The spectrum of a locally coherent category

Henning Krause

Abstract: A topology on the spectrum of a locally coherent Grothendieck category is introduced. The closed subsets are related to certain localizing subcategories which are characterized in terms of Serre subcategories of the full subcategory of finitely presented objects.

A Grothendieck category A is said to be locally coherent provided that A has a generating set of finitely presented objects and the full subcategory fp(A) of finitely presented objects in A is abelian. The spectrum sp(A) of A is the set of isomorphism classes of indecomposable injective objects in A. We show that this set carries a natural topology and it is the purpose of this paper to establish a natural and bijective correspondence between the following structures which arise for each locally coherent category A:

– Serre subcategories of fp(A),

– hereditary torsion theories of finite type for A,

– closed subsets of sp(A). This analysis is motivated by some construction which reduces the theory of purity for a locally finitely presented category with products to the study of injectives in a locally coherent category (e.g. [3]). Model theory of modules is a classical example of this theory. For instance, given a ring Λ the category (mod(Λ^{op}), Ab) of additive functors from the category of finitely presented Λ^{op}-modules to the category of abelian groups is locally coherent, and the functor

Mod(Λ) → (mod(Λ^{op}), Ab), M ↦ M ⊗_{Λ} −

identifies the pure-injective Λ-modules with the injective objects of (mod(Λ^{op}), Ab).

Central parts of this paper were presented at a workshop in Bielefeld in November 1993 which was devoted to connections between model theory and representation theory of finite dimensional algebras. I wish to thank all participants for exciting discussions and encouragement. In particular I would like to mention I. Herzog and M. Prest. Finally, I should refer to a recent paper of I. Herzog [5] which also discusses locally coherent categories.

1. Locally finitely presented abelian categories

We recall some terminology and some well-known facts about locally finitely presented categories and Grothendieck categories. Let A be an additive category with direct limits. An object X ∈ A is finitely presented provided that Hom(X, ) commutes with direct limits and we denote by fp(A) the full subcategory of finitely presented objects in A. The category A is said to be locally finitely presented if the isomorphism classes of fp(A) form a set and every object in A is a direct limit of objects in fp(A). An abelian category A is locally finitely presented if and only if it is a Grothendieck category with a generating set of finitely presented objects [1, Satz 1.5], [2, 2.4].

Fix now a locally finitely presented abelian category A. An object in A is finitely generated if it is a quotient of some finitely presented object and it is easily seen that any object in A is a direct limit of finitely generated subobjects. In particular, one has the following lemma.

Lemma 1.1 An object X ∈ A is finitely generated iff for any epimorphism φ: Y → X there is a finitely generated subobject U of Y such that φ(U) = X.

Proof : Straightforward.

The category A is completely determined by fp(A). In fact, fp(A) is additive, has cokernels and the functor

A → Lex(fp(A)^{op}, Ab), X ↦ Hom( , X)|_{fp(A)}

from A into the category of additive left exact functors from fp(A)^{op} to Ab is an equivalence [1, Satz 2.4]. Given a full additive subcategory C of fp(A) we denote by C^{→} the full subcategory of A which consists of direct limits ⟶ lim X_{i} with X_{i} ∈ C for all i. Note that this subcategory is closed under direct limits in A and that the finitely presented objects in C^{→} are precisely the direct summands of objects in C [2, Theorem 4.1]. Finally, we recall that A is locally coherent provided that finitely generated subobjects of finitely presented objects are finitely presented, equivalently if fp(A) is abelian [5, Proposition 2.2].

2. Torsion theories and localization

In this section we discuss certain localizing subcategories which arise naturally for any locally coherent category. We begin with some definitions. Let A be an abelian category. A pair (T , F) of full subcategories of A is said to be a torsion theory for A provided that

(1) Hom(T , F) = 0;

(2) Hom(T , X) = 0 implies X ∈ F for all X ∈ A;

(3) Hom(X, F) = 0 implies X ∈ T for all X ∈ A;

(4) for all X ∈ A there exists a subobject Y ⊆ X such that Y ∈ T and X/Y ∈ F. We denote by t: A → T the functor which assigns to X ∈ A the largest subobject t(X) of X belonging to T . A torsion theory (T , F) is called hereditary if T is closed under subobjects, equivalently if t is left exact. A full subcategory C of A is called a Serre subcategory provided that for every exact sequence 0 → X → Y → Z → 0 in A the object Y is in C iff both X and Z are in C.

We fix a Serre subcategory C of A. The quotient category A/C of A relative to C is defined as follows. The objects of A/C are those of A and Hom_{A/C}(X, Y) = ⟶ lim Hom_{A}(X′, Y/Y′) with X′ ⊆ X, Y′ ⊆ Y and X/X′, Y′ ∈ C. Again A/C is abelian and the canonical quotient functor q: A → A/C with q(X) = X is exact. A Serre subcategory C is called localizing provided that q admits a right adjoint s: A/C → A which is called section functor. The adjointness gives a natural morphism ξ_{X}: X → s∘q(X) for each X ∈ A with Ker(ξ_{X}), Coker(ξ_{X}) ∈ C, and Ker(ξ_{X}) is the maximal subobject of X which belongs to C. An object X ∈ A is said to be C-closed (C-torsionfree) provided that ξ_{X} is an isomorphism (a monomorphism). Thus the section functor induces an equivalence between A/C and the full subcategory of C-closed objects in A. The following lemma collects some well-known facts about the connection between localizing subcategories and torsion theories.

Lemma 2.1 Let A be an abelian category and C a Serre subcategory of A. Consider the following conditions.

(1) (C, {X ∈ A | Hom(C, X) = 0}) is a hereditary torsion theory.

(2) The inclusion C → A admits a right adjoint t: A → C.

(3) C is a localizing subcategory.

Then (1) and (2) are equivalent and (3) implies (2). If A has injective envelopes, then also (2) implies (3).

Proof : (1) ⇒ (2) Take for t: A → C the functor which assigns to X ∈ A the largest subobject t(X) of X belonging to C.

(2) ⇒ (1) The right adjoint t: A → C induces a morphism ξ_{X}: t(X) → X for every X ∈ A which is a monomorphism since C is closed under quotients. It is easily checked that Hom(C, Coker(ξ_{X})) = 0, and it follows that (C, {X ∈ A | Hom(C, X) = 0}) is a hereditary torsion theory.

(2) ⇒ (3) follows from [3, III.3, Corollaire 1].

(3) ⇒ (2) Let s be a right adjoint of the quotient functor q: A → A/C. Take for t: A → C the functor which assigns to X ∈ A the kernel of the canonical morphism ξ_{X}: X → s∘q(X).

Given a collection C of objects in an abelian category A, recall that the right perpendicular category C^{⊥} of C is the full subcategory of all objects M ∈ A satisfying Hom(X, M) = 0 and Ext^{1}(X, M) = 0 for all X ∈ C.

Lemma 2.2 If C is a localizing subcategory, then the section functor induces an equivalence between A/C and C^{⊥}.

Proof : The right perpendicular category C^{⊥} coincides with the full subcategory of C-closed objects since a C-torsionfree object X ∈ A is C-closed iff Ext^{1}(C, X) = 0 [3, III.2, Lemme 1].

Let A be an abelian category and (T , F) a torsion theory for A. If A has direct limits, then the isomorphism Hom(⟶ lim X_{i}, Y) ≅ ⟵ lim Hom(X_{i}, Y) shows that T is closed under direct limits in A. A torsion theory is said to be of finite type provided that the corresponding right adjoint t of the inclusion T → A commutes with direct limits. If direct limits in A are exact, then (T , F) is of finite type iff F is closed under direct limits.

Lemma 2.3 Let A be a locally coherent category. Suppose that the pair (T , F) is a hereditary torsion theory of finite type for A. Then S = T ∩ fp(A) is a Serre subcategory of fp(A) with T = S^{→} and F = {X ∈ A | Hom(S, X) = 0}.

Proof : We have S^{→} ⊆ T since T is closed under direct limits. Now let X ∈ T and write X = ⟶ lim X_{i} as direct limit of finitely presented objects. We have t(Y) ∈ S^{→} for all Y ∈ fp(A) since t(Y) can be written as direct limit of finitely generated subobjects which belong to S. Thus X = t(X) ≅ ⟶ lim t(X_{i}) ∈ S^{→} since S^{→} is closed under direct limits and the equality T = S^{→} is shown. It is clear that F ⊆ {X ∈ A | Hom(S, X) = 0}. On the other hand, Hom(S, X) = 0 implies Hom(S^{→}, X) = 0 and therefore X ∈ F since T = S^{→}. This completes the proof.

Lemma 2.4 Let A be an abelian category with exact direct limits and suppose that C is a localizing subcategory of A. Then C and A/C have direct limits. Consider the following conditions.

(1) t commutes with direct limits.

(2) s commutes with direct limits. Then (2) implies (1) and if A is locally coherent, then also (1) implies (2).

Proof : Direct limits exist in C since C is closed under direct limits taken in A. For a direct limit ⟶ lim X_{i} in A/C take q(⟶ lim_{λ∈I} s(X_{i})).

(2) ⇒ (1) We use the natural exact sequence 0 → t(X) → X →^{ξ_{X}} s∘q(X) which is given for each X ∈ A. By assumption s commutes with direct limits and therefore also s∘q since q is a left adjoint. From the exactness of direct limits in A we obtain therefore an isomorphism ⟶ lim t(X_{i}) ≅ t(⟶ lim X_{i}) for each direct limit ⟶ lim X_{i} in A.

(1) ⇒ (2) Using the adjointness we have for a direct limit ⟶ lim X_{i} in A/C the isomorphism ⟶ lim X_{i} ≅ ⟶ lim q∘s(X_{i}) ≅ q(⟶ lim s(X_{i})). Therefore it suffices to show that ⟶ lim s(X_{i}) is C-closed. We have t(⟶ lim s(X_{i})) ≅ ⟶ lim t∘s(X_{i}) = 0 since t commutes with direct limits. Thus ξ_{⟶ lim s(X_{i})} induces an exact sequence 0 → ⟶ lim s(X_{i}) → s∘q(⟶ lim s(X_{i})) → C → 0 for some C ∈ C. We need to show that C = 0. Therefore assume C ≠ 0. We can choose a non-zero morphism Z → C for some Z ∈ S = C ∩ fp(A) since C = S^{→} by Lemma 2.3. This morphism induces an exact sequence 0 → ⟶ lim s(X_{i}) → Y → Z → 0 and we obtain from Lemma 1.1 a morphism V → Y with V ∈ fp(A) which give the following commutative diagram with exact rows.

0 ⟶  U  ⟶  V  ⟶ Z ⟶ 0
↓  ↓  ∥
0 ⟶ ⟶ lim s(X_{i}) ⟶  Y  ⟶ Z ⟶ 0
∥  ↓  ↓
0 ⟶ ⟶ lim s(X_{i}) ⟶ s∘q(⟶ lim s(X_{i})) ⟶ C ⟶ 0

The object U is finitely presented since fp(A) is abelian and therefore U → ⟶ lim s(X_{i}) factorizes as U →^{φ} s(X_{j}) → ⟶ lim s(X_{i}) for some j. The morphism φ induces an element in Ext^{1}(Z, s(X_{j})) which is zero since Ext^{1}(C, X) = 0 for all C-closed objects X ∈ A [3, III.2, Lemme 1]. Thus 0 → ⟶ lim s(X_{i}) → Y → Z → 0 splits and therefore the morphism Z → C is zero since Hom(Z, s∘q(⟶ lim s(X_{i}))) = 0. This contradiction finishes the proof.

Lemma 2.5 Let f: A → B be a functor between categories with direct limits. Suppose there exists a right adjoint g which commutes with direct limits. Then f(X) is finitely presented, if X is a finitely presented object in A.

Proof : Let X ∈ A and ⟶ lim Y_{i} ∈ B. We have the following sequence of morphisms

⟶ lim Hom(f(X),Y_{i})≅⟶ lim Hom(X,g(Y_{i}))→^{φ}Hom(X,⟶ lim g(Y_{i}))≅Hom(X,g(⟶ lim Y_{i}))≅Hom(f(X),⟶ lim Y_{i})

since f and g is a pair of adjoint functors and g commutes with direct limits. Now the assertion follows since φ is an isomorphism if X is finitely presented.

We say that a localizing subcategory C of an abelian category A with direct limits is of finite type if the right adjoint t: A → C of the inclusion commutes with direct limits. The next result summarizes our discusssion and it shows that this type of localizing subcategories is most natural for locally coherent categories.

Theorem 2.6 Let A be a locally coherent category and suppose that C is a Serre subcategory of A. Then the following are equivalent.

(1) The inclusion functor C → A admits a right adjoint t: A → C.

(2) The quotient functor q: A → A/C admits a right adjoint s: A/C → A. If the above conditions (1) – (2) are satisfied, then the following are equivalent.

(3) t commutes with direct limits.

(4) s commutes with direct limits. If the above conditions (1) – (4) are satisfied, then C and A/C are locally coherent and the following diagram of functors commutes where the unlabeled functors are inclusions.

fp(C) ⟶ fp(A) ⟶^{fp(q)} fp(A/C)
↓ ↓ ↓
C ⟶ A ⟶^{q} A/C

Moreover, there is a unique functor f: fp(A)/ fp(C) → fp(A/C) such that fp(q) = f ∘p, where p: fp(A) → fp(A)/ fp(C) denotes the quotient functor. The functor f is an equivalence.

Proof : The equivalence (1) ⇔ (2) follows from Lemma 2.1 and (3) ⇔ (4) from Lemma 2.4. Now suppose (1) – (4). It follows from Lemma 2.1 and Lemma 2.3 that C is locally coherent. To show that A/C is locally coherent first observe that A/C has direct limits by Lemma 2.4. Moreover, q sends finitely presented objects to finitely presented objects by Lemma 2.5 since s is a right adjoint of q commuting with direct limits. Furthermore, any object X in A/C can be written as a direct limit X = q(X) = ⟶ lim q(X_{i}) of objects in q(fp(A)). Thus A/C is locally finitely presented and the commutativity of the diagram is also shown. It remains to check that there exists a canonical equivalence between fp(A)/ fp(C) and fp(A/C). In particular this would imply that A/C is locally coherent since fp(A)/ fp(C) is abelian. The existence of a unique functor f: fp(A)/ fp(C) → fp(A/C) such that fp(q) = f ∘p follows from the universal property of the quotient functor p: fp(A) → fp(A)/ fp(C) [3, III.1, Corollaire 2] since fp(q)(X) = 0 for all X ∈ fp(C). In fact f is induced from the canonical morphism

Hom_{fp(A)/ fp(C)}(X, Y) = ⟶ lim Hom_{fp(A)}(X′′, Y/Y′′) → ⟶ lim Hom_{A}(X′, Y/Y′) = Hom_{A/C}(X, Y)

for objects X, Y ∈ fp(A)/ fp(C) with X′, X′′ ⊆ X, Y′, Y′′ ⊆ Y, X/X′′, Y′′ ∈ fp(C) and X/X′, Y′ ∈ C. This is an isomorphism since for each pair X′ ⊆ X, Y′ ⊆ Y in A with X/X′, Y′ ∈ C there exists a subobject X′′ ⊆ X in fp(A) with X/X′′ ∈ fp(C) such that X′′ ⊆ X′. Furthermore, one uses that Hom(X′′, Y/Y′) ≅ ⟶ lim Hom(X′′, Y/Y′′) where Y′′ ⊆ Y′ with Y′′ ∈ fp(C). Thus the functor f is fully faithful. Finally, f is also dense since any object in fp(A/C) is a direct limit of objects in the image of f and therefore a direct summand of some object in the image of f. This completes the proof.

Having shown that any finite type localizing subcategory of a locally coherent category A is of the form S^{→} for some Serre subcategory of fp(A) it is natural to ask whether the converse is true. We shall need the following lemma.

Lemma 2.7 Let A and B be a pair of locally coherent categories. Any functor f: fp(A) → fp(B) extends, up to isomorphism, uniquely to a functor f^{∗}: A → B which commutes with direct limits. This functor has the following properties.

(1) f is exact iff f^{∗} is exact.

(2) f is right exact iff f^{∗} has a right adjoint f_{∗}: B → A. A right adjoint of f^{∗} commutes with direct limits.

(3) If f exact, then f is faithful iff f^{∗} is faithful.

Proof : We shall identify A = Lex(fp(A)^{op}, Ab) and B = Lex(fp(B)^{op}, Ab). The functor f induces the functor

f_{∗}: (fp(B)^{op}, Ab) → (fp(A)^{op}, Ab), X ↦ X∘f

and this has a left adjoint f^{∗}: (fp(A)^{op}, Ab) → (fp(B)^{op}, Ab) which is determined by the fact that it sends Hom( , X) to Hom( , f(X)) and preserves coproducts and cokernels. Therefore f^{∗} induces the following commutative diagram where the vertical arrows represent inclusions, since the objects in Lex(fp(A)^{op}, Ab) are precisely the direct limits ⟶ lim Hom( , X_{i}) of representable functors.

fp(A)  ⟶^{f}  fp(B)
↓  ↓
Lex(fp(A)^{op}, Ab) ⟶ Lex(fp(B)^{op}, Ab)
↓ ↓
(fp(A)^{op}, Ab) ⟶^{f^{∗}} (fp(B)^{op}, Ab)

(1) Any exact sequence 0 → X → Y → Z → 0 in A can be written as a direct limit of exact sequences 0 → X_{i} → Y_{i} → Z_{i} → 0 in fp(A). The assertion follows from this fact since f^{∗} commutes with direct limits which are exact.

(2) If f is right exact, then the functor Lex(fp(B)^{op}, Ab) → Lex(fp(A)^{op}, Ab), X ↦ X∘f is a right adjoint for f^{∗}. Conversely, the existence of a right adjoint implies that f^{∗} is right exact. Thus f is right exact.

(3) Suppose that f is faithful. To show that f^{∗} is faithful assume that f^{∗}(φ) = 0 for some morphism φ ≠ 0. It follows that f^{∗}(Im(φ)) = 0 and therefore f^{∗}(X/U) = 0 for some X ∈ fp(A) and some proper subobject U. Writing U = ∑ U_{i} as sum of finitely generated subobjects we obtain ∑ f(U_{i}) = f^{∗}(U) = f^{∗}(X) = f(X) and therefore f(U_{j}) = f(X) for some j since f(X) is finitely generated. Thus U_{j} = X since f is faithful. Contradiction.

Theorem 2.8 Let A be a locally coherent category and suppose that S is a Serre subcategory of fp(A). Then S^{→} is a localizing subcategory of finite type of A.

Proof : The category S^{→} is locally coherent since fp(S^{→}) = S and therefore the inclusion S^{→}→ A admits a right adjoint commuting with direct limits by Lemma 2.7. We conclude from Lemma 2.1 that S^{→} is a localizing subcategory of finite type iff S^{→} is a Serre subcategory of A. Thus it remains to show that S^{→} is a Serre subcategory. To this end consider the quotient functor q: fp(A) → fp(A)/S which induces an exact functor q^{∗}: A → B for B = Lex((fp(A)/S)^{op}, Ab) by Lemma 2.7. The kernel Ker(q^{∗}) is a Serre subcategory of A and we claim that S^{→} = Ker(q^{∗}). Clearly, S^{→} ⊆ Ker(q^{∗}) since q^{∗} commutes with direct limits. To check the converse we use the following criterion. An object X ∈ A belongs to S^{→} iff any morphism Y → X with Y ∈ fp(A) factors through some object in S [3, Lemma 4.1]. Now let φ: Y → X be such a morphism with X ∈ Ker(q^{∗}). Write X = ⟶ lim X_{i} with canonical morphisms θ_{i}: X_{i} → X and X_{i} ∈ fp(A) for all i. We obtain

0 = q^{∗}(φ) ∈ Hom(q^{∗}(Y), q^{∗}(X)) ≅ ⟶ lim Hom(q(Y), q(X_{i}))

and it follows directly that there is some j and some morphism α: Y → X_{j} such that φ = θ_{j}∘α and q(α) = 0. Thus Im(α) ∈ S and therefore φ factors through some object in S. This finishes the proof.

Some consequences are as follows.

Corollary 2.9 Let f: A → B be an exact functor between locally coherent categories and suppose that f preserves direct limits and finitely presented objects. If S = Ker(f) ∩ fp(A), then Ker(f) = S^{→} and this is a localizing subcategory of finite type.

Proof : The subcategory S^{→} is localizing of finite type by the preceding theorem and the argument used in its proof shows that Ker(f) = S^{→} since S is the kernel of the restricted functor fp(A) → fp(B).

Corollary 2.10 Let A be a locally coherent category. There is a bijective correspondence between Serre subcategories of fp(A) and hereditary torsion theories of finite type for A. The correspondence is given by

S ↦ (S^{→}, {X ∈ A | Hom(S, X) = 0})  and  (T , F) ↦ T ∩ fp(A).

Proof : Combine Lemma 2.3 and Theorem 2.8.

Corollary 2.11 Let A be a locally coherent category. If S is a Serre subcategory of fp(A), then the right perpendicular category S^{⊥} of S in A is locally coherent and coincides with (S^{→})^{⊥}. In particular the section functor A/S^{→} → A induces an equivalence between A/S^{→} and S^{⊥}.

Proof : It suffices to show that an object in A belongs to S^{⊥} iff it is S^{→}-closed, since the S^{→}- closed objects are precisely those of (S^{→})^{⊥} by Lemma 2.2. The rest of the assertion then follows from Theorem 2.6. Clearly, any S^{→}-closed object belongs to S^{⊥}. To verify the converse denote by q: A → A/S^{→} the quotient functor and let s be a right adjoint. Let X ∈ S^{⊥}. It follows immediately that Hom(S^{→}, X) = 0 and we obtain therefore the canonical exact sequence 0 → X →^{ξ_{X}} s∘q(X) → C → 0 with C ∈ S^{→}. We need to show that C = 0. Write C = ⟶ lim C_{i} with C_{i} ∈ S for all i. Each canonical morphism φ_{i}: C_{i} → C factors through s∘q(X) since Ext^{1}(S, X) = 0, and therefore φ_{i} = 0 for all i since Hom(S, s∘q(X)) = 0. Thus C = 0 and X is S^{→}-closed.

We finish this section with an example of a localizing subcategory of finite type. To this end let A be a locally finitely presented abelian category and let B = (fp(A)^{op}, Ab)

Proposition 2.12 There exists a finite type localizing subcategory C of B such that the composition of A → B, X ↦ Hom( , X)|_{fp(A)} with the quotient functor B → B/C is an equivalence. The category A is locally coherent if and only if B is locally coherent, equivalently if fp(A) has pseudo-kernels.

Proof : Recall that φ: X → Y is a pseudo-kernel for ψ: Y → Z if the induced sequence of functors Hom( , X) → Hom( , Y) → Hom( , Z) is exact. The first part of the assertion is well-known (e.g. [1, Satz 2.7]). The fact that C is of finite type follows from Lemma 2.4 since the functor A → B commutes with direct limits. If B is locally coherent, then A is locally coherent by Theorem 2.6. The converse follows from the well-known fact that for any skeletally small additive category D the category fp(D^{op}, Ab) is abelian iff D has pseudo-kernels [1, p. 315].

3. The spectrum

Let A be a Grothendieck category with a generating set G of objects. Recall that the isomorphism classes of indecomposable injective objects in A form a set since any indecomposable injective object in A occurs as the injective envelope of some quotient X/U with X ∈ G. We denote by sp(A) the set of isomorphism classes of indecomposable injective objects in A and call sp(A) the spectrum of A. It will be convenient to identify each isomorphism class in sp(A) with a representative belonging to it.

Lemma 3.1 Let A be a locally finitely presented Grothendieck category and let U = ∏_{M∈sp(A)} M. Any object X ∈ A embeds in some product of copies of U.

Proof : First observe that Hom(X, U) ≠ 0 for every finitely generated object X ≠ 0 since there exists a maximal subobject Y ⊆ X so that an injective envelope X/Y → M provides a non-zero morphism X → M with M ∈ sp(A). Given an arbitrary object X, let δ: X → ∏_{φ∈Hom(X,U)} U be the map defined by δ_{φ} = φ for all φ ∈ Hom(X, U). We claim that Ker(δ) = 0. To show this assume that Ker(δ) ≠ 0 and choose a finitely generated non-zero object X′ ⊆ Ker(δ). There is a non-zero morphism X′ → U which extends to a non-zero morphism φ: X → U. But φ factors through δ and therefore φ(X′) = 0. This contradiction finishes the proof.

Let A be a Grothendieck category and C a localizing subcategory of A. It is well-known that C and A/C are also Grothendieck categories [3, III.4, Proposition 9]. Denote by s the section functor A/C → A and for each object X ∈ A let E(X) be its injective envelope in A. The assignment X ↦ E(X) induces an injective map sp(C) → sp(A) and X ↦ s(X) induces an injective map sp(A/C) → sp(A). We consider both maps as identification. They satisfy sp(A) = sp(C) ∪ sp(A/C) and sp(C) ∩ sp(A/C) = ∅ [3, III.3, Corollaire 2].

Proposition 3.2 Let A be a Grothendieck category and (T , F) be a hereditary torsion theory for A. If A/T is locally finitely presented, then U = F ∩sp(A) = sp(A/C) cogenerates (T , F) in the following sense.

(1) X ∈ F iff X embeds in some product of copies of U = ∏_{M∈U} M.

(2) X ∈ T iff Hom(X, U) = 0.

Proof : (1) We use the fact that the left exact functor s induces an equivalence between the full subcategory of injective objects inj(A/C) and inj(A) ∩ F. Given X ∈ F let X → M be an injective envelope. It is clear that M ∈ F. Using Lemma 3.1 we find a monomorphism q(M) → ∏_{i∈I} q(U) for some set I. The composition X → M → ∏_{i∈I} U gives the desired monomorphism. For the converse use that F is closed under products and subobjects.

(2) Given X ∈ T , clearly Hom(X, U) = 0. For the converse suppose that X ∉ T . By definition there is a non-zero morphism φ: X → Y for some Y ∈ F. Using part (1) it follows that Hom(X, U) ≠ 0 and the proof is finished.

4. A topology for the spectrum

In [7] Ziegler introduces a topology on the isomorphism classes of indecomposable pure injective Λ-modules for a ring Λ. We extend this concept of a topology on sp(mod(Λ^{op}), Ab) to the spectrum of an arbitrary locally coherent category. Note that this is independent from all previous results in this paper.

Let A be a locally coherent category. For a subset U of sp(A) denote by Σ(U) the Serre subcategory of fp(A) formed by the objects X ∈ fp(A) satisfying Hom(X, U) = 0. For a subcategory S of fp(A) let Υ(S) = {M ∈ sp(A) | Hom(S, M) = 0}.

Lemma 4.1 The assignment

U ↦ \overline{U} = Υ∘Σ(U)

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

Proof : Following Kuratowski’s axiomatization of a topological space we need to verify that (i) \overline{∅} = ∅; (ii) U ⊆ \overline{U} for every subset U; (iii) \overline{\overline{U}} = \overline{U} for every subset U; (iv) \overline{U_{1} ∪ U_{2}} = \overline{U_{1}} ∪ \overline{U_{2}} for every pair of subsets U_{1} and U_{2}.

The conditions (i) – (iii) are easily checked and it remains to show (iv). 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(A) such that M ∉ \overline{U_{1}} ∪ \overline{U_{2}}. We claim that this implies M ∉ \overline{U_{1} ∪ U_{2}}. From the definitions we may choose 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 uses Lemma 1.1 to find finitely generated subobjects 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 monomorphisms.

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 epimorphisms we find finitely generated subobjects 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 ∈ fp(A) since fp(A) 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 and M. Prest that Ziegler’s closed sets in sp(mod(Λ^{op}), Ab) are in bijective correspondence to Serre subcategories of fp(mod(Λ^{op}), Ab). In our context this observation takes the following form.

Theorem 4.2 Let A be a locally coherent category. There is a bijective inclusion reversing correspondence between Serre subcategories of fp(A) and closed subsets of sp(A). 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 fp(A), the pair (T , F) with T = S^{→} and F = {X | Hom(S, X) = 0} forms a hereditary torsion theory of finite type by Corollary 2.10. Thus A/T is locally finitely presented by Theorem 2.6 and it follows from Proposition 3.2 that Σ∘Υ(S) = S since Υ(S) = F ∩ sp(A). In particular this shows that Υ(S) is closed. Conversely, Υ∘Σ(U) = U is clear since U is closed.

Corollary 4.3 Let A be a locally coherent category. There is a bijective correspondence between closed subsets of sp(A) and hereditary torsion theories of finite type for A given by

U ↦ ({X | Hom(X, U) = 0}, {X | X ⊆ ∏_{I}( ∏_{M∈U}M) for some I}) and

(T , F) ↦ sp(A) ∩ F.

Proof : Combine Corollary 2.10 and Theorem 4.2.

Let C be a finite type localizing subcategory of A. Then it is shown in Theorem 2.6 that C and A/C are locally coherent and it is therefore natural to ask how the topologies of sp(C) and sp(A/C) are related to that of sp(A).

Corollary 4.4 The topologies of sp(C) and sp(A/C) coincide with their topologies induced from sp(A).

Proof : Viewing sp(C) and sp(A/C) in the natural way as subsets of sp(A) the assertion follows from Theorem 4.2.

Recall that a topological space X is quasi-compact provided that for every every family (U_{i})_{i∈I} of open subsets X = ∪_{i∈I}U_{i} implies X = ∪_{i∈J}U_{i} for some finite subset J of I. A subset of X is quasi-compact if it is quasi-compact with respect to the induced topology.

Rephrasing Theorem 4.2 the assignment S ↦ {M ∈ sp(A) | Hom(S, M) ≠ 0} gives a bijective correspondence between Serre subcategories of fp(A) and open subsets of sp(A). Note that for every subcategory C of fp(A) one has {M ∈ sp(A) | Hom(C, M) ≠ 0} = {M ∈ sp(A) | Hom(S, M) ≠ 0} where S denotes the smallest Serre subcategory of fp(A) containing C. This has the following consequence.

Corollary 4.5 Any open subset U of sp(A) is of the form {M ∈ sp(A) | Hom(S, M) ≠ 0} for some Serre subcategory S of fp(A). For such an open set U the following are equivalent.

(1) U is quasi-compact.

(2) U = {M ∈ sp(A) | Hom(X, M) ≠ 0} for some X ∈ S.

(3) S is the smallest Serre subcategory containing X for some X ∈ S.

Writing (X) = {M ∈ sp(A) | Hom(X, M) ≠ 0} for each X ∈ fp(A) we obtain a basis of open sets as follows.

Corollary 4.6 The family of all subsets of the form (X) for some X ∈ fp(A) forms a basis of quasi-compact open sets for the topology on sp(A).

We conclude this paper with two examples. Let C be a skeletally small additive category and suppose that the category (C, Ab) of additive functors from C to Ab is locally coherent. The following proposition is an immediate consequence of the preceding result and Yoneda’s lemma.

Proposition 4.7 The spectrum of (C, Ab) is quasi-compact if and only if there exists an object X ∈ C such that M(X) ≠ 0 for all M ∈ sp(C, Ab).

Let A be a skeletally small abelian category. The Serre subcategories of A form a small partially ordered set which we denote by L(A). If (C_{i})_{i∈I} is a family in L(A), then their intersection ∩_{i∈I}C_{i} is again a Serre subcategory and one obtains ∪_{i∈I}C_{i} as the intersection of all C ∈ L(A) with C_{i} ⊆ C for all i ∈ I. Now observe that Lex(A^{op}, Ab) is a locally coherent category. It follows from Theorem 4.2 that the assignment

C ↦ {M ∈ sp(Lex(A^{op}, Ab)) | M(C) ≠ 0}

gives a lattice isomorphism between L(A) and the lattice of open subsets of sp(Lex(A^{op}, Ab)).

Proposition 4.8 L(A) is isomorphic to the lattice of open subsets of some topological space.

References

[1] M. Auslander, I. Reiten, Stable equivalence of dualizing R-varieties I, Adv. in Math. 12 (1974) 306–366.

[2] S. Breitsprecher, Lokal endlich präsentierbare Grothendieck-Kategorien, Mitt. Math. Sem. Giessen, 85 (1970) 1–25.

[3] W. Crawley-Boevey, Locally finitely presented additive categories, Comm. Algebra 22 (1994) 1644–1674.

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

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

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

[7] 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