-R, Ab) is homeomorphic to the open subset O(\underline{mod}-R, Ab) of the covariant Gabriel spectrum of R. The equivalence of categories Tr_{∗} : ((R-\underline{mod})op, Ab) → (\underline{mod}-R, Ab) induced by the Auslander-Bridger Transpose yields a homeomorphism Sp(Tr_{∗}) of the respective Gabriel spectra of stable functor categories. It is part of the commutative diagram

[figure: Sp(\underline{mod}-R, Ab); Sp(E); Sp(mod-R, Ab); Sp(Tr_{∗}); Ξ_{R}; Sp((R-\underline{mod})op, Ab); Sp(E); Sp((R-\underline{mod})op, Ab)]

Associated to a hereditary torsion class T ⊆ C is another Grothendieck category, the codomain of the localization of C at T . This localization is an exact functor (−)_{T} : C → C/T to the Grothendieck category C/T satisfying certain universal properties (cf. [17, Ch. 3]). A section functor σ_{T} : C/T ⊆ C of the localization functor allows one to identify the category C/T with the subcategory of C consisting of T -torsion-free and T -closed objects; recall that an object M ∈ C is T -closed if for every object T ∈ T , Ext^{1}(T, M) = 0. Because the section functor σ_{T} is the right adjoint of the exact functor (−)_{T} , it preserves injective objects. Furthermore, the category C/T viewed as a subcategory of C is closed under direct summands, so the section functor yields a continuous function that preserves [31, Prop. IV.9.5] indecomposable injective objects and thus induces a continuous function

Sp(σ_{T}) : Sp(C/T ) → Sp(C)

that consitutes a homeomorphism from Sp(C/T ) to its image O(T )^{c}, the complement of the open subset associated to T . We describe this state of affairs by expressing the Gabriel spectrum Sp(C) as a disjoint union

Sp(C) = Sp(T ) ∪^{·} Sp(C/T )

of Gabriel spectra, homeomorphic to an open and closed subset of Sp(C), respectively.

For example, the subcategory T = ((R-\underline{mod})op, Ab) is a hereditary torsion class of Grothendieck category C = ((R-mod)op, Ab). The localization of C at T is given by the evaluation functor

Ev : ((R-mod)op, Ab) → R-Mod,

defined by F ↦ F (R). We noted earlier how the hereditary torsion theory with torsion class T is cogenerated by the injective objects of the form (−, E), as E ranges over the injective R-modules. Using the fact that this class of injective objects is closed under products, the T -torsion-free, T -closed objects of ((R-mod)op, Ab) may be classified as those objects G that possess an injective resolution of the form

[figure: 0; G; (−, E_{0}); η; (−, E_{1}),]

where E_{0} and E_{1} are injective R-modules. There is a morphism f : E_{0} → E_{1} such that η = (−, f), so if we let M = Ker f, then G = (−, M). This shows that the subcategory C/T of C is nothing more than the category Flat((R-mod)op, Ab) of flat functors, and that the section functor σ_{T} is equivalent to the functor Υ : R-Mod → ((R-mod)op, Ab) given by M ↦ (−, M). The induced continuous function of Gabriel spectra

Sp(Υ) : Sp(R-Mod) → Sp((R-mod)op, Ab)

is given by E ↦ (−, E).

Similarly, the subcategory T = (\underline{mod}-R, Ab) is a hereditary torsion class of Grothendieck category C = (mod-R, Ab). The localization of C at T is also given by the evaluation functor Ev : (mod-R, Ab) → R-Mod. We noted earlier that this hereditary torsion theory with torsion class T is cogenerated by the injective objects of the form − ⊗ E, as E ranges over the injective R-modules. Using the fact that this class of injective objects of (mod-R, Ab) is closed under products, the T -torsion-free, T -closed objects of (mod-R, Ab) may be classified as those objects G that possess an injective resolution of the form

[figure: 0; G; − ⊗ E_{0}; η; − ⊗ E_{1},]

where E_{0} and E_{1} are injective R-modules. There is a morphism f : E_{0} → E_{1} such that η = − ⊗ f, so if we let M = Ker f, then the section functor, which we will denote by σ_{R} in this case, is given by M ↦ G. The point is that the induced continuous function of Gabriel spectra

Sp(σ_{R}) : Sp(R-Mod) → Sp(mod-R, Ab)

is given by E ↦ − ⊗ E.

These two situations are brought together by the following commutative diagram of continuous functions,

[figure: Sp(R-Mod); Sp(Υ); Sp(mod-R, Ab); Ξ_{R}; Sp(R-Mod); Sp(σ_{R}); Sp((R-mod)op, Ab).]

Let us summarize these findings as follows.

Theorem 36. There are partitions of the covariant and contravariant Gabriel spectra of R,

Sp(mod-R, Ab) = Sp(\underline{mod}-R, Ab) ∪^{·} Sp(R-Mod) and
Sp((R-mod)op, Ab) = Sp((R-\underline{mod})op, Ab) ∪^{·} Sp(R-Mod),

each into an open subset and a closed subset. The canonical bijective correspondence Ξ_{R} : Sp(mod-R, Ab) → Sp((R-mod)op, Ab) respects the partitions and induces a homeomorphism of the open subsets and closed subsets, respectively.

The contravariant functor category ((R-mod)op, Ab) and the covariant functor category (mod-R, Ab) give a natural example of a pair of Grothendieck categories with equivalent hereditary torsion classes ((R-\underline{mod})op, Ab) and (\underline{mod}-R, Ab) whose localizations are also equivalent. The ambient categories ((R-mod)op, Ab) and (mod-R, Ab) are themselves rarely equivalent. Indeed, an equivalence of these two categories exists if and only if there is an equivalence between R-mod ≅ proj((R-mod)op, Ab) and (mod-R)op ≅ proj(mod-R, Ab), that is, if and only if there a duality between R-mod and mod-R.

4. Purity in R-\overline{Mod}

If the ring R is Quasi-Frobenius, then the category R-\overline{Mod} may be equipped with the structure of a triangulated category [11, 33, 34]. A theory of purity has been developed for triangulated categories by Beligiannis [8, 9] and H. Krause [24]. In this chapter, we use the results proved so far to show that if m : M → PE(M) is the pure-injective envelope of M, then the induced morphism \overline{m} : \overline{M} → \overline{PE(M)} on the stable classes is the R-\overline{Pinj}-envelope of \overline{M}. In case R is Quasi-Frobenius, the objects of R-\overline{Pinj} are the pure-injective objects of R-\overline{Mod}, considered as a triangulated category.

6.1 Quasi-Frobenius rings.

The ring R is called Quasi-Frobenius (QF) [28] if it is left and right artinian and left and right self-injective. QF rings are really nice. If R is QF, then it is left perfect, so that every flat left R-module is projective. The category R-\underline{Mod}, which we have defined in general to be the category R-Mod modulo morphisms that factor through a flat module, coincides with the usual definition of R-\underline{Mod} as the category of left R-modules modulo morphisms that factor through a projective. If R is QF, then it is left noetherian, so that every fp-injective left R-module is injective. Thus the category R-\overline{Mod}, which we have defined to be the category R-Mod modulo morphisms that factor through an fp-injective module, coincides with the usual definition of R-\overline{Mod} as the category of left R-modules modulo morphisms that factor through an injective module. If M is an arbitrary R-module, we may decompose it as a direct sum

M = Mif ⊕ E_{M},

where Mif is injective-free and E_{M} is injective. Thus is done by applying Zorn’s Lemma to find a maximal fp-injective submodule of M, which is necessarily injective and yields E_{M}. Furthermore, if R is a QF ring, then a left R-module is projective if and only if it is injective. Thus

R-\underline{Mod} = R-\overline{Mod},

and the decomposition above has the property that E_{M} is projective and that Mif is projective-free.

If R is QF, then every module M has a projective cover c : P → M, whose kernel is denoted Ω(M). The assignment M ↦ Ω(M) induces a functor Ω : R-\underline{Mod} → R-\underline{Mod} which is a selfequivalence of R-\underline{Mod} [11, Thm. 11.4]. The equivalence inverse Ω^{−1} : R-\underline{Mod} → R-\underline{Mod} is induced by the rule that associates to M the cokernel of the injective envelope e : M → E. If M is a finitely presented module, then Ω(M) and Ω^{−1}(M) are also finitely presented, so the restriction of Ω to R-\underline{mod} is also a self-equivalence with equivalence inverse given by the restriction of Ω^{−1} to R-\underline{mod}. These self-equivalences of R-\underline{mod} induce self-equivalences of the functor category ((R-\underline{mod})op, Ab), defined by Ω_{∗}(F) = F (Ω(−)) and Ω_{∗}^{−1}(F) = F (Ω^{−1}(−)). Given modules M and N, there is an isomorphism,

(5)  Ext^{1}(M, N) ≅ \underline{Hom}(Ω(\underline{M}), \underline{N}),

natural in both variables (cf. [11, Thm. 5.1]). Restricting the equivalence to finitely presented modules M implies that the functors Ext^{1}(−, N) and Ω_{∗}[\underline{Hom}(−, \underline{N})] are isomorphic as objects of ((R-\underline{mod})op, Ab).

When R is a QF ring, then R-\underline{Mod} = R-\overline{Mod} has the structure of a triangulated category. A morphism \underline{f} : \underline{M} → \underline{N} is said to be a pure-monomorphism in R-\underline{Mod} = R-\overline{Mod} provided that the morphism \underline{Hom}(−, \underline{f}) is a monomorphism in ((R-\underline{mod})op, Ab). This is equivalent to the definition of Belligianis [9, §4] and H. Krause [24, §1] using [23, Lemma 9.6.9]. The naturality of the isomorphisms (5) yields the equation Ext^{1}(−, f) = Ω_{∗}[\underline{Hom}(−, \underline{f})], which implies that \underline{f} is a pure-monomorphism if and only if Ext^{1}(−, f) is a monomorphism in ((R-\underline{mod})op, Ab).

6.2 Ext-monomorphisms.

A morphism \overline{f} : \overline{X} → \overline{Y} in R-\overline{Mod} will be called an Ext-monomorphism if the morphism Ext^{1}(−, f) : Ext^{1}(−, X) → Ext^{1}(−, Y) is a monomorphism in ((R-\underline{mod})op, Ab). By Proposition 15, a monomorphism f : X → Y induces an Ext-monomorphism \overline{f} : \overline{X} → \overline{Y} if and only if it is a pure-monomorphism. Evidently, the composition of two Ext-monomorphisms is again such, and if \overline{f} = \overline{g}\overline{h} is an Ext-monomorphism, then so is \overline{h}. Similarly, a morphism \overline{f} : \overline{X} → \overline{Y} will be called an Ext-phantom if Ext^{1}(−, f) = 0. The two concepts are related as follows.

Proposition 37. The functor R-\overline{Mod} → ((R-\underline{mod})op, Ab), given by M ↦ Ext^{1}(−, M) is faithful if and only if every Ext-monomorphism is a monomorphism in R-\overline{Mod}.

Proof. Let f : M → N be a morphism in R-Mod. The fp-injective preenvelope a : M → A of M exists by [16, Prop. 6.2.4]. Consider the pushout diagram

[figure: M; a; A; f; q; N; p; K.]

It gives rise to a short exact sequence

[figure: 0; M; (f, −a); N ⊕ A; ( \begin{matrix}p \\ q \end{matrix} ); K; 0,]

which induces a long exact sequence of functors. Part of that long exact sequence is

[figure: Ext^{1}(−, M); Ext^{1}(−, f); Ext^{1}(−, N); Ext^{1}(−, p); Ext^{1}(−, K).]

If \overline{f} is an Ext-phantom, then \overline{p} is an Ext-monomorphism. Furthermore, the composition pf factors through the fp-injective module A, so that \overline{p}\overline{f} = 0 in R-\overline{Mod}. If every Extmonomorphism is a monomorphism, then \overline{f} = 0.

For the converse, suppose that every Ext-phantom is 0, and let \overline{f} : \overline{N} → \overline{K} be an Extmonomorphism. If \overline{g} : \overline{M} → \overline{N} is such that \overline{f}\overline{g} = 0, then Ext^{1}(−, f)Ext^{1}(−, g) = 0. Since \overline{f} is an Ext-monomorphism, \overline{g} is an Ext-phantom, and so \overline{g} = 0. □

Suppose that \overline{f} : \overline{M} → \overline{N} is a morphism in R-\overline{Mod} and let \overline{n} : \overline{N} → \overline{PE(N)} be the morphism induced by the pure-injective envelope of N. Because \overline{n} is an Ext-monomorphism, \overline{f} is an Ext-phantom if and only if \overline{n}\overline{f} is an Ext-phantom. But by Theorem 16, there are no nonzero Ext-phantoms whose codomain is the class of a pure-injective module. Thus \overline{f} is an Ext-phantom if and only if \overline{n}\overline{f} = 0.

Define an object \overline{M} ∈ R-\overline{Mod} to be Ext-injective if for every Ext-monomorphism \overline{f} : \overline{X} → \overline{Y} and morphism \overline{g} : \overline{X} → \overline{M}, there is a morphism \overline{h} : \overline{Y} → \overline{M} such that the diagram

[figure: \overline{X}; \overline{f}; \overline{Y}; \overline{g}; \overline{h}; \overline{M}]

commutes. If R is a QF ring, and R-\overline{Mod} is regarded as a triangulated category, then the Ext-injective objects are precisely the pure-injective objects of R-\overline{Mod} [24, Prop. 1.15]. If M is a pure-injective R-module and \overline{f} : \overline{X} → \overline{Y} and \overline{g} : \overline{X} → \overline{M} are given as in the definition, then, because Ext^{1}(−, M) is an injective object of ((R-\underline{mod})op, Ab), there is a morphism χ : Ext^{1}(−, Y) → Ext^{1}(−, M) such that the diagram

[figure: Ext^{1}(−, X); Ext^{1}(−, f); Ext^{1}(−, Y); Ext^{1}(−, g); χ; Ext^{1}(−, M)]

commutes. By Theorem 16, there is a morphism h : Y → M such that χ = Ext^{1}(−, h). Similarly, Theorem 16 implies that \overline{h}\overline{f} = \overline{g}. It follows that every object \overline{M} ∈ R-\overline{Pinj} is Ext-injective.

Proposition 38. An object \overline{M} of R-\overline{Mod} is Ext-injective if and only if it belongs to R-\overline{Pinj}.

Proof. One direction has just been proved, so assume that \overline{M} is Ext-injective and consider the morphism \overline{m} : \overline{M} → \overline{PE(M)} in R-\overline{Mod} induced by the pure-injective envelope of M in R-Mod. There is a morphism \overline{g} : \overline{PE(M)} → \overline{M} that makes the diagram

[figure: \overline{M}; \overline{m}; \overline{PE(M)}; \overline{g}; \overline{M}]

commute. Let us show that \overline{m} is an isomorphism with inverse \overline{g}. By the commutativity of the diagram, \overline{g} is the left inverse of \overline{m}. Consider the corresponding commutative diagram

[figure: Ext^{1}(−, M); Ext^{1}(−, m); Ext^{1}(−, PE(M)); Ext^{1}(−, g); Ext^{1}(−, M)]

in ((R-\underline{mod})op, Ab). By the commutativity of the diagram, Ext^{1}(−, g) is an epimorphism. Since Ext^{1}(−, m) : Ext^{1}(−, M) → Ext^{1}(−, PE(M)) is the injective envelope of Ext^{1}(−, M), and the restriction of Ext^{1}(−, g) to Ext^{1}(−, M) is the identity, the morphism Ext^{1}(−, g) is also a monomorphism. Thus Ext^{1}(−, g) and consequently Ext^{1}(−, m) are both isomorphism. Because PE(M) is pure-injective, Theorem 16 implies that \overline{m}\overline{g} = 1\overline{M}. □

The proof of the proposition also shows that if \overline{M} ∈ R-\overline{Pinj} and \overline{f} : \overline{M} → \overline{N} is an Ext-monomorphism, then it is a split monomorphism.

Theorem 39. If \overline{M} belongs to R-\overline{Mod}, then the R-\overline{Pinj}-envelope of \overline{M} is given by the morphism \overline{m} : \overline{M} → \overline{PE(M)} induced by the pure-injective envelope of M. Regarded as a morphism, the R-\overline{Pinj}-envelope \overline{M} is an Ext-monomorphism.

Proof. The second statement follows from Corollary 31. Let \overline{f} : \overline{M} → \overline{N} be a morphism in R-\overline{Mod} with \overline{N} in R-\overline{Pinj}. Because we may take N pure-injective, there is a morphism g : PE(M) → N such that the diagram

[figure: Ext^{1}(−, M); Ext^{1}(−, m); Ext^{1}(−, PE(M)); Ext^{1}(−, f); Ext^{1}(−, g); Ext^{1}(−, N)]

in ((R-\underline{mod})op, Ab) commutes. By Theorem 16, \overline{f} = \overline{g}\overline{m}, and so \overline{f} factors through \overline{m}. This shows that \overline{m} : \overline{M} → \overline{PE(M)} is an R-\overline{Pinj}-preenvelope of \overline{M}.

To verify that \overline{m} : \overline{M} → \overline{PE(M)} is the R-\overline{Pinj}-envelope of \overline{M}, suppose that \overline{f} : \overline{PE(M)} → \overline{PE(M)} is an endomorphism of \overline{PE(M)} over \overline{M}. Then the diagram

[figure: Ext^{1}(−, M); Ext^{1}(−, m); Ext^{1}(−, PE(M)); Ext^{1}(−, m); Ext^{1}(−, f); Ext^{1}(−, PE(M))]

in ((R-\underline{mod})op, Ab) commutes. Since Ext^{1}(−, PE(M)) is the injective envelope of Ext^{1}(−, M), the morphism Ext^{1}(−, f) is an isomorphism. By Theorem 16, the morphism \overline{f} is an isomorphism in ((R-\underline{mod})op, Ab). □

5. Hereditary torsion classes of finite type

Notation: Throughout this section, C will denote a locally finitely generated Grothendieck category. The advantage of this hypothesis is that the category C is cogenerated by the indecomposable injective objects. In particular the open subset O(T ) associated to a nonzero hereditary torsion class T ⊆ C is nonempty.

An object C ∈ C is finitely presented if it is finitely generated and every epimorphism g : B → C with B finitely generated has a finitely generated kernel. The subcategory of finitely presented objects of C is denoted by fp(C). A hereditary torsion class T ⊆ C is of finite type if it is generated by a set of finitely presented objects of C,

T = Gen(T ∩ fp(C)).

Proposition 40. If T ⊆ C is a hereditary torsion theory of finite type, and F ∈ C is finitely generated, then the localization F_{T} is finitely generated in C/T . The localization C/T is locally finitely generated.

Proof. The second statement follows from the first together with the fact that the localization functor preserves direct limits. If C ∈ C and T (C) denotes the T -subobject of C, then there is a short exact sequence [17, Ch. III.1]

[figure: 0; C′; C_{T}; G; 0,]

where C′ = C/T (C) and G ∈ T . If C is finitely generated, then so is C′. To prove that C_{T} is finitely generated in C/T , it must be shown that whenever a directed union

∑_{i∈I} F_{i} ⊆ C_{T}

of T -torsion-free, T -closed subobjects of C_{T} has the property that the quotient C_{T} /Σ_{i}F_{i} belongs to T , then there is a k ∈ I such that C_{T} /F_{k} ∈ T .

Since C_{T} /C′ ∈ T , it follows that (C′ + Σ_{i}F_{i})/Σ_{i}F_{i} is an object of T , which is finitely generated, because it is a quotient of C′. Now

(C′ + Σ_{i}F_{i})/Σ_{i}F_{i} = C′/(C′ ∩ Σ_{i}F_{i}) = C′/Σ_{i}(C′ ∩ F_{i}).

As T is of finite type, there is an object T ∈ T , finitely presented in C, and an epimorphism f : T ↦ C′/Σ_{i}(C′ ∩ F_{i}). As T is finitely presented, there is a j ∈ I such that the morphism f factors according to the commutative diagram

[figure: C′/(C′ ∩ F_{j}); g; p; T; f; C′/Σ_{i}(C′ ∩ F_{i}).]

Working in C′/(C′ ∩F_{j}), we have the equality [Img+Σ_{i≥j} (C′ ∩F_{i})]/(C′ ∩F_{j}) = C′/(C′ ∩F_{j}). Because this object is finitely generated, there is a k ≥ j such that

[Im g + (C′ ∩ F_{k})]/(C′ ∩ F_{j}) = C′/(C′ ∩ F_{j}).

As Im g belongs to T , so does C′/(C′ ∩ F_{k}). But C_{T} /C′ is also an object of T , so that C_{T} /(C′ ∩ F_{k}) ∈ T and hence C_{T} /F_{k} ∈ T . □

One consequence of Proposition 40 is that the Gabriel spectrum of C detects propriety of finite type hereditary torsion classes.

Proposition 41. Let T ⊆ T′ be a proper inclusion of hereditary torsion classes of C with T of finite type, then the inclusion O(T ) ⊆ O(T′) of open subsets in the Gabriel spectrum Sp(C) is proper.

Proof. Let C ∈ T′ be an object that does not belong to T . Then there is a finitely generated subobject A ⊆ C that does not belong to T . By the proof of Proposition 40, the localization A_{T} is a nonzero finitely generated object of the locally finitely generated Grothendeick category C/T . The finitely generated object A_{T} has a simple quotient object X ∈ C/T . Then E(X) is a point in O(T′) that does not belong to O(T ). □

For example, suppose that R is a simple von Neumann regular ring that is not artinian. Then the hereditary torsion class T ⊆ R-Mod of semi-artinian modules is proper, but it is not known whether the open subset O(T ) is always proper in Sp(R-Mod) (cf. [32]).

Proposition 42. Let T ⊆ C be a hereditary torsion theory of finite type. If M ∈ C is fp-injective, then the T -torsion subobject of M is fp-injective in T .

Proof. Denote by T (M) the T -torsion subobject of M and let C ∈ T be finitely presented. Consider an extension of T (M) by C and take the pushout with the embedding T (M) ⊆ M,

[figure: 0; T (M); N; C; 0; g; M.]

Because T is of finite type, the object C is finitely presented in C, and M is fp-injective, the embedding of T (M) into M extends to the morphism g : N → M as indicated by the dotted arrow. Now N ∈ T , so that the image of g must lie in T (M). □

5.1 The Ziegler topology.

An object of C is coherent if it is finitely presented and every finitely generated subobject is also finitely presented. The category C is said to be locally coherent [21, 25] if there is a generating set of coherent objects. In that case, every finitely presented object is coherent and the category fp(C) is abelian. Conversely, if C is locally finitely presented and fp(C) is abelian, then every finitely presented object is coherent. A ring R is left coherent if the category R-Mod is a locally coherent Grothendieck category. One advantage of the category (mod-R, Ab) of covariant functors on mod-R is that it is a locally coherent Grothendieck category for any associative ring R.

If the Grothendieck category C is locally coherent, then the subsets O(T ), as T ⊆ C ranges over the hereditary torsion classes of finite type, satisfy [21, Thm. 3.4] the axioms for an algebra of open subsets of a topology on Sp(C). The set of indecomposable injective objects of C endowed with this topology is known as the Ziegler spectrum of C, and is denoted by Zg(C). The open subsets of Zg(C) are in bijective correspondence T ↦ O(T ) (cf. [21, Thm. 2.8]) with the hereditary torsion classes T ⊆ C of finite type.

If C is locally coherent and T ⊆ C is of finite type, then the Grothendieck categories T and C/T are also locally coherent [21, §3]. They yield a partition of the Ziegler spectrum of C into the open subset O(T ) and its complement, which are canonically homeomorphic to the Ziegler spectrum of T and that of the localization C/T , respectively,

Zg(C) = Zg(T ) ∪^{·} Zg(C/T ).

Proposition 43. [2, §III.2] The following are equivalent for a ring R :

(1) the ring R is left coherent;

(2) the category ((R-mod)op, Ab) is locally coherent;

(3) the hereditary torsion class (\underline{mod}-R, Ab) ⊆ (mod-R, Ab) is of finite type.

Hereditary torsion theories of finite type play a prominent role in the theory of locally coherent Grothendieck categories. The argument just before Proposition 14 implies that the hereditary torsion class ((R-\underline{mod})op, Ab) ⊆ ((R-mod)op, Ab) is of finite type, whether or not the ring R is left coherent.

Corollary 44. If R is a left coherent ring, then for every R-module, the object Ext^{1}(−, M) is fp-injective in ((R-\underline{mod})op, Ab).

Proof. Since R is left coherent, the hereditary torsion class (\underline{mod}-R, Ab) ⊆ (mod-R, Ab) is of finite type. The functor − ⊗ M is fp-injective in (mod-R, Ab), so Theorem 42 implies that the torsion subobject t(− ⊗ M) is fp-injective in (\underline{mod}-R, Ab). By Theorem 29, the equivalence Tr_{∗} : (\underline{mod}-R, Ab) → ((R-\underline{mod})op, Ab) associates to t(− ⊗ M) the contravariant functor Ext^{1}(−, M). □

Given any associative ring R, there is a covariant Ziegler spectrum Zg(mod-R, Ab), because the category (mod-R, Ab) is locally coherent. The Ziegler spectrum Zg((R-mod)op, Ab) of the category of contravariant functors on R-mod only makes sense if the ring R is left coherent. In that case, there is a partition of the category of covariant functors

Zg(mod-R, Ab) = Zg(\underline{mod}-R, Ab) ∪^{·} Zg(R-Mod)

as a disjoint union of a Ziegler open subset, because (\underline{mod}-R, Ab) is of finite type, and a Ziegler closed subset. Similarly, there is a partition of the category of contravariant functors

Zg((R-mod)op, Ab) = Zg((R-\underline{mod})op, Ab) ∪^{·} Zg(R-Mod).

The canonical bijective correspondence Ξ_{R} of Gabriel spectra given by (4) respects these partitions and induces by restriction a homeomorphism of the respective open subsets, as well as a homeomorphism of the closed subsets.

5.2 Serre subcategories of fp(C).

If C is locally coherent and T ⊆ C is a hereditary torsion class, then the intersection S(T ) = T ∩ fp(C) is a Serre subcategory of fp(C) in the sense that whenever

[figure: 0; A; B; C; 0]

is a short exact sequence in fp(C), then B ∈ S if and only if A and C belong to S. If T is of finite type, then S(T ) provides a set of generators of T . In the other direction we have that if S ⊆ fp(C) is a Serre subcategory, then the torsion theory Gen(S) of objects generated by S is hereditary [21, Thm. 2.5]. Thus the Ziegler interior of a Gabriel open subset O(T ) is the open subset associated to the hereditary torsion class Gen(S(T )), and T is of finite type if and only if

T = Gen(S(T )).

If M ∈ C is an fp-injective object, then applying the functor (−, M) to the short exact sequence in fp(C) above yields a short exact sequence

[figure: 0; (C, M); (B, M); (A, M); 0]

of abelian groups. Thus the subcategory S(M) ⊆ fp(C) defined by

S(M) := {A ∈ fp(C) : (A, M) = 0}

is a Serre subcategory of fp(C). If S ⊆ S(M), then M is Gen(S)-torsion-free and Gen(S)- closed [21, Prop. 3.10], so that M = M_{Gen(S)} belongs to the localization C/Gen(S). It is also fp-injective (ibid) as an object of the localization.

The next result, which does not require that C be locally coherent, will be useful in obtaining an upper bound on the injective dimension of an fp-injective object M ∈ C when C is locally coherent.

Proposition 45. If M is an fp-injective object of C and A is a finitely presented noetherian object of C, then

Hom[A, Ω^{−1}(M)] = 0.

Proof. Let f : A → Ω^{−1}(M) be a morphism. Since M is fp-injective, this morphism lifts to the injective envelope E of M as indicated by the dotted arrow,

[figure: A; g; f; 0; M; E; Ω^{−1}(M); 0.]

Every subobject of A is finitely generated, so every quotient of A is finitely presented. Thus the object K in the short exact sequence

[figure: 0; M; M + g(A); K; 0]

is finitely presented. Since M is fp-injective, the sequence must split. But M ⊆ M+g(A) ⊆ E is an essential extension of M. It follows that g(A) ⊆ M, and hence that f = 0. □

Proposition 46. If C is locally coherent and M ∈ C an fp-injective object, then Ω^{−1}(M) is an fp-injective object of C.

Proof. To begin, let us show that if a monomorphism f : A → B is given, with A finitely presented, then any morphism g : A → Ω^{−1}(M) extends to h : B → Ω^{−1}(M) as indicated by the dotted arrow

[figure: 0; A; f; B; k; g; h; 0; M; e; E; p; Ω^{−1}(M); 0.]

The morphism k : A → E exists, because M is fp-injective. As E is injective, the morphism k : A → E extends to a morphism q : B → E (not shown) such that k = qf. Let h = pq. Then g = pk = pqf = hf.

Now suppose that there is a short exact sequence

[figure: 0; Ω^{−1}(M); X; p; C; 0,]

with C finitely presented. Because the category C is locally coherent, there is a morphism g : B → X from a coherent object B such that the composition pg : B → C is an epimorphism. As C is finitely presented, the kernel A = Ker (pg) is finitely generated. As B is coherent, A is finitely presented. One obtains a morphism of short exact sequences

[figure: 0; A; k; B; pg; C; 0,; g; 0; Ω^{−1}(M); X; p; C; 0.]

The dotted arrow in the left commutative square exists due to the argument from the previous paragraph. It induces the dotted arrow in the right commutative square, which yields a section of p : X → C. □

Let us note that if C is locally coherent and M ∈ C is fp-injective, then

S(M) ⊆ S(Ω^{−1}(M)).

If A ∈ C is finitely presented and f : A → Ω^{−1}(M) is nonzero then, because M is fp-injective, this morphism will factor through p : E → Ω^{−1}(M) as indicated by the dotted arrow,

[figure: A; g; f; 0; M; e; E; p; Ω^{−1}(M); 0.]

Since g : A → E is nonzero, we obtain S(E) ⊆ S(Ω^{−1}(M)). But because Gen(S(M)) is hereditary of finite type, S(E) = S(M).

Let S ⊆ fp(mod-R, Ab) be a Serre subcategory. Define S′ ⊇ S to be the subcategory of fp(mod-R, Ab) that consists of the objects A ∈ fp(mod-R, Ab) for which the localized object A_{Gen(S)} is noetherian in the localization of (mod-R, Ab) at Gen(S). Because A_{Gen(S)} is finitely presented in the localization (mod-R, Ab)/Gen(S), the subcategory S′ ⊆ fp(mod-R, Ab) is also a Serre subcategory. Define S^{(n)} for n > 1, by recursion as S^{(n+1)} := (S^{(n)})′. An R-module N has finite elementary Krull dimension [29, p. 217] if there is a natural number n such that [S(− ⊗_{R} N)]^{(n)} = fp(mod-R, Ab). In that case, define the elementary Krull dimension of M, eK-dim(M), to be the least whole number n such that [S(− ⊗ M)]^{(n+1)} = fp(mod-R, Ab).

Theorem 47. Let C be locally coherent and M ∈ C an fp-injective object. Then

[S(M)]^{(n)} ⊆ S[Ω^{−n}(M)].

So if M has finite elementary Krull dimension, then eK-dim(M) ≥ dim(M), the injective dimension of M.

Proof. The inclusion is proved by induction on n. To prove the case n = 1, let S = S(M) and suppose that A ∈ fp(C) is such that A_{Gen(S)} is noetherian in C/Gen(S). All the terms in the short exact sequence

[figure: 0; M; E; Ω^{−1}(M); 0]

are fp-injective. Furthermore, S = S(E) ⊆ S(Ω^{−1}(M)), so that this sequence may be thought of a short exact sequence of fp-injective objects in the localization C/Gen(S). We need to prove that (A, Ω^{−1}(M)) = 0. Since Ω^{−1}(M) belongs to the localization C/Gen(S), the universal property of localization implies that (A, Ω^{−1}(M)) = (A_{Gen(S)}, Ω^{−1}(M)). Now A_{Gen(S)} is a noetherian finitely presented object of the locally coherent Grothendieck category C/Gen(S), so that Proposition 46 applies to yield (A_{Gen(S)}, Ω^{−1}(M)) = 0.

To prove the induction step, assume that [S(M)]^{(n)} ⊆ S[Ω^{−n}(M)]. Then

[S(M)]^{(n+1)} ⊆ (S[Ω^{−n}(M)])′ ⊆ S(Ω^{−1}(Ω^{−n}(M))) = S[Ω^{−(n+1)}(M)].

The first inclusion is obvious, while the second follows from the case n = 1 together with Proposition 46. If eK-dim(M) = n, then

S[Ω^{−(n+1)}(M)] = [S(M)]^{(n+1)} = fp(C).

It follows that Ω^{−(n+1)}(M) = 0 and hence that dim(M) ≤ n. □

If C = (mod-R, Ab) and N is an R-module, then the object − ⊗ N is fp-injective in C and its injective dimension is the pure-injective dimension of N (cf. [30]). The elementary Krull dimension of −⊗N, if finite, is therefore a bound of the pure-injective dimension of N. Recall that the ring R is called left pure-semisimple if every left R-module is pure-injective. Equivalently, every fp-injective object −⊗M of (mod-R, Ab) is injective. If R is left coherent, then the analogous condition on the category (\underline{mod}-R, Ab) may be characterized as follows.

Theorem 48. Consider the following conditions on a ring R :

(1) for every module M, the quotient PE(M)/M is fp-injective;

(2) for every module M, the object Ext^{1}(−, M) is injective in ((R-\underline{mod})op, Ab);

(3) the category ((R-\underline{mod})op, Ab) is locally noetherian. Then (1) ⇒ (2) ⇒ (3). If R is left coherent, the conditions are equivalent.

Proof. (1) ⇒ (2). Consider the long exact sequence of functors induced by the short exact sequence

[figure: 0; M; PE(M); Z; 0,]

where the monomorphism is the pure-injective envelope of M and Z = PE(M)/M. The sequence is pure-exact, so the first connecting map is 0,

[figure: (−, Z); 0; Ext^{1}(−, M); Ext^{1}(−, PE(M)); Ext^{1}(−, Z),]

and because Z is fp-injective, Ext^{1}(−, Z) = 0. It follows that Ext^{1}(−, M) ≅ Ext^{1}(−, PE(M)) is injective.

(2) ⇒ (3). By [12, Prop. 1.1.2], ⨿_{i} Ext^{1}(−, M_{i}) ≅ Ext^{1}(−, ⊕_{i}M_{i}). The hypothesis thus implies that a coproduct of injective objects in ((R-\underline{mod})op, Ab) is injective. By [31, Prop. V.4.3], the category ((R-\underline{mod})op, Ab) is locally noetherian.

(3) ⇒ (1), assuming R is left coherent. The hypothesis is equivalent to the statement that (\underline{mod}-R, Ab) is locally noetherian. Since R is left coherent, the hereditary torsion class (\underline{mod}-R, Ab) is of finite type, and so every noetherian object in it is finitely presented in (mod-R, Ab). If M is an R-module, then − ⊗ M is an fp-injective object of (mod-R, Ab) whose injective envelope is given by the monomorphism in the short exact sequence

[figure: 0; − ⊗ M; − ⊗ PE(M); − ⊗ Z; 0,]

where Z = PE(M)/M. By Theorem 45, Hom[A, − ⊗ Z] = 0 for every A ∈ (\underline{mod}-R, Ab). By Theorem 29, t(− ⊗ Z) = Tr_{∗}[Ext^{1}(−, Z)] = 0, and so Z is fp-injective. □

6. Examples.

In this section, some aspects of the covariant and contravariant Gabriel spectra of R are described for specific classes of rings. For example, it is shown that if R is left semihereditary, then the closed subset Sp(R-Mod) is open in both the covariant as well as the contravariant Gabriel spectrum of R. Since the canonical bijective correspondence Ξ_{R} is a homeomorphism restricted to both Sp(R-Mod) and its open complement, it is a homeomorphism of the two Gabriel spectra of R.

Proposition 49. Let I ⊆ R be a maximal left ideal and E = E(R/I) the injective envelope of the simple R-module R/I. Then the indecomposable injective functor (−, E) is the injective envelope in ((R-mod)op, Ab) of a simple object. The point (−, E) is therefore isolated in the Gabriel spectrum of ((R-mod)op, Ab).

Proof. If (−, E) is the injective envelope of a simple object X, then it is the only injective indecomposable that belongs to the open subset associated to the minimum hereditary torsion theory containing X; this is the class of objects all of whose quotients have a copy of X in the socle. To find such a simple functor, denote the inclusion morphism by i : I → R and let X = Coker (−, i). The exact sequence

[figure: 0; I; i; R; E]

in R-Mod yields an exact sequence

[figure: 0; (−, I); (−, i); (−, R); (−, E),]

in ((R-mod)op, Ab). The functor X is isomorphic to a subfunctor of (−, E). Furthermore, it is obvious that X(R) = R/I is simple as a left R-module. To see that X is simple, suppose that Y ⊆ X is a proper nonzero subfunctor. The epimorphism (−, R) → X composed with the quotient map X → X/Y yields a nonzero morphism from (−, R) to X/Y. By Yoneda’s Lemma, (X/Y)(R) = X(R)/Y (R) ≠ 0. Now X(R) = R/I is a simple R-module, which forces Y (R) = 0. Since Y ⊆ X is torsion-free, it leaves Y = 0 as the only possibility. □

If E = E(S) is the injective envelope of a simple module S, then the point − ⊗ E of the covariant Gabriel spectrum Sp(mod-R, Ab) is also isolated. Let p : E → E/S be the natural quotient map and denote by Y the kernel of the morphism − ⊗ f : − ⊗ E → − ⊗ E/S. It is a simple object of the category (mod-R, Ab). For, suppose that μ : − ⊗ E → F were a morphism with nonzero kernel. Composing μ with the injective envelope η : F → − ⊗ M yields a morphism − ⊗ g : − ⊗ E → − ⊗ M which is not a split monomorphism, since its kernel is also nonzero. But then g : E → M is also not a split monomorphism and so its kernel must contain S. It follows that the kernel of − ⊗ g contains Y, which implies that Y must be simple.

Example 1: The ring R is left semiartinian if every nonzero R-module has nonzero socle. In particular, every indecomposable injective R-module is the envelope of a simple module. Then Sp(R-Mod) is an open discrete subset of both the contravariant and covariant Gabriel spectra. The two spectra are therefore homeomorphic.

If R is left coherent and the maximal left ideal I is finitely generated, then the simple functor Coker (−, i) in the proof of Proposition 49 is finitely presented. The injective indecomposable (−, E) is then isolated in the contravariant Ziegler spectrum of R. It is therefore also isolated in the contravariant Gabriel spectrum.

Example 2: Let R = Λ be an artin algebra. Such a ring is both left coherent and left semiartinian. If E is an indecomposable injective module, then the point (−, E) is isolated in the contravariant Ziegler spectrum of Λ. By duality, E is a finitely presented module so the point −⊗_{Λ}E is isolated ([29, Cor. 13.4]) in the covariant Ziegler spectrum of Λ. It follows that the canonical bijective correspondence Ξ_{Λ} between the covariant and contravariant spectra of R is a homeomorphism with respect to the Ziegler topology as well as the Gabriel topology.

There is another homeomorphism between the Gabriel (resp., Ziegler) spectra of Λ, namely, that induced by the duality D : (Λ-mod)op → mod-Λ. In general, it is not the same as Ξ_{Λ}.

Example 3: Let R = ℤ be the ring of integers; it is a coherent ring. The indecomposable injective ℤ-modules that are envelopes of simple modules are the Prüfer groups ℤ(p^{∞}). It is well-known (cf. [29, Example 1, p.104]) that the corresponding points − ⊗_{ℤ} ℤ(p^{∞}) are not isolated in the Ziegler spectrum of ℤ. Therefore, the canonical bijective correspondence Ξ_{ℤ} is not a homeomorphism of Ziegler spectra.

Example 4: Let R be a von Neumann regular ring. This means that R satisfies any (all) of the following equivalent conditions:

(1) every short exact sequence of R-modules is pure-exact;

(2) every module is flat;

(3) every module is fp-injective;

(4) every pure-injective module is injective;

(5) every finitely presented module is projective. The last condition of the proposition is equivalent to the equation R-\underline{mod} = 0, which the Auslander-Bridger transpose shows to be left-right symmetric. Thus ((R-\underline{mod})op, Ab) = 0 and the evaluation functor Ev : ((R-mod)op, Ab) → R-Mod is an equivalence. The section morphism Υ : R-Mod → ((R-mod)op, Ab) given by M ↦ (−, M) is its equivalence inverse. That the morphism Υ is an equivalence implies that for a von Neumann regular ring R the functorial perspective considered in this article coincides with the classical approach to the study of R-modules.

Example 5: Let R be a ring with the property that every submodule of a flat module is flat. Equivalently, the category R-Mod has flat global dimension at most 1. This is a left-right symmetric condition, because it is equivalent to Tor_{2}^{R}(M, N) = 0 for any right R-module M and left R-module N.

Proposition 50. The category R-Mod has flat global dimension at most 1 if and only if the torsion class Gen (−, R) is hereditary.

Proof. Suppose that R-Mod has flat global dimension at most 1. It suffices to prove that every subfunctor F of a coproduct (−, R)^{(α)} ≅ (−, R^{(α)}) itself belongs to Gen (−, R). So let η : (−, M) → F be an epimorphism from a projective functor (−, M). Composing the epimorphism with the inclusion F ⊆ (−, R^{(α)}) yields a morphism of the form (−, f) : (−, M) → (−, R^{(α)}), where f : M → R^{(α)} is morphism of R-modules. By assumption, the image of f is a flat module Z, so that F = Im(−, f) ≅ (−, Imf) = (−, Z). But (−, Z) belongs to Gen (−, R), since any epimorphism g : R^{(β)} → Z from a free module to a flat module induces an epimorphism (−, g) : (−, R)^{(β)} → (−, Z).

To prove the converse, consider an inclusion Y ⊆ Z of R-modules, with Z flat. Since (−, Z) belongs to Gen (−, R), and (−, Y) ⊆ (−, Z), the assumption that Gen (−, R) is hereditary implies that (−, Y) also belongs to Gen (−, R). Thus there is an epimorphism η : (−, R^{(β)}) ≅ (−, R)^{(β)} → (−, Y), which by Proposition 2 implies η is of the form (−, g) where g : R^{(β)} → Y is a pure-epimorphism. But then Y is flat. □

Let us note that the open subset of the contravariant Gabriel spectrum Sp((R-mod)op, Ab) that corresponds to the hereditary torsion class Gen (−, R) is the subset Sp(R-Mod). It is clear that if E is an indecomposable injective R-module, then there is a nonzero morphism from (−, R) to (−, E). Thus Sp(R-Mod) is contained in the open subset associated to Gen (−, R). To prove that equality holds, suppose that I = E(Ext^{1}(−, M)), is a point of the contravariant Gabriel spectrum of R that does not belong to Sp(R-Mod). If there were a nonzero morphism from (−, R) to I, then, as Gen (−, R) is hereditary and Ext^{1}(−, M) is essential in I, it would lead to the contradiction that there exists a nonzero morphism from (−, R) to Ext^{1}(−, M).

Example 6: Let R be a left semi-hereditary ring, that is, suppose that every finitely generated left ideal of R is projective. Then (cf. [31, Prop. I.6.9]), every finitely generated submodule of a finitely generated free module is projective. Equivalently, every finitely presented R-module has projective dimension at most 1. This is expressible by the equation in ((R-mod)op, Ab),

Ext^{2}(−, M) = 0,

for every R-module M. Since every finitely presented module A has projective dimension at most 1, it also has flat dimension at most 1. For any right R-module W, we thus obtain that Tor_{2}^{R}(W, A) = 0. As every left R-module is a direct limit of finitely presented modules, and the functor Tor_{R}^{2}(W, −) : R-Mod → Ab commutes with direct limits, it follows that for every left semi-hereditary ring R, the category R-Mod has flat global dimension at most 1. Indeed, it is not difficult to see that a ring R is left semi-hereditary if and only if it is left coherent and R-Mod has flat global dimension at most 1.

Proposition 51. If R is a left semi-hereditary ring, then the subset Sp(R-Mod) of the covariant Gabriel spectrum Sp(mod-R, Ab) is open.

Proof. First, note that any quotient of an injective module E is fp-injective. For if M ⊆ E is a submodule, then a routine dimension shift argument implies that Ext^{1}(−, E/M) ≅ Ext^{2}(−, M) = 0 in ((R-mod)op, Ab).

Now suppose that there is an indecomposable injective module E such that the point − ⊗ E of the covariant Gabriel spectrum belongs to the closure of Sp(\underline{mod}-R, Ab). There is an embedding − ⊗ m : − ⊗ E → ∏_{i} − ⊗ U_{i}, where each − ⊗ U_{i} belongs to Sp(\underline{mod}-R, Ab). The R-morphism m : E → ∏_{i} U_{i} is an embedding, so that one of the component functions m_{i} : E → U_{i} is nonzero. The image of m_{i} is fp-injective, which implies that U_{i} must be the pure-injective envelope of an fp-injective module. But then U_{i} would have to be injective, contradicting that − ⊗ U_{i} belongs to Sp(\underline{mod}-R, Ab). □

References

[1] Auslander, M., Coherent Functors, in Proc. Conf. on Categorical Algebra (La Jolla, 1965), 189-231, Springer, New York, 1966.

[2] Auslander, M., Representation dimension of artin algebras, Queen Mary College, Mathematics Notes, University of London, 1971.

[3] Auslander, M., Functors and morphisms determined by objects, in Representation Theory of Algebras; Proceedings of the Conference at Temple University, 1976, Lecture Notes in Pure and Applied Mathematics 37, Dekker, 1978, 1-111.

[4] Auslander, M., Isolated singularities and existence of almost split sequences, in Representation Theory II (ICRA IV, Ottawa, 1984), 194-241, Lecture Notes in Math 1178, Springer, Berlin-New York, 1986.

[5] Auslander, M., and Bridger, M., Stable Module Theory, Memoirs of the AMS 94, American Mathematical Society, 1969.

[6] Auslander, M., and Reiten, I., Stable equivalence of artin algebras, in Proceedings of the Conference on Orders, Groups rings and related topics (Ohio, 1972), Lecture Notes in Mathematics 353, Springer, 1973, 8-71.

[7] Auslander, M., and Reiten, I., Representation theory of artin algebras III: Almost split sequences, Communications in Algebra 3 (1975), 239-294.

[8] Beligiannis, A., Relative homological algebra and purity in triangulated categories, J. Algebra 227 (2000), 268-361.

[9] Beligiannis, A., Purity and almost split morphisms in abstract Homotopy categories: A unified approach via Brown Representability, Alg. and Rep. Theory 5 (2002), 483-525.

[10] Benson, D., and Gnacadja, G., Phantom maps and modular representation theory, II, Algebras and Representation Theory 4 (2001), 395-404.

[11] Carlson, J., Modules and Group Algebras, Lecture Notes in Mathematics, ETH Zürich, Birkäuser, 1996.

[12] Colby, R. and Fuller, K., Equivalence and Duality for Module Categories, Cambridge Tracts in Mathematics 161, Cambridge University Press, 2004.

[13] Crawley-Boevey, W.W., Modules of finite length over their endomorphism rings, in Representations of Algebras and Related Topics, Tachikawa, H. and Brenner, Sh., eds., London Math Soc Lecture Notes Series 168, Cambridge University Press, 1992.

[14] Crawley-Boevey, W.W., Locally finitely presented additive categories, Communications in Algebra 22(5), (1994), 1641-1674.

[15] Enochs, E., Estrada, S., Garcia Rozas, J.R., and Oyonarte, L., Flat covers in the category of quasicoherent sheaves over the projective line, Communications in Algebra 32 (2004), 1497-5008.

[16] Enochs, E.E., and Jenda, O.M.G., Relative Homological Algebra, De Gruyter Expositions in Mathematics 30, Walter DeGruyter, 2000.

[17] Gabriel, P., Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323-448.

[18] Gruson, L. and Jensen, C.U., Modules algébriquement compacts et foncteurs lim_{→}^{i}, C.R. Acad. Sci. Paris 276 (1973), 1651-1653.

[19] Guil Asensio, P.A., and Herzog, I., Left Cotorsion Rings, Bulletin of the London Mathematical Society 36 (2004), 303-309.

[20] Herzog, I., A test for finite representation type, Journal of Pure and Applied Algebra 95 (1994), 151-182.

[21] Herzog, I., The Ziegler spectrum of a locally coherent grothendieck category, Proc. London Math. Soc. 74(3)(1997), 503-558.

[22] Herzog, I., Pure-injective envelopes, Journal of Algebra and its Applications 2(4) (Dec. 2003), 397-402.

[23] Hovey, M., Palmieri, J. and Strickland, N., Axiomatic Stable Homotopy Theory, Memoirs of the AMS 610, American Mathematical Society, July 1997.

[24] Krause, H., Smashing subcategories and the telescope conjecture - an algebraic approach, Invent. Math. 139 (2000), 99-133.

[25] Krause, H., The spectrum of a module category, Memoirs of the AMS 149 American Mathematical Society, 2001.

[26] Mohamed, S.H. and Müller, B.J., Continuous and Discrete Modules, London Mathematical Society Lecture Notes Series 147, Cambridge University Press, 1990.

[27] Nicholson, W.K., Lifting Idempotents and Exchange Rings, Transactions of the American Mathematical Society 229 (1977), 269-278.

[28] Nicholson, W.K. and Yousif, M., Quasi-Frobenius Rings, Cambridge tracts in mathematics 158, Cambridge University Press, 2003.

[29] Prest, M., Model Theory and Modules, London Math. Soc. Lecture Notes Series 130, Cambridge University Press, 1988.

[30] Simson, D., On pure global dimension of locally finitely presented Grothendieck categories, Fund. Math. 96 (1977), 91-116.

[31] Stenström, B., Rings of Quotients, Springer-Verlag, 1975.

[32] Trlifaj, J., Two problems of Ziegler and uniform modules over regular rings, in Abelian Groups and Modules, Marcel Dekker (1996), 373-383.

[33] Verdier, J.-L., Des Catégories Dérivées des Catégories Abéliennes, Astérisque 239, Société Mathémathique de France, 1996.

[34] Weibel, Ch., An Introduction to Homological Algebra, Cambridge studies in advanced mathematics 38 Cambridge University Press, 1994.

[35] Xu, J., Flat Covers of Modules, Lecture Notes in Mathematics 1634, Springer, 1996.

The Ohio State University (Lima), Lima, OH 45804 USA