Recognizing modules over a Dedekind domain

S. Paul Smith and James J. Zhang

March 2, 2000

Abstract Let A be a locally noetherian category and M a 1-critical noetherian weak generator of A. Suppose that D = End_{A} M is a Dedekind domain, and that Hom_{A}(M, −) and −⊗_{D} M set up a bijection between the isomorphism classes of simple objects in A and simple D-modules. Then M is a progenerator of A and Hom_{A}(M, −) implements an equivalence between A and the category of D-modules.

1 Introduction

It is always useful to recognize when a given abelian category is equivalent to the category of modules over a ring. Mitchell’s theorem says that a Grothendieck category [6, p.63] is equivalent to the category of modules over a ring if and only if it contains a progenerator. However, recognizing whether one has a progenerator is not always easy. There is a need for results giving criteria which imply the existence of a progenerator. Proposition 1.1, due to O. Gabber, is an example of such a result.

First we need a definition. Let A be an abelian category. Objects in A will be called A-modules. An A-module M is a weak generator of A if every A-module is a subquotient of a (possibly infinite) direct sum of copies of M.

Proposition 1.1 [3, 2.14] Let k be a field, and let A be a k-linear abelian category such that every A-module has finite length, and dim_{k} Hom_{k}(M, N) < ∞ for all A-modules M and N. If A has a weak generator, then it has a progenerator, and A is equivalent to the category of finite dimensional right modules over a finite dimensional k-algebra.

In the same spirit, a result of Nastasescu [5, 3.3] says that a Grothendieck category with an artinian generator is equivalent to a category of modules over a right artinian ring. This was extended by Albu and Wisbauer [1] to the case of a Grothendieck category having a generator whose endomorphism ring is right perfect. Artinian rings have Krull dimension zero and perfect rings are close to artinian rings in some sense because a noetherian perfect ring is artinian.

In this paper we prove the following result which gives a criterion for a Grothendieck category to be equivalent to the category of modules over a commutative Dedekind domain.

Theorem 1.2 Let A be a locally noetherian category. Suppose that

1. M is a 1-critical, noetherian, weak generator of A;

2. End_{A} M is a Dedekind domain D;

3. Hom_{A}(M, −) and − ⊗_{D} M set up a bijection between the isomorphism classes of simple A-modules and simple D-modules.

Then M is a progenerator for A, and Hom_{A}(M, −) : A → ModD is an equivalence of categories.

A locally noetherian category A is a Grothendieck category having a set of noetherian generators. Krull dimension, as defined by Gabriel, makes sense in any locally noetherian category. It is denoted by Kdim M. We say that M is dcritical if Kdim M = d and every proper quotient of M has strictly smaller Krull dimension. A d-critical module has no nonzero submodule of Krull dimension less than d. A noetherian module is 1-critical if it is not of finite length, but every proper quotient of it has finite length.

If M is a noetherian A-module, then M is a weak generator if the set of submodules of all finite direct sums of M generates A. If M is an A-module, then M is a weak generator of the smallest full subcategory of A containing M that is closed under direct sums and subquotients.

In [8] we use 1.3 to show that certain subspaces of some non-commutative spaces are affine lines. In that paper we need to know that the full subcategory weakly generated by a module M over a ring is equivalent to Modk[X]. The following theorem, which is a consequence of Theorem 1.2, gives us the result we need in [8].

Theorem 1.3 Let A be a locally noetherian category. Suppose that

1. M is a 1-critical, noetherian, weak generator of A and every nonzero submodule of M is isomorphic to M;

2. D := End_{A} M is a commutative PID (principal ideal domain);

3. if S is a simple A-module, then Hom_{A}(M, S) is a simple D-module.

Then M is a progenerator, whence Hom_{A}(M, −) : A → ModD is an equivalence.

The proof of 1.2 (given in Section 3) depends crucially on the structure of noetherian modules over Dedekind domains. We don’t know how to extend it to other commutative rings. However, Section 4 gives a partial extension of 1.2 and 1.3 when End_{A} M is a commutative ring of higher Krull dimension.

2 Preliminaries

Let R be a ring, and write ModR for the category of right R-modules. Let A be a Grothendieck category.

Definition 2.1 Let M an A-module. An R-module in A is a pair (M, φ) consisting of an R-module M and a ring homomorphism φ : R → Hom_{A}(M, M). We will usually just write M for the module.

If R is commutative and A is R-linear, then every A-module has an R-module structure in a canonical way.

Let M be an R-module in A. For each A-module N, composition of morphisms Hom_{A}(M, N) × Hom_{A}(M, M) → Hom_{A}(M, N) gives Hom_{A}(M, N) the structure of a right R-module. The functor

g_{∗} = Hom_{A}(M, −) : A → ModR

is left exact and commutes with products, so has a left adjoint

g^{∗} = − ⊗_{R} M : ModR → A

(cf. [6, p.108]). In particular, − ⊗_{R} M commutes with colimits. If M is finitely presented, then Hom_{A}(M, −) also commutes with direct limits (this can be taken as either a definition or result depending on one’s point of view). The basic properties of this tensor product are given in [2], although in [2] morphisms in A are viewed as acting on the right of an object, so there one works with the category of left R-modules.

We denote the natural transformations associated to the adjoint pair (g^{∗}, g_{∗}) by

τ : id_{ModR} → g_{∗}g^{∗}

and

η : g^{∗}g_{∗} → id_{A} .

The adjunction isomorphism

Hom_{R}(R, Hom_{A}(M, M)) ⟶^{∼} Hom_{A}(R ⊗_{R} M, M)

sends φ : R → Hom_{A}(M, M) to a distinguished map ν : R ⊗_{R} M → M. Using the Yoneda lemma, it is not difficult to show that this map is an isomorphism. If δ : R → R is left multiplication by x, then the induced map νδν^{−1} : M → M coincides with the action of φ(x) as an endomorphism of M. Hence if one uses ν to identify R ⊗_{R} M with M, then the action of x on R ⊗_{R} M agrees with the action of φ(x) on M. We write xM for the image of M under φ(x). If I is a right ideal of R, then there is an exact sequence

I ⊗_{R} M → R ⊗_{R} M → R/I ⊗_{R} M → 0.

By the preceding remarks, the image of the composition

I ⊗_{R} M ⟶ R ⊗_{R} M ⟶^{ν} M.

is ∑_{x∈I} xM. We denote this by IM. It follows that R/I ⊗_{R} M ≅ M/IM.

In the rest of this section we prove some lemmas which will be used in the next sections.

Let D be a Dedekind domain. Every noetherian D-module is isomorphic to a direct sum of various ideals of D and quotients of the form D/𝔪^{n} for various maximal ideals 𝔪 and positive integers n. We call this the structure theorem for D-modules.

Lemma 2.2 Let D be a Dedekind domain. Suppose that M is a 1-critical Amodule, and that End_{A} M = D. Then

1. g^{∗} = − ⊗_{D} M is exact;

2. g^{∗} is faithful;

3. if I and J are ideals of D such that I ⊂ J but I ≠ J, then IM is properly contained in JM. In particular, if d ∈ D\I, then dM ⊄ IM.

Proof. (1) First, we show that if I is a non-zero ideal of D, then I ⊗_{D} M is 1-critical. There is an ideal J such that I ⊕J ≅ D ⊕D. Thus I ⊗_{D} M ⊕J ⊗_{D} M is isomorphic to M ⊕ M. Since M is 1-critical and since I ⊗_{D} M and J ⊗_{D} M are non-zero, I ⊗_{D} M is 1-critical.

The map I ⊗_{D} M → D ⊗_{D} M ≅ M is monic because it is a non-zero map between 1-critical modules. Therefore I ⊗_{D} M ≅ IM and the sequence

0 → I ⊗_{D} M → D ⊗_{D} M → D/IM → 0

is exact.

The functor Tor_{1}^{D}(−, M) commutes with direct limits by [2, C1.7], so it suffices to show this vanishes on noetherian D-modules. By the structure theorem for noetherian D-modules and the long exact sequence for Tor^{D}(−, M), it is enough to prove that Tor_{1}^{D}(−, M) vanishes on cyclic D-modules and on nonzero ideals. Since ideals of D are projective, Tor_{1}^{D}(−, M) vanishes on ideals. By the previous paragraph, and [2, C1.7], Tor_{1}^{D}(D/I, M) = 0 for all non-zero ideals I.

(2) Since g^{∗} is exact, it suffices to show that g^{∗}V ≠ 0 for any simple Dmodule V = D/I. Let α : I → D be the inclusion. By adjointness, the following diagram commutes

Hom_{A}(g^{∗}D, M) ⟶^{Hom(g^{∗}(α),M)} Hom_{A}(g^{∗}I, M)
≅↓  ↓≅
Hom_{D}(D, g_{∗}M) ⟶^{Hom(α,g_{∗}M)} Hom_{D}(I, g_{∗}M).

The bottom arrow is the natural inclusion Hom(α, D) : D → I^{−1}. This is not an epimorphism, so neither is the top arrow. Therefore Ext_{A}^{1}(g^{∗}(D/I), M) ≠ 0. It follows that g^{∗}(D/I) ≠ 0.

(3) Follows from (1) and (2). □

Lemma 2.3 Suppose that the hypotheses of 2.2 hold.

1. If g^{∗} sends simple modules to simple modules, then g^{∗} preserves the length of finite length modules.

2. If g_{∗} sends simple modules to simple modules, then g_{∗} does not increase the length of finite length modules.

Proof. Part (1) follows by induction and 2.2, and part (2) follows by induction on the length. □

Lemma 2.4 If a simple A-module S is a subquotient of a direct sum ⊕_{i=1}^{n}M_{i}, then it is a subquotient of some M_{i}.

Proof. Let S be a quotient of a submodule N ⊂ ⊕_{i=1}^{n}M_{i}. Consider the chain of submodules

0 = N_{0} ⊂ N_{1} ⊂ ⋯ ⊂ N_{n} = N

where N_{j} = N ∩ (⊕_{i=1}^{j}M_{i}). Hence N_{j}/N_{j−1} is a submodule of M_{j}. Since Hom_{A}(N, S) ≠ 0, Hom_{A}(N_{j}/N_{j−1}, S) ≠ 0 for some j and S is a subquotient of M_{j}. □

3 The proofs of 1.1 and 1.2

In this section we assume

Hypothesis 3.1

1. A is a locally noetherian category;

2. M is a 1-critical noetherian A-module, and End_{A} M is a Dedekind domain D;

3. Hom_{A}(M, −) and − ⊗_{D} M set up a bijection between the isomorphism classes of simple A-modules and simple D-modules.

We use the notation g_{∗} = Hom_{A}(M, −) and g^{∗} = − ⊗_{D} M.

Proposition 3.2 Suppose that hypothesis 3.1 holds. Then the natural transformation τ : id_{ModD} → g_{∗}g^{∗} is a natural isomorphism.

Proof. Since g^{∗} is a right adjoint it commutes with direct limits. Since M is a noetherian module in a locally noetherian category it is finitely presented, so Hom_{A}(M, −) commutes with direct limits. Therefore g_{∗}g^{∗} commutes with direct limits. Since A and ModD are locally noetherian, and since id_{ModD} and g_{∗}g^{∗} commute with direct limits, it suffices to show that τ_{V} : V → g_{∗}g^{∗}V is an isomorphism whenever V is noetherian. By the structure theorem for noetherian D-modules it suffices to prove this when V is an ideal of D and when V is a quotient of D.

Consider the case when V = I is an ideal. There is a commutative diagram

I  ⟶  D  ⟶  D/I
τ_{I} ↓  ↓^{τ_{D}}  ↓^{τ_{D/I}}
Hom_{A}(M, I ⊗ M) ⟶ Hom_{A}(M, D ⊗ M) ⟶ Hom_{A}(M, D/I ⊗ M)
≅↓  ↓≅  ↓≅
Hom_{A}(M, IM) ⟶ Hom_{A}(M, M) = D ⟶ Hom_{A}(M, M/IM)

in ModD. Hence if d ∈ I, then τ_{I}(d) : M → IM is simply the map d : M → M considered as a map to IM. In particular, if d is non-zero, so is τ_{I}(d). Thus τ_{I} is injective. To see that τ_{I} epic, let θ be an element of Hom_{A}(M, I ⊗_{D} M). Then θ may be considered as a map from M to M whose image is contained in IM. It follows from 2.2(3) that θ is given by an element of I; that is, θ = τ_{I}(d) for some d ∈ I. Hence τ_{I} is an isomorphism.

Now suppose V = D/I for some non-zero ideal I. The commutative diagram shows that τ_{D/I}([d + I]) is the composition

M ⟶^{d} M ⟶ M/IM.

Therefore, to show that τ_{D/I} is an isomorphism, we must show that the map

D = Hom_{A}(M, M) → Hom_{A}(M, M/IM)

is surjective and that its kernel is no larger than I. 2.2(3) shows that the kernel can be no larger than I. Hence τ_{D/I} provides an injective map D/I → Hom_{A}(M, M/IM) = g_{∗}g^{∗}(D/I). However, the length of g_{∗}g^{∗}(D/I) is at most the length of D/I (see 2.3), so the map D/I → g_{∗}g^{∗}(D/I) is an isomorphism. Thus τ_{D/I} is an isomorphism. □

Corollary 3.3 Suppose that hypothesis 3.1 holds. Then g^{∗} is full and faithful.

Proof. By the dual version of [4, Theorem 1, p.88], this is equivalent to the fact that g_{∗}g^{∗} ≅ id_{ModD}. □

Proposition 3.4 Suppose that hypothesis 3.1 holds. Then every submodule of a finite direct sum of copies of JM, for various ideals J ⊂ D, is isomorphic to a direct sum of copies of IM for various ideals I of D.

Proof. First we show that every simple A-module is of the form g^{∗}V for some simple D-module V. If S is a simple A-module, it follows from adjointness and 2.3 that η_{S} : g^{∗}g_{∗}S → S is a non-zero map of simple modules. Therefore S ≅ g^{∗}V where V = g_{∗}S.

Let E be a submodule of N = N_{1} ⊕ … ⊕ N_{r} where each N_{i} is isomorphic to J_{i}M for some ideal J_{i} of D. Thus N ≅ g^{∗}N′ where N′ = J_{1} ⊕ … ⊕ J_{r}. If some N_{i} ∩ E is zero, then E embeds in N/N_{i}. Hence we can assume that E ∩ N_{i} ≠ 0 for all i. Therefore N/E has finite length. Let E = E_{0} ⊂ E_{1} ⊂ … ⊂ E_{s} = N be a chain of A-submodules such that each E_{j}/E_{j−1} is simple. By a downwards induction argument it suffices to prove the result, namely that E is isomorphic to a direct sum of various IM, when N/E is simple.

Hence we assume that S = N/E is simple. Let θ : N → S be an A-module map with kernel E. By the first paragraph, S ≅ g^{∗}V for some simple D-module V. Since g^{∗} is fully faithful, there is an isomorphism

g^{∗} : Hom_{D}(N′, V) → Hom_{A}(N, S).

In particular, there is a map α : N′ → V such that g^{∗}(α) = θ. Since g^{∗} is exact, E, which is equal to ker θ, is isomorphic to g^{∗}(ker α). But ker α is a submodule of N′, so is isomorphic to a direct sum of various ideals of D (by the structure theorem of D-modules). Hence E is isomorphic to a direct sum of various IM. □

Proof of 1.2. Since g^{∗} commutes with direct limits, it suffices to show that g^{∗} implements an equivalence of categories between the full subcategories of noetherian modules. Since g^{∗} is full and faithful (see 3.3), it suffices to show that every noetherian A-module L is isomorphic to g^{∗}L′ for some D-module L′.

Since L is noetherian, it is a subquotient of a finite direct sum of copies of M. Thus, by 3.4, L is the cokernel of a map θ : P → Q, where P and Q are finite direct sums of IM for various ideals I of D. In particular, there are finitely generated D-modules, say P′ and Q′, such that P = g^{∗}P′ and Q = g^{∗}Q′. Since g^{∗} is full, θ = g^{∗}(α) for some α ∈ Hom_{D}(P′, Q′) and, since g^{∗} is exact, L = coker θ = g^{∗}(coker α). □

Theorem 1.3 is a special case of the next result.

Corollary 3.5 Let A be a locally noetherian category. Suppose that

1. M is a 1-critical, noetherian, weak generator of A.

2. D := End_{A} M is a Dedekind domain.

3. Every nonzero submodule of M is isomorphic to IM for some ideal I ⊂ D.

4. If S is a simple A-module, then Hom_{A}(M, S) is a simple D-module.

Then Hom_{A}(M, −) : A → ModD is an equivalence.

Proof. The result will follow from 1.2 once we show that g^{∗} and g_{∗} set up a bijection between isomorphism classes of simple modules.

Hypothesis (4) ensures that g_{∗} sends simple modules to simple modules. Now we show that g^{∗} sends simple modules to simple modules. Let U = D/J

be a simple D-module for some maximal ideal J ⊂ D. Then g^{∗}U ≅ M/JM.

We must show that JM is a maximal submodule of M. If not, then there is a submodule M′ strictly between JM and M. By hypothesis (3), there is an ideal I ⊂ D such that M′ ≅ IM. Hence the evaluation map

Hom_{A}(M, M′) ⊗_{D} M → M′

is epic. Since I′ := Hom_{A}(M, M′) is an ideal of D, M′ = I′M. Therefore by 2.2 I′ is not contained in J and

M = (I′ + J)M = I′M + JM = M′ + JM = M′,

which yields a contradiction. □

4 A partial generalization

In this section we prove an ungraded version of a part of the main result of [7]. This result is an analogue of 1.3 when End_{A} M is a commutative ring of Krull dimension > 1.

The basic tool we use is localization, which we review next. We refer to [2] for the basic definitions and properties of localization and base change for Grothendieck categories.

Let R be a commutative noetherian ring, and let A denote an R-linear Grothendieck category. Let T be another commutative ring and R → T a ring homomorphism. The T -modules in A form a Grothendieck category, which is denoted by A_{T}. By [2, B2.2], A_{T} is T -linear and the restriction functor A_{T} → A is an exact functor commuting with limits and colimits. The restriction functor has a left adjoint, namely the tensor functor T ⊗_{R} − : A → A_{T}.

Let S be a multiplicative set in R, and let T be the localization RS^{−1}. The tensor product T ⊗_{R} − is a direct limit with respect to a filtering system [2, B6.1]. Because T_{R} is flat, the functor T ⊗_{R} − : A → A_{T} is exact [2, B6.2]. An A-module M can be given a T -module structure if and only if x : M → M is an isomorphism for all x ∈ S, and there is at most one T -module structure on M. Note that Hom_{A_{T}} (M, N) = Hom_{A}(M, N) for all M, N ∈ A_{T}.

Let B be another R-linear category and F : A → B an R-linear functor. Then for every T -module M in A, F (M) ∈ B has a unique T -module structure because the maps F (x) : F (M) → F (M) are isomorphisms for all x ∈ S. Therefore F induces a functor A_{T} → B_{T}, which we denote by F_{T}. Note that F_{T} (M) = F (M).

If F commutes with direct limits, then F commutes with the localization T ⊗_{R} − because the localization can be interpreted as a direct limit [2, B6.1]. Therefore we have

F_{T} (T ⊗_{R} M) = F (T ⊗_{R} M) ≅ T ⊗_{R} F (M)

for all M ∈ A. Consequently, F_{T} commutes with direct limits.

Let F and F′ be R-linear functors from A to B. If τ : F → F′ is a natural transformation, then τ induces a natural transformation τ_{T} : F_{T} → F′_{T}. If F and F′ commute with direct limits, then τ_{T ⊗_{R}M} = T ⊗_{R} τ_{M}.

For any prime ideal 𝔭 ⊂ R, we write A_{𝔭}, F_{𝔭}, and τ_{𝔭}, for A_{R_{𝔭}}, F_{R_{𝔭}}, and τ_{R_{𝔭}} respectively.

Let Φ be a set of prime ideals of R. We say that A is Φ-faithful if for every non-zero A-module M, the localization M_{𝔭} := R_{𝔭} ⊗_{R} M is nonzero for some 𝔭 ∈ Φ.

Lemma 4.1 Let A and B be R-linear locally noetherian categories. Let F and F′ be R-linear functors from A to B that commute with direct limits. Let τ : F → F′ be a natural transformation. Suppose that B is Φ-faithful for some Φ ⊂ Spec R. If τ_{𝔭} : F_{𝔭} → F′_{𝔭} is an isomorphism for all 𝔭 ∈ Φ, then τ is an isomorphism.

Proof. Let M be an A-module. Consider the exact sequence

0 → K → F (M) ⟶^{τ_{M}} F′(M) → C → 0.

Tensoring with R_{𝔭} yields an exact sequence

0 → K_{𝔭} → F_{𝔭}(M_{𝔭}) → F′_{𝔭}(M_{𝔭}) → C_{𝔭} → 0.

Since τ_{𝔭} is an isomorphism for all 𝔭 ∈ Φ, K_{𝔭} = C_{𝔭} = 0. Since B is Φ-faithful, K = C = 0. □

Lemma 4.2 If (F, G) is an adjoint pair of functors between R-linear categories A and B, then (F_{𝔭}, G_{𝔭}) is an adjoint pair of functors between A_{𝔭} and B_{𝔭} for all 𝔭 ∈ Spec R.

Proof. For every M ∈ A_{𝔭} and N ∈ B_{𝔭}, we have

Hom_{B_{𝔭}}(F_{𝔭}M, N) = Hom_{B}(F M, N) ≅ Hom_{A}(M, GN) = Hom_{A_{𝔭}}(M, G_{𝔭}N).

□

Lemma 4.3 Let (F, G) be a pair of adjoint functors between two locally noetherian categories A and B. If F preserves noetherian modules, then G commutes with direct limits.

Proof. To prove G(colim N_{i}) ≅ colim G(N_{i}) for any direct limit colim N_{i} in B, it suffices to show that

Hom_{A}(M, G(colim N_{i})) ≅ Hom_{A}(M, colim G(N_{i}))

for all noetherian modules M ∈ A. But this follows from the adjointness of (F, G) and the fact that Hom_{A}(M, −) and Hom_{B}(F M, −) commute with direct limits (because M and F M are noetherian). □

Proposition 4.4 Let A and B be R-linear locally noetherian categories. Let F : A → B be an R-linear functor that commutes with direct limits and preserves noetherian modules. Let Φ ⊂ Spec R, and suppose that A and B are Φ-faithful. If the induced functor F_{𝔭} : A_{𝔭} → B_{𝔭} is an equivalence for every 𝔭 ∈ Φ, then F is an equivalence.

Proof. First we show that F is exact. Let

C : 0 → L → M → N → 0

be a short exact sequence in A. Since F commutes with direct limits, F_{𝔭}(R_{𝔭} ⊗_{R} C) ≅ R_{𝔭} ⊗_{R} F (C). Now localization R_{𝔭} ⊗_{R} − is exact [2, B6.2] and F_{𝔭} is an equivalence, so R_{𝔭} ⊗_{R} F (C) is exact for all 𝔭 ∈ Φ. Since B is Φ-faithful, F (C) is exact.

Since F commutes with direct limits it has a right adjoint, say G : B → A [9, 2.1]. Since F preserves noetherian modules, G commutes with direct limits by 4.3.

There are natural transformations τ : id_{A} → GF and η : F G → id_{B}. By 4.2, (F_{𝔭}, G_{𝔭}) is an adjoint pair of functors between A_{𝔭} and B_{𝔭}, and τ_{𝔭} and η_{𝔭} are the induced natural transformations. By the hypothesis, F_{𝔭} is an equivalence, so τ_{𝔭} and η_{𝔭} are natural isomorphisms. Since A and B are Φ-faithful, 4.1 shows that τ and η are natural isomorphisms. Therefore F is an equivalence. □

We now introduce a category V_{R} of Krull dimension 1 that is a ungraded version of the category 𝕍_{n}^{1} studied in [7].

Let R be a commutative noetherian domain of global dimension d ≥ 1. Define Tor to be the full Grothendieck subcategory of ModR generated by noetherian modules whose associated primes have height at least 2. Let

V_{R} := ModR/ Tor

denote the quotient category. If Φ is the set of all height one primes in R, then V_{R} is Φ-faithful. Using the ideas in [7, 3.4], one can prove that the Krull and global dimensions of V_{R} are one.

Theorem 4.5 Let R be a commutative noetherian domain of global dimension d ≥ 1. Let A be an R-linear locally noetherian category, and let M be an Amodule. Suppose that

1. M is a 1-critical, noetherian, weak generator of A;

2. End_{A} M = R;

3. Every nonzero submodule of M is isomorphic to IM for some ideal I ⊂ R;

4. If S is a simple module in A, then End S ≅ Q(R/𝔭), the field of fractions of R/𝔭 for some height one prime 𝔭.

Then A is equivalent to V_{R}.

We need two lemmas before proving 4.5.

Lemma 4.6 Suppose that the hypotheses of 4.5 hold. If N is a 1-critical Amodule, then N contains a submodule which is isomorphic to M.

Proof. We may assume N is noetherian. So N is a subquotient of ⊕_{i=1}^{r}M for some integer r. Let P be a submodule of ⊕_{i=1}^{r}M mapping epicly to N. We have a filtration of P,

0 = P_{0} ⊂ P_{1} ⊂ ⋯ ⊂ P_{r} = P

where P_{j} = P ∩ ⊕_{i=1}^{j}M. Let j_{0} be the minimal j such that P_{j} → N is nonzero. Since P_{j_{0}}/P_{j_{0}−1} is isomorphic to a non-zero submodule of M, it is 1-critical. Hence the map P_{j_{0}}/P_{j_{0}−1} → N is injective, or equivalently, N contains a nonzero submodule which is a submodule of M. By hypothesis (3) of 4.5, every submodule of M is isomorphic to IM for some ideal I ⊂ R. Finally, IM contains a submodule xM ≅ M for every nonzero element x ∈ I. □

Lemma 4.7 Suppose that the hypotheses of 4.5 hold. Then the hypotheses of Theorem 1.3 hold for the A_{𝔭}-module M_{𝔭} for every height one prime 𝔭 in R. Therefore A_{𝔭} is equivalent to ModR_{𝔭} via Hom_{A_{𝔭}}(M_{𝔭}, −) and − ⊗_{R_{𝔭}} M_{𝔭}.

Proof. Since R has finite global dimension, R_{𝔭} has global dimension and Krull dimension 1. Hence R_{𝔭} is a DVR (discrete valuation ring) and hence a PID. Since localization commutes with Hom [2, C3.1(iv)], we have End_{A_{𝔭}}(M_{𝔭}) = R_{𝔭}. This verifies hypothesis (2) of 1.3. This also shows that M_{𝔭} is not simple.

Since localization is exact, it follows that M_{𝔭} is a weak generator of A_{𝔭}. By [2, B6.2], M_{𝔭} is noetherian of Krull dimension ≤ 1. Since M_{𝔭} is not

simple, it has a proper submodule N_{𝔭} for some nonzero 1-critical A-submodule N ⊂ M. By 4.6, N contains a submodule that is isomorphic to M. Hence M_{𝔭} ⊂ N_{𝔭} ⊂ M_{𝔭}. Thus M is not artinian. Hence Kdim M = 1. Every proper quotient of M_{𝔭} is a localization of a proper quotient of M, so is artinian. Thus M_{𝔭} is 1-critical.

By hypothesis (3) of 4.5, every submodule N of M is isomorphic to IM for some ideal I ⊂ R. Hence N_{𝔭} ≅ (IM)_{𝔭}, which is an image of I_{𝔭} ⊗_{R_{𝔭}} M. Since R_{𝔭} is a DVR, I_{𝔭} ≅ R_{𝔭} and I_{𝔭} ⊗_{R_{𝔭}} M ≅ M. Therefore there is an epimorphism from M_{𝔭} to N_{𝔭}. This is an isomorphism because both M_{𝔭} and N_{𝔭} are 1-critical. Thus hypothesis (1) of 1.3 holds. Similarly one can show that N_{𝔭} is 1-critical for every noetherian 1-critical A-module N.

Let S be a simple A_{𝔭}-module. Then S = R_{𝔭} ⊗_{R} S′ for any noetherian critical A-submodule S′ ⊂ S. If S′ is 1-critical, then by 4.6 we may take S′ ≅ M. This contradicts the fact that M_{𝔭} is 1-critical. Hence S′ is a simple module. Then hypothesis (4) of 4.5 shows that End_{A} S = R_{𝔭}/𝔭R_{𝔭}. We must show that Hom_{A_{𝔭}}(M_{𝔭}, S) = R_{𝔭}/𝔭R_{𝔭}.

First, since M_{𝔭} is a weak generator, S is a subquotient of M_{𝔭} by 2.4. We have already shown that every submodule of M_{𝔭} is isomorphic to M_{𝔭}. This implies that S is a quotient of M_{𝔭}. If M_{𝔭} contains a unique maximal submodule, then Hom_{A_{𝔭}}(M, S) = Hom_{A_{𝔭}}(S, S) = R_{𝔭}/𝔭R_{𝔭} as required. If not, then there are at least two maximal submodules M_{1} and M_{2} such that M_{𝔭}/M_{i} ≅ S for i = 1, 2. By the last paragraph, there are isomorphisms f_{i} : M_{𝔭} → M_{i}. Each f_{i} can be viewed as an element in R_{𝔭} so we can write M_{i} = f_{i}M_{𝔭}. By 2.2, − ⊗_{R_{𝔭}} M_{𝔭} is exact and faithful, which implies that ideals (f_{i}) are maximal and distinct. But this contradicts the fact that R is a DVR so we conclude that Hom(M_{𝔭}, S) = R_{𝔭}/𝔭R_{𝔭} is a simple module in ModR_{𝔭}, thus verifying hypothesis (3) of 1.3. □

Note that if d > 1, we don’t know a priori whether M is R-flat, or equivalently whether g^{∗} = − ⊗_{R} M : ModR → A is exact. Hence even if g^{∗} maps modules in Tor to zero, we don’t know if it induces a functor from V_{R} to A (because g^{∗} is not exact). So, in the proof of 4.5, we work with the functor g_{∗} = Hom_{A}(M, −) instead .

Proof. The proof is based on 4.4. We write Φ for the set of height one primes of R.

Let F be the composition of Hom_{A}(M, −) : A → ModR followed by the quotient functor ModR → V_{R}. We will show that F is an equivalence. By 4.4, it suffices to show the following:

(i) F commutes with direct limits;

(ii) F_{𝔭} is an equivalence for every 𝔭 ∈ Φ;

(iii) A and V_{R} are Φ-faithful.

(iv) F preserves noetherian modules;

(i) Since M is noetherian, Hom_{A}(M, −) commutes with direct limits. The subcategory Tor is closed under direct sums in ModR, so the quotient functor ModR → V_{R} has a right adjoint, and hence it commutes with direct limits. Therefore F commutes with direct limits.

(ii) For every 𝔭 ∈ Φ, (Tor)_{𝔭} = 0, so the quotient functor (ModR)_{𝔭} = ModR_{𝔭} → (V_{R})_{𝔭} is an equivalence. Hence F_{𝔭} ≅ Hom_{A_{𝔭}}(M_{𝔭}, −). By 4.7, F_{𝔭} is an equivalence.

(iii) It is clear that V_{R} is Φ-faithful, and hypothesis (4) ensures that A is Φ-faithful.

(iv) By exact sequence arguments, this reduces to showing that Hom_{A}(M, N) is noetherian in V_{R} when N is a noetherian 1-critical and when N is simple. If N is noetherian 1-critical, by 4.6 we may assume N = M and the assertion follows from hypothesis (2). Suppose N = S is simple. By hypothesis (4), End_{A} S ≅ Q(R/𝔭) for some 𝔭 ∈ Φ. So Hom_{A}(M, S) ≅ Q(R/𝔭)^{r} for some r. We need to show that r is finite. Localizing at 𝔭, we have Hom_{A_{𝔭}}(M_{𝔭}, S_{𝔭}) ≅ Q(R/𝔭)^{r}. It follows from (ii) that

Hom_{A_{𝔭}}(M_{𝔭}, S_{𝔭}) ≅ Hom_{R_{𝔭}}(R_{𝔭}, R_{𝔭}/(𝔭)) ≅ Q(R/𝔭).

So r = 1. □

Acknowledgment

Both authors were supported by the National Science Foundation and J. J. Zhang was also supported by a Sloan Research Followship.

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Address: Department of Mathematics, University of Washington, Seattle, WA 98195, USA

  Emails: smith@math.washington.edu, zhang@math.washington.edu