THE TELESCOPE CONJECTURE FOR HEREDITARY RINGS VIA EXT-ORTHOGONAL PAIRS

HENNING KRAUSE AND JAN ŠŤOVÍČEK

Abstract. For the module category of a hereditary ring, the Ext-orthogonal pairs of subcategories are studied. For each Ext-orthogonal pair that is generated by a single module, a 5-term exact sequence is constructed. The pairs of finite type are characterized and two consequences for the class of hereditary rings are established: homological epimorphisms and universal localizations coincide, and the telescope conjecture for the derived category holds true.

1. Introduction

In this paper, we prove the telescope conjecture for the derived category of any hereditary ring. To achieve this, we study Ext-orthogonal pairs of subcategories for hereditary module categories.

The telescope conjecture for the derived category of a module category is also called smashing conjecture. It is the analogue of the telescope conjecture from stable homotopy theory which is due to Bousfield and Ravenel [5, 22]. In each case one deals with a compactly generated triangulated category. The conjecture then claims that a localizing subcategory is generated by compact objects provided it is smashing, that is, the localizing subcategory arises as the kernel of a localization functor that preserves arbitrary coproducts [18].

In this general form, the telescope conjecture seems to be wide open. For the stable homotopy category, we refer to the work of Mahowald, Ravenel, and Shick [17] for more details. For the derived category of a module category, only two results seem to be known so far. Neeman proved the conjecture for the derived category of a commutative noetherian ring [19], essentially by classifying all localizing subcategories. On the other hand, Keller gave an explicit example of a commutative ring where the conjecture does not hold [13]. In fact, an analysis of Keller’s argument [14] shows that there are such examples having global dimension 2. The approach for hereditary rings presented here is completely different from Neeman’s. In particular, we are working in a non-commutative setting and without using any noetherianess assumption.

A useful concept for proving the telescope conjecture in our context is the notion of an Ext-orthogonal pair. This concept seems to be new, but it is based on the notion of a perpendicular category which is one of the fundamental tools for studying hereditary categories arising in representation theory [27, 11].

Given any abelian category A, we call a pair (X , Y) of full subcategories Ext-orthogonal if X and Y are orthogonal to each other with respect to the bifunctor ∐_{n≥0} Ext_{A}^{n}(−, −).

Version from October 21, 2008.

The second author is supported by the Research Council of Norway through the Storforsk-project “Homological and geometric methods in algebra”.

This concept is the analogue of a torsion pair and a cotorsion pair where one considers instead the bifunctors Hom_{A}(−, −) and ∐_{n>0} Ext_{A}^{n}(−, −) respectively [8, 24].

Torsion and cotorsion pairs are most interesting when they are complete. For a torsion pair this means that each object M in A admits a short exact sequence 0 → X_{M} → M → Y^{M} → 0 with X_{M} ∈ X and Y^{M} ∈ Y. In the second case this means that each object M admits short exact sequences 0 → Y_{M} → X_{M} → M → 0 and 0 → M → Y^{M} → X^{M} → 0 with X_{M}, X^{M} ∈ X and Y_{M}, Y^{M} ∈ Y.

It turns out that there is also a reasonable notion of completeness for Ext-orthogonal pairs. In that case each object M in A admits a 5-term exact sequence

0 → Y_{M} → X_{M} → M → Y^{M} → X^{M} → 0

with X_{M}, X^{M} ∈ X and Y_{M}, Y^{M} ∈ Y.

In this work we study Ext-orthogonal pairs for the module category of a hereditary ring. The assumption on the ring implies a close connection between the module category and its derived category. It is this connection which we exploit in both directions. We use Bousfield localization functors which exist for the derived category to establish the completeness of certain Ext-orthogonal pairs for the module category; see §2. On the other hand, we are able to prove the telescope conjecture for the derived category by showing first a similar result for Ext-orthogonal pairs; see §5 and §7.

At a first glance the telescope conjecture seems to be a rather abstract statement about unbounded derived categories. However in the context of a fixed hereditary ring, it turns out that smashing localizing subcategories are in bijective correspondence to various natural structures. We mention here extension closed abelian subcategories of finitely presented modules and homological ring epimorphisms; see §8. Note that Extorthogonal pairs provide a useful link between these structures.

The telescope conjecture and its proof are related to interesting recent work by some other authors. In [28], Schofield describes for any hereditary ring its universal localizations in terms of appropriate subcategories of finitely presented modules. This is a consequence of the present work since we show that homological epimorphisms and universal localizations coincide for any hereditary ring; see §6. In [21], Nicolás and Saorín establish for a differential graded algebra a correspondence between recollements for its derived category and differential graded homological epimorphisms. This correspondence specializes for a hereditary ring to the above mentioned bijection between smashing localizing subcategories and homological epimorphisms.

Specific examples of Ext-orthogonal pairs arise in the representation theory of finite dimensional algebras via perpendicular categories; see §4. Note that a perpendicular category is just one half of an Ext-othogonal pair. Schofield introduced perpendicular categories for representations of quivers [27] and this fits into our set-up because the path algebra of any quiver is hereditary. In fact, the concept of a perpendicular category is fundamental for studying hereditary categories arising in representation theory [11]. It is therefore somewhat surprising that the 5-term exact sequence for a complete Extorthogonal pair seems to appear for the first time in this work.

The authors would like to thank Lidia Angeleri Hügel and Manolo Saorín for helpful discussions concerning this work.

2. Ext-orthogonal pairs

Let A be an abelian category. Given a pair of objects X, Y ∈ A, set

Ext_{A}^{∗}(X, Y) = ∐_{n∈ℤ}Ext_{A}^{n}(X, Y).

For a subcategory C of A we consider its full Ext-orthogonal subcategories

⊥C = {X ∈ A | Ext_{A}^{∗}(X, C) = 0 for all C ∈ C},
C^{⊥} = {Y ∈ A | Ext_{A}^{∗}(C, Y) = 0 for all C ∈ C}.

Definition 2.1. An Ext-orthogonal pair for A is a pair (X , Y) of full subcategories such that X^{⊥} = Y and X = ^{⊥}Y. An Ext-orthogonal pair (X , Y) is called complete if there exists for each object M ∈ A an exact sequence

ε_{M}: 0 → Y_{M} → X_{M} → M → Y^{M} → X^{M} → 0

with X_{M}, X^{M} ∈ X and Y_{M}, Y^{M} ∈ Y. The pair (X , Y) is generated by a subcategory C of A if Y = C^{⊥}.

The definition can be extended to the derived category D(A) of A if we put for each pair of complexes X, Y ∈ D(A) and n ∈ ℤ

Ext_{A}^{n}(X, Y) = Hom_{D(A)}(X, Y [n]).

Thus an Ext-orthogonal pair for D(A) is a pair (X , Y) of full subcategories of D(A) such that X^{⊥} = Y and X = ^{⊥}Y.

Recall that an abelian subcategory of A is a full subcategory C such that the category C is abelian and the inclusion functor C → A is exact. Suppose A is hereditary, that is, Ext_{A}^{n}(−, −) vanishes for all n > 1. Then a simple calculation shows that for any subcategory C of A, the subcategories C^{⊥} and ^{⊥}C are extension closed abelian subcategories; see [11, Proposition 1.1].

The following result establishes the completeness for certain Ext-orthogonal pairs. Recall that an abelian category is a Grothendieck category if it has a set of generators and admits colimits that are exact when taken over filtered categories.

Theorem 2.2. Let A be a hereditary abelian Grothendieck category and X an object in A. Set Y = X^{⊥} and let X denote the smallest extension closed abelian subcategory of A that is closed under taking coproducts and contains X. Then (X , Y) is a complete Ext-orthogonal pair for A. Thus there exists for each object M ∈ A a natural exact sequence

0 → Y_{M} → X_{M} → M → Y^{M} → X^{M} → 0

with X_{M}, X^{M} ∈ X and Y_{M}, Y^{M} ∈ Y. The sequence induces bijections Hom_{A}(X, X_{M}) → Hom_{A}(X, M) and Hom_{A}(Y^{M}, Y) → Hom_{A}(M, Y) for all X ∈ X and Y ∈ Y.

The proof uses derived categories and Bousfield localization functors. Thus we need to collect some basic facts about hereditary abelian categories and their derived categories.

The derived category of a hereditary abelian category. Let A be a hereditary abelian category and let D(A) denote its derived category. We assume that A admits coproducts and that the coproduct of any set of exact sequences is again exact. Thus the category D(A) admits coproducts, and for each integer n these coproducts are preserved by the functor H^{n}: D(A) → A which takes a complex to its cohomology in degree n.

It is well-known that each complex is quasi-isomorphic to its cohomology.

Lemma 2.3. Given a complex X in D(A), there are (non-canonical) isomorphisms

∐_{n∈ℤ}(H^{n}X)[−n] ≅ X ≅ ∏_{n∈ℤ}(H^{n}X)[−n].

Proof. See for instance [15, §1.6]. □

A full subcategory C of D(A) is called thick if C is a triangulated subcategory and closed under taking direct summands. A thick subcategory is localizing if it is closed under taking coproducts. Note that for each subcategory C the subcategories C^{⊥} and ⊥C are thick.

To a subcategory C of D(A) we assign the full subcategory

H^{0}C = {M ∈ A | M = H^{0}X for some X ∈ C},

and given a subcategory X of A, we define the full subcategory

D_{X} (A) = {X ∈ D(A) | H^{n}X ∈ X for all n ∈ ℤ}.

Both assignments induce mutually inverse bijections between appropriate subcategories. This is a useful fact which we recall from [6, Theorem 6.1].

Proposition 2.4. The functor H^{0} : D(A) → A induces a bijection between the localizing subcategories of D(A) and the extension closed abelian subcategories of A that are closed under coproducts. The inverse map sends a subcategory X of A to D_{X} (A). □

Remark 2.5. The bijection in Proposition 2.4 has an analogue for thick subcategories. Given any abelian category B, the functor H^{0} : D^{b}(B) → B induces a bijection between the thick subcategories of D^{b}(B) and the extension closed abelian subcategories of B.

Next we extend these maps to bijections between Ext-orthogonal pairs.

Proposition 2.6. The functor H^{0} : D(A) → A induces a bijection between the Extorthogonal pairs for D(A) and the Ext-orthogonal pairs for A. The inverse map sends a pair (X , Y) for A to (D_{X} (A), D_{Y}(A)).

Proof. First observe that for each pair of complexes X, Y ∈ D(A), we have Ext_{A}^{∗}(X, Y) = 0 if and only if Ext_{A}^{∗}(H^{p}X, H^{q}Y) = 0 for all p, q ∈ ℤ. This is a consequence of Lemma 2.3. It follows that H^{0} and its inverse send Ext-orthogonal pairs to Extorthogonal pairs. Each Ext-orthogonal pair is determined by its first half, and therefore an application of Proposition 2.4 shows that both maps are mutually inverse. □

Localization functors. Let T be a triangulated category. A localization functor L: T → T is an exact functor that admits a natural transformation η: Id_{T} → L such that Lη_{X} is an isomorphism and Lη_{X} = η_{LX} for all objects X ∈ T . Basic facts about localization functors one finds, for example, in [4, §3].

Proposition 2.7. Let A be a hereditary abelian category. For a full subcategory X of A the following are equivalent.

(1) There exists a localization functor L: D(A) → D(A) such that Ker L = D_{X} (A).

(2) There exists a complete Ext-orthogonal pair (X , Y) for A.

Proof. (1) ⇒ (2): The kernel Ker L and the essential image Im L of a localization functor L form an Ext-orthogonal pair for D(A); see for instance [4, Lemma 3.3]. Then it follows from Proposition 2.6 that the pair (X , Y) = (H^{0} Ker L, H^{0} Im L) is Ext-orthogonal for A.

The localization functor L comes equipped with a natural transformation η: Id_{D(A)} → L, and for each complex M we complete the morphism η_{M}: M → LM to an exact triangle

Γ M → M → LM → Γ M[1].

Note that Γ M ∈ Ker L and LM ∈ Im L since Lη_{M} is an isomorphism and L is exact. Now suppose that M is concentrated in degree zero. Applying H^{0} to this triangle yields an exact sequence

0 → Y_{M} → X_{M} → M → Y^{M} → X^{M} → 0

with X_{M}, X^{M} ∈ X and Y_{M}, Y^{M} ∈ Y.

(2) ⇒ (1): Let (X , Y) be an Ext-orthogonal pair for A. This pair induces an Extorthogonal pair (D_{X} (A), D_{Y}(A)) for D(A) by Proposition 2.6. In order to construct a localization functor L: D(A) → D(A) such that Ker L = D_{X} (A), it is sufficient to construct for each object M in D(A) an exact triangle X → M → Y → X[1] with X ∈ D_{X} (A) and Y ∈ D_{Y}(A). Then one defines LM = Y and the morphism M → Y induces a natural transformation η: Id_{D(A)} → L having the required properties. In view of Lemma 2.3 it is sufficient to assume that M is a complex concentrated in degree zero.

Suppose that M admits an approximation sequence

ε_{M}: 0 → Y_{M} → X_{M} → M → Y^{M} → X^{M} → 0

with X_{M}, X^{M} ∈ X and Y_{M}, Y^{M} ∈ Y. Let M′ denote the image of X_{M} → M and M′′ the image of M → Y^{M}. Then ε_{M} induces the following three exact sequences

α_{M}: 0 → M′ → M → M′′ → 0,
β_{M}: 0 → Y_{M} → X_{M} → M′ → 0,
γ_{M}: 0 → M′′ → Y^{M} → X^{M} → 0.

In D(A) these three exact sequence give rise to the following square

\xymatrix{X^{M}[−2] \ar[r]^{γ_{M}} \ar[d]^{0} & M′′[−1] \ar[d]^{α_{M}} \\ X_{M} \ar[r]^{β_{M}} & M′}

which is commutative since Hom_{D(A)}(U[−2], V) = 0 for any U, V ∈ A. An application of the octahedral axiom shows that this square can be extended as follows to a diagram where each row and each column is an exact triangle.

\xymatrix{X^{M}[−2] \ar[r] \ar[d]^{0} & M′′[−1] \ar[r] \ar[d] & Y^{M}[−1] \ar[r] \ar[d]^{0} & X^{M}[−1] \ar[d]^{0} \\ X_{M} \ar[r] \ar[d] & M′ \ar[r] \ar[d] & Y_{M}[1] \ar[r] \ar[d] & X_{M}[1] \ar[d] \\ X_{M} ⊕ X^{M}[−1] \ar[r] \ar[d] & M \ar[r] \ar[d] & Y_{M}[1] ⊕ Y^{M} \ar[r] \ar[d] & X_{M}[1] ⊕ X^{M} \ar[d] \\ X^{M}[−1] \ar[r] & M′′ \ar[r] & Y^{M} \ar[r] & X^{M}}

The first and third column are split exact triangles, and this explains the objects appearing in the third row. In particular, this yields the desired exact triangle X → M → Y → X[1] with X ∈ D_{X} (A) and Y ∈ D_{Y}(A). □

Next we formulate the functorial properties of the 5-term exact sequence constructed in Proposition 2.7.

Lemma 2.8. Let A be an abelian category and (X , Y) an Ext-orthogonal pair for A. Suppose there is an exact sequence

ε_{M}: 0 → Y_{M} → X_{M} → M → Y^{M} → X^{M} → 0

in A with X_{M}, X^{M} ∈ X and Y_{M}, Y^{M} ∈ Y.

(1) The sequence ε_{M} induces for all X ∈ X and Y ∈ Y bijections Hom_{A}(X, X_{M}) → Hom_{A}(X, M) and Hom_{A}(Y^{M}, Y) → Hom_{A}(M, Y).

(2) Let ε_{N}: 0 → Y_{N} → X_{N} → N → Y^{N} → X^{N} → 0 be an exact sequence in A with X_{N}, X^{N} ∈ X and Y_{N}, Y^{N} ∈ Y. Then each morphism M → N extends uniquely to a morphism ε_{M} → ε_{N} of exact sequences.

(3) Any exact sequence 0 → Y′ → X′ → M → Y′′ → X′′ → 0 in A with X′, X′′ ∈ X and Y′, Y′′ ∈ Y is uniquely isomorphic to ε_{M}.

Proof. We prove part (1). Then part (2) and (3) are immediate consequences.

Fix an object X ∈ X. The map μ: Hom_{A}(X, X_{M}) → Hom_{A}(X, M) is injective because Hom_{A}(X, Y_{M}) = 0. Any morphism X → M factors through the kernel M′ of M → Y^{M} since Hom_{A}(X, Y^{M}) = 0. The induced morphism X → M′ factors through X_{M} → M′ since Ext_{A}^{1}(X, Y_{M}) = 0. Thus μ is surjective. The argument for the other map Hom_{A}(Y^{M}, Y) → Hom_{A}(M, Y) is dual. □

Ext-orthogonal pairs for Grothendieck categories. We give the proof of Theorem 2.2. The basic idea is to establish a localization functor for D(A) and to derive the exact approximation sequence in A by taking the cohomology of some appropriate exact triangle.

Proof of Theorem 2.2. Let X denote the smallest extension closed abelian subcategory of A that contains X and is closed under coproducts. Then Proposition 2.4 implies that D_{X} (A) is the smallest localizing subcategory of D(A) containing X. Thus there exists a localization functor L: D(A) → D(A) with Ker L = D_{X} (A). This is a result which goes back to Bousfield. In the context of derived categories we refer to [2, Theorem 5.7].

Now apply Proposition 2.7 to get the 5-term exact sequence for each object M in A. The properties of this sequence follow from Lemma 2.8. □

Remark 2.9. We do not know an example of an Ext-orthogonal pair (X , Y) for a hereditary abelian Grothendieck category such that the pair (X , Y) is not complete.

Further examples of Ext-orthogonal pairs arise as follows. Let A be any abelian Grothendieck category and X a localizing subcategory. Thus X is a full subcategory closed under taking coproducts such that for any exact sequence 0 → M′ → M → M′′ → 0 in A we have M ∈ X if and only if M′, M′′ ∈ X. Set Y = X^{⊥} and let Y_{inj} denote the full subcategory of injective objects in Y. Then X = ^{⊥}Y_{inj} and therefore (X , Y) is an Ext-orthogonal pair for A; see [9, III.4] for details.

Torsion and cotorsion pairs. Let A be an abelian category and (X , Y) an Extorthogonal pair. We sketch an interpretation of the pair (X , Y) in terms of torsion and cotorsion pairs. Here, a pair (U, V) of full subcategories of A is called torsion pair if U and V are orthogonal to each other with respect to Hom_{A}(−, −). Analogously, a pair of full subcategories is a cotorsion pair if both categories are orthogonal to each other with respect to ∐_{n>0} Ext_{A}^{n}(−, −).

The subcategory X generates a torsion pair (X_{0}, Y_{0}) and a cotorsion pair (X_{1}, Y_{1}) for A, if one defines the corresponding full subcategories of A as follows:

Y_{0} = {Y ∈ A | Hom_{A}(X, Y) = 0 for all X ∈ X },
X_{0} = {X ∈ A | Hom_{A}(X, Y) = 0 for all Y ∈ Y_{0}},
Y_{1} = {Y ∈ A | Ext_{A}^{n}(X, Y) = 0 for all X ∈ X , n > 0},
X_{1} = {X ∈ A | Ext_{A}^{n}(X, Y) = 0 for all Y ∈ Y_{1}, n > 0}.

Note that X = X_{0} ∩ X_{1} and Y = Y_{0} ∩ Y_{1}. In particular, one recovers the pair (X , Y) from (X_{0}, Y_{0}) and (X_{1}, Y_{1}).

Suppose an object M ∈ A admits an approximation sequence

ε_{M}: 0 → Y_{M} → X_{M} → M → Y^{M} → X^{M} → 0

with X_{M}, X^{M} ∈ X and Y_{M}, Y^{M} ∈ Y. We give the following interpretation of this sequence. Let M′ denote the image of X_{M} → M and M′′ the image of M → Y^{M}. Then there are three short exact sequences:

α_{M}: 0 → M′ → M → M′′ → 0,
β_{M}: 0 → Y_{M} → X_{M} → M′ → 0,
γ_{M}: 0 → M′′ → Y^{M} → X^{M} → 0.

The sequence α_{M} is the approximation sequence of M with respect to the torsion pair (X_{0}, Y_{0}), that is, M′ ∈ X_{0} and M′′ ∈ Y_{0}. On the other hand, β_{M} and γ_{M} are approximation sequences of M′ and M′′ respectively, with respect to the cotorsion pair (X_{1}, Y_{1}), that is, X_{M}, X^{M} ∈ X_{1} and Y_{M}, Y^{M} ∈ Y_{1}. Thus the 5-term exact sequence ε_{M} is obtained by splicing together three short exact approximation sequences.

Suppose that the Ext-orthogonal pair (X , Y) is complete. Then the associated torsion pair (X_{0}, Y_{0}) has an explicit description: we have X_{0} = Fac X and Y_{0} = Sub Y, where

Fac X = {X/U | U ⊆ X, X ∈ X } and Sub Y = {U | U ⊆ Y, Y ∈ Y}.

3. Homological epimorphisms

From now on we study Ext-orthogonal pairs for module categories. Thus we fix a ring A and denote by Mod A the category of (right) A-modules. The full subcategory formed by all finitely presented A-modules is denoted by mod A.

Most of our result require the ring A to be hereditary. This means the category of A-modules is hereditary, that is, Ext_{A}^{n}(−, −) vanishes for all n > 1.

Ext-orthogonal pairs for module categories over hereditary rings are closely related to homological epimorphisms. Recall that a ring homomorphism A → B is a homological epimorphism if

B ⊗_{A} B ≅ B and Tor_{n}^{A}(B, B) = 0 for all n > 0,

equivalently, if restriction induces isomorphisms

Ext_{B}^{∗}(X, Y) ⟶^{∼} Ext_{A}^{∗}(X, Y)

for all B-modules X, Y; see [11] for details.

Proposition 3.1. Let A be a hereditary ring and f: A → B a homological epimorphism. Denote by Y the category of A-modules which are restrictions of modules over B. Set X = ^{⊥}Y and Y^{⊥} = Z. Then (X , Y) and (Y, Z) are complete Ext-orthogonal pairs for Mod A with Y = (Ker f ⊕ Coker f)^{⊥} and Z = B^{⊥}.

Proof. We wish to apply Theorem 2.2 which provides a construction for complete Extorthogonal pairs.

First observe that Y is the smallest extension closed abelian subcategory of Mod A closed under coproducts and containing B. This yields Z = B^{⊥}.

Next we show that Y = (Ker f ⊕ Coker f)^{⊥}. In fact, an A-module Y is the restriction of a B-module if and only if f induces an isomorphism Hom_{A}(B, Y) → Hom_{A}(A, Y). Using the assumptions on A and f, a simple calculation shows that this implies Y = (Ker f ⊕ Coker f)^{⊥}.

It remains to apply Theorem 2.2. Thus (X , Y) and (Y, Z) are complete Ext-orthogonal pairs. □

Next we use a theorem of Gabriel and de la Peña. It identifies the full subcategories of a module category Mod A that arise as the image of the restriction functor Mod B → Mod A for a ring epimorphism A → B.

Proposition 3.2. Let A be a hereditary ring and Y an extension closed abelian subcategory of Mod A that is closed under taking products and coproducts. Then there exists a homological epimorphism f: A → B such that the restriction functor Mod B → Mod A induces an equivalence Mod B ⟶^{∼} Y.

Proof. It follows from [10, Theorem 1.2] that there exists an epimorphism f: A → B such that the restriction functor Mod B → Mod A induces an equivalence Mod B ⟶^{∼} Y. To be more specific, one constructs a left adjoint F: Mod A → Y for the inclusion Y → Mod A. Then F A is a small projective generator for Y, because A has this property for Mod A and the inclusion of Y is an exact functors that preserves coproducts. Thus one takes for f the induced map A ≅ End_{A}(A) → End_{A}(F A).

We claim that restriction via f induces an isomorphism

Ext_{B}^{n}(X, Y) ⟶^{∼} Ext_{A}^{n}(X, Y)

for all B-modules X, Y and all n ≥ 0. This is clear for n = 0, 1 since Y is extension closed. On the other hand, the isomorphism for n = 1 implies that Ext_{B}^{1}(X, −) is right exact since A is hereditary. It follows that Ext_{B}^{n}(−, −) vanishes for all n > 1. □

We will use the fact that each homological epimorphism A → B induces a pair of localization functors D(Mod A) → D(Mod A).

Lemma 3.3. Let A → B be a homological epimorphism and denote by Y the category of A-modules which are restrictions of modules over B.

(1) The functor D(Mod A) → D(Mod A) sending a complex X to X ⊗_{A}^{L} B is a localization functor with essential image equal to D_{Y}(Mod A).

(2) The functor D(Mod A) → D(Mod A) sending a complex X to the cone of the natural morphism RHom_{A}(B, X) → X is a localization functor with kernel equal to D_{Y}(Mod A).

Proof. Restriction along f: A → B identifies Mod B with Y. The functor induces an isomorphism

Ext_{B}^{n}(X, Y) ⟶^{∼} Ext_{A}^{n}(X, Y)

for all B-modules X, Y and all n ≥ 0, because f is a homological epimorphism. This isomorphism implies that the induced functor f_{∗}: D(Mod B) → D(Mod A) is fully faithful with essential image D_{Y}(Mod A).

(1) The functor f_{∗} admits a left adjoint f^{∗} = − ⊗_{A}^{L} B and we have therefore a localization functor L: D(Mod A) → D(Mod A) sending a complex X to f_{∗}f^{∗}(X); see [4, Lemma 3.1]. It remains to note that the essential images of L and f_{∗} coincide.

(2) The functor f_{∗} admits a right adjoint f^{!} = RHom_{A}(B, −) and we have therefore a colocalization functor Γ: D(Mod A) → D(Mod A) sending a complex X to f_{∗}f^{!}(X). Note that the adjunction morphism Γ X → X is an isomorphism if and only if X belongs to D_{Y}(Mod A). Completing Γ X → X to a triangle yields a well defined localization functor D(Mod B) → D(Mod A) with kernel D_{Y}(Mod A); see [4, Lemma 3.3]. □

Corollary 3.4. Let A be a hereditary ring and Y an extension closed abelian subcategory of Mod A that is closed under taking products and coproducts. Set X = ^{⊥}Y and Z = Y^{⊥}. Then (X , Y) and (Y, Z) are both complete Ext-orthogonal pairs.

Proof. The assertion is an immediate consequence of Propositions 3.1 and 3.2. However, we prefer to give an alternative proof because it is more explicit.

There exists a homological epimorphism f: A → B such that restriction identifies Mod B with Y; see Proposition 3.2. Then Lemma 3.3 produces two localization functors L_{1}, L_{2} : D(Mod A) → D(Mod A) with Im L_{1} = D_{Y}(Mod A) = Ker L_{2}. Thus

Ker L_{1} = ^{⊥}(Im L_{1}) = D_{X} (Mod A) and Im L_{2} = (Ker L_{2})^{⊥} = D_{Z}(Mod A),

where in both cases the first equality follows from [4, Lemma 3.3] and the second from Proposition 2.6. It remains to apply Proposition 2.7 which yields in both cases for each A-module the desired 5-term exact sequence. □

Remark 3.5. The proof of Corollary 3.4 yields for any A-module M an explicit description of some terms of the 5-term exact sequence ε_{M}, using the homological epimorphism A → B. In the first case, we have

ε_{M}: 0 → Tor_{1}^{A}(M, B) → X_{M} → M → M ⊗_{A} B → X^{M} → 0,

and in the second case, we have

ε_{M}: 0 → Z_{M} → Hom_{A}(B, M) → M → Z^{M} → Ext_{A}^{1}(B, M) → 0.

The following result reflects the recollement of the derived category D(Mod A) which arises from a homological epimorphism A → B; it is an immediate consequence of Corollary 3.4.

Corollary 3.6. Let A be a hereditary ring and (X , Y) an Ext-orthogonal pair for the category of A-modules.

(1) There is an Ext-orthogonal pair (W, X) if and only if X is closed under products.

(2) There is an Ext-orthogonal pair (Y, Z) if and only if Y is closed under coproducts.

4. Examples

We present a number of examples of Ext-orthogonal pairs which illustrate the results of this work. The first example is classical and provides one of the motivations for studying perpendicular categories in representation theory of finite dimensional algebras. We refer to Schofield’s work [26, 27] which contains some explicit calculations; see also [11, 12].

Example 4.1. Let A be a finite dimensional hereditary algebra over a field k and X a finite dimensional A-module. Then X^{⊥} = Y identifies via a homological epimorphism A → B with the category of modules over a k-algebra B and this yields a complete Ext-orthogonal pair (X , Y). If X is exceptional, that is, Ext_{A}^{1}(X, X) = 0, then B is finite dimensional and can be constructed explicitly. Note that in this case for each finite dimensional A-module M the corresponding 5-term exact sequence ε_{M} consists of finite dimensional modules. Moreover, the category X is equivalent to the module category of another finite dimensional algebra. We do not know of a criterion on X that characterizes the fact that B is finite dimensional; see however the following proposition.

Proposition 4.2. Let A be a finite dimensional hereditary algebra over a field and (X , Y) a complete Ext-ortghogonal pair such that Y is closed under coproducts. Fix a homological epimorphism A → B inducing an equivalence Mod B ⟶^{∼} Y. Then the following are equivalent.

(1) There exists an exceptional module X ∈ mod A such that Y = X^{⊥}.

(2) The algebra B belongs to mod A when viewed as an A-module.

(3) For each M ∈ mod A, the 5-term exact sequence ε_{M} belongs to mod A.

Proof. (1) ⇒ (2): This follows, for example, from [11, Proposition 3.2].

(2) ⇒ (3): This follows from Remark 3.5.

(3) ⇒ (1): Let X_{fp} = X ∩ mod A and Y_{fp} = Y ∩ mod A. The assumption on (X , Y) implies that (X_{fp}, Y_{fp}) is a complete Ext-orthogonal pair for mod A. Moreover, every object in X is a filtered colimit of objects in X_{fp}; see Lemma 5.3. Now choose an injective cogenerator Q in mod A and let X = X_{Q} be the module from the 5-term exact sequence ε_{Q}. This module is the image of Q under a right adjoint of the inclusion X_{fp} → mod A. Note that a right adjoint of an exact functor preserves injectivity. It follows that X is an exceptional object and that X_{fp} is the smallest extension closed abelian subcategory of mod A containing X. Thus X^{⊥} = X_{fp}^{⊥} = X^{⊥} = Y, since every object in X is a filtered colimit of objects in X_{fp}. □

Any finitely generated projective module generates an Ext-orthogonal pair that can be described explicitly; see [11, §5].

Example 4.3. Let A be a hereditary ring and e^{2} = e ∈ A an idempotent. Let X denote the category of A-modules M such that the natural map Me ⊗_{eAe} eA → M is an isomorphism, and let Y = eA^{⊥} = {M ∈ Mod A | Me = 0}. Thus − ⊗_{eAe} eA identifies Mod eAe with X and restriction via A → A/AeA identifies Mod A/AeA with Y. Then (X , Y) is a complete Ext-orthogonal pair for Mod A, and for each A-module M the 5-term exact sequence ε_{M} is of the form

0 → Tor_{1}^{A}(M, A/AeA) → Me ⊗_{eAe} eA → M → M ⊗_{A} A/AeA → 0.

The next example^{1} arises from the work of Reiten and Ringel on infinite dimensional representations of canonical algebras; see [23] which is our reference for all concepts and results in the following discussion. Note that these algebras are not necessarily hereditary. The example shows the interplay between Ext-orthogonal pairs and (co)torsion pairs.

Example 4.4. Let A be a finite dimensional canonical algebra over a field k. Take for example a tame hereditary algebra, or, more specifically, the Kronecker algebra [ \begin{matrix}k & k^{2} \\ 0 & k \end{matrix} ]. For such algebras, there is the concept of a separating tubular family. We fix such a family and denote by T the category of finite dimensional modules belonging to this family. There is also a particular generic module over A which depends in some cases on the choice of T ; it is denoted by G. Then the full subcategory X = \underrightarrow{lim} T consisting of all filtered colimits of modules in T and the full subcategory Y = Add G consisting of all coproducts of copies of G form an Ext-orthogonal pair (X , Y) for Mod A. Note that the endomorphism ring D = End_{A}(G) of G is a division ring and that the canonical map A → B with B = End_{D}(G) is a homological epimorphism which induces an equivalence

Mod B ⟶^{∼} Y.

The category of A-modules which are generated by T and the category of A-modules which are cogenerated by G form a torsion pair (Fac X , Sub Y) for Mod A which equals the torsion pair (X_{0}, Y_{0}) generated by X. On the other hand, let C denote the category of A-modules which are cogenerated by X, and let D denote the category of A-modules M satisfying Hom_{A}(M, T ) = 0. Then the pair (C, D) forms a cotorsion pair for Mod A which identifies with the cotorsion pair (X_{1}, Y_{1}) generated by X.

If A is hereditary, then the Ext-orthogonal pair (X , Y) is complete by Corollary 3.4; see also Remark 3.5 for an explicit description of the 5-term approximation sequence ε_{M} for each A-module M. Alternatively, one obtains the sequence ε_{M} by splicing together appropriate approximation sequences which arise from (X_{0}, Y_{0}) and (X_{1}, Y_{1}).

The following example of an Ext-orthogonal pair arises from a localizing subcategory; it provides a simple model for the previous example.

Example 4.5. Let A be an integral domain with quotient field Q. Let X denote the category of torsion modules and Y the category of torsion free divisible modules. Note that the modules in Y are precisely the coproducts of copies of Q. Then (X , Y) is a complete Ext-orthogonal pair for Mod A, and for each A-module M the 5-term exact sequence ε_{M} is of the form

0 → tM → M → M ⊗_{A} Q → M̄ → 0.

1The first author is grateful to Lidia Angeleri Hügel for suggesting this example.

There are examples of abelian categories that admit only trivial Ext-orthogonal pairs.

Example 4.6. Let A be a local artinian ring and set A = Mod A. Then Hom_{A}(X, Y) ≠ 0 for any pair X, Y of non-zero A-modules. Thus if (X , Y) is an Ext-orthogonal pair for A, then X = A or Y = A.

5. Ext-orthogonal pairs of finite type

We characterize for a hereditary ring the Ext-orthogonal pairs of finite type.

Theorem 5.1. Let A be a hereditary ring and (X , Y) an Ext-orthogonal pair for the module category of A. Then the following are equivalent.

(1) The subcategory Y is closed under taking coproducts.

(2) Every module in X is a filtered colimit of finitely presented modules from X.

(3) There exists a category C of finitely presented modules such that C^{⊥} = Y.

We need some preparations for the proof of this result. The first lemma is a slight modification of [3, Proposition 2.1].

Lemma 5.2. Let A be a ring and Y a subcategory of its module category. Denote by X the category of A-modules X of projective dimension at most 1 satisfying Ext_{A}^{1}(X, Y) = 0 for all Y ∈ Y. Then any module in X is a filtered colimit of finitely presented modules from X.

Proof. Let X ∈ X. Choose an exact sequence 0 → P ⟶^{ϕ} Q → X → 0 such that P is free and Q is projective. Note that Ext_{A}^{1}(X, Y) = 0 is equivalent to the fact that every morphism P → Y factors through ϕ. The commuting squares of A-module morphisms

\xymatrix{P_{i} \ar[r]^{ϕ_{i}} \ar[d] & Q_{i} \ar[d] \\ P \ar[r]^{ϕ} & Q}

with P_{i} and Q_{i} finitely generated projective form a filtered system such that \underrightarrow{lim} ϕ_{i} = ϕ. There is a cofinal system such that each morphism P_{i} → P is a split monomorphism, since P is free and therefore a filtered colimit of finitely generated direct summands. Thus we may assume that each morphism P_{i} → P is a split monomorphism. Now set X_{i} = Coker ϕ_{i}. Then \underrightarrow{lim} X_{i} = X, and each morphism P_{i} → Y with Y ∈ Y factors through ϕ_{i}. Thus Ext_{A}^{1}(X_{i}, Y) = 0 for all i. □

Lemma 5.3. Let A be a hereditary ring and (X , Y) a complete Ext-orthogonal pair for Mod A. Let M be an A-module and ε_{M} the corresponding 5-term exact sequence.

(1) If Ext_{A}^{1}(M, Y) = 0, then Y_{M} = 0.

(2) Suppose that Y is closed under coproducts and let M = \underrightarrow{lim} M_{i} be a filtered colimit of A-modules M_{i}. Then ε_{M} = \underrightarrow{lim} ε_{M_{i}}.

Proof. If Ext_{A}^{1}(M, Y) = 0, then the image of the morphism X_{M} → M belongs to X. Thus X_{M} → M is a monomorphism and this yields (1).

To prove (2), one uses that X and Y are closed under taking colimits and that taking filtered colimits is exact. Thus \underrightarrow{lim} ε_{M_{i}} is an exact sequence with middle term M and all other terms in X or Y. Now the uniqueness of ε_{M} implies that ε_{M} = \underrightarrow{lim} ε_{M_{i}}; see Lemma 2.8. □

The following lemma is needed for hereditary rings which are not noetherian.

Lemma 5.4. Let M be a finitely presented module over a hereditary ring and N ⊆ M any submodule. Then N is a direct sum of finitely presented modules.

Proof. We combine two results. Over a hereditary ring, any submodule of a finitely presented module is a direct sum of a finitely presented module and a projective module; see [7, Theorem 5.1.6]. In addition, one uses that any projective module is a direct sum of finitely generated projective modules; see [1]. □

Proof of Theorem 5.1. (1) ⇒ (2): Suppose that Y is closed under taking coproducts. We apply Corollary 3.4 and obtain for each module M the natural exact sequence ε_{M}. Now suppose that M belongs X. Then one can write M = \underrightarrow{lim} M_{i} as a filtered colimit of finitely presented modules with Ext_{A}^{1}(M_{i}, Y) = 0 for all i; see Lemma 5.2. Next we apply Lemma 5.3. Thus

\underrightarrow{lim} X_{M_{i}} ⟶^{∼} X_{M} ⟶^{∼} M,

and each X_{M_{i}} is a submodule of the finitely presented module M_{i}. Finally, each X_{M_{i}} is a filtered colimit of finitely presented direct summands by Lemma 5.4. Thus M is a filtered colimit of finitely presented modules from X.

(2) ⇒ (3): Let X_{fp} denote the full subcategory that is formed by all finitely presented modules in X. Observe that ^{⊥}Y is closed under taking colimits for each module Y, because ^{⊥}Y is closed under taking coproducts and cokernels. Thus X_{fp}^{⊥} = X^{⊥} = Y provided that X = \underrightarrow{lim} X_{fp}.

(3) ⇒ (1): Use that for each finitely presented A-module X, the functor Ext_{A}^{∗}(X, −) preserves all coproducts. □

Theorem 5.1 gives rise to a bijection between subcategories of finitely presented modules and Ext-orthogonal pairs of finite type. This is a consequence of the following proposition.

Proposition 5.5. Let A be a hereditary ring and C a category of finitely presented Amodules. Then ^{⊥}(C^{⊥}) ∩ mod A equals the smallest extension closed abelian subcategory of mod A containing C.

Proof. Let D denote the smallest extension closed abelian subcategory of mod A containing C. We claim that the category \underrightarrow{lim} D which is formed by all filtered colimits of modules in D is an extension closed abelian subcategory of Mod A.

On the other hand, Theorem 2.2 implies that X = ^{⊥}(C^{⊥}) equals the smallest extension closed abelian subcategory of Mod A closed under coproducts and containing C. Thus the first claim implies X = \underrightarrow{lim} D and therefore X ∩ mod A = D.

To prove the claim, observe that every morphism in \underrightarrow{lim} D can be written as a filtered colimit of morphisms in D. Using that taking filtered colimits is exact, it follows immediately that \underrightarrow{lim} D is closed under kernels and cokernels in Mod A.

It remains to show that \underrightarrow{lim} D is closed under extensions. To this end let 0 → L → M → N → 0 be an exact sequence with L and N in \underrightarrow{lim} D. We can without loss of generality assume that N belongs to D, because the sequence is a filtered colimit of the pull-back exact sequences with the last term in D. We choose a morphism ϕ: M′ → M with M′ finitely presented and need to show that ϕ factors through an object in D; see [16]. We may assume that the composite with M → N is an epimorphism and denote by L′ its kernel which is finitely presented. Thus the induced map L′ → L factors through an object L′′ in D since L belongs to \underrightarrow{lim} D. Forming the push-out exact sequence of 0 → L′ → M′ → N → 0 along the morphism L′ → L′′ gives an exact sequence 0 → L′′ → M′′ → N → 0. Now ϕ factors through M′′ which belongs to D. □

6. Universal localizations

A ring homomorphism A → B is called universal localization if there exists a set Σ of morphisms between finitely generated projective A-modules such that

(1) σ ⊗_{A} B is an isomorphism of B-modules for all σ ∈ Σ, and

(2) every ring homomorphism A → B′ such that σ ⊗_{A} B′ is an isomorphism of B-modules for all σ ∈ Σ factors uniquely through A → B.

Let A be a ring and Σ a set of morphisms between finitely generated projective Amodules. Then there exists a universal localization inverting Σ and this is unique up to a unique isomorphism; see [25] for details. The universal localization is denoted by A → A_{Σ} and restriction identifies Mod A_{Σ} with the full subcategory consisting of all A-modules M such that Hom_{A}(σ, M) is an isomorphism for all σ ∈ Σ. Note that Hom_{A}(σ, M) is an isomorphism if and only if M belongs to {Ker σ, Coker σ}^{⊥}, provided that A is hereditary.

Theorem 6.1. Let A be a hereditary ring. A ring homomorphism f: A → B is a homological epimorphism if and only if f is a universal localization.

Proof. Suppose first that f: A → B is a homological epimorphism. This gives rise to an Ext-orthogonal pair (X , Y) for Mod A, if we identify Mod B with a full subcategory Y of Mod A; see Proposition 3.1. Let X_{fp} denote the subcategory that is formed by all finitely presented modules in X. It follows from Theorem 5.1 that X_{fp}^{⊥} = Y. Now fix for each X ∈ X_{fp} an exact sequence

0 → P_{X} ⟶^{σ_{X}} Q_{X} → X → 0

such that P_{X} and Q_{X} are finitely generated projective, and let Σ = {σ_{X} | X ∈ X_{fp}}. Then

Mod B = X_{fp}^{⊥} = Mod A_{Σ},

and therefore f: A → B is a universal localization.

Now suppose f: A → B is a universal localization. Then restriction identifies the category of B-modules with an extension closed subcategory of Mod A. Thus we have induced isomorphisms

Ext_{B}^{∗}(X, Y) ⟶^{∼} Ext_{A}^{∗}(X, Y)

for all B-modules X, Y, since A is hereditary. It follows that f is a homological epimorphism. □

Remark 6.2. Neither implication in Theorem 6.1 is true if one drops the assumption on the ring A to be hereditary. In [13], Keller gives an example of a Bézout domain A and a non-zero ideal I such that the canonical map A → A/I is a homological epimorphism, but any map σ between finitely generated projective A-modules needs to be invertible if σ ⊗_{A} A/I is invertible. On the other hand, Neeman, Ranicki, and Schofield use finite dimensional algebras to construct in [20] examples of universal localizations that are not homological epimorphisms.

7. The telescope conjecture

Let A be a ring. A complex of A-modules is called perfect if it is a bounded complex of finitely generated projective modules. Note that a complex X is isomorphic to a perfect complex if and only if the functor Hom_{D(Mod A)}(X, −) preserves coproducts.

A localizing subcategory C of D(Mod A) is generated by perfect complexes if C admits no proper localizing subcategory containing all perfect complexes from C.

Theorem 7.1. Let A be a hereditary ring. For a localizing subcategory C of D(Mod A) the following conditions are equivalent:

(1) There exists a localization functor L: D(Mod A) → D(Mod A) that preserves coproducts such that C = Ker L.

(2) The localizing subcategory C is generated by perfect complexes.

(3) There exists a localizing subcategory D of D(Mod A) that is closed under products such that C = ^{⊥}D.

Proof. (1) ⇒ (2): The kernel Ker L and the essential image Im L of a localization functor L form an Ext-orthogonal pair for D(Mod A); see [4, Lemma 3.3]. We obtain an Ext-orthogonal pair (X , Y) for Mod A by taking X = H^{0} Ker L and Y = H^{0} Im L; see Proposition 2.6. The fact that L preserves coproducts implies that Y is closed under taking coproducts. It follows from Theorem 5.1 that X is generated by finitely presented modules. Each finitely presented module is isomorphic in D(Mod A) to a perfect complex, and therefore Ker L is generated by perfect complexes.

(2) ⇒ (3): Suppose that C is generated by perfect complexes. Then there exists a localization functor L: D(Mod A) → D(Mod A) such that Ker L = C. Thus we have an Ext-orthogonal pair (C, D) for D(Mod A) with D = Im L; see [4, Lemma 3.3]. Now observe that D = C^{⊥} is closed under coproducts, since for any perfect complex X the functor Hom_{D(Mod A)}(X, −) preserves coproducts. It follows that D is a localizing subcategory.

(3) ⇒ (1): Let D be a localizing subcategory that is closed under products such that C = ^{⊥}D. Then Y = H^{0}D is an extension closed abelian subcategory of Mod A that is closed under products and coproducts; see Proposition 2.4. In the proof of Corollary 3.4 we have constructed a localization functor L: D(Mod A) → D(Mod A) such that C = Ker L. More precisely, there exists a homological epimorphism A → B such that L = − ⊗_{A}^{L} B. It remains to notice that this functor preserves coproducts. □

Remark 7.2. The implication (1) ⇒ (2) is known as “telescope conjecture”. Let us sketch the essential ingredients of the proof of this implication. In fact, the proof is not as involved as one might expect from the references to preceding results of this work.

We need the 5-term exact sequence ε_{M} for each module M which one gets immediately from the the localization functor L; see Proposition 2.7. The perfect complexes generating C are constructed in the proof of Theorem 5.1, where the relevant implication is (1) ⇒ (2). For this proof, one uses Lemmas 5.2 – 5.4, but this is all.

8. A bijective correspondence

In this final section we summarize our findings by stating explicitly the correspondence between various structures arising from Ext-orthogonal pairs for hereditary rings.

Theorem 8.1. For a hereditary ring A there are bijections between the following sets:

(1) Ext-orthogonal pairs (X , Y) for Mod A such that Y is closed under coproducts.

(2) Ext-orthogonal pairs (Y, Z) for Mod A such that Y is closed under products.

(3) Extension closed abelian subcategories of Mod A that are closed under products and coproducts.

(4) Extension closed abelian subcategories of mod A.

(5) Homological epimorphisms A → B (up to isomorphism).

(6) Universal localizations A → B (up to isomorphism).

(7) Localizing subcategories of D(Mod A) that are closed under products.

(8) Localization functors D(Mod A) → D(Mod A) preserving coproducts (up to natural isomorphism).

(9) Thick subcategories of D^{b}(mod A).

Proof. We state the bijections explicitly in the following table and give the references to the places where these bijections are established.

Direction  Map  Reference
(1) ↔ (3)  (X , Y) ↦ Y  Corollary 3.4
(2) ↔ (3)  (Y, Z) ↦ Y  Corollary 3.4
(3) → (4) Y ↦ (^{⊥}Y) ∩ mod A Thm. 5.1 & Prop. 5.5
(4) → (3) C ↦ C^{⊥} Thm. 5.1 & Prop. 5.5
(3) → (5) Y ↦ (A → End_{A}(F A))^{2} Proposition 3.2
(5) → (3) f ↦ (Ker f ⊕ Coker f)^{⊥} Proposition 3.1
(5) ↔ (6)  f ↦ f  Theorem 6.1
(3) → (7)  Y ↦ D_{Y}(Mod A)  Proposition 2.4
(7) → (3)  C ↦ H^{0}C  Proposition 2.4
(7) → (8)  C ↦ (X ↦ GX)^{3}  Theorem 7.1
(8) → (7)  L ↦ Im L  Theorem 7.1
(4) → (9)  X ↦ D_{X}^{b} (mod A)  Remark 2.5
(9) → (4)  C ↦ H^{0}C  Remark 2.5
□

Let us mention that this correspondence is related to recent work of some other authors. In [28], Schofield establishes for any hereditary ring the bijection (4) ↔ (6). In [21], Nicolás and Saorín establish for a differential graded algebra A a correspondence between recollements for the derived category D(A) and differential graded homological epimorphisms A → B. This correspondence specializes for a hereditary ring to the bijection (5) ↔ (8).^{4}

A finiteness condition. Given an Ext-orthogonal pair for the category of A-modules as in Theorem 8.1, it is a natural question to ask when its restriction to the category of finitely presented modules yields a complete Ext-orthogonal pair for mod A. This finiteness condition we characterize in terms of finitely presented modules for any finite dimensional algebra; see also Proposition 4.2.

2The functor F denotes a left adjoint of the inclusion Y → Mod A 3The functor G denotes a left adjoint of the inclusion C → D(Mod A)

4The first author is grateful to Manolo Saorín for pointing out this bijection.

Proposition 8.2. Let A be a finite dimensional hereditary algebra over a field and C an extension closed abelian subcategory of mod A. Then the following are equivalent.

(1) There exists a complete Ext-orthogonal pair (C, D) for mod A.

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

(3) There exists an exceptional object X ∈ C such that C is the smallest extension closed abelian subcategory of mod A containing X.

(4) Let (X , Y) be the Ext-orthogonal pair for Mod A generated by C. Then for each M ∈ mod A the 5-term exact sequence ε_{M} belongs to mod A.

Proof. (1) ⇒ (2): For M ∈ mod A let 0 → D_{M} → C_{M} → M → D^{M} → C^{M} → 0 be its 5-term exact sequence. Sending a module M to C_{M} induces a right adjoint for the inclusion C → mod A; see Lemma 2.8.

(2) ⇒ (3): Choose an injective cogenerator Q in mod A and let X denote its image under the right adjoint of the inclusion of C. A right adjoint of an exact functor preserves injectivity. It follows that X is an exceptional object and that C is the smallest extension closed abelian subcategory of mod A containing X.

(3) ⇒ (4): See Proposition 4.2.

(4) ⇒ (1): The property of the pair (X , Y) implies that (X ∩ mod A, Y ∩ mod A) is a complete Ext-orthogonal pair for mod A. An application of Proposition 5.5 yields the equality X ∩ mod A = C. Thus there exists a complete Ext-orthogonal pair (C, D) for mod A. □

Remark 8.3. There is a dual result which is obtained by applying the duality between modules over the algebra A and its opposite A^{op}. Note that condition (3) is self-dual.

References

[1] F. Albrecht, On projective modules over semi-hereditary rings, Proc. Amer. Math. Soc. 12 (1961), 638–639.

[2] L. Alonso Tarrío, A. Jeremías López and M. J. Souto Salorio, Localization in categories of complexes and unbounded resolutions, Canad. J. Math. 52 (2000), no. 2, 225–247.

[3] S. Bazzoni, P. C. Eklof and J. Trlifaj, Tilting cotorsion pairs, Bull. London Math. Soc. 37 (2005), no. 5, 683–696.

[4] D. Benson, S. Iyengar and H. Krause, Local cohomology and support for triangulated categories, to appear in Ann. Sci. École Norm. Sup., arXiv:math/0702610v2.

[5] A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257–281.

[6] K. Brüning, Thick subcategories of the derived category of a hereditary algebra, Homology, Homotopy Appl. 9 (2007), no. 2, 165–176.

[7] P. M. Cohn, Free ideal rings and localization in general rings, Cambridge Univ. Press, Cambridge, 2006.

[8] S. E. Dickson, A torsion theory for Abelian categories, Trans. Amer. Math. Soc. 121 (1966), 223– 235.

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

[10] P. Gabriel and J. A. de la Peña, Quotients of representation-finite algebras, Comm. Algebra 15 (1987), no. 1-2, 279–307.

[11] W. Geigle and H. Lenzing, Perpendicular categories with applications to representations and sheaves, J. Algebra 144 (1991), no. 2, 273–343.

[12] D. Happel, Partial tilting modules and recollement, in Proceedings of the International Conference on Algebra, Part 2 (Novosibirsk, 1989), 345–361, Contemp. Math., Part 2, Amer. Math. Soc., Providence, RI, 1992.

[13] B. Keller, A remark on the generalized smashing conjecture, Manuscripta Math. 84 (1994), no. 2, 193–198.

[14] H. Krause, Cohomological quotients and smashing localizations, Amer. J. Math. 127 (2005), no. 6, 1191–1246.

[15] H. Krause, Derived categories, resolutions, and Brown representability, in Interactions between homotopy theory and algebra, 101–139, Contemp. Math., 436, Amer. Math. Soc., Providence, RI, 2007.

[16] H. Lenzing, Homological transfer from finitely presented to infinite modules, in Abelian group theory (Honolulu, Hawaii, 1983), 734–761, Lecture Notes in Math., 1006, Springer, Berlin, 1983.

[17] M. Mahowald, D. Ravenel and P. Shick, The triple loop space approach to the telescope conjecture, in Homotopy methods in algebraic topology (Boulder, CO, 1999), 217–284, Contemp. Math., 271, Amer. Math. Soc., Providence, RI, 2001.

[18] A. Neeman, The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 547–566.

[19] A. Neeman, The chromatic tower for D(R), Topology 31 (1992), no. 3, 519–532.

[20] A. Neeman, A. Ranicki and A. Schofield, Representations of algebras as universal localizations, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 1, 105–117.

[21] P. Nicolás and M. Saorín, Parametrizing recollement data, arXiv:math/0801.0500v1.

[22] D. C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984), no. 2, 351–414.

[23] I. Reiten and C. M. Ringel, Infinite dimensional representations of canonical algebras, Canad. J. Math. 58 (2006), no. 1, 180–224.

[24] L. Salce, Cotorsion theories for abelian groups, in Symposia Mathematica, Vol. XXIII (Conf. Abelian Groups and their Relationship to the Theory of Modules, INDAM, Rome, 1977), 11–32, Academic Press, London, 1979.

[25] A. H. Schofield, Representation of rings over skew fields, Cambridge Univ. Press, Cambridge, 1985.

[26] A. H. Schofield, Universal localisation for hereditary rings and quivers. Ring theory (Antwerp, 1985), 149–164, Lecture Notes in Math., 1197, Springer, Berlin, 1986.

[27] A. H. Schofield, Semi-invariants of quivers, J. London Math. Soc. (2) 43 (1991), no. 3, 385–395.

[28] A. H. Schofield, Universal localisations of hereditary rings, arXiv:math/0708.0257v1.

Henning Krause, Institut für Mathematik, Universität Paderborn, D-33095 Paderborn, Germany.

   E-mail address: hkrause@math.upb.de

Jan Šťovíček, Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway.

   E-mail address: jan.stovicek@math.ntnu.no