A NOTE ON AUSLANDER-REITEN TRIANGLES AND A THEOREM OF ZIMMERMANN

HENNING KRAUSE

Abstract. A classical theorem of Zimmermann describes the relation between almost split sequences in the category of finitely presented modules and those in the category of all modules over some fixed ring [19]. We prove an analogue for Auslander-Reiten triangles in triangulated categories. This is used to explain the relation between different existence results for Auslander-Reiten triangles which are based either on Brown’s representability theorem [12], or on the existence of Serre functors [16].

1. Introduction

This note is a continuation of [12] where Brown representability was introduced as a foundation for a general Auslander-Reiten theory in triangulated categories. More recently, Auslander-Reiten triangles played an important role in work of Reiten, Van den Bergh [16], and Jørgensen [10]. In [16], the existence of Auslander-Reiten triangles is translated into the existence of a Serre functor in the sense of Bondal and Kapranov [4]. This is used for a classification of the noetherian hereditary abelian categories satisfying Serre dualtity. In [10], the singular cochain differential graded algebra of a simply connected space is studied. The existence of Auslander-Reiten triangles over this algebra characterizes Poincaré duality spaces.

Our aim in this note is to explain some relation between the approach in [12] which is based on Brown representability, and the existence theorems for Auslander-Reiten triangles which are based on variants of Serre duality. It was Neeman [14] who noticed that Brown representability can be applied to prove Grothendieck’s duality theorem which contains Serre duality as a special case. The extra ingredient which is relevant for the existence of Auslander-Reiten triangles, is the analogue of a theorem of Zimmermann [19] about the existence of almost split sequences in module categories. We present Zimmermann’s result in a form which is different from his own account. In particular, some unnecessary assumptions are removed, and this might be of independent interest. An alternative discussion of Zimmermann’s result can be found in work of Herzog [8], who uses some ideas from model theory, in particular those of Ziegler [17]. More recently, Beligiannis [3] analyses the Ziegler spectrum of a triangulated category and establishes a connection with its Auslander-Reiten theory.

2. A theorem of Zimmermann

Let A be an associative ring with identity. We consider the category Mod A of (right) A-modules and denote by mod A the full subcategory which is formed by the finitely presented A-modules. In this section we present a theorem of Zimmerman on the existence of almost split sequences. For the definition and the basic properties of almost split maps and sequences, we refer to [1, 2]. Recall that a map α: X → Y of A-modules is a pure monomorphism if it induces an exact sequence

0 ⟶ Hom_{A}(C, X) ⟶ Hom_{A}(C, Y) ⟶ Hom_{A}(C, Coker α) ⟶ 0

for every finitely presented A-module C. For basic facts about purity in module categories, we refer to [9].

Theorem (Zimmermann). Let Z be a finitely presented and non-projective module over some associative ring A. Suppose Z has a local endomorphism ring, and let 0 → X → Y → Z → 0 be an almost split sequence in the category of A-modules (which always exists). Then there exists an almost split sequence 0 → X′ → Y′ → Z → 0 in the category of finitely presented A-modules if and only if X has a finitely presented non-zero pure submodule.

This result appears as Theorem 1 in [19], where the ring A is assumed to be semiperfect and the pure submodule of X is assumed to have a local endomorphism ring. We deduce Zimmermann’s theorem from the following result which generalizes an earlier result of Zimmermann; see [18, Proposition 3].

Proposition 2.1. Suppose there is given the following map between short exact sequences in the category of A-modules.

(2.1)  [figure: δ′ :; 0; X′; Y′; Z; 0; δ :; 0; X; Y; Z; 0]

Let the modules in the first row be finitely presented and the map X′ → X be a pureinjective envelope. Then the following are equivalent:

(1) δ is an almost split sequence in the category of all A-modules.

(2) δ′ is an almost split sequence in the category of finitely presented A-modules.

Proof. It follows from classical Auslander-Reiten theory that an exact sequence 0 → L → M → N → 0 is almost split if and only if the map L → M is left almost split and the endomorphism ring of N is local; see [1, Proposition II.4.4]. Recall that α: L → M is left almost split if α is not a section and if every map L → M′ which is not a section factors through α.

(1) ⇒ (2): We assume that X → Y is left almost split and need to show that X′ → Y′ is left almost split. Clearly, X′ → Y′ is not a section since X → Y is not a section. Now let X′ → M′ be a map in mod A which is not a section, and form the following push-out.

[figure: 0; X′; X; X/X′; 0; 0; M′; M; X/X′; 0]

If X → M is a section, then the map X′ → M′ is a pure monomorphism and is therefore a section since the cokernel of X′ → M′ is finitely presented. Thus X → M factors through X → Y and therefore X′ → M′ factors through X′ → Y′. It follows that X′ → Y′ is left almost split.

(2) ⇒ (1): We assume that X′ → Y′ is left almost split and need to show that X → Y is left almost split. If X → Y were a section, then X′ → X would factor through X′ → Y′. Thus X′ → Y′ would be a pure monomorphism, contradicting the fact that X′ → Y′ is not a section. Thus X → Y is not a section. Now let X → M be a map in Mod A which is not a section. We consider the composite α: X′ → M with X′ → X. Note that α is not a pure monomorphism since X′ → X is a pure-injective envelope. The maps α_{i}: X′ → M_{i} in mod A such that α factors through α_{i} form a filtered system, and its colimit is α. There is an index j such that α_{j} is not a section because a filtered colimit of sections is a pure monomorphism. Thus α_{j} factors through X′ → Y′. We conclude that X → M factors through X → Y because the diagram (2.1) is a push-out. Thus X → Y is left almost split, and this completes the proof. □

The proof of Proposition 2.1 shows the following.

Corollary 2.2. Let X′ → Y′ be a map between finitely presented A-modules and let the following diagram be the push-out with a pure-injective envelope X′ → X.

\xymatrix{X′ \ar[r]^{α′} \ar[d] & Y′ \ar[d] \\ X \ar[r]^{α} & Y}

Then α is left almost split in Mod A if and only if α′ is left almost split in mod A.

Proof of Zimmermann’s theorem. Let X′ ≠ 0 be a finitely presented pure submodule of X. The projection X → X/X′ factors through X → Y because X → Y is a left almost split map. Thus we obtain the following commutative diagram

[figure: δ′ :; 0; X′; Y′; Z; 0; δ :; 0; X; Y; Z; 0; X/X′; X/X′; 0; 0]

with exact rows and columns. We conclude from Proposition 2.1 that the sequence δ′ is almost split. To prove the converse, suppose there is an almost split sequence 0 → X′ → Y′ → Z → 0 in the category of finitely presented A-modules. Let X′ → X′′ be a pure-injective envelope which induces the following commutative diagram

[figure: δ′ :; 0; X′; Y′; Z; 0; δ′′ :; 0; X′′; Y′′; Z; 0]

with exact rows. It follows from Proposition 2.1 that the sequence δ′′ is almost split. Thus X′′ ≅ X and we have therefore a finitely presented pure submodule of X. This completes the proof. □

3. Auslander-Reiten triangles for compact objects

In this section we prove the analogue of Zimmermann’s theorem for triangulated categories. We fix a compactly generated triangulated category T . Thus T has arbitrary coproducts and is generated by a set of compact objects. Recall that an object X in T is compact if for every family (Y_{i})_{i∈I} in T , the canonical map

∐_{i} Hom(X, Y_{i}) ⟶ Hom(X, ∐_{i} Y_{i})

is an isomorphism. We denote by T_{c} the full subcategory of compact objects in T and observe that T_{c} is a triangulated subcategory of T . For basic properties of compactly generated triangulated categories, we refer to [14]. The analogue of an almost split sequence for a triangulated category was introduced by Happel [5].

Definition 3.1. A triangle X →^{α} Y →^{β} Z →^{γ} X[1] is called Auslander-Reiten triangle if the following conditions hold:

(1) every map X → Y′ which is not a section factors through α;

(2) every map Y′ → Z which is not a retraction factors through β;

(3) γ ≠ 0.

Note that the end terms X and Z of an Auslander-Reiten triangle are indecomposable objects with local endomorphism rings. Moreover, each end term determines an Auslander-Reiten triangle up to isomorphism.

Now fix a compact object Z. Suppose that the endomorphism ring Γ = End(Z) is local and denote by E a minimal injective cogenerator in the category of Γ-modules. Brown’s representability theorem [14] provides a representing object T Z in T such that

(3.1)  Hom_{Γ}(Hom(Z, −), E) ≅ Hom(−, T Z).

This yields an Auslander-Reiten triangle

(T Z)[−1] ⟶ Y ⟶ Z ⟶ T Z

in T , where the map Z → T Z corresponds under (3.1) to a non-zero map Hom(Z, Z) → E annihilating the radical of Γ; see [12, Theorem 2.2].

Next we recall from [11] the notion of purity in T . A map X → Y in T is a pure monomorphism if the induced map Hom(C, X) → Hom(C, Y) is a monomorphism for every compact object C. An object X is pure-injective if every pure monomorphism X → Y is a split monomorphism. Finally, a map α: X → Y is a pure-injective envelope if Y is pure-injective and if the composite β^{∘} α with a map β: Y → Z is a pure monomorphism if and only if β is a pure monomorphism. We have the following analogue of Proposition 2.1.

Proposition 3.2. Suppose there is given the following map between triangles in a compactly generated triangulated category.

(3.2)  [figure: δ′ :; X′; Y′; Z; X[1]; δ :; X; Y; Z; X[1]]

Let the objects in the first triangle be compact and the map X′ → X be a pure-injective envelope. Then the following are equivalent:

(1) δ is an Auslander-Reiten triangle in the category of all objects.

(2) δ′ is an Auslander-Reiten triangle in the category of compact objects.

Recall that a map α: X → Y is left almost split if it is not a section and if every map X → X′ which is not a section factors through α. We need the following simple reformulation of this definition.

Lemma 3.3. For a triangle X →^{α} Y →^{β} Z →^{γ} X[1] the following are equivalent: (1) Every map X → X′ which is not a section factors through α.

(2) The map γ[−1] factors through every non-zero map X′′ → X.

Proof. Clear, since a map X → X′ is not a section if and only if its fibre X′′ → X is non-zero. □

Proof of Proposition 3.2. We use the fact that a triangle X → Y → Z → X[1] is an Auslander-Reiten triangle if and only if End(Z) is local and the map X → Y is left almost split; see [12, Lemma 2.6]. Rotating the triangles δ and δ′, we obtain from (3.2) the following commutative diagram.

[figure: Z[−1]; γ′[−1]; X′; ϕ; α′; Y′; ψ; β′; Z; Z[−1]; γ[−1]; X; α; Y; β; Z]

Note that our assumption on ϕ implies that α is not a section if and only if α′ is not a section.

(1) ⇒ (2): We assume that α is left almost split and need to show that α′ is left almost split. We want to apply Lemma 3.3 and choose a non-zero map σ: C → X′ from a compact object C. The composite ϕ^{∘} σ is non-zero since ϕ is a pure mono. Thus γ[−1] factors through ϕ^{∘} σ since α is left almost split. We obtain γ[−1] = ϕ^{∘} σ^{∘} ρ for some map ρ: Z[−1] → C. This implies γ′[−1] = σ^{∘} ρ, since γ[−1] = ϕ^{∘} γ′[−1] and ϕ is a pure mono. We conclude that α′ is left almost split.

(2) ⇒ (1): We assume that α′ is left almost split and need to show that α is left almost split. To this end choose a map ρ: X → M which is not a section. Thus ρ is not a pure mono because X is pure-injective. This implies ρ^{∘} ϕ is not a pure mono since ϕ is a pure-injective envelope. So there exists a non-zero map σ: C → X′ from a compact object C such that the composite with ρ^{∘} ϕ is zero. The map γ′[−1] factors through σ by Lemma 3.3 since α′ is left almost split by our assumption. This implies ρ^{∘} γ[−1] = 0 since ρ^{∘} ϕ^{∘} σ = 0. Thus ρ factors through α. This completes the proof. □

We leave it to the reader to formulate the analogue of Corollary 2.2. The following consequence is the analogue of Zimmermann’s existence result for almost split sequences in the category of finitely presented modules.

Corollary 3.4. Let Z be a compact object with local endomorphism ring and let X → Y → Z → X[1] be an Auslander-Reiten triangle in the category of all objects (which always exists). Then there exists an Auslander-Reiten triangle X′ → Y′ → Z → X′[1] in the category of compact objects if and only if X has a compact non-zero pure subobject.

Proof. Suppose first there is a pure monomorphism X′ → X such that X′ is a non-zero compact object. Complete the map to a triangle X′ → X → X′′ → X′[1] and observe that X → X′′ factors through the map X → Y since X → Y is left almost split. Using the octahedral axiom, we obtain a commutative diagram of the form (3.2). We conclude from Proposition 3.2 the existence of an Auslander-Reiten triangle X′ → Y′ → Z → X′[1] in the category of compact objects.

To prove the converse, suppose that we have an Auslander-Reiten triangle X′ → Y′ → Z → X′[1] in the category of compact objects. Choose a pure-injective envelope X′ → X′′ which exists by [11, Corollary 1.13]. Compose this map with Z[−1] → X′ and complete the composite to a triangle Z[−1] → X′′ → Y′′ → Z. This gives rise to the following map

[figure: δ :; X′; Y′; Z; X[1]; δ′′ :; X′′; Y′′; Z; X′′[1]]

between triangles. It follows from Proposition 3.2 that δ′′ is an Auslander-Reiten triangle. Thus X′′ ≅ X and we have therefore a compact pure subobject of X. This completes the proof. □

Examples of Auslander-Reiten triangles in the category of compact objects, which are not Auslander-Reiten triangles in the category of all objects, can be obtained from Zimmerman’s work [19]. He constructs almost split sequences in the category of finitely presented modules over some hereditary artinian ring A which are not almost split in the category of all modules. The canonical embedding

Mod A ⟶ D(Mod A)

identifies such a sequence with an Auslander-Reiten triangle in the bounded derived category D^{b}(mod A) which is not an Auslander-Reiten triangle when viewed as a triangle in the unbounded derived category D(Mod A). This follows for example from the fact that an A-module is pure-injective if and only if it is pure-injective when viewed as a complex in D(Mod A) which is concentrated in a single degree. Note that the inclusion

D^{b}(mod A) ⟶ D(Mod A)

identifies D^{b}(mod A) with the compact objects in D(Mod A) since A has finite global dimension.

4. Brown representability and Serre duality

Brown’s representability theorem has been used to construct an Auslander-Reiten triangle for each compact object having a local endomorphism ring. In this section, we make this construction functorial and obtain in this way the connection with the Serre functors arising in the work of Bondal and Kapranov [4], and also in recent work of Reiten and Van den Bergh [16].

Throughout this section we fix a commutative noetherian ring k which is complete local, that is, k = \underleftarrow{lim} k/𝔪^{i} where 𝔪 denotes the unique maximal ideal of k. We assume that T is a compactly generated k-linear category such that Hom(X, Y) is a finitely generated k-module for every pair of compact objects X, Y. We denote by D = Hom_{k}(−, E): Mod k → Mod k the duality which is induced by the injective envelope E = E(k/𝔪).

Lemma 4.1. There exists a fully faithful functor T: T_{c} → T together with a natural isomorphism

(4.1)  D Hom(X, Y) ≅ Hom(Y, T X)

for every compact object X and every object Y in T .

Proof. Given a compact object X in T , we apply Brown’s representability theorem [14] and obtain for D Hom(X, −) a representing object T X in T such that

D Hom(X, −) ≅ Hom(−, T X).

The assignment X ↦ T X induces a fully faithful functor T: T_{c} → T since we have a natural map

Hom(X, Y) ⟶^{∼} D^{2} Hom(X, Y) ⟶^{∼} D Hom(Y, T X) ⟶^{∼} Hom(T X, T Y)

where the first map is an isomorphism by our assumption on Hom(X, Y) to be finitely generated over k. □

The functor T provides the Auslander-Reiten translation in T .

Proposition 4.2. Let Z be a compact object which is indecomposable. Then there exists an Auslander-Reiten triangle (T Z)[−1] → Y → Z → T Z.

Proof. First observe that End(Z) is local because it is a noetherian k-algebra. Now apply Theorem 2.2 from [12]. □

The k-linear structure of T implies the pure-injectivity of the compact objects.

Lemma 4.3. Every compact object in T is pure-injective.

Proof. Fix a compact object X in T . We use the category F = (T_{c}^{op}, Ab) of additive functors from compact objects to abelian groups. The duality D: Mod k → Mod k induces a natural map F → D^{2}F for each F in F, which is an isomorphism for H_{X} = Hom(−, X)|_{T_{c}}. Thus H_{X} is a pure-injective object in F; see [1, Proposition I.3.8]. It follows from [11, Theorem 1.8] that X is a pure-injective object in T . □

We say that a triangulated category has Auslander-Reiten triangles if for every indecomposable object X, there are Auslander-Reiten triangles of the form X → Y → Z → X[1] and V → W → X → V [1].

Theorem 4.4. The following are equivalent for T .

(1) The category T_{c} of compact objects in T has Auslander-Reiten triangles.

(2) An object in T is compact if and only it it is isomorphic to some object in the image of T.

(3) The functor T: T_{c} → T defined by (4.1) induces an equivalence T_{c} → T_{c}.

Proof. (1) ⇒ (2): We need to show that an object X in T is compact if and only if X ≅ T X′ for some compact object X′. Suppose first that X is compact. Let X → Y → Z → X[1] be the corresponding Auslander-Reiten triangle. Then X is a pure subobject of (T Z)[−1] by Corollary 3.4 and Proposition 4.2. However, X is pure-injective by Lemma 4.3, and therefore isomorphic to (T Z)[−1] ≅ T (Z[−1]). The same argument shows that every object in the image of T is compact.

(2) ⇒ (1): Apply Proposition 4.2.

(2) ⇔ (3): Clear, since T is fully faithful. □

Let us comment on how to apply this result. Suppose we have given a triangulated category C and we ask if C has Auslander-Reiten triangles. In many cases, there is a natural choice for a compactly generated triangulated category T such T_{c} is equivalent to C; see the examples further below. In fact, many interesting examples are of the form C = D^{b}(A) for some exact category A in the sense of Quillen [15], where D^{b}(A) denotes the bounded derived category of A. One can show [13] that A carries a DG structure such that C is equivalent to the category of compact objects in the unbounded derived category of DG modules over A. Thus Theorem 4.4 can be applied to characterize the existence of Auslander-Reiten triangles in C.

We end this paper by giving some examples which illustrate our approach towards the existence of Auslander-Reiten triangles.

(1) Let A be a k-algebra and denote by D(Mod A) the unbounded derived category of Mod A. Recall that a complex of R-modules is perfect if it is quasi-isomorphic to a bounded complex of finitely generated projectives. The perfect complexes are precisely the compact objects in D(Mod A), and we identify them with the homotopy category K^{b}(proj A) of finitely generated projective modules.

Now fix complexes X and Y in D(Mod A) and suppose X is perfect. Note that we have isomorphisms

RHom_{A}(X, Y) ≅ Y ⊗_{A}^{L} X^{∗} and X ≅ X^{∗∗}

where X^{∗} = RHom_{A}(X, A). Using the adjointness of RHom and ⊗^{L}, and viewing the injective k-module E as a complex concentrated in degree zero, we obtain the following isomorphism

(4.2)  D Hom_{D(A)}(X, Y) ≅ Hom_{k}(H^{0}(RHom_{A}(X, Y)), E)
≅ Hom_{D(k)}(RHom_{A}(X, Y), E)
≅ Hom_{D(k)}(Y ⊗_{A}^{L} X^{∗}, E)
≅ Hom_{D(A)}(Y, RHom_{k}(X^{∗}, E))
≅ Hom_{D(A)}(Y, RHom_{k}(A ⊗_{A}^{L} X^{∗}, E))
≅ Hom_{D(A)}(Y, RHom_{A}(X^{∗}, RHom_{k}(A, E)))
≅ Hom_{D(A)}(Y, X ⊗_{A}^{L} Hom_{k}(A, E))

which is natural in X and Y. We might view this as an analogue of Serre duality where the dualizing complex is the A^{op}-module DA = Hom_{k}(A, E) concentrated in degree zero. Now suppose the algebra A is noetherian over k. Then the functor T defined via (4.1) identifies with the restriction of

− ⊗_{A}^{L} DA: D(Mod A) ⟶ D(Mod A)

to K^{b}(proj A). This functor preserves compactness if and only if the right adjoint RHom_{A}(DA, −) preserves coproducts; see [14, Theorem 5.1]. By definition, this happens if and only if DA is a compact object in D(Mod A). Note that an A-module, when viewed as a complex concentrated in a single degree, is compact if and only if it is finitely generated and has finite projective dimension. Also, observe that DA is artinian over k. It follows that − ⊗_{A}^{L} DA induces a functor K^{b}(proj A) → K^{b}(proj A) if and only if A as an A^{op}-module has finite composition length and is of finite injective dimension. Of course, we have the same result for the opposite algebra A^{op}, and combining both we conclude that T induces an equivalence K^{b}(proj A) → K^{b}(proj A) if and only A is an artin k-algebra which is Gorenstein, that is, A has finite injective dimension when viewed as an A-module and also as an A^{op}-module. In particular, we obtain Happel’s criterion [5] for the existence of Auslander-Reiten triangles in K^{b}(proj A) if we apply Theorem 4.4. This result carries over to differential graded algebras; see [10].

(2) Let A be an artin k-algebra and denote by K(Proj A) the homotopy category of complexes of projective A-modules. This category is compactly generated and the bounded derived category D^{b}(mod A) of mod A is equivalent to the category of compact objects in K(Proj A). The functor

T: D^{b}(mod A) ⟶ K(Proj A)

induces an equivalence D^{b}(mod A) → D^{b}(mod A) if and only if A has finite global dimension; see [6].

(3) Let k be a field and let 𝕏 be a projective scheme over k, that is, we have a projective morphism

f: 𝕏 ⟶ 𝕐 = Spec k.

The derived category D(Qcoh 𝕏) of quasi coherent sheaves on 𝕏 is compactly generated, and the subcategory of compact objects is equivalent to the bounded derived category D^{b}(Vect 𝕏) of vector bundles on 𝕏; see [14, Example 1.10]. In [14], Neeman shows that Grothendieck’s duality theorem is a formal consequence of Brown’s representability theorem, that is, the functor

Rf_{∗}: D(Qcoh 𝕏) ⟶ D(Qcoh 𝕐)

has a right adjoint

f^{!} : D(Qcoh 𝕐) ⟶ D(Qcoh 𝕏).

This yields Serre duality for 𝕏.

Fix complexes X and Y in D(Qcoh 𝕏) and suppose X is perfect. Note that we have isomorphisms

RHom_{𝕏}(X, Y) ≅ Y ⊗_{O_{𝕏}}^{L} X^{∗} and X ≅ X^{∗∗}

where X^{∗} = RHom_{𝕏}(X, O_{𝕏}). Using Grothendieck duality, we obtain the following isomorphism

(4.3)  D Hom_{D(𝕏)}(X, Y) ≅ Hom_{D(𝕐)}(Rf_{∗} RHom_{𝕏}(X, Y), k)
≅ Hom_{D(𝕏)}(RHom_{𝕏}(X, Y), f^{!}k)
≅ Hom_{D(𝕏)}(Y ⊗_{O_{𝕏}}^{L} X^{∗}, f^{!}k)
≅ Hom_{D(𝕏)}(Y, RHom_{𝕏}(X^{∗}, f^{!}k))
≅ Hom_{D(𝕏)}(Y, X ⊗_{O_{𝕏}}^{L} f^{!}k)

which is natural in X and Y. Notice the analogy between (4.2) and (4.3). There is a notion for a scheme to be Gorenstein (see [7, page 144]), which means that the dualizing complex f^{!}k is isomorphic to an invertible sheaf and therefore inducing an equivalence

− ⊗_{O_{𝕏}}^{L} f^{!}k: D(Qcoh 𝕏) ⟶ D(Qcoh 𝕏).

For instance, if 𝕏 is smooth of dimension n, then this equivalence takes a familiar form since f^{!}k = Ω_{𝕏/k}^{n}[n]. In particular, the category D^{b}(Vect 𝕏) has Auslander-Reiten triangles.

Acknowledgement. I would like to thank Amnon Neeman for his help with the algebraic geometry in this paper. Additional thanks to Apostolos Beligiannis for helpful comments on this material.

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Henning Krause, Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom.

E-mail address: henning@maths.leeds.ac.uk