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DUALITY FOR CATEGORIES OF FINITELY PRESENTED MODULES

G. A. GARKUSHA AND A. I. GENERALOV

Abstract. With the help of localizing subcategories and the corresponding torsion functors of the functor category (mod-R, Ab), duality is studied for the categories of finitely presented modules R-mod and mod-R. Also, the classes of absolutely pure and of almost regular rings and the classes of fp-injective and of fp-flat modules are investigated, as well as their relationship with the properties of the Ziegler spectrum of a ring.

§0. Introduction

The quasi-Frobenius rings can be characterized as the rings R that determine an R-duality between the categories of finitely generated right and left R-modules. In the present paper we investigate rings yielding an R-duality between the categories of finitely presented modules (such rings will be called weakly quasi-Frobenius rings). It turns out that such rings can be described as (two-sided) absolutely pure coherent rings. This class of rings occurred earlier in [1, 2].

The key technical tools employed in the paper are furnished by the so-called categories of generalized modules _{R}C = (mod-R, Ab) and C_{R} = (R-mod, Ab) that consist of covariant additive functors defined on the corresponding categories of finitely presented modules (Ab stands for the category of Abelian groups). Using these tools, we find a new characterization of the class of (one-sided) absolutely pure rings (Theorem 2.5; cf. [2, 3, 4]). In turn, this characterization implies various descriptions of weakly quasi-Frobenius rings (Theorem 2.11). Furthermore, we introduce and investigate the class of almost regular rings, which generalize both the regular rings and the “indiscrete” rings; the latter class arose in [5] in connection with the study of the Ziegler spectrum of a ring.

§1. Preliminaries

Consider a Grothendieck category C. A full subcategory S of C is called a Serre subcategory if for any exact sequence 0 → X → Y → Z → 0 in C the object Y belongs to C if and only if so do X and Z. A Serre subcategory S is said to be localizing if S is closed under taking direct limits (see [6, 15.11]) or, equivalently, the inclusion functor i: S → C admits a right adjoint t = t_{S}: C → S, which is called the torsion functor. The functor t takes each object X ∈ C to the largest subobject t(X) ⊆ X belonging to S (see [7, 2.1]).

If S is a localizing subcategory, then the quotient category C/S is the full subcategory of C consisting of S-closed objects, i.e., X ∈ C/S if and only if _{C}(C, X) = 0 and Ext_{C}^{1}(C, X) = 0 for any C ∈ S (cf. [7, Subsection 2.11; 8, p. 517]). The proof of the following assertion is similar to that of [9, X.1.4].

\begin{matrix}c \\ ◯ \end{matrix}2000 American Mathematical Society

1991 Mathematics Subject Classification. Primary 18E15.

Key words and phrases. U-duality, Ziegler spectrum, absolutely pure, fp-injective, fp-flat modules.

The first author was supported by the St. Petersburg Government (grant no. 98-2.1K-48).

Lemma 1.1. Let E ∈ C/S; then E is C/S-injective if and only if E is C-injective.

An object X of the Grothendieck category C is said to be finitely generated, if whenever X = ∑_{i∈I} X_{i} for some subobjects X_{i} ⊆ X, there is a finite subset J ⊂ I such that X = ∑_{i∈J} X_{i}. The full subcategory of finitely generated objects is denoted by fg C. A finitely generated object X is said to be finitely presented if for every finitely generated object Y and every epimorphism γ: Y → X, the kernel ker γ is also finitely generated. We denote by fp C the full subcategory of finitely presented objects of C. The corresponding subcategories of finitely presented R-modules are denoted by R-mod = fp(R-Mod) and mod-R = fp(Mod-R), respectively. A finitely presented object C ∈ C is said to be coherent if every finitely generated subobject B ⊆ C is finitely presented. The subcategory of coherent objects of C will be denoted by coh C.

Let Ab be the category of Abelian groups. The main category we work with is the category of additive covariant functors _{R}C = (mod-R, Ab) (respectively, C_{R} = (R-mod, Ab)). The category _{R}C has enough finitely generated projectives, and these are precisely the objects of the form (M, −) with M ∈ mod-R (see [10, 2.2; 9, IV.7.5]). The functor −⊗_{R}?: R-Mod → _{R}C, M ↦ − ⊗_{R} M, is fully faithful, and if M ∈ R-mod, then the functor − ⊗_{R} M is a coherent object (see [8, §2]). An object M ∈ _{R}C is said to be cohinjective if for every C ∈ coh_{R}C we have Ext1_{_{R}C}(C, M) = 0. The coh-injective objects of the category _{R}C are precisely the objects of the form − ⊗_{R} M, M ∈ R-Mod (see [8, §2.1]).

Proposition 1.2 (see [11, 1.2]). An object E ∈ _{R}C is injective if and only if it is isomorphic to one of the functors − ⊗_{R} Q where Q is a pure-injective left R-module.

The Abelian subcategory coh_{R}C of coherent objects has also enough injectives, and these are precisely the objects of the form − ⊗_{R} M, M ∈ R-mod (see [12, 7.3′]). Thus, any coherent object C ∈ coh_{R}C has both a projective presentation

(1.1)  (K, −) ⟶^{(α,−)} (L, −) ⟶ C ⟶ 0

and an injective presentation in coh_{R}C

(1.2)  0 ⟶ C ⟶ − ⊗_{R} M ⟶^{−⊗μ} − ⊗_{R} N.

Here K, L ∈ mod-R and M, N ∈ R-mod.

Let P be a localizing subcategory of _{R}C, and let t = t_{P} : _{R}C → P be the corresponding torsion functor. We say that P is of finite type if t commutes with the direct limits (see [7, §2]), or, equivalently, the inclusion functor _{R}C/P → _{R}C commutes with the direct limits (see [7, Subsection 2.4]). In _{R}C, any subcategory P of finite type is of the form S⃗, where S⃗ denotes the subcategory of C formed by the direct limits of objects of the Serre subcategory S = P ∩ coh_{R}C of coh C (see [7, §2]). If S⃗ is of finite type, then the corresponding torsion functor t will be denoted by t_{S}.

Let Zg_{R}C denote the set of isomorphism classes of indecomposable injective objects in _{R}C. This is indeed a set by [8, §3]. Let X be an arbitrary subcategory of coh_{R}C. Consider the following subset of Zg_{R}C:

(X) = { − ⊗_{R} Q ∈ Zg_{R}C | (C, − ⊗_{R} Q) ≠ 0 for some C ∈ X }.

Then (see [8, 3.4]) the collection

{ (S) | S is a Serre subcategory in coh _{R}C }

of subsets of Zg_{R}C satisfies the axioms for the open sets of a topology on Zg_{R}C; the resulting topological space will be referred to as the (left) Ziegler spectrum of the ring R. Moreover, there is a bijection between the Serre subcategories of coh_{R}C and the open subsets of Zg_{R}C (see [8, 3.8]).

§2. Absolutely pure and flat modules

A left R-module M is said to be absolutely pure (or FP-injective) if for every F ∈ R-mod we have Ext_{R}^{1}(F, M) = 0, or equivalently, any monomorphism μ: M → N is pure (see [3, 2.6]). The ring R is left absolutely pure if the module _{R}R is absolutely pure. M is fp-injective if for any monomorphism μ: K → L in R-mod the morphism (μ, M) is an epimorphism. Clearly, any absolutely pure module is fp-injective, and any finitely presented fp-injective module is absolutely pure. M is fp-flat if for any monomorphism μ: K → L in mod-R the morphism μ ⊗_{R} M is also a monomorphism. Obviously, any flat module is fp-flat. In the sequel, we use the following notation: P^{R} = { F ∈ _{R}C | F (R_{R}) = 0 }, S^{R} = { C ∈ coh_{R}C | C(R_{R}) = 0 }, S_{R} = { C ∈ coh_{R}C | _{_{R}C}(C, − ⊗_{R} R) = 0 }. The definition of the subcategories ^{R}S and _{R}S of coh C_{R} is similar to that of S^{R} and S_{R}, respectively. From (1.2) it easily follows that

S^{R} = { ker(− ⊗ μ) | μ: M → N is a monomorphism, M, N ∈ R-mod }.

In a similar way,

(2.1)  S_{R} = { coker(μ, −) | μ: L → K is a monomorphism in mod-R }.

Clearly, the subcategories S⃗^{R}, S⃗_{R}, ^{R}S⃗, _{R}S⃗ are of finite type. We also note that any finitely generated projective object of _{R}C is S_{R}-torsion-free.

Theorem 2.1 ([13]). The category R-Mod of R-modules is equivalent to the quotient category _{R}C/P^{R}. Moreover, P^{R} = S⃗^{R} if and only if the ring R is left coherent.

Proposition 2.2. If K ∈ R-Mod, then:

(1) K is absolutely pure if and only if t_{P^{R}} (− ⊗_{R} K) = 0;

(2) K is fp-injective if and only if t_{S^{R}} (− ⊗_{R} K) = 0;

(3) K is fp-flat if and only if t_{S_{R}} (− ⊗_{R} K) = 0.

Proof. (1) Suppose that K is absolutely pure. If μ: K → E is an injective envelope of the module K, then, by assumption, − ⊗ μ is an _{R}C-monomorphism. By Lemma 1.1 and Theorem 2.1, − ⊗_{R} E is P^{R}-torsion-free. Since − ⊗_{R} K is a subobject of − ⊗_{R} E, it follows that − ⊗_{R} K is P^{R}-torsion-free. Conversely, for any monomorphism μ: K → N we have ker(− ⊗ μ) ∈ P^{R}; hence, if t_{P^{R}} (− ⊗_{R} K) = 0, then − ⊗ μ is a monomorphism.

(2) Since −⊗K is a coh-injective object, application of the functor (−, − ⊗_{R} K) to an exact sequence of the form (1.2) with C ∈ S^{R}, M, N ∈ R-mod yields an exact sequence of Abelian groups

RC(− ⊗ N, − ⊗ K) ⟶^{(−⊗μ,−⊗K)} _{_{R}C}(− ⊗ M, − ⊗ K) ⟶ _{_{R}C}(C, − ⊗ K) ⟶ 0.

Therefore, if − ⊗ K is S^{R}-torsion-free, then (μ, K) is an epimorphism. It remains to observe that the above argument can easily be converted.

(3) The proof is similar to that of (2), with the use of (1.1) and the Yoneda lemma.□

Proposition 2.3. (1) For any family {M_{i}}_{I} of left R-modules, the module ∏_{I} M_{i} is absolutely pure (respectively, fp-injective, fp-flat) if and only if every M_{i} is absolutely pure (respectively, fp-injective, fp-flat).

(2) The direct limit \underrightarrow{lim} M_{i} of fp-injective (respectively, fp-flat) modules M_{i} is an fp-injective (respectively, fp-flat) module.

Proof. (1) By [9, I.13.2], in _{R}C we have isomorphism ∏_{I}(− ⊗_{R} M_{i}) ≈ − ⊗_{R} ( ∏_{I} M_{i}). Thus, if S is a localizing subcategory of_{R}C, then t_{S} (− ⊗_{R} (∏_{I} M_{i})) ≈ t_{S} (∏_{I}(− ⊗_{R} M_{i})) ≈ ∏_{I} t_{S} (− ⊗_{R} M_{i}). Now statement (1) follows from Proposition 2.2.

(2) Since \underrightarrow{lim}(− ⊗_{R} M_{i}) ≈ − ⊗_{R} ( \underrightarrow{lim} M_{i}), and the torsion functors t_{S^{R}} and t_{S_{R}} commute with the direct limits, statement (2) is also a consequence of Proposition 2.2. □

The following theorem extends the list of properties characterizing the coherent rings (see also [3, 14]).

Theorem 2.4. For a ring R, the following statements are equivalent:

(1) R is right coherent;

(2) in _{R}C there is a localizing subcategory S such that a left R-module M is flat if and only if the functor − ⊗_{R} M is S-torsion-free;

(3) every right fp-injective R-module is absolutely pure;

(4) every left fp-flat R-module is flat;

(5) a direct limit of absolutely pure right R-modules is absolutely pure.

Proof. (1) ⟹ (4). Over a right coherent ring, any short exact sequence 0 → K → L → N → 0 of right R-modules is the direct limit of short exact sequences 0 → K_{i} → L_{i} → N_{i} → 0, where K_{i}, L_{i}, N_{i} ∈ mod-R (see [15, 5.9]). This implies the claim.

(4) ⟹ (2). We may take S = S⃗_{R}.

(2) ⟹ (1). For _{R}R_{i} = R, i ∈ I, we have

t_{S} (− ⊗_{R} ( ∏_{I} R_{i})) ≈ t_{S} ( ∏_{I}(− ⊗_{R} R_{i})) ≈ ∏_{I} t_{S} (− ⊗_{R} R_{i}) = 0,

which implies that the module ∏_{I} R_{i} is flat. Hence, the ring R is right coherent (see [16, 2.1; 9, I.13.3]).

(1) ⟺ (5). This is a consequence of [3, 3.2].

(1) ⟹ (3). This follows from Theorem 2.1 and Proposition 2.2.

(3) ⟹ (5). By Proposition 2.3, the direct limit of absolutely pure modules is fp-injective. □

Theorem 2.5. For a ring R, the following conditions are equivalent:

(1) R is left absolutely pure;

(2) S^{R} ⊂ S_{R};

(3) _{R}S ⊂ ^{R}S;

(4) every fp-flat left R-module is fp-injective;

(5) every indecomposable pure-injective fp-flat left R-module is fp-injective;

(6) every pure-injective fp-flat left R-module is fp-injective;

(7) every fp-injective right R-module is fp-flat;

(8) every indecomposable pure-injective fp-injective right R-module is fp-flat;

(9) every pure-injective fp-injective right R-module is fp-flat.

Proof. (1) ⟹ (2). Let C ∈ S^{R}. In coh_{R}C we consider an exact sequence (1.2) in which μ: M → N is a monomorphism and M, N ∈ R-mod. Applying the functor (−, − ⊗ R) to it, we obtain an exact sequence

RC(− ⊗ N, − ⊗ R) ⟶^{(−⊗μ,−⊗R)} _{_{R}C}(− ⊗ M, − ⊗ R) ⟶ _{_{R}C}(C, − ⊗ R) ⟶ 0.

Since R is left absolutely pure, μ^{∗} = (μ, R) is an epimorphism, whence _{_{R}C}(C, −⊗R) = 0.

(2) ⟹ (1). Consider a monomorphism μ in R-mod. Since ker(− ⊗ μ) ∈ S^{R} ⊂ S_{R}, it follows that (μ, R) is an epimorphism. So, the module _{R}R is fp-injective, and, consequently, absolutely pure.

(1) ⟺ (3). The proof is similar to that of the implications (1) ⟹ (2) and (2) ⟹ (1) (we must use exact sequences of the form (1.1)).

(2) ⟹ (4). This follows from Proposition 2.2.

(4) ⟹ (6) ⟹ (5). This is obvious.

(5) ⟹ (2).

Let I(S) = Zg_{R}C\(S), where S is a Serre subcategory of coh_{R}C, and (S) is an open subset of Zg_{R}C. By assumption, we have I(S_{R}) ⊂ I(S^{R}). Therefore, if C ∈ S^{R}, then RC(C, I(S_{R})) = 0. From [7, 4.3] it follows that C ∈ S_{R}, whence S^{R} ⊂ S_{R}.

(7) ⟹ (9) ⟹ (8). This is obvious.

(8) ⟹ (3). The proof is similar to that of the implication (5) ⟹ (2).

(3) ⟹ (7). This follows from Proposition 2.2. □

Corollary 2.6. The ring R is left absolutely pure and left Noetherian if and only if it is quasi-Frobenius.

Proof. If the ring R is left absolutely pure and left Noetherian, then, by the Baer criterion, it is left self-injective, and [17, Subsection 24.5] shows that R is quasi-Frobenius. The converse statement is trivial. □

Corollary 2.7 ([4, 4.3]). The ring R is left absolutely pure and left semihereditary if and only if it is regular.

Proof. Let R be a left absolutely pure and left semihereditary ring, and let I be a finitely generated left ideal of R. Then, by Theorems 2.5(4) and 2.4(3), _{R}I is absolutely pure. Hence, − ⊗ I is an injective (in coh_{R}C) subobject of the object − ⊗ R, and, therefore, a direct summand of − ⊗ R. Thus, I is a direct summand of R, which proves that R is regular. The converse statement is trivial. □

We say that a module M ∈ R-Mod is an fp-cogenerator if for any nonzero μ: K → L in R-mod there exists ϕ: L → M such that ϕμ ≠ 0. Observe that for such M and every K ∈ R-mod, K ≠ 0, we have Hom_{R}(K, M) ≠ 0.

Lemma 2.8. If the ring R is right absolutely pure, then the module _{R}R is an fpcogenerator.

Proof. By Theorem 2.5, S_{R} ⊂ S^{R} in _{R}C. We consider μ: K → L, μ ≠ 0, K, L ∈ R-mod. If C = im(−⊗μ), then C ∈ coh_{R}C. Suppose that αμ = 0 for any α ∈ L^{∗} = Hom_{R}(L, R). Let λ: C → − ⊗ R be an arbitrary morphism. Since − ⊗ R is a coh_{R}C-injective object, there exists a morphism − ⊗ η: − ⊗L → − ⊗ R such that (− ⊗ η)|_{C} = λ. By assumption we have ημ = 0, so that λ = 0. Therefore, C ∈ S_{R}, whence C ∈ S^{R}. We see that C(R) = 0, which yields μ = 0, a contradiction. □

Theorem 2.9. Let R be a left absolutely pure ring satisfying one of the following conditions:

(1) R is left coherent, and for any nonzero M ∈ R-mod the module M^{∗} = Hom_{R}(M, R) is nonzero;

(2) for any nonzero cyclic M ∈ R-Mod the module M^{∗} is nonzero. Then the ring R is right absolutely pure.

Proof. Let 0 ≠ M ∈ R-mod, I = Hom_{R}(M, R). Consider a homomorphism μ: M → R^{I} = ∏_{I} R, μ(x) = (ϕ(x))_{ϕ∈I}. If x is a nonzero element of M, then, clearly, M/Rx ∈ R-mod. By assumption (if the ring R is left coherent, the module Rx is finitely presented), there exists a nonzero homomorphism α: Rx → R. Since the ring R is left absolutely pure, it follows that Ext_{R}^{1}(M/Rx, R) = 0, and we see that α extends to a nonzero γ: M → R. Consequently, μ(x) ≠ 0, and μ is a monomorphism.

Let F = ker(− ⊗ μ) = ∑ X_{i}, where X_{i} ∈ fg_{R}C. Being a subobject of the coherent object − ⊗ M, X_{i} is coherent itself, which implies the relation F ∈ S⃗^{R} ⊂ S⃗_{R}. Since t_{S_{R}} (− ⊗ R) = t_{S_{R}} ((R, −)) = 0, it follows that

t_{S_{R}} (− ⊗ R^{I}) ≈ t_{S_{R}} ( ∏_{I}(− ⊗ R)) ≈ ∏_{I} t_{S_{R}} (− ⊗ R) = 0.

Applying the left exact functors t_{S_{R}}, t_{S^{R}} to the exact sequence

0 ⟶ F ⟶ − ⊗ M ⟶^{−⊗μ} − ⊗ R^{I},

we get

t_{S^{R}} (− ⊗ M) = F = t_{S_{R}} (− ⊗ M).

Consider an arbitrary module M ∈ R-Mod. Let M = \underrightarrow{lim} M_{i}, M_{i} ∈ R-mod. Then we have the following isomorphisms:

t_{S_{R}} (− ⊗ M) = t_{S_{R}} (\underrightarrow{lim} − ⊗ M_{i}) ≈ \underrightarrow{lim} t_{S_{R}} (− ⊗ M_{i}) = \underrightarrow{lim} t_{S^{R}} (− ⊗ M_{i})
≈ t_{S^{R}} (\underrightarrow{lim} − ⊗ M_{i}) = t_{S^{R}} (− ⊗ M).

Now, using Proposition 2.2, we deduce that every left fp-injective module is fp-flat. By Theorem 2.5, the ring R is right absolutely pure. □

Lemma 2.10. Suppose that a pair of contravariant functors

(2.2)  G: R-mod ⟶ mod-R,  H: mod-R ⟶ R-mod

determines a duality between the categories R-mod and mod-R of finitely presented modules. Then the ring R is left and right coherent.

Proof. Let I be a finitely generated left ideal of the ring R. Then R/I ∈ R-mod and G(R/I) ∈ mod-R. We consider an exact sequence 0 → G(R/I) → G(R) → J → 0 in mod-R and then the commutative diagram

0 ⟶ H(J) ⟶ HG(R) ⟶ HG(R/I) ⟶ 0
↓  ↓
0 ⟶ I ⟶ R ⟶ R/I ⟶ 0

in which the vertical arrows are isomorphisms. We obtain I ≈ H(J), whence R is left coherent. By symmetry, R is right coherent. □

By the Morita theorem (see [17, 23.29]), any duality of the form (2.2) between categories of finitely presented modules is determined by the (R, R)-bimodule U = G(R). Moreover, there are natural equivalences of functors G ≈ Hom_{R}(−, U), H ≈ Hom_{R}(−, U). In this case, we shall say that the bimodule U gives a U-duality. Obviously, _{R}U ∈ R-mod and U_{R} ∈ mod-R. Since the functor (−, U) is exact both on R-mod and on mod-R, the module U is left and right fp-injective, and, therefore, left and right absolutely pure.

A ring R determining an R-duality between the categories R-mod and mod-R will be called a weakly quasi-Frobenius ring.

The equivalence (2) ⟺ (3) in the next theorem was obtained earlier in [1] (see also [2]).

Theorem 2.11. For a ring R, the following statements are equivalent:

(1) R is weakly quasi-Frobenius;

(2) R is (left and right) absolutely pure and (left and right) coherent;

(3) the classes of flat left R-modules and of absolutely pure left R-modules coincide;

(4) R is right absolutely pure and right coherent, and any flat left R-module is absolutely pure;

(5) R is left absolutely pure and left coherent, and any absolutely pure left R-module is flat.

The statements (3′)–(5′) obtained from (3)–(5), by interchanging the words “left” and “right” are also equivalent.

Proof. (1) ⟹ (2). Since the functor Hom_{R}(−, R) is exact (see, e.g., [17, p. 290]), the modules _{R}R, R_{R} are fp-injective, and, consequently, absolutely pure. The fact that R is coherent follows from Lemma 2.10.

(2) ⟹ (1). The reflexivity of every finitely presented module M, i.e., the fact that the natural homomorphism σ_{M}: M → M^{∗∗} = Hom_{R}((M, R), R) is an isomorphism, is a consequence of [3, 4.9]. Given a finitely presented module M, consider an exact sequence

P_{1} ⟶ P_{0} ⟶ M ⟶ 0,

where P_{1}, P_{0} are finitely generated free modules. By assumption, the functor (−, R) is exact (both on R-mod and on mod-R); so, the module M^{∗} = Hom_{R}(M, R), which is a kernel of the homomorphism P_{0}^{∗} → P_{1}^{∗}, is finitely presented. Finally, since f^{∗∗}σ_{M} = σ_{N}f for any f ∈ Hom_{R}(M, N) (in R-mod or in mod-R), the ring R indeed determines the duality in question.

(2) ⟹ (3), (2) ⟹ (4). This follows from Theorems 2.4 and 2.5.

(3) ⟹ (5).

It suffices to show that R is left coherent. For this, consider a direct system {M_{i}}_{i∈I} of absolutely pure left R-modules. Since each M_{i} is flat by assumption, the module \underrightarrow{lim} M_{i} is flat, and, hence, absolutely pure. Therefore, R is left coherent (see [3, 3.2]).

(4) ⟹ (3).

By Theorem 2.5(7), any absolutely pure left R-module is fp-flat; hence, it is flat by Theorem 2.4(4).

(5) ⟹ (2).

Since the module _{R}R is absolutely pure, so is ∏_{I} R; therefore, the latter module is flat. By [16, Subsection 2.1; 9, Subsection I.13.3], the ring R is right coherent. By Theorem 2.4, any fp-injective left R-module is absolutely pure, and, hence, flat. From Theorem 2.5(7) it follows that R is right absolutely pure. □

The equivalence (1) ⟺ (8) in the next result was also proved in [3, 4.2] and [4, 3.10].

Corollary 2.12. For a two-sided coherent ring R, the following statements are equivalent:

(1) R is weakly quasi-Frobenius;

(2) S^{R} = S_{R};

(3) _{R}S = ^{R}S;

(4) R is left absolutely pure, and for every nonzero M ∈ R-mod we have M^{∗} = Hom_{R}(M, R) ≠ 0;

(5) R is right absolutely pure, and every indecomposable pure-injective flat left Rmodule is injective;

(6) R is right absolutely pure, and every pure-injective flat left R-module is injective;

(7) R is left absolutely pure, and every indecomposable injective right R-module is flat; (8) R is left absolutely pure, and every injective right R-module is flat. The statements (4′)–(8′) obtained from (4)–(8) by interchanging the words “left” and

“right” are also equivalent.

Proof. The implications (1) ⟹ (2) ⟹ (3) ⟹ (1) follow from Theorem 2.5, the implication (1) ⟹ (4) follows from Lemma 2.8, and the implication (4) ⟹ (1) follows from Theorem 2.9.

Next, using Theorem 2.4(3), we see that every pure-injective fp-injective left R-module is injective. Now the implications (1) ⟹ (6) ⟹ (5) ⟹ (1), (1) ⟹ (8) ⟹ (7) ⟹ (1) can easily be deduced from Theorems 2.4(4), 2.5, and 2.11. □

Suppose that an (R, R)-bimodule U determines a U-duality between categories of finitely presented R-modules. Let S^{U} denote the subcategory { C ∈ coh_{R}C | C(U_{R}) = 0 }, and S_{U} the subcategory { C ∈ coh_{R}C | _{_{R}C}(C, − ⊗_{R} U) = 0 }.

Proposition 2.13. Suppose that an (R, R)-bimodule U determines a U-duality between categories of finitely presented R-modules. Then S^{U} = S_{R} and S_{U} = S^{R}.

Proof. Given C ∈ coh_{R}C, consider an exact sequence of the form (1.2) with M, N ∈ R-mod. If C ∈ S^{R}, then μ is a monomorphism, whence (μ, U) is an epimorphism. Since the sequence

(− ⊗ M, − ⊗ U) ⟶^{(−⊗μ,−⊗U)} (− ⊗ N, − ⊗ U) ⟶ (C, − ⊗ U) ⟶ 0

is exact, we obtain C ∈ S_{U}. On the other hand, if C ∈ S_{U}, then (μ, U) is an epimorphism. Since U determines a duality, ((μ, U), U) is a monomorphism. Thus, μ is a monomorphism, i.e., C ∈ S^{R}.

The second relation is proved in a similar way, with the use of a projective presentation of the form (1.1). □

We recall that a module M ∈ R-Mod is said to be faithfully projective if it is a finitely generated projective generator.

Theorem 2.14. Suppose that an (R, R)-bimodule U determines a duality between the categories R-mod and mod-R. Then the ring R is weakly quasi-Frobenius if and only if U is left and right faithfully projective.

Proof. Suppose that R is weakly quasi-Frobenius. Since the module U determines a duality between the categories R-mod and mod-R, for any M ∈ R-mod we have an exact sequence of the form

0 ⟶ M ⟶ U^{k} ⟶ U^{l}.

On the other hand, for M we have an exact sequence of the form

(2.3)  0 ⟶ M ⟶ R^{n} ⟶ R^{m}.

As was mentioned before Theorem 2.11, the module U is absolutely pure. Therefore, taking M = U in (2.3), we see that − ⊗ U is a subobject of − ⊗ R^{n}. Since − ⊗ U is injective in coh_{R}C, U is a direct summand of R^{n}. In a similar way, R is a direct summand of U^{k}, whence it easily follows that U is left faithfully projective. By symmetry, U is right faithfully projective.

Conversely, suppose that U is faithfully projective. Then the module _{R}R (respectively, R_{R}) is absolutely pure, as a direct summand of the absolutely pure module _{R}U^{k} (respectively, U_{R}^{k}) with k ∈ ℕ. The coherence of the ring R follows from Lemma 2.10.□

§3. Almost regular rings

A ring R is said to be left almost regular if the subcategories S^{R} and S_{R} are zero (the right almost regular rings are defined in a symmetric way). Theorem 2.5 implies that the almost regular rings (left or right) are left and right absolutely pure, so that ^{R}S = _{R}S.

Theorem 3.1. For a ring R, the following conditions are equivalent:

(1) R is left almost regular;

(2) R is right almost regular;

(3) every short exact sequence of (left and right) finitely presented R-modules splits;

(4) every left R-module is fp-flat and fp-injective; (4′) every finitely presented left R-module is fp-flat and absolutely pure;

(5) every (left and right) R-module is fp-injective; (5′) every (left and right) finitely presented R-module is absolutely pure;

(6) every (left and right) R-module is fp-flat; (6′) every (left and right) finitely presented R-module is fp-flat;

(7) R is right absolutely pure, and every finitely presented left R-module is absolutely pure;

(8) R is left absolutely pure, and every finitely presented left R-module is fp-flat.

Proof. (1) ⟹ (2). It suffices to prove that ^{R}S = 0. In coh C_{R}, consider an exact sequence of the form

0 ⟶ C ⟶ M ⊗ − ⟶^{μ⊗−} N ⊗ −,

where M, N ∈ mod-R. If C ∈ ^{R}S, then μ is a monomorphism. Invoking (2.1), we see that coker(μ, −) ∈ S_{R} = 0, i.e., (μ, −) is an epimorphism. Since (M, −) is a projective object of _{R}C, it follows that (μ, −) splits, so that μ splits, and C = 0. The implication (2) ⟹ (1) is proved by symmetry.

(1), (2) ⟹ (3). If μ is a monomorphism in mod-R or in R-mod, then coker(μ, −) belongs to S_{R} or to _{R}S, respectively. As above, we see that μ splits.

(3) ⟹ (4′), (5′), (6′). These implications follows from the definitions.

(4′) ⟹ (4), (5′) ⟹ (5), (6′) ⟹ (6).

It suffices to recall that every R-module is a direct limit of finitely presented modules.

(4) ⟹ (4′), (5) ⟹ (5′), (6) ⟹ (6′). Obvious.

(4) ⟹ (7), (4) ⟹ (8). It suffices to apply Theorem 2.5(7).

(7) ⟹ (1).

By Theorem 2.5(3), we have S_{R} ⊂ S^{R}. Since every C ∈ coh_{R}C is a subobject of − ⊗_{R} M for some M ∈ R-mod, and, by Proposition 2.2(2), − ⊗_{R} M is S^{R}-torsion-free, it follows that S^{R} = 0.

(8) ⟹ (4′).

Since R is left absolutely pure, we have S^{R} ⊂ S_{R} by Theorem 2.5. Then any M ∈ R-mod is fp-injective by Proposition 2.2; hence, it is absolutely pure.

(5′) ⟹ (7). Obvious.

(6) ⟹ (8). This is a consequence of Theorem 2.5(7). □

Since the property of almost regularity has turned out to be two-sided, the rings with this property are said to be almost regular.

We say that a ring R is left indiscrete if the topology of Zg_{R}C is trivial, i.e., Zg_{R}C has only two open sets. If R is a left indiscrete ring, then (S^{R}) = (S_{R}) = ∅ in Zg_{R}C, whence S^{R} = S_{R} = 0 (see [8, Subsection 3.5]). Therefore, the indiscrete rings are almost regular. Since there is a bijection between the algebras of open sets of the left and right Ziegler spectra of R (see [8, Subsection 5.5]), the indiscreteness of a ring is a two-sided property. We preface a criterion for indiscreteness with the following statement.

Lemma 3.2. The map

Υ : I ↦ S_{I} = { C ∈ coh_{R}C | (C, − ⊗ (R/I, ℚ/ℤ)) = 0 }

of the set of two-sided ideals I of the ring R to the set of Serre subcategories of coh_{R}C is injective.

Proof. Given an R-module M, we put M̂ = (M, ℚ/ℤ). For arbitrary N ∈ R-Mod and M ∈ Mod-R we have the following natural isomorphisms:

(3.1)  (− ⊗_{R} N, − ⊗_{R} M̂) ≈ Hom_{R}(N, M̂) ≈ (M ⊗_{R} N) .

For an ideal I ⊂ R, the exact sequence 0 → I →^{i} R →^{σ} R/I → 0 induces the exact sequence

(3.2)  0 ⟶ Ĩ ⟶ − ⊗_{R} R ⟶^{−⊗σ} − ⊗_{R} R/I ⟶ 0,

where Ĩ = ker(− ⊗ σ) = im(− ⊗ i). Since im(R/I ⊗ i) ≈ R/I · I = 0, we can apply the functor (−, − ⊗_{R} (R/I) ) to (3.2) and use (3.1) to show that Ĩ ∈ S⃗_{I}.

Now, consider two ideals I_{1}, I_{2} such that S_{I_{1}} = S_{I_{2}}. Then Ĩ_{1} ∈ S⃗_{I_{2}}; consequently, (Ĩ_{1}, − ⊗_{R} (R/I_{2}) ) = 0. Taking the isomorphisms (3.1) into account and using the fact that ℚ/ℤ is a cogenerator of Ab, we see that in the exact sequence

R/I_{2} ⊗_{R} I_{1} ⟶^{ν} R/I_{2} ⊗_{R} R ⟶^{ρ} R/I_{2} ⊗_{R} R/I_{1} ⟶ 0

ρ is an isomorphism. Then im ν ≈ R/I_{2} · I_{1} = 0, i.e., I_{1} ⊂ I_{2}. By symmetry, I_{2} ⊂ I_{1}. □

Theorem 3.3. A ring R is indiscrete if and only if it is simple and almost regular.

Proof. If R is indiscrete then, by Lemma 3.2, it is simple.

Conversely, suppose that R is an almost regular ring. Let S be a Serre subcategory of coh_{R}C. Then either the ideal t_{S} (− ⊗ R)(R) coincides with R (and then it is easy to show that S = coh_{R}C), or this ideal is zero. In the latter case, any finitely generated subobject X of t_{S} (− ⊗ R) ⊂ − ⊗ R is coherent, whence X ∈ S^{R} = 0. Hence, t_{S} (− ⊗ R) = 0, which implies that S ⊂ S_{R} = 0. □

Corollary 3.4 ([5, 2.4]). A ring R is indiscrete if and only if it is simple and all its (left and right) finitely presented modules are absolutely pure.

Theorems 3.1 and 2.4 show that the almost regular left coherent rings are regular. Combining this observation with Theorem 3.3, we obtain the following statement.

Corollary 3.5. Let R be a left coherent ring. Then R is indiscrete if and only if R is regular and simple.

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St. Petersburg State University, Department of Mathematics and Mechanics, Bibliotechnaya pl. 2, 198904, St. Petersburg, Russia

E-mail address: ggarkusha@hotmail.com

Received 7/DEC/1998

Translated by THE FIRST AUTHOR