MORPHISMS DETERMINED BY OBJECTS AND FLAT COVERS

HENNING KRAUSE

Abstract. We describe a procedure for constructing morphisms in additive categories, combining Auslander’s concept of a morphism determined by an object with the existence of flat covers. Also, we show how flat covers are turned into projective covers, and we interprete these constructions as right adjoints.

Introduction

Functors and morphisms determined by objects were introduced by Auslander in his Philadelphia notes [2]. These concepts provide a method to construct and organise morphisms in additive categories, generalising previous work of Auslander and Reiten on almost split sequences [5]. More recently, Ringel presented a survey of these results, rearranged them as lattice isomorphisms (the Auslander bijections), and added a host of interesting examples [13].

The starting point for our work is the following natural question: Is there a procedure for constructing morphisms ending at a fixed object in an additive category? More precisely, we are looking for

– invariants of morphisms ending at some fixed object, and

– constructions for universal morphisms with respect to these invariants. An answer to this question is presented in Theorem 1.1. This combines Aus-

lander’s concept of a morphism determined by an object with a deep result about functor categories, which says that every additive functor admits a flat cover [6].

The second part of this note is inspired by the first. We show in Theorem 2.2 that every flat cover is in fact a projective cover, when viewed in an appropriate abelian category. Building on another of Auslander’s paradigms [1], we use the Yoneda embedding

A ⟶ Fp(A^{op}, Ab), X ↦ Hom_{A}(−, X)

of an additive category A into the category of finitely presented functors A^{op} → Ab. Take for instance the category A = Mod Λ of modules over a ring Λ. It is somewhat surprising that any functor A^{op} → Ab preserving filtered colimits in A belongs to Fp(A^{op}, Ab) and admits a projective cover, even a minimal projective presentation. This is precisely what we exploit in Theorem 1.1 and explain in Theorem 2.2.

The final section discusses the connection with almost split sequences, and we complement Theorem 1.1 by a non-existence result for an almost split sequence ending at a module which is not finitely presented (Proposition 3.1).

1. Morphisms determined by objects

Invariants of morphisms. We fix an additive category A and let C be a set of objects. We shall view C as a full subcategory of A. A C-module is by definition an additive functor C^{op} → Ab into the category Ab of abelian groups, and a morphism between two C-modules is a natural transformation. The C-modules form an abelian category which we denote by (C^{op}, Ab). For example, if C consists of one object C, then (C^{op}, Ab) is the category of modules over the endomorphism ring of C. Note that (co)kernels and (co)products in (C^{op}, Ab) are computed pointwise: for instance, a sequence X → Y → Z of morphisms between C-modules is exact if and only if the sequence X(C) → Y (C) → Z(C) is exact in Ab for all C in C.

Version from March 23, 2014.

Every object X in A gives rise to a C-module

Hom_{A}(C, X) = Hom_{A}(−, X)|_{C} : C^{op} ⟶ Ab

and every morphism α: X → Y in A yields a morphism

Hom_{A}(C, α): Hom_{A}(C, X) ⟶ Hom_{A}(C, Y).

The image Im Hom_{A}(C, α) of Hom_{A}(C, α) is the invariant we shall use. In fact, we construct a ‘right adjoint’ which takes a submodule of Hom_{A}(C, Y) to a morphism ending at Y.

Constructing morphisms. We work in an additive category A which is locally finitely presented [7]. This means the full subcategory fp A of finitely presented objects is essentially small and each object in A is a filtered colimit of objects in fp A. Recall that an object X is finitely presented if the functor Hom_{A}(X, −): A → Ab preserves filtered colimits.

For example, the category Mod Λ of modules over a ring Λ is locally finitely presented. Then mod Λ denotes the full subcategory of finitely presented Λ-modules.

A morphism α: X → Y is called right minimal if every endomorphism ϕ: X → X with αϕ = α is invertible.

The following theorem is the main result of this work.

Theorem 1.1. Let A be a locally finitely presented additive category and C a set of finitely presented objects. For an object Y in A and a submodule H ⊆ Hom_{A}(C, Y), there exists a morphism α: X → Y in A (unique up to non-unique isomorphism) such that the following holds:

(1) Im Hom_{A}(C, α) = H and any morphism α′: X′ → Y with Im Hom_{A}(C, α′) ⊆ H factors through α.

(2) α is right minimal.

The proof will be given later in this section. A second and more elementary proof for A = Mod Λ can be found in an appendix.

Morphisms determined by objects. Following [2], a morphism α: X → Y in A is called right determined by C (or simply right C-determined) if any morphism α′: X′ → Y satisfying

Im Hom_{A}(C, α′) ⊆ Im Hom_{A}(C, α)

factors through α. In fact, Auslander established Theorem 1.1 for module categories with C consisting of a single object [2, Theorem I.3.19], generalising previous work of Auslander and Reiten on almost split sequences [5].

An obvious question to ask is when a morphism is right determined by some set of finitely presented objects. We give a general answer in Proposition 1.13. For an Artin algebra, every morphism between finitely presented modules is right determined by some finitely presented module [3, Theorem 2.6].

Functoriality. The assignment (C, H, Y) ↦ α in Theorem 1.1 is functorial. This has been pointed out by Ringel in his survey of Auslander’s results [13].

Let us formulate the functoriality. We fix an object Y ∈ A. The morphisms ending at Y are preordered. This means for morphisms α: X → Y and α′: X′ → Y that α′ ≤ α when α′ factors through α. We obtain a poset by identifying α and α′ when α′ ≤ α and α ≤ α′. Let us denote this poset by [A/Y] because it is derived from the slice category A/Y.^{1}

For a pair C ⊆ fp A and H ⊆ Hom_{A}(C, Y) let α_{C,H}: X_{C,H} → Y denote the right minimal and right C-determined morphism such that Im Hom_{A}(C, α_{C,H}) = H; it exists and is well-defined up to a non-unique isomorphism by Theorem 1.1. Note that α_{C,H} is unique when viewed as an element of [A/Y].

Lemma 1.2. Let ϕ: Y′ → Y be a morphism in A, C ⊆ C′ ⊆ fp A, H ⊆ Hom_{A}(C, Y), and H′ ⊆ Hom_{A}(C′, Y′). Then Hom_{A}(C, ϕ)(H′) ⊆ H implies ϕα_{C′,H′} ≤ α_{C,H}.

Proof. The assumptions imply

Im Hom_{A}(C, ϕα_{C′,H′}) = Hom_{A}(C, ϕ)(Im Hom_{A}(C, α_{C′,H′})) ⊆ H.

Thus ϕα_{C′,H′} factors through α_{C,H}. □

For an object X in an abelian category let sub(X) denote its lattice of subobjects.

Remark 1.3. Viewing a poset as a category, the map

sub(Hom_{A}(C, Y)) ⟶ [A/Y], H ↦ α_{C,H}

is right adjoint to

[A/Y] ⟶ sub(Hom_{A}(C, Y)), α ↦ Im Hom_{A}(C, α).

There are some natural choices of triples (C, H, Y). For example, right almost split morphisms arise from triples (Y, rad End_{A}(Y), Y) for a finitely presented object Y with local endomorphism ring [2, §II.2]. Another obvious choice is H = 0; not much seems to be known in this case.

Problem 1.4 (Auslander [4]). Describe the morphism α_{C,0} for Y ∈ A and C ⊆ fp A.

Note the extremes: α_{∅,0} = id_{Y} is the identity morphism and α_{C,0} = 0 when C contains a generator of fp A.

The assignment (C, H) ↦ α_{C,H} has been studied in some detail for modules over Artin algebras. We include Ringel’s formulation of the Auslander bijection as an example.

Example 1.5 (Ringel [13]). Let Λ be an Artin algebra. For Y ∈ mod Λ the assignment (C, H) ↦ α_{C,H} induces an isomorphism^{2}

colim_{C∈mod Λ} sub(Hom_{Λ}(C, Y)) ⟶^{∼} [mod Λ/Y].

1In [13], the poset [A/Y] is called right factorisation lattice for Y and is denoted by [→ Y ⟩.

2The colimit is taken over the collection of finite subsets C ⊆ mod Λ, identifying C = {C_{1}, …, C_{n}} with C = C_{1} ⊕ … ⊕ C_{n}.

Functors determined by objects. Let A be an additive category. We consider additive functors A^{op} → Ab into the category of abelian groups; they form an abelian category which we denote by (A^{op}, Ab).

Fix an additive functor F: A^{op} → Ab and a set C of objects in A (viewed as a full subcategory). We write F |_{C} for the restriction C^{op} → Ab. Following Auslander [2], a subfunctor F′ ⊆ F is called C-determined if for any subfunctor F′′ ⊆ F

F′′ ⊆ F′ ⟺ F′′|_{C} ⊆ F′|_{C}.

For a subfunctor H ⊆ F |_{C} define a subfunctor F_{H} ⊆ F by

(1.1)  F_{H}(X) = ⋂_{{α: C→X \atop C∈C}} F (α)^{−1}(H(C))  for X ∈ A.

Lemma 1.6 (Auslander [2, Proposition I.1.2]). The subfunctor F_{H} ⊆ F is Cdetermined and F_{H}|_{C} = H.

Proof. Let F′ ⊆ F be a subfunctor and F′|_{C} ⊆ H. This implies for each morphism α: C → X with C ∈ C that F′(X) ⊆ F (α)^{−1}(H(C)). Thus F′ ⊆ F_{H}. □

Remark 1.7. Let F ∈ (A^{op}, Ab). Viewing a poset as a category, the map

sub(F |_{C}) ⟶ sub(F), H ↦ F_{H}

is right adjoint to

sub(F) ⟶ sub(F |_{C}), G ↦ G|_{C}.

We include an explicit example.

Example 1.8. Let Λ be a ring and consider the forgetful functor F: mod Λ → Ab which takes a module to its underlying abelian group.

(1) For C ∈ mod Λ and an End_{Λ}(C)-submodule H ⊆ C, the C-determined subfunctor F_{H} is given by

F_{H}(X) = ⋂_{α: X→C}{x ∈ X | α(x) ∈ H} for X ∈ mod Λ.

(2) For an Artin algebra Λ, let sub_{fg}(F) denote the poset of finitely generated subfunctors of F. Then the assignment H ↦ F_{H} induces an isomorphism

colim_{C∈mod Λ} sub(C) ⟶^{∼} sub_{fg}(F).

(3) Consider the algebra Λ = 𝔽_{2}[ε] of dual numbers (ε^{2} = 0) and the Λ-module C = 𝔽_{2}[ε] ⊕ 𝔽_{2}. Then the poset sub(C) is isomorphic to sub(F) and its Hasse diagram is the following.

\begin{matrix}𝔽_{2}[ε] ⊕ 𝔽_{2} \\ 𝔽_{2}[ε] \\ (ε) ⊕ 𝔽_{2} \\ (ε) \\ (ε + 1) \\ 𝔽_{2} \\ (0) \end{matrix}

Flat functors and flat covers. Let C be an essentially small additive category. We consider the category (C^{op}, Ab) of additive functors F: C^{op} → Ab. Recall that F is flat if it can be written as a filtered colimit of representable functors.

The following result establishes a connection between locally finitely presented additive categories and categories of flat functors.

Theorem 1.9 (Crawley-Boevey [7, §1.4]). Let A be a locally finitely presented category. Then the Yoneda functor

h: A ⟶ ((fp A)^{op}, Ab), X ↦ Hom_{A}(−, X)|_{fp A}

identifies A with the full subcategory of flat functors (fp A)^{op} → Ab. □

A morphism π: F → G in (C^{op}, Ab) is called a flat cover of G if

(1) F is flat, every morphism F′ → G with F′ flat factors through π, and (2) π is right minimal.

A minimal flat presentation of G is an exact sequence

F_{1} ⟶ F_{0} ⟶^{π} G ⟶ 0

such that F_{0} → G and F_{1} → Ker π are flat covers. A projective cover and a minimal projective presentation are defined analogously, replacing the term flat by projective.^{3}

Theorem 1.10 (Bican–El Bashir–Enochs [6]). Every additive functor C^{op} → Ab admits a flat cover. □

Proof of the main theorem. We are ready to prove Theorem 1.1. The basic idea ist to identify A with the category of flat functors (fp A)^{op} → Ab and to employ the existence of flat covers.

Proof of Theorem 1.1. We apply Theorems 1.9 and 1.10. Consider the subfunctor H ⊆ h(Y)|_{C} and choose a flat cover π: h(X) → h(Y)_{H} in ((fp A)^{op}, Ab). The composite h(X) → h(Y)_{H} → h(Y) is of the form h(α) for some morphism α: X → Y in A. Now apply Lemma 1.6. Thus Im Hom_{A}(C, α) = h(Y)_{H}|_{C} = H. For a morphism α′: X′ → Y in A with Im Hom_{A}(C, α′) ⊆ H, we have Im h(α′)|_{C} ⊆ H and therefore Im h(α′) ⊆ h(Y)_{H} = Im h(α). Using that π is a flat cover, we obtain a morphism ϕ: X′ → X in A satisfying αϕ = α′. The fact that π is right minimal implies that α is right minimal. □

Corollary 1.11. Let α: X → Y be a morphism which is right determined by a set of finitely presented objects. Then there is an essentially unique decomposition X = X′ ⊕ X′′ such that α|_{X′} is right minimal and α|_{X′′} = 0.

Proof. Suppose that α is right C-determined and let H = Im Hom_{A}(C, α). Then there exists a right minimal and right C-determined morphism α′: X′ → Y with Im Hom_{A}(C, α′) = H by Theorem 1.1. The minimality of α′ yields a decomposition X = X′ ⊕ X′′ such that α|_{X′} = α′ and α|_{X′′} = 0. □

3This definition of a projective cover is equivalent to the usual one which requires the kernel to be superfluous.

Weak kernels. Let A be an additive category. A morphism X → Y is a weak kernel of a morphism Y → Z in A if the induced sequence

Hom_{A}(−, X) ⟶ Hom_{A}(−, Y) ⟶ Hom_{A}(−, Z)

is exact. A weak kernel is called minimal if it is a right minimal morphism. Note that a minimal weak kernel is unique up to a non-unique isomorphism; it is a kernel if a kernel exists.

Proposition 1.12. In a locally finitely presented additive category every morphism admits a minimal weak kernel.

Proof. We apply Theorems 1.9 and 1.10. Fix a morphism β: Y → Z and choose a flat cover h(X) → Ker h(β) in ((fp A)^{op}, Ab). The composite h(X) → Ker h(β) → h(Y) is of the form h(α) for some morphism α: X → Y in A, which is a minimal weak kernel for β. □

Morphisms determined by finitely presented objects. Let A be a locally finitely presented additive category. Recall that a sequence 0 → X → Y → Z → 0 of morphisms in A is pure-exact if for each C ∈ fp A the induced sequence

0 → Hom_{A}(C, X) → Hom_{A}(C, Y) → Hom_{A}(C, Z) → 0

is exact. An object X in A is pure-injective if every pure-exact sequence 0 → X → Y → Z → 0 is split exact.

The following proposition characterises the morphisms which are right determined by a set of finitely presented objects. The relevance of pure-injectives was noticed by Auslander in some special cases [2, §I.10].

Proposition 1.13. Let A be a locally finitely presented additive category.

(1) If a morphism in A is right minimal and right determined by a set of finitely presented objects in A, then its minimal weak kernel is pure-injective.

(2) If a morphism in A has a pure-injective kernel, then it is right determined by a set of finitely presented objects in A.

Proof. We set C = fp A and recall that a functor F ∈ (C^{op}, Ab) is cotorsion if Ext^{1}(E, F) = 0 for each flat E ∈ (C^{op}, Ab). Clearly, X ∈ A is pure-injective if and only if h(X) is cotorsion.

Observe that the kernel of a flat cover in (C^{op}, Ab) is cotorsion. Also, the flat cover of a cotorsion functor is cotorsion; see [6, §2] for details.

Now fix a sequence X ⟶^{α} Y ⟶^{β} Z of morphisms such that α is the minimal weak kernel of β. If β is right minimal and right determined by any set of finitely presented objects, then h(Y) → Im h(β) is a flat cover; this follows from the proof of Theorem 1.1. Thus Im h(α) = Ker h(β) is cotorsion. The proof of Proposition 1.12 shows that h(X) → Im h(α) is a flat cover. Thus h(X) is cotorsion and therefore X is pure-injective. Conversely, if α is a kernel of β and X is pure-injective, then every morphism h(Y′) → Im h(β) factors through h(Y) → Im h(β). Thus every morphism β′: Y′ → Z with Im Hom_{A}(C, β′) ⊆ Im Hom_{A}(C, β) factors through β. This means that β is right C-determined. □

In view of Corollary 1.11 we obtain the following consequence.

Corollary 1.14. Let α: X → Y be a morphism which admits a kernel. Then α is right determined by a set of finitely presented objects if and only if there is a decomposition X = X′ ⊕X′′ such that Ker α|_{X′} is pure-injective and α|_{X′′} = 0. □

2. Flat versus projective covers

In this section we show how flat covers are turned into projective covers. In fact, we prove a general result about locally finitely presented additive categories which is tantamount to the the existence of flat covers in functor categories.

Finitely presented functors. Let A be an additive category. We denote by Fp(A^{op}, Ab) the category of finitely presented functors F: A^{op} → Ab. Recall that F is finitely presented (or coherent) if it fits into an exact sequence

Hom_{A}(−, X) ⟶ Hom_{A}(−, Y) ⟶ F ⟶ 0.

We call such a presentation minimal if the morphisms Hom_{A}(−, Y) → F and Hom_{A}(−, X) → Hom_{A}(−, Y) are right minimal.

The following lemma is well-known and easily proved (using Yoneda’s lemma).

Lemma 2.1. The category Fp(A^{op}, Ab) is abelian if and only A admits weak kernels. In this case the Yoneda embedding X ↦ Hom_{A}(−, X) identifies the idempotent completion of A with the category of projective objects in Fp(A^{op}, Ab). □

If A is locally finitely presented then we consider the evaluation functor

(2.1)  Fp(A^{op}, Ab) ⟶ ((fp A)^{op}, Ab), F ↦ F |_{fp A}.

The following theorem establishes a right adjoint which identifies flat covers with projective covers.

Theorem 2.2. Let A be a locally finitely presented category.

(1) The category Fp(A^{op}, Ab) of finitely presented functors A^{op} → Ab is abelian.

(2) For an additive functor F: (fp A)^{op} → Ab, the unique functor F̃: A^{op} → Ab extending F and preserving filtered colimits in A is finitely presented and admits a minimal projective presentation in Fp(A^{op}, Ab).

(3) The assignment F ↦ F̃ provides a fully faithful right adjoint to the evaluation functor (2.1).

We postpone the proof and illustrate the theorem by a couple of examples. The first one shows how flat covers of modules over a ring are derived from this result.

Example 2.3. Let Λ be a ring and denote by A the category of flat Λ-modules. Then fp A equals the category of finitely generated projective Λ-modules and evaluation at Λ yields an equivalence ((fp A)^{op}, Ab) ⟶^{∼} Mod Λ (which we view as identification). For a Λ-module Y, the theorem yields a projective cover Hom_{A}(−, X) → Ỹ. Evaluation at Λ then gives a flat cover X → Y.

Example 2.4. Not all functors in Fp(A^{op}, Ab) admit a projective cover. For instance, take A = Mod ℤ and consider the canonical morphism ℤ → ℤ/p for any prime p. Then the image of the induced morphism Hom_{ℤ}(−, ℤ) → Hom_{ℤ}(−, ℤ/p) admits no projective cover, because a projective cover Hom_{ℤ}(−, X) → F would give a projective cover X → ℤ/p in A (which is known not to exist).

Proof of Theorem 2.2. The category A has weak kernels by Proposition 1.12. Thus Fp(A^{op}, Ab) is abelian by Lemma 2.1.

Now fix F ∈ ((fp A)^{op}, Ab) and observe that F̃ ≅ Hom(h−, F) since h preserves filtered colimits. This yields for X ∈ A a functorial isomorphism

(2.2)  Hom(Hom_{A}(−, X), F̃) ≅ F̃(X) ≅ Hom(Hom_{A}(−, X)|_{fp A}, F)

which extends to an isomorphism

Hom(E, F̃) ≅ Hom(E|_{fp A}, F)

for all E ∈ Fp(A^{op}, Ab). The adjointness property of the assignment F ↦ F̃ then follows. Also, the functor is fully faithful since F̃|_{fp C} = F, and it identifies the flat functors in ((fp A)^{op}, Ab) with the projective objects in Fp(A^{op}, Ab).

Next we show that F̃ is finitely presented. In fact, we obtain a minimal projective presentation of F̃ in Fp(A^{op}, Ab) by choosing a minimal flat presentation

h(X) ⟶ h(Y) ⟶^{π} F ⟶ 0

in ((fp A)^{op}, Ab); see Theorem 1.10. From (2.2) it follows that the corresponding sequence

Hom_{A}(−, X) ⟶ Hom_{A}(−, Y) \underrightarrow{π̃} F̃ ⟶ 0

is a minimal projective presentation. □

Corollary 2.5. The evaluation functor (2.1) is exact and admits both adjoints; it induces an equivalence

\frac{Fp(A^{op}, Ab)}{\{F | F |_{fp A} = 0\}} ⟶^{∼} ((fp A)^{op}, Ab).

Proof. The right adjoint exists by Theorem 2.2, and the left adjoint is the unique colimit preserving functor sending Hom_{A}(−, X) to Hom_{A}(−, X) for all X ∈ fp A.

Let S denote the kernel of the evaluation functor. A quasi-inverse for the functor Fp(A^{op}, Ab)/S → ((fp A)^{op}, Ab) is obtaind by composing the left (or right) adjoint of the evaluation functor with the quotient functor Fp(A^{op}, Ab) → Fp(A^{op}, Ab)/S. □

Auslander’s formula. Let A be an abelian category. A somewhat hidden result in Auslander’s account on coherent functors [1, p. 205] shows that the Yoneda functor A → Fp(A^{op}, Ab) admits an exact left adjoint which sends a representable functor Hom_{A}(−, X) to X and yields Auslander’s formula [11]

\frac{Fp(A^{op}, Ab)}{\{F | F is exactly presented\}} ⟶^{∼} A.

Here, a functor F: A^{op} → Ab is exactly presented if it fits into an exact sequence

(2.3)  0 → Hom_{A}(−, X) → Hom_{A}(−, Y) → Hom_{A}(−, Z) → F → 0

such that the corresponding sequence 0 → X → Y → Z → 0 in A is exact.

One may think of Auslander’s formula as a prototype for Corollary 2.5. In particular, we see that the kernel of the evaluation functor is given by the functors F: A^{op} → Ab with a presentation (2.3) such that the corresponding sequence 0 → X → Y → Z → 0 in A is pure-exact.

3. Almost split sequences

The concept of a morphism determined by an object generalises that of an almost split morphism. In fact, a morphism α: X → Y in an additive category A is right almost split (that is, α is not a retraction and every morphism X′ → Y that is not a retraction factors through α) if and only if Γ = End_{A}(Y) is a local ring, α is right determined by Y, and Im Hom_{A}(Y, α) = rad Γ [2, §II.2].

Now let A be an abelian category. Recall that an exact sequence 0 → X ⟶^{α} Y ⟶^{β} Y → 0 is almost split if the morphism α is left almost split and β is right almost split. This is equivalent to β being a right minimal and right almost split morphism [2, §II.4].

The construction of morphisms in Theorem 1.1 requires the determining objects to be finitely presented. The following proposition shows that this assumption is necessary. A similar non-existence result for finite dimensional representations of infinite quivers is due to Paquette [12].

Proposition 3.1. The category of modules over ℤ does not admit an almost split sequence 0 → X → Y → Z → 0 for Z = ℚ.

Thus the category Mod ℤ does not admit a right minimal and right determined morphism for the triple (C, H, Y) = (ℚ, 0, ℚ).

The end terms of an almost split sequence determine each other, and this correpondence enjoys some weak functoriality. The proof of Proposition 3.1 uses this argument, and I am grateful to Helmut Lenzing for suggesting it.

Lemma 3.2 ([10, Lemma A.10]). Let 0 → X → Y → Z → 0 be an almost split sequence in any abelian category A. Then there is an isomorphism

End_{A}(X)/ rad End_{A}(X) ≅ End_{A}(Z)/ rad End_{A}(Z).  □

Proof of Proposition 3.1. Suppose there exists an almost split sequence 0 → X → Y → ℚ → 0 in Mod ℤ. Then the lemma implies

End_{ℤ}(X)/ rad End_{ℤ}(X) ≅ ℚ.

It follows that X is divisible since multiplication with any non-zero integer induces an endomorphism of X which is invertible. Thus the sequence splits which is impossible. □

Very little seems to be known about the non-existence of almost split sequences. We conjecture the following.

Conjecture 3.3. Given an almost split sequence 0 → X → Y → Z → 0 in any module category, the module Z is finitely presented.

Appendix A. Functors on module categories

The proof of Theorem 1.1 uses the existence of flat covers. In this appendix we provide an elementary argument for the case of a module category A = Mod Λ. We proceed in three steps.

Minimal presentations. The existence of flat covers in ((mod Λ)^{op}, Ab) can be phrased as follows.

Theorem A.1. Let Λ be a ring. An additive functor F: (Mod Λ)^{op} → Ab preserving filtered colimits in Mod Λ admits a minimal presentation

Hom_{Λ}(−, X) ⟶ Hom_{Λ}(−, Y) ⟶ F ⟶ 0.

Remark A.2. (1) The evaluation functor Fp((Mod Λ)^{op}, Ab) → ((mod Λ)^{op}, Ab) sends a minimal projective presentation to a minimal flat presentation.

(2) The theorem complements a result of Crawley-Boevey [8]: An additive functor F: Mod Λ → Ab preserves filtered colimits and products if and only if it admits a presentation

Hom_{Λ}(Y, −) ⟶ Hom_{Λ}(X, −) ⟶ F ⟶ 0

with X, Y ∈ mod Λ.

Proof. Any Λ-module can be written as a filtered colimit of finitely presented modules. It follows that F is determined by its restriction F |_{mod Λ}. The functors

I_{C} = Hom_{ℤ}(Hom_{Λ}(C, −), ℚ/ℤ)

with C ∈ mod Λ form a set of injective cogenerators for ((mod Λ)^{op}, Ab) and this yields an injective copresentation

0 ⟶ F |_{mod Λ} ⟶ ∏_{i} I_{C_{i}} ⟶ ∏_{j} I_{C_{j}}

with C_{i}, C_{j} ∈ mod Λ. Thus we obtain an exact sequence

0 ⟶ F ⟶ ∏_{i} I_{C_{i}} ⟶ ∏_{j} I_{C_{j}}

in ((Mod Λ)^{op}, Ab). It suffices to show for each C ∈ mod Λ that the functor I_{C} is finitely presented. Then one uses that the finitely presented functors are closed under taking products and kernels. Choose a presentation

P_{1} ⟶ P_{0} ⟶ C ⟶ 0

such that each P_{i} is finitely generated projective. This yields a presentation

I_{P_{1}} ⟶ I_{P_{0}} ⟶ I_{C} ⟶ 0.

We have

Hom_{ℤ}(Hom_{Λ}(P_{i}, −), ℚ/ℤ) ≅ Hom_{ℤ}(− ⊗_{Λ} P_{i}^{∗}, ℚ/ℤ)
≅ Hom_{Λ}(−, Hom_{ℤ}(P_{i}^{∗}, ℚ/ℤ))

where P_{i}^{∗} = Hom_{Λ}(P_{i}, Λ). Thus I_{C} is finitely presented.

The morphisms Hom_{Λ}(−, Y) → F and Hom_{Λ}(−, X) → Hom_{Λ}(−, Y) can be chosen to be right minimal; this follows from a standard argument [9, §7]. □

Coinduction. Let A be an additive category. For a full subcategory C ⊆ A consider the evaluation functor

ev_{C} : (A^{op}, Ab) ⟶ (C^{op}, Ab), F ↦ F |_{C}

and its right adjoint, the coinduction functor

coind_{C} : (C^{op}, Ab) ⟶ (A^{op}, Ab)

given by

coind_{C} I(X) = Hom(Hom_{A}(−, X)|_{C}, I) for I ∈ (C^{op}, Ab), X ∈ A.

For F ∈ (A^{op}, Ab) and I ∈ (C^{op}, Ab), the isomorphism

(A.1)  Hom(F, coind_{C} I) ⟶^{∼} Hom(F |_{C}, I), η ↦ η̄

is given by η̄ = η|_{C}, where we identify (coind_{C} I)|_{C} ⟶^{∼} I.

The coinduction functor assists in understanding the assignment (1.1)

F |_{C} ⊇ H ⟼ F_{H} ⊆ F

for an additive functor F: A^{op} → Ab.

Lemma A.3. Let F: A^{op} → Ab be an additive functor and H ⊆ F |_{C} a subfunctor. For a morphism η̄: F |_{C} → I with kernel H we have F_{H} = Ker η, where η and η̄ are related via (A.1).

Proof. Let F′ ⊆ F be a subfunctor. The isomorphism (A.1) is functorial and the inclusion F′ → F gives η|_{F′} = 0 iff η̄|_{F′|_{C}} = 0. Thus F′ ⊆ Ker η iff F′|_{C} ⊆ H. This means that Ker η is C-determined, and Ker η|_{C} = H by construction. □

Morphisms determined by objects. The following lemma provides the connection between morphisms and functors determined by objects.

Lemma A.4. Let A be an additive category and C ⊆ A. A morphism α: X → Y in A is right C-determined if and only if the subfunctor

Im Hom_{A}(−, α) ⊆ Hom_{A}(−, Y)

is C-determined.

Proof. This is clear because a morphism α′: X′ → Y factors through α iff

Im Hom_{A}(−, α′) ⊆ Im Hom_{A}(−, α).  □

We are now ready for a second proof of Theorem 1.1 for A = Mod Λ.

Proof of Theorem 1.1. Set F = Hom_{A}(−, Y). The subfunctor F_{H} arises as a kernel of a morphism F → coind_{C} I for some I ∈ (C^{op}, Ab) by Lemma A.3. The functor coind_{C} I preserves filtered colimits in A since C consists of finitely presented objects. Also F preserves filtered colimits in A, and therefore F_{H} has this property. Thus F_{H} admits a projective cover π: Hom_{A}(−, X) → F_{H} by Theorem A.1. The composite Hom_{A}(−, X) → F_{H} → F is represented by a morphism α: X → Y which is right C-determined by Lemma A.4, and right minimal since π is a projective cover. □

References

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Henning Krause, Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany.

E-mail address: hkrause@math.uni-bielefeld.de