The endocategory of a module

Henning Krause

Dedicated to the memory of Maurice Auslander

Given a ring Λ we introduce the endocategory E_{M} of a Λ-module M. It is an abelian subcategory of Mod(Γ) where Γ = End_{Λ}(M)^{op}, and E_{M} is constructed in such a way that it is the smallest abelian subcategory of Mod(Γ) containing M regarded in the natural way as a Γ-module and all the endomorphisms of M induced by multiplication with an element from Λ. The first aim of this paper is to discuss some basic properties of this category. For instance, we show that E_{M} reflects various properties of the module M which are related to purity. Another aim is to give a functorial description of the Ziegler spectrum of Λ which is, by definition, a representative set of indecomposable pure-injective Λ-modules, together with a topology introduced by Ziegler [14]. We shall also discuss the relation between certain right and left Λ-modules which arises from the well-known duality between the categories of finitely presented functors from mod(Λ^{op}) and mod(Λ), respectively, into the category of abelian groups. In fact, this duality induces a bijection M ↦ DM between certain subsets of the Ziegler spectra of Λ and Λ^{op}, respectively, which has been studied by Herzog [6] using positive primitive formulas, and by Crawley-Boevey [1] using characters. The functoriality of this bijection is expressed by dualities between the endocategories E_{M} and E_{DM}. A final example illustrates these dualities as well as their limitations.

This paper is dedicated to the memory of Maurice Auslander. In fact, the material presented here depends in an essential way on homological and categorical techniques and ideas that indelibly bear his mark.

1. Preliminaries

Let C be a skeletally small pre-additive category. A (right) C-module is an additive functor from C^{op} into the category Ab of abelian groups and we denote by Mod(C) the category of all C-modules. Recall that M ∈ Mod(C) is said to be finitely presented provided that there exists an exact sequence ∐_{i=1}^{n} Hom( , X_{i}) → ∐_{j=1}^{m} Hom( , Y_{j}) → M → 0 in Mod(C), and M is finitely generated if it is a quotient of a finitely presented module. The full subcategory of all finitely presented C-modules is denoted by mod(C). It is an additive catgory with cokernels, and C is called coherent if mod(C) is abelian. Given a ring Λ we shall view Λ as a category with one object and we define C_{Λ} = mod(Λ^{op})^{op}. Note that this category is coherent. The fully faithful functor

Mod(Λ) → Mod(C_{Λ}), M ↦ M ⊗_{Λ} −

will play an important role in our considerations. An exact sequence 0 → L → M → N → 0 in Mod(Λ) is said to be pure-exact if its image

0 ⟶ L ⊗_{Λ} − ⟶ M ⊗_{Λ} − ⟶ M ⊗_{Λ} − ⟶ 0

under this functor is exact and M ∈ Mod(Λ) is pure-injective if M ⊗_{Λ} − is injective. Finally, recall that there is a well-known duality d: mod(C_{Λ}) → mod(C_{Λ^{op}}) given by

d(M)(X) = Hom(M, X ⊗_{Λ} −).

Thus mod(C_{Λ}) has sufficently many projectives which are the functors Hom_{Λ^{op}}(Y, ), Y ∈ mod(Λ^{op}), and sufficently many injectives which are the functors X ⊗_{Λ} −, X ∈ mod(Λ) [5].

We continue with some general facts. Recall that a full subcategory S of an abelian category A is a Serre subcategory provided that for any exact sequence 0 → X → Y → Z → 0 in A the object Y belongs to S iff X and Z belong to S. For any Serre subcategory S of A one can form the quotient category A/S which is abelian and admits an exact quotient functor q: A → A/S with Ker(q) = S [4]. The functor q is characterized by the property that any exact functor f: A → B between abelian categories with Ker(f) = S induces, up to isomorphism, a unique faithful and exact functor f̄: A/S → B such that f = f̄∘q. The subcategory S is called localizing if q has a right adjoint.

Suppose now that C is a skeletally small pre-additive category which is coherent, e.g. C = C_{Λ}. Let S be a Serre subcategory of mod(C) and denote by S^{→} the full subcategory of Mod(C) which consists of all direct limits ⟶ lim X_{i} with X_{i} ∈ S for all i.

Lemma 1.1 The subcategory S^{→} is localizing and the quotient functor q: Mod(C) → Mod(C)/S^{→} has the following properties.

(1) Let M ∈ Mod(C) with Hom(S, M) = 0. Then M is injective iff q(M) is injective. In that case q induces an isomorphism Hom(X, M) → Hom(q(X), q(M)) for every X ∈ Mod(C).

(2) The composition of q with the inclusion mod(C) → Mod(C) induces a fully faithful and exact functor mod(C)/S → Mod(C)/S^{→} which makes the following diagram of canonical functors commutative.

mod(C) ⟶ mod(C)/S
↓  ↓
Mod(C) ⟶^{q} Mod(C)/S^{→}

Proof : For a proof see [9].

We shall also need the well-known fact that Mod(C) has injective envelopes. Moreover, Mod(C)/S has injective envelopes provided that S is a localizing subcategory.

2. The endocategory

Let M be a Λ-module and let Γ = End_{Λ}(M)^{op}. Identifying End(M ⊗_{Λ} −)^{op} in Mod(C_{Λ}) with Γ we obtain a contravariant functor

h_{M}: mod(C_{Λ}) → Mod(Γ), X ↦ Hom(X, M ⊗_{Λ} −)

which is easily seen to be exact. We denote by S_{M} the kernel of h_{M} which is a Serre subcategory. The endocategory E_{M} of M is, by definition, the image of the induced functor mod(C_{Λ})/S_{M} → Mod(Γ) and we shall assume that this subcategory of Mod(Γ) is closed under isomorphisms. The next lemma is a reformulation of this definition.

Lemma 2.1 The funcor h_{M} induces a duality mod(C_{Λ})/S_{M} → E_{M} which makes the following diagram of functors commutative.

mod(C_{Λ}) ⟶ mod(C_{Λ})/S_{M}  ⟶  E_{M}
↓  ↓  ↓
Mod(C_{Λ}) ⟶^{q} Mod(C_{Λ})/S_{M}^{→} ⟶^{Hom( ,q(M⊗_{Λ}−))} Mod(Γ)

Proof : We use Lemma 1.1. The isomorphism End(M ⊗_{Λ} −) ≅ End(q(M ⊗_{Λ} −)) allows to identify Γ and End(q(M ⊗_{Λ} −))^{op}. The assertion then follows from this lemma and the definition of E_{M}.

Recall that an abelian subcategory of an abelian category is a subcategory which is abelian and has an exact inclusion functor.

Proposition 2.2 The endocategory E_{M} has the following properties.

(1) E_{M} is an abelian subcategory of Mod(Γ) which is closed under isomorphisms.

(2) E_{M} contains M and End_{E_{M}} (M) contains the image of the canonical morphism Λ → End_{Γ}(M). Moreover, E_{M} is a subcategory of any other category E which satisfies (1) – (2).

Proof : mod(C_{Λ}) is the free abelian category over Λ, i.e. any functor Λ → E to an abelian category extends uniquely to an exact functor mod(C_{Λ}) → E [5, Lemma 1]. Applying this fact to the canonical functor Λ → Mod(Γ), Λ ↦ M, the assertion follows.

We list some further properties which follow immediately from the definition.

Proposition 2.3 (1) E_{M} consists precisely of all cokernels of morphisms Hom_{Λ}(φ, M) with φ in mod(Λ).

(2) E_{M} consists precisely of all kernels of morphisms M ⊗_{Λ} ψ with ψ in mod(Λ^{op}).

(3) Every object in E_{M} is a subquotient in Mod(Γ) of some finite coproduct M^{n}.

Proof : Use the fact that mod(C_{Λ}) has sufficently many injectives and projectives.

Our discussion suggests the following definition. A pair M and N of Λ-modules is called purely equivalent provided that S_{M} = S_{N}.

Lemma 2.4 The following are equivalent.

(1) M and N are purely equivalent.

(2) Hom_{Λ}(φ, M) is an epi iff Hom_{Λ}(φ, N) is an epi for every φ in mod(Λ).

(3) M ⊗_{Λ} ψ is a mono iff N ⊗_{Λ} ψ is a mono for every ψ in mod(Λ^{op}).

Proof : Use the fact that mod(C_{Λ}) has sufficently many injectives and projectives.

Proposition 2.5 Let M and N be Λ-modules which are purely equivalent. Then there is an equivalence e: E_{M} → E_{N} which is, up to isomorphism, uniquely determined by functorial isomorphisms

e(Hom_{Λ}(X, M)) ≅ Hom_{Λ}(X, N)  and  e(M ⊗_{Λ} Y) ≅ N ⊗_{Λ} Y

for all X ∈ mod(Λ) and Y ∈ mod(Λ^{op}).

Proof : The assertion follows from Lemma 2.1.

The endocategory E_{M} is intimately related to the collection of subgroups of finite definition which correspond to M. Recall that for X ∈ mod(Λ) a subgroup of finite definition of Hom_{Λ}(X, M) is the image M_{φ} of a map Hom_{Λ}(φ, M) arising from a morphism φ: X → Y in mod(Λ). The subgroups of finite definition form a lattice in Hom_{Λ}(X, M) which we denote by Latt_{Λ}(X, M).

Proposition 2.6 (1) The lattice of subobjects of Hom_{Λ}(X, M) in E_{M} coincides with Latt_{Λ}(X, M).

(2) Every object in E_{M} is isomorphic to Hom_{Λ}(X, M)/M_{φ} for some X ∈ mod(Λ) and some M_{φ} ∈ Latt_{Λ}(X, M).

Proof : Use Proposition 2.3.

The next few results show that the endocategory E_{M} reflects various properties of the module M which are related to purity.

Proposition 2.7 M is pure-injective if and only if for every X ∈ E_{M} and for every codirected family (X_{i})_{i∈I} of subobjects of X in E_{M} the canonical map X → ⟵ lim X/X_{i} in Mod(Γ) is an epi.

Proof : Adapt the argument of [8, Corollary 7.4].

Recall that M is Σ-pure-injective if any coproduct ∐_{I} M is pure-injective. It is wellknown that M is Σ-pure-injective iff the descending chain condition holds in Latt_{Λ}(Λ, M). Thus we obtain the following characterization.

Corollary 2.8 M is Σ-pure-injective if and only if E_{M} is artinian, i.e. any object in E_{M} is artinian.

A module M is called endofinite if M is of finite length in Mod(Γ). It has been shown by Crawley-Boevey that M is endofinite iff ascending and descending chain condition hold in Latt_{Λ}(Λ, M) [1, Proposition 4.1]. This has the following consequence.

Corollary 2.9 M is endofinite if and only if E_{M} is a length category, i.e. any object in E_{M} is of finite length.

The next result gives a condition which guarantees that E_{M} is a full subcategory of Mod(Γ).

Theorem 2.10 If M is endofinite, then E_{M} is the full subcategory of all subquotients in Mod(Γ) of finite coproducts M^{n}.

Proof : Suppose that M is endofinite. Using the argument of [1, Proposition 4.1] it can be shown that every Γ-submodule of Hom_{Λ}(X, M) is a subgroup of finite definition. Thus E_{M} consists of all subquotients of finite coproducts M^{n}. It remains to prove that E_{M} is a full subcategory. We shall use the quotient functor q: Mod(C_{Λ}) → Mod(C_{Λ})/S_{M}^{→}. By Lemma 1.1 and Lemma 2.1 the subcategory E_{M} is full iff the functor

Hom( , q(M ⊗_{Λ} −)): Mod(C_{Λ})/S_{M}^{→} ⟶ Mod(Γ)

is full when it is restricted to the objects q(X) with X ∈ mod(C_{Λ}). Applying the argument of [12, Lemma 4.1] the assertion follows from the fact that q(M ⊗_{Λ}−) is an injective cogenerator and that Hom(q(X), q(M ⊗_{Λ} −)) ≅ Hom(X, M ⊗_{Λ} −) is finitely generated over End(q(M ⊗_{Λ} −))^{op} ≅ Γ for each X ∈ mod(C_{Λ}).

3. Ziegler’s topology

Let C be a skeletally small pre-additive category. A set of indecomposable injective C-modules which meets each isomorphism class exactly once is called the spectrum of C and is denoted by sp(C). The Ziegler spectrum Zsp(Λ) of a ring Λ is by definition a set of indecomposable pure-injective Λ-modules which meets each isomorphism class exactly once. Thus we may assume that the functor M ↦ M ⊗_{Λ} − identifies Zsp(Λ) with sp(C_{Λ}) since any injective C_{Λ}-module is isomorphic to M ⊗_{Λ} − for some M ∈ Mod(Λ). In [14] Ziegler introduces a topology on the set Zsp(Λ). We obtain this topology as follows.

Let C be a skeletally small pre-additive category which is coherent, e.g. C = C_{Λ}. For a subset U of sp(C) denote by Σ(U) the Serre subcategory of mod(C) formed by the objects X ∈ mod(C) satisfying Hom(X, U) = 0. For a subcategory S of mod(C) let Υ(S) = {M ∈ sp(C) | Hom(S, M) = 0}.

Lemma 3.1 The assignment

U ↦ \overline{U} = Υ∘Σ(U)

is a closure operator on the spectrum sp(C) of A, i.e. the subsets U ⊆ sp(C) satisfying \overline{U} = U form the closed sets of a topology on sp(C).

Proof : It is easily checked that \overline{∅} = ∅, U ⊆ \overline{U} and \overline{\overline{U}} = \overline{U}. It remains to show that \overline{U_{1} ∪ U_{2}} = \overline{U_{1}} ∪ \overline{U_{2}}. From Σ(U_{1} ∪ U_{2}) ⊆ Σ(U_{1}) ∩ Σ(U_{2}) it follows that \overline{U_{1}} ∪ \overline{U_{2}} ⊆ \overline{U_{1} ∪ U_{2}}. Now choose M ∈ sp(C) such that M ∉ \overline{U_{1}} ∪ \overline{U_{2}}. We claim that this implies M ∉ \overline{U_{1} ∪ U_{2}}. From the definitions one obtains non-zero morphisms φ_{i}: X_{i} → M such that X_{i} ∈ Σ(U_{i}). We have Im(φ_{1}) ∩ Im(φ_{2}) ≠ 0 since M is indecomposable injective. Choosing U ⊆ Im(φ_{1}) ∩ Im(φ_{2}) finitely generated one can find finitely generated submodules Y_{i} ⊆ X_{i} such that φ_{i}(Y_{i}) = U. We obtain the following exact commutative diagram where the vertical morphisms are the canonical monos.

0 → W \underrightarrow{\left[{ψ_{1} \atop ψ_{2}}\right]} Y_{1} ∐ Y_{2} → U → 0
↓  ↓  ↓
0 → V → X_{1} ∐ X_{2} →^{[φ_{1} φ_{2}]} M

The morphisms ψ_{i} being epis we find finitely generated submodules W_{i} of W such that ψ_{i}(W_{i}) = Y_{i}. Let X = Y_{1} ∐ Y_{2}/ \left[{ψ_{1} \atop ψ_{2}}\right] (W_{1} + W_{2}). We have X ∈ mod(C) since mod(C) is abelian and it is easily checked that Hom(X, M) ≠ 0. On the other hand X ∈ Σ(U_{1} ∪ U_{2}) since X is a quotient of each Y_{i}. Therefore M ∉ \overline{U_{1} ∪ U_{2}} and the proof is complete.

It has been observed by I. Herzog [7] and M. Prest that Ziegler’s closed sets in Zsp(Λ) are in bijective correspondence to Serre subcategories of mod(C_{Λ}). In our context this observation takes the following form.

Theorem 3.2 There is a bijective inclusion reversing correspondence between Serre subcategories of mod(C) and closed subsets of sp(C). The correspondence is given by

S ↦ Υ(S)  and  U ↦ Σ(U).

Proof : We check that the assignments are inverse to each other. Given a Serre subcategory S of mod(C) the subcategory T = S^{→} of Mod(C) is localizing by Lemma 1.1 and therefore the pair (T , F) with F = {X | Hom(S, X) = 0} forms a hereditary torsion theory for Mod(C).

Moreover, the set Υ(S) = F ∩ sp(C) cogenerates F since the quotient Mod(C)/S^{→} is locally finitely presented [9]. It follows that Σ∘Υ(S) = S. Conversely, Υ∘Σ(U) = U is clear since U is closed.

4. Duality

We combine the correspondence U ↦ Σ(U) between closed subsets of Zsp(Λ) and Serre subcategories of mod(C_{Λ}) with the duality d: mod(C_{Λ}) → mod(C_{Λ^{op}}).

Proposition 4.1 There is a unique inclusion preserving bijection U ↦ D(U) between the closed subsets of Zsp(Λ) and Zsp(Λ^{op}) such that Σ(D(U)) = d(Σ(U)).

We shall use this fact to construct a homeomorphism between certain subsets of the Ziegler spectra of Λ and Λ^{op}, respectively.

Suppose there is given a pair of topological spaces X and Y and an inclusion preserving bijection U ↦ D(U) between the closed subsets of X and Y, respectively. Recall that a point M is isolated provided that {M} is open. We call a point M in X or Y reflexive provided that M is isolated in its closure \overline{\{M\}} and D(\overline{\{M\}}) contains also an isolated point, say N. The point N is uniquely determined by M and we call DM = N the reflection of M. Denote by R(X) and R(Y) the set of reflexive points in X and Y, respectively, equipped with their induced topology.

Lemma 4.2 Reflection, the map which sends M to DM, is a homeomorphism between R(X) and R(Y) satisfying D^{2} = id.

Proof : The reflection of a reflexive point M is again reflexive and satisfies DDM = M. Thus D is a bijection between R(X) and R(Y) satisfying D^{2} = id. Let U be a closed subset of R(X), say U = V ∩ R(X) for some closed subset V of X. It is easily checked that {DM | M ∈ U} = D(V) ∩ R(Y) since M ∈ U iff the closure of M in X is contained in V. This shows that D is continuous and the proof is complete.

We call a pair of modules M ∈ Mod(Λ) and N ∈ Mod(Λ^{op}) purely opposed provided that S_{N} = d(S_{M}).

Lemma 4.3 The following are equivalent.

(1) M and N are purely opposed.

(2) Hom_{Λ}(φ, M) is an epi iff φ ⊗_{Λ} N is a mono for every φ in mod(Λ).

(3) Hom_{Λ^{op}}(ψ, N) is an epi iff M ⊗_{Λ} ψ is a mono for every ψ in mod(Λ^{op}).

Proof : The proof is analogous to that of Lemma 2.4.

Proposition 4.4 Let M ∈ Mod(Λ) and N ∈ Mod(Λ^{op}) be modules which are purely opposed. Then there is a duality e: E_{M} → E_{N} which is, up to isomorphism, uniquely determined by functorial isomorphisms

e(Hom_{Λ}(X, M)) ≅ X ⊗_{Λ} N  and  e(M ⊗_{Λ} Y) ≅ Hom_{Λ^{op}}(Y, N)

for all X ∈ mod(Λ) and Y ∈ mod(Λ^{op}).

Proof : The duality d: mod(C_{Λ}) → mod(C_{Λ^{op}}) induces a duality mod(C_{Λ})/S_{M} → mod(C_{Λ^{op}})/S_{N}. The assertion then follows with Lemma 2.1.

We mention two properties which are preserved by the above duality. Here ann_{Λ}(M) denotes the annihilator of a Λ-module M.

Proposition 4.5 Let M ∈ Mod(Λ) and N ∈ Mod(Λ^{op}) be modules which are purely opposed.

(1) Latt_{Λ^{op}}(Λ, N) ≅ Latt_{Λ}(Λ, M)^{op}.

(2) ann_{Λ^{op}}(N) = ann_{Λ}(M).

Proof : Use Proposition 4.4 and Proposition 2.6.

Our discussion suggests the following definition. An indecomposable pure-injective Λmodule is called finitely reflexive provided that M is isomorphic to a point in Zsp(Λ) which is reflexive with respect to the topologies of Zsp(Λ) and Zsp(Λ^{op}). We include the following property.

Lemma 4.6 Let M ∈ Zsp(Λ) and suppose that M is isolated in its closure \overline{\{M\}}. Then any point N ∈ Zsp(Λ) is purely equivalent to M if and only if N = M.

Proof : If N and M are purely equivalent, then \overline{\{N\}} = \overline{\{M\}}. In particular, N ∈ \overline{\{M\}}. On the other hand, M ∉ \overline{\{N\}} for N ≠ M since M is isolated in \overline{\{M\}}. Thus N = M.

The next result summarizes our discussion. It extends Herzog’s elementary duality [6] (see also [11]).

Corollary 4.7 The map which sends M to DM is a homeomorphism between the sets of reflexive points in Zsp(Λ) and Zsp(Λ^{op}), respectively, satisfying D^{2} = id. A reflexive point M ∈ Zsp(Λ) has the following properties.

(1) Any point N ∈ Zsp(Λ) is purely equivalent to M if and only if N = M.

(2) Any point N ∈ Zsp(Λ^{op}) is purely opposed to M if and only if N = DM.

(3) There is a duality e: E_{M} → E_{DM} which is, up to isomorphism, uniquely determined by functorial isomorphisms

e(Hom_{Λ}(X, M)) ≅ X ⊗_{Λ} DM  and  e(M ⊗_{Λ} Y) ≅ Hom_{Λ^{op}}(Y, DM)

for all X ∈ mod(Λ) and Y ∈ mod(Λ^{op}).

In view of the preceding result there are two natural questions.

– Which points in Zsp(Λ) are reflexive?

– Which properties of M and DM, respectively, are preserved by the duality E_{M} → E_{DM}? We devote the rest of this paper to a discussion of these problems. Note that a first answer is given in Proposition 4.5.

5. Simply reflexive modules

We use the endocategory to introduce a class of indecomposable pure-injective modules. We start with some technical lemmas. To this end fix an indecomposable pure-injective Λ-module M with Γ = End_{Λ}(M)^{op}.

Lemma 5.1 Let X ∈ mod(C_{Λ}) and suppose that h_{M}(X) is simple in E_{M}. Then the quotient functor q: Mod(C_{Λ}) → Mod(C_{Λ})/S_{M}^{→} sends X to a simple object and M ⊗_{Λ} − to an injective envelope of q(X).

Proof : The object q(X) is simple precisely if for any exact sequence 0 → X′ → X → X′′ → 0 either X′ ∈ S_{M}^{→} or X′′ ∈ S_{M}^{→}. Writing X′ = ⟶ lim X_{i} as a direct limit of all its finitely generated submodules we obtain induced sequences 0 → X_{i} → X → X/X_{i} → 0 in mod(C_{Λ}) since C_{Λ} is coherent. Now, for any i either X_{i} ∈ S_{M} or X/X_{i} ∈ S_{M} since q(X) is simple in mod(C_{Λ})/S_{M} by assumption and Lemma 2.1. If X/X_{i} ∈ S_{M} for some i, then X′′ ∈ S_{M}^{→} since X′′ is a quotient of X/X_{i}. Otherwise all X_{i} ∈ S_{M} and therefore X′ ∈ S_{M}^{→}. Thus q(X) is simple. If φ ∈ Hom(X, M ⊗_{Λ} −) is a non-zero morphism, then q(φ) ≠ 0 by Lemma 1.1. Thus q(M ⊗_{Λ} −) is an injective envelope of q(X) since q(M ⊗_{Λ} −) is indecomposable injective.

Lemma 5.2 Let N be an injective envelope of a simple object X in any abelian category and let Σ = End(N)^{op}.

(1) Y = Hom(X, N) is a simple Σ-module.

(2) The functor Hom( , N) induces an isomorphism

End(N)/ rad(End(N)) ≅ End(X) ≅ End_{Σ}(Y)^{op}.

Proof : Straightforward.

Lemma 5.3 The following are equivalent for an object S in E_{M}.

(1) S is simple in E_{M}.

(2) S is simple in Mod(Γ).

Proof : Clearly, S = h_{M}(X) is simple in E_{M} if it is simple in Mod(Γ). The other direction follows from Lemma 5.1 and Lemma 5.2.

We call an indecomposable pure-injective module M simply reflexive provided that E_{M} contains a simple object.

Proposition 5.4 The following are equivalent for an indecomposable pure-injective Λ-module M.

(1) M is simply reflexive.

(2) There exists a map X → Y in mod(Λ) such that the cokernel of the induced map Hom(Y, M) → Hom(X, M) is a simple End_{Λ}(M)^{op}-module.

Proof : The assertion follows from Lemma 5.3 since the objects in E_{M} are precisely the cokernels of maps Hom(φ, M) arising from a map φ in mod(Λ) by Proposition 2.3.

Proposition 5.5 Any simply reflexive module is finitely reflexive.

Proof : We shall identify Zsp(Λ) = sp(C_{Λ}). Also, we keep the notation of Lemma 5.1. Now suppose that M ∈ Zsp(Λ) is simply reflexive. Consider the open set U = {N ∈ Zsp(Λ) | Hom(X, N) ≠ 0} and note that Υ(S_{M}) is the closure of {M}. Using the fact that an injective envelope of q(X) is unique up to isomorphism it follows from Lemma 5.1 that U ∩ Υ(S_{M}) = {M}. Thus M is isolated in its closure. Applying the same argument one shows that the intersection of {N ∈ Zsp(Λ^{op}) | Hom(d(X), N) ≠ 0} with D(Υ(S_{M})) contains precisely one point. Thus M is reflexive.

Proposition 5.6 Any indecomposable Σ-pure-injective module is simply reflexive.

Proof : Let M be Σ-pure-injective. The endocategory E_{M} is artinian by Corollary 2.8 and therefore E_{M} contains a simple object.

Having shown in Proposition 2.2 that the endocategory of a module M is determined by End_{Λ}(M) and its action on M, we obtain now some converse.

Proposition 5.7 Let M be an indecomposable pure-injective Λ-module and suppose that M is simply reflexive. Then the endomorphism ring End_{Λ}(M) is, up to isomorphism, uniquely determined by the endocategory E_{M}.

Proof : We use the fact that Mod(C_{Λ})/S_{M}^{→} is equivalent to the category Lex((mod(C_{Λ})/S_{M})^{op}, Ab) of left exact functors (mod(C_{Λ})/S_{M})^{op} → Ab [9]. Thus Mod(C_{Λ})/S_{M}^{→} is equivalent to Lex(E_{M}, Ab) by Lemma 2.1. Taking a simple object S ∈ E_{M} the functor Hom(S, ) is simple in Lex(E_{M}, Ab) and therefore an injective envelope N of Hom(S, ) corresponds to q(M ⊗_{Λ}−) under an equivalence Lex(E_{M}, Ab) → Mod(C_{Λ})/S_{M}^{→} by Lemma 5.1. Using Lemma 1.1 we deduce that End(N) and End_{Λ}(M) are isomorphic and the assertion follows.

Example 5.8 (1) A point M ∈ Zsp(Λ) is simply reflexive iff it is reflexive in the sense of Herzog [6]. It follows from Corollary 4.7 that M ↦ DM induces a bijection between the isomorphism classes of simply reflexive right and left Λ-modules which coincides with the duality studied by Herzog [6].

(2) The preceding result shows that any endofinite point M ∈ Zsp(Λ) is reflexive. It follows from Corollary 4.7 that M ↦ DM induces a bijection between the isomorphism classes of indecomposable endofinite right and left Λ-modules which coincides with the duality studied by Crawley-Boevey [1].

(3) Let Λ be an artin algebra which has Krull-Gabriel dimension in the sense of Geigle, e.g. a tame hereditary algebra [3]. Then it can be shown that any point in Zsp(Λ) is simply reflexive. Thus Zsp(Λ) and Zsp(Λ^{op}) are homeomorphic.

(4) Let M ∈ Zsp(Λ) be simply reflexive. Suppose also that M is a Λ-Γ-bimodule and that I ∈ Mod(Γ) is an injective cogenerator. Then it is shown in [10] that DM is isomorphic to a direct summand of the Λ^{op}-module Hom_{Γ}(M, I).

To formulate the next result we denote for an indecomposable pure-injective Λ-module M by Δ(M) the factor ring End_{Λ}(M)/ rad(End_{Λ}(M)). Note that Δ(M) is a division ring since End_{Λ}(M) is local.

Theorem 5.9 If M is simply reflexive and S ∈ E_{M} is any simple object, then

Δ(M)^{op} ≅ End_{E_{M}} (S) = End_{Γ}(S).

Proof : We use Lemma 5.1 and its notation. In particular we suppose that S = h_{M}(X) is simple. Thus the radical factor of End(q(M ⊗_{Λ} −)) is isomorphic to End(q(X)) by Lemma 5.2 since q(M ⊗_{Λ} −) is an injective envelope of the simple object q(X). The following isomorphisms End_{Λ}(M) ≅ End(M ⊗_{Λ} −) ≅ End(q(M ⊗_{Λ} −)) then show that Δ(M)^{op} ≅ End_{E_{M}} (S) since End(q(X))^{op} ≅ End_{E_{M}} (S) by Lemma 1.1 and Lemma 2.1. Inparticular End_{E_{M}} (S) = End_{Γ}(S) follows from Lemma 5.2.

Corollary 5.10 If M ∈ Zsp(Λ) is simply reflexive, then Δ(DM) ≅ Δ(M)^{op}.

Remark 5.11 The preceding result can be derived from a result of Herzog [6]. It shows that Crawley-Boevey’s notion of generical wildness is right-left symmetric [1].

Theorem 5.12 If M ∈ Zsp(Λ) is endofinite, then

End_{End_{Λ^{op}}(DM)^{op}}(DM) ≅ End_{End_{Λ}(M)^{op}}(M)^{op}.

In particular, End_{Λ^{op}}(DM) and End_{Λ}(M) have isomorphic centers.

Proof : The endocategories E_{M} and E_{DM} are full subcategories of Mod(End_{Λ}(M)^{op}) and Mod(End_{Λ^{op}}(DM)^{op}), respectively, by Theorem 2.10. Using the duality E_{M} → E_{DM} the assertion follows. The statement about the centers is a direct consequence since for any Λ-module M the center of End_{Λ}(M) and End_{End_{Λ}(M)^{op}}(M) coincide.

6. An example

We present a local two-sided artinian ring and describe explicitly the endocategory of the indecomposable endofinite module M = _{Λ}Λ and its dual DM. The author is indebted to M. Schmidmeier for suggesting this example. It shows that even for an indecomposable endofinite Λ-module M the endomorphism rings End_{Λ^{op}}(DM) and End_{Λ}(M)^{op} are not necessarily isomorphic.

Let T be a skew field and B a T -T^{op}-bimodule. Denote by X^{∗} = Hom_{T} (X, T) the dual of any X ∈ mod(T). Let Λ = T ^{|}× B be the trivial extension which is a local ring, and let Γ = T ^{|}× B^{∗∗}. Then I = (_{Λ}Λ)^{∗} is a minimal injective cogenerator for Mod(Λ) and End_{Λ}(I) ≅ Γ [2]. Now suppose that Λ is two-sided artinian. It follows that M = ΛΛ is endofinite and DM = Hom_{Λ}(Λ, I) = I since End_{Λ^{op}}(M) ≅ Λ^{op} [1]. It is easily checked that the endocategory of M is mod(Λ) and that the duality E_{M} → E_{DM} is given by X ↦ Hom_{Λ}(X, I). Returning to the bimodule B let us assume that ℓ(B_{T}) = 1 and ℓ((B^{∗∗})_{T}) = 2. Such a bimodule exists according to a result of Schofield [13]. We obtain ℓ(Λ_{Λ}) = ℓ(T_{T}) + ℓ(B_{T}) = 2 and ℓ(Γ_{Γ}) = ℓ(T_{T}) + ℓ((B^{∗∗})_{T}) = 3. Therefore Γ ≅ End_{Λ}(DM) and Λ ≅ End_{Λ^{op}}(M)^{op} are non-isomorphic rings and we have shown that, in general, the duality M ↦ DM does not share the usual properties of a duality between right and left modules.

References

[1] W. Crawley-Boevey, Modules of finite endolength over their endomorphism ring, in: Representations of algebras and related topics, eds. S. Brenner and H. Tachikawa, London Math. Soc. Lec. Note Series 168 (1992) 127–184.

[2] R. Fossum, P. Griffith and I. Reiten, Trivial extensions of abelian categories, Springer Lec. Notes 456 (1975).

[3] W. Geigle, The Krull-Gabriel dimension of the representation theory of a tame hereditary artin algebra and applications to the structure of exact sequences, manuscripta math., 54 (1985), 83–106.

[4] P. Gabriel, Des catégories abéliennes, Bull. Soc. math. France, 90 (1962), 323–448.

[5] L. Gruson, Simple coherent functors, in: Representations of algebras, Springer Lec. Notes 488 (1975) 156–159.

[6] I. Herzog, Elementary duality for modules, Trans. Am. Math. Soc. 340 (1993) 37–69.

[7] I. Herzog, The Ziegler spectrum of a locally coherent Grothendieck category, preprint.

[8] C.U. Jensen and H. Lenzing, Model theoretic algebra, Gordon and Breach, New York (1989).

[9] H. Krause, The spectrum of a locally coherent category, preprint.

[10] H. Krause, Elementary duality and local duality for modules, preprint.

[11] M. Prest, Remarks on elementary duality, Ann. of Pure and Appl. Logic. 62 (1993) 183–205.

[12] J.-E. Roos, Locally noetherian categories, in: Category Theory, Homology Theory and their Applications II, Springer Lec. Notes 92 (1969) 197–277.

[13] A. H. Schofield, Representations of rings over skew fields, London Math. Soc. Lec. Note Series 92 (1985).

[14] M. Ziegler, Model theory of modules, Ann. of Pure and Appl. Logic 26 (1984) 149–213.

Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

E-mail address: henning@mathematik.uni-bielefeld.de