THE AUSLANDER-REITEN FORMULA FOR COMPLEXES OF MODULES

HENNING KRAUSE AND JUE LE

Abstract. An Auslander-Reiten formula for complexes of modules is presented. This formula contains as a special case the classical Auslander Reiten formula. The Auslander-Reiten translate of a complex is described explicitly, and various applications are discussed.

1. Introduction

The classical Auslander-Reiten formula for modules over a noetherian algebra Λ says that

D Ext_{Λ}^{1}(M, N) ≅ \overline{Hom}_{Λ}(N, D Tr M)

whenever M is finitely generated [2]. Here, D denotes the duality over a fixed commutative ground ring k. In this paper, we extend this to a formula for complexes of modules. We take as morphisms chain maps up to homotopy and obtain the formula for D Ext_{Λ}^{1}(M, N) as a special case by applying it to injective resolutions iM and iN.

Various authors noticed the analogy between the Auslander-Reiten formula and Serre duality for categories of sheaves; see for instance [17, 7], and see [5] for a formulation of Serre duality in terms of extension groups. Passing from abelian categories to their derived categories, further analogies have been noticed, in particular in connection with the existence of Auslander-Reiten triangles [12, 16].

There is the following common setting for proving such duality formulas. Let T be a klinear triangulated category which is compactly generated. Then one can apply Brown’s representability theorem and has for any compact object X a representing object tX such that

D Hom_{T} (X, −) ≅ Hom_{T} (−, tX).

In this paper, we take for T the category K(Inj Λ) of complexes of injective Λ-modules up to homotopy. Then we can prove that tX = pX ⊗_{Λ} DΛ, where pX denotes the projective resolution of X.

There is a good reason to consider the category of complexes K(Inj Λ). The injective resolutions of all finitely generated modules generate the full subcategory of compact objects, which therefore is equivalent to the bounded derived category D^{b}(mod Λ) of the category mod Λ of finitely generated Λ-modules.

Our identification of the translation t has various interesting consequences. For instance, we can describe Auslander-Reiten triangles in K(Inj Λ), and we get a simple recipe for computing almost split sequences in the category Mod Λ of Λ-modules which seems to be new.

Version from December 1, 2004.

There is another method for computing Auslander-Reiten triangles in K(Inj Λ). This is based on the construction of an adjoint for Happel’s functor

D^{b}(mod Λ) ⟶ \underline{mod} Λ̂

into the stable module category of the repetitive algebra Λ̂ [8]. To be precise, we extend Happel’s functor to a functor K(Inj Λ) → \underline{Mod} Λ̂ on unbounded complexes, and this admits a right adjoint which preserves Auslander-Reiten triangles.

The Auslander-Reiten theory for complexes has been initiated by Happel. In [8, 9], he introduced Auslander-Reiten triangles and characterized their existence in the derived category D^{b}(mod Λ). This pioneering work has been extended by various authors. More recently, Bautista et al. establish in [3] the existence of almost split sequences in some categories of complexes of fixed size.

The methods in this paper might be of interest not only for studying module categories. In fact, they can be applied to other more general settings. To illustrate this point, we include an Auslander-Reiten formula for computing Ext_{A}^{1}(−, −) in any locally noetherian Grothendieck category A.

2. The homotopy category of injectives

Let k be a commutative noetherian ring which is complete and local. Throughout this paper, we fix a noetherian k-algebra Λ, that is, a k-algebra which is finitely generated as a module over k.

We consider the category Mod Λ of (right) Λ-modules and the following full subcategories:

mod Λ = the finitely presented Λ-modules,

Inj Λ = the injective Λ-modules, Proj Λ = the projective Λ-modules, proj Λ = the finitely generated projective Λ-modules.

Note that the assumptions on Λ imply that every finitely generated Λ-module decomposes essentially uniquely into a finite coproduct of indecomposable modules with local endomorphism rings.

In addition, we fix an injective envelope E = E(k/𝔪), where 𝔪 denotes the unique maximal ideal of k. We obtain a duality

D = Hom_{k}(−, E): Mod k ⟶ Mod k

which induces a duality between Mod Λ and Mod Λ^{op}.

Given any additive category A, we denote by C(A) the category of cochain complexes in A, and we write K(A) for the category of cochain complexes up to homotopy. If A is abelian, the derived category is denoted by D(A). We refer to [19] for further notation and terminology concerning categories of complexes.

We denote by

p: D(Mod Λ) ⟶ K(Proj Λ)

the left adjoint of the composite

K(Proj Λ) ⟶^{inc} K(Mod Λ) ⟶^{can} D(Mod Λ)

which sends a complex X to its projective resolution pX. Dually, we denote by

i: D(Mod Λ) ⟶ K(Inj Λ) the right adjoint of the composite

K(Inj Λ) ⟶^{inc} K(Mod Λ) ⟶^{can} D(Mod Λ)

which sends a complex X to its injective resolution iX. For the existence of p and i, see [18, 4].

We shall work in the category K(Inj Λ). This is a triangulated category with arbitrary coproducts. We denote by K^{c}(Inj Λ) the full subcategory which is formed by all compact objects. Recall that an object X is compact if every map X → ∐_{i∈I} Y_{i} factors through ∐_{i∈J} Y_{i} for some finite J ⊆ I. Let us collect from [13] the basic properties of K(Inj Λ).

Lemma 2.1. An object in K(Inj Λ) is compact if and only if it is isomorphic to a complex X satisfying

(1) X^{n} = 0 for n ≪ 0,

(2) H^{n}X is finitely generated over Λ for all n, and

(3) H^{n}X = 0 for n ≫ 0.

Lemma 2.2. The canonical functor K(Inj Λ) → D(Mod Λ) induces an equivalence

K^{c}(Inj Λ) ⟶ D^{b}(mod Λ).

Lemma 2.3. The triangulated category K(Inj Λ) is compactly generated, that is, it coincides with the smallest full triangulated subcategory closed under all coproducts and containing all compact objects.

3. The Auslander-Reiten formula

In this section, the Auslander-Reiten formula for complexes is proved. We begin with a number of simple lemmas. Given a pair of complexes X, Y of modules over Λ or Λ^{op}, we denote by Hom_{Λ}(X, Y) and X ⊗_{Λ} Y the total Hom and the total tensor product respectively, which are complexes of k-modules.

Lemma 3.1. Let X, Y be complexes in C(Mod Λ). Then we have in C(Mod k) a natural map

(3.1)  Y ⊗_{Λ} Hom_{Λ}(X, Λ) ⟶ Hom_{Λ}(X, Y),

which is an isomorphism if X ∈ C^{−}(proj Λ) and Y ∈ C^{+}(Mod Λ).

Proof. Given Λ-modules M and N, we have a map

σ: N ⊗_{Λ} Hom_{Λ}(M, Λ) ⟶ Hom_{Λ}(M, N)

which is defined by

σ(n ⊗ ϕ)(m) = nϕ(m).

This map is an isomorphism if M is finitely generated projective and extends to an isomorphism of complexes provided X and Y are bounded in the appropriate direction.

□

Lemma 3.2. Let M, N be Λ-modules and suppose that M is finitely presented. Then we have an isomorphism

(3.2)  M ⊗_{Λ} Hom_{k}(N, E) ≅ Hom_{k}(Hom_{Λ}(M, N), E).

Proof. We have the isomorphism for M = Λ and therefore whenever M has a presentation Λ^{n} → Λ^{m} → M → 0, since − ⊗_{Λ} Hom_{k}(N, E) and Hom_{k}(Hom_{Λ}(−, N), E) are both right exact. □

Lemma 3.3. Let X, Y be complexes of Λ-modules. Then we have an isomorphism

(3.3)  H^{0} Hom_{Λ}(X, Y) ≅ Hom_{K(Mod Λ)}(X, Y).

Proof. Taking cycles of Hom_{Λ}(X, Y) in degree zero picks the chain maps X → Y, and boundaries in degree zero form the subgroup of null-homotopic chain maps. Thus H^{0} Hom_{Λ}(X, Y) equals the set of chain maps X → Y up to homotopy. □

Let us consider the following commutative diagram

[figure: D^{−}(Mod Λ); K^{−}(Proj Λ); ∼; K^{c}(Inj Λ); ∼; D^{b}(mod Λ); K^{−,b}(proj Λ); ∼; K^{+}(Inj Λ); ∼; D^{+}(Mod Λ)]

in which all horizontal functors are obtained by restricting the localization functor K(Mod Λ) → D(Mod Λ) to appropriate subcategories. We denote by

π: K^{c}(Inj Λ) ⟶ K^{−,b}(proj Λ)

the composite of the equivalence K^{c}(Inj Λ) → D^{b}(mod Λ) with a quasi-inverse of the equivalence K^{−,b}(proj Λ) → D^{b}(mod Λ). Note that πX ≅ pX.

Theorem 3.4. Let X and Y be complexes of injective Λ-modules. Suppose that X^{n} = 0 for n ≪ 0, that H^{n}X is finitely generated over Λ for all n, and that H^{n}X = 0 for n ≫ 0. Then we have an isomorphism

(3.4)  D Hom_{K(Inj Λ)}(X, Y) ≅ Hom_{K(Inj Λ)}(Y, (πX) ⊗_{Λ} DΛ)

which is natural in X and Y.

Proof. We use the fact that K(Inj Λ) is compactly generated. Therefore it is sufficient to verify the isomorphism for every compact object Y. This follows from the subsequent Lemma 3.5. Thus we suppose that Y is a compact object in K(Inj Λ). Note that this implies Y^{n} = 0 for n ≪ 0, and in particular Y ≅ iY. We obtain the following sequence of isomorphisms, where short arguments are added on the right hand side.

D Hom_{K(Inj Λ)}(X, Y) ≅ Hom_{k}(Hom_{K(Inj Λ)}(X, iY), E)  Y compact
≅ Hom_{k}(Hom_{D(Mod Λ)}(X, Y), E)  adjunction
≅ Hom_{k}(Hom_{K(Mod Λ)}(πX, Y), E)  adjunction
≅ Hom_{k}(H^{0} Hom_{Λ}(πX, Y), E)  from (3.3)
≅ H^{0} Hom_{k}(Hom_{Λ}(πX, Y), E)  E injective
≅ H^{0} Hom_{k}(Y ⊗_{Λ} Hom_{Λ}(πX, Λ), E)  from (3.1)
≅ H^{0} Hom_{Λ}(Y, Hom_{k}(Hom_{Λ}(πX, Λ), E))  adjunction
≅ H^{0} Hom_{Λ}(Y, (πX) ⊗_{Λ} Hom_{k}(Λ, E))  from (3.2)
≅ Hom_{K(Inj Λ)}(Y, (πX) ⊗_{Λ} DΛ)  from (3.3)

This isomorphism completes the proof. □

Lemma 3.5. Let X, X′ be objects in a k-linear compactly generated triangulated category. Suppose that X is compact. If there is a natural isomorphism

D Hom_{T} (X, Y) ≅ Hom_{T} (Y, X′)

for all compact Y ∈ T , then D Hom_{T} (X, −) ≅ Hom_{T} (−, X′).

Proof. We shall use Theorem 1.8 in [11], which states the following equivalent conditions for an object Y in T .

(1) The object Y is pure-injective.

(2) The object H_{Y} = Hom_{T} (−, Y)|_{T^{c}} is injective in the category Mod T^{c} of contravariant additive functors T^{c} → Ab.

(3) The map Hom_{T} (Y′, Y) → Hom_{T^{c}}(H_{Y′}, H_{Y}) sending ϕ to H_{ϕ} is bijective for all Y′ in T .

Here, T^{c} denotes the full subcategory of compact objects in T .

We apply Brown’s representability theorem (see [10, 5.2] or [14, Theorem 3.1]) and obtain an object X′′ such that

D Hom_{T} (X, −) ≅ Hom_{T} (−, X′′),

since X is compact. Condition (2) implies that both objects X′ and X′′ are pureinjective, since Hom_{T^{c}}(X, −) is a projective object in the category of covariant additive functors T^{c} → Ab, by Yoneda’s lemma. We have an isomorphism

H_{X′} = Hom_{T} (−, X′)|_{T^{c}} ≅ D Hom_{T} (X, −)|_{T^{c}} ≅ Hom_{T} (−, X′′)|_{T^{c}} = H_{X′′},

and (3) implies that this isomorphism is induced by an isomorphism X′ → X′′ in T . We conclude that

D Hom_{T} (X, −) ≅ Hom_{T} (−, X′).
□

4. The Auslander-Reiten translation

In this section, we investigate the properties of the Auslander-Reiten translation for complexes of Λ-modules. The Auslander-Reiten translation D Tr for modules is obtained from the translation for complexes. In particular, we deduce the classical Auslander-Reiten formula.

Proposition 4.1. The functor

t: K(Inj Λ) ⟶^{can} D(Mod Λ) ⟶^{p} K(Proj Λ) ⟶^{−⊗_{Λ}DΛ} K(Inj Λ)

has the following properties.

(1) t is exact and preserves all coproducts.

(2) For compact objects X, Y in K(Inj Λ), the natural map

Hom_{K(Inj Λ)}(X, Y) ⟶ Hom_{K(Inj Λ)}(tX, tY)

is bijective.

(3) For X, Y in K(Inj Λ) with X compact, there is a natural isomorphism

D Hom_{K(Inj Λ)}(X, Y) ≅ Hom_{K(Inj Λ)}(Y, tX).

(4) t admits a right adjoint which is i Hom_{Λ}(DΛ, −).

Proof. (1) is clear and (3) follows from (3.4). Now observe that for each pair X, Y of compact objects, the k-module Hom_{K(Inj Λ)}(X, Y) is finitely generated. Therefore (2) follows from (3), since we have the isomorphism

Hom_{K(Inj Λ)}(X, Y) ≅ D^{2} Hom_{K(Inj Λ)}(X, Y)
≅ D Hom_{K(Inj Λ)}(Y, tX)
≅ Hom_{K(Inj Λ)}(tX, tY).

To prove (4), let X, Y be objects in K(Inj Λ). Then we have

Hom_{K(Mod Λ)}(pX ⊗_{Λ} DΛ, Y) ≅ Hom_{K(Mod Λ)}(pX, Hom_{Λ}(DΛ, Y))
≅ Hom_{D(Mod Λ)}(X, Hom_{Λ}(DΛ, Y))
≅ Hom_{K(Mod Λ)}(X, i Hom_{Λ}(DΛ, Y)).

Thus t and i Hom_{Λ}(DΛ, −) form an adjoint pair. □

Let us continue with some definitions. We denote by \underline{Mod} Λ the stable module category modulo projectives which is obtained by forming for each pair of Λ-modules M and N the quotient

\underline{Hom}_{Λ}(M, N) = Hom_{Λ}(M, N)/\{M → P → N | P projective\}.

Analogously, the stable module category \overline{Mod} Λ modulo injectives is defined. Recall that a Λ-module M is finitely presented if it admits a projective presentation

P_{1} ⟶ P_{0} → M → 0

such that P_{0} and P_{1} are finitely generated. The transpose Tr M relative to this presentation is the Λ^{op}-module which is defined by the exactness of the induced sequence

Hom_{Λ}(P_{0}, Λ) ⟶ Hom_{Λ}(P_{1}, Λ) → Tr M → 0.

Note that the presentation of M is minimal if and only if the corresponding presentation of Tr M is minimal. The construction of the transpose is natural up to maps factoring through a projective and induces a duality \underline{mod} Λ → \underline{mod} Λ^{op}.

Proposition 4.2. The functor

a: Mod Λ ⟶^{inc} D(Mod Λ) ⟶^{i} K(Inj Λ) ⟶^{t} K(Inj Λ) ⟶^{Z^{−1}} \overline{Mod} Λ

has the following properties.

(1) aM ≅ D Tr M for every finitely presented Λ-module M.

(2) a preserves all coproducts.

(3) a annihilates all projective Λ-modules and induces a functor \underline{Mod} Λ → \overline{Mod} Λ.

(4) Each exact sequence 0 → L → M → N → 0 of Λ-modules induces a sequence

0 → aL → aM → aN → L ⊗_{Λ} DΛ → M ⊗_{Λ} DΛ → N ⊗_{Λ} DΛ → 0

of Λ-modules which is exact.

Proof. (1) The functor t sends an injective resolution iM of M to pM ⊗_{Λ} DΛ. Using (3.2), we have

pM ⊗_{Λ} DΛ ≅ D Hom_{Λ}(pM, Λ).

This implies

(4.1)  Z^{−1}(pM ⊗_{Λ} DΛ) ≅ D Tr M.

(2) First observe that ∐_{i}(iM_{i}) ≅ i(∐_{i} M_{i}) for every family of Λ-modules M_{i}. Clearly, t and Z^{−1} preserve coproducts. Thus a preserves coproducts.

(3) We have t(iΛ) = DΛ and therefore aΛ = 0 in \overline{Mod} Λ. Thus a annihilates all projectives since it preserves coproducts.

(4) An exact sequence 0 → L → M → N → 0 induces an exact triangle pL → pM → pN → (pL)[1]. This triangle can be represented by a sequence 0 → pL → pM → pN → 0 of complexes which is split exact in each degree. Now apply − ⊗_{Λ} DΛ and use the snake lemma. □

We are now in the position to deduce the classical Aulander-Reiten formula for modules [2] from the formula for complexes.

Corollary 4.3 (Auslander/Reiten). Let M and N be Λ-modules and suppose that M is finitely presented. Then we have an isomorphism

(4.2)  D Ext_{Λ}^{1}(M, N) ≅ \overline{Hom}_{Λ}(N, D Tr M).

Proof. Let iM and iN be injective resolutions of M and N, respectively. We apply the Auslander-Reiten formula (3.4) and the formula (4.1) for the Auslander-Reiten translate. Thus we have

D Ext_{Λ}^{1}(M, N) ≅ D Hom_{K(Inj Λ)}(iM, (iN)[1]) ≅ Hom_{K(Inj Λ)}(iN, (pM ⊗_{Λ} DΛ)[−1]),

and the map

Hom_{K(Inj Λ)}(iN, (pM ⊗_{Λ} DΛ)[−1]) ⟶ \overline{Hom}_{Λ}(N, D Tr M), ϕ ↦ Z^{0}ϕ,

is clearly surjective. The composite is also an injective map, since a map N → D Tr M factoring through an injective module N′ comes from an element in D Ext_{Λ}^{1}(M, N′) which vanishes. □

5. A general Auslander-Reiten formula for Ext_{A}^{1}(−, −)

In this section, we extend the classical Auslander-Reiten formula for modules to a formula for a more general class of abelian categories. Thus we fix a locally noetherian Grothendieck category A, that is, A is an abelian Grothendieck category having a set of generators which are noetherian objects in A.

Theorem 5.1. Let M and N be objects in A. Suppose that M is noetherian and let Γ = End_{A}(M). Given an injective Γ-module I, there is an object a_{I}M in A and an isomorphism

Hom_{Γ}(Ext_{A}^{1}(M, N), I) ≅ \overline{Hom}_{A}(N, a_{I}M)

which is natural in I and N.

Proof. The category K(Inj A) is compactly generated and an injective resolution iM of M is a compact object; see [13, Proposition 2.3]. The functor

Hom_{Γ}(Hom_{K(Inj A)}(iM, −), I)

is cohomological and sends coproducts in K(Inj A) to products of abelian groups. Using Brown representability (see [10, 5.2] or [14, Theorem 3.1]), we have a representing object t_{I}(iM) in K(Inj A) such that

Hom_{Γ}(Hom_{K(Inj A)}(iM, −), I) ≅ Hom_{K(Inj A)}(−, t_{I}(iM)).

Now put a_{I}M = Z^{0}t_{I}(iM) and adapt the proof of the classical Auslander-Reiten formula (4.2) from the previous section. □

Remark 5.2. If A is a k-linear category, then we can take I = DΓ and obtain

D Ext_{A}^{1}(M, N) ≅ \overline{Hom}_{A}(N, a_{DΓ}M).

6. Auslander-Reiten triangles

In [2], Auslander and Reiten used the formula

D Ext_{Λ}^{1}(M, N) ≅ \overline{Hom}_{Λ}(N, D Tr M)

to establish the existence of almost split sequences. More precisely, for a finitely presented indecomposable and non-projective Λ-module M, there exists an almost split sequence

0 → D Tr M → L → M → 0

in the category of Λ-modules.

In this section, we produce Auslander-Reiten triangles in the category K(Inj Λ), using the Auslander-Reiten formula for complexes. In addition, we show that almost split sequences can be obtained from Auslander-Reiten triangles. This yields a simple recipe for the construction of an almost split sequence.

Let us recall the relevant definitions from Auslander-Reiten theory. A map α: X → Y is called left almost split, if α is not a section and if every map X → Y′ which is not a section factors through α. Dually, a map β: Y → Z is right almost split, if β is not a retraction and if every map Y′ → Z which is not a retraction factors through β.

Definition 6.1. (1) An exact sequence 0 → X ⟶^{α} Y ⟶^{β} Z → 0 in an abelian category is called almost split sequence, if α is left almost split and β is right almost split.

(2) An exact triangle X ⟶^{α} Y ⟶^{β} Z ⟶^{γ} X[1] in a triangulated category is called Auslander-Reiten triangle, if α is left almost split and β is right almost split.

Happel introduced Auslander-Reiten triangles and studied their existence in D^{b}(mod Λ) [8]. There is a general existence result for Auslander-Reiten triangles in compactly generated triangulated categories; see [12]. This yields the following.

Proposition 6.2. Let Z be a compact object in K(Inj Λ) which is indecomposable. Then there exists an Auslander-Reiten triangle

(6.1)  (pZ ⊗_{Λ} DΛ)[−1] ⟶^{α} Y ⟶^{β} Z ⟶^{γ} pZ ⊗_{Λ} DΛ.

Proof. First observe that Γ = End_{K(Inj Λ)}(Z) is local because it is a noetherian k-algebra. Now apply Theorem 2.2 from [12] and use the formula (3.4). Note that D = Hom_{k}(−, E) is isomorphic to the functor Hom_{Γ}(−, I) with I = E(Γ/ rad Γ) which is used in [12]. □

Let us mention that the exact triangle (6.1) is determined by the map

γ: Z ⟶ pZ ⊗_{Λ} DΛ

which corresponds under the isomorphism

D Hom_{K(Inj Λ)}(Z, Z) ≅ Hom_{K(Inj Λ)}(Z, pZ ⊗_{Λ} DΛ)

to a non-zero map End_{K(Inj Λ)}(Z) → E annihilating the radical of End_{K(Inj Λ)}(Z).

An Auslander-Reiten triangle ending in the injective resolution of a finitely presented indecomposable non-projective module induces an almost split sequence as follows.

Theorem 6.3. Let N be a finitely presented Λ-module which is indecomposable and non-projective. Then there exists an Auslander-Reiten triangle

(pN ⊗_{Λ} DΛ)[−1] ⟶^{α} Y ⟶^{β} iN ⟶^{γ} pN ⊗_{Λ} DΛ

in K(Inj Λ) which the functor Z^{0} sends to an almost split sequence

0 → D Tr N ⟶^{Z^{0}α} Z^{0}Y ⟶^{Z^{0}β} N → 0

in the category of Λ-modules.

Proof. The Auslander-Reiten triangle for iN is obtained from the triangle (6.1) by taking Z = iN. Let us assume that the projective resolution pN is minimal. Note that we have a sequence

(6.2)  0 → (pN ⊗_{Λ} DΛ)[−1] ⟶^{α} Y ⟶^{β} iN → 0

of chain maps which is split exact in each degree. This sequence is obtained from the mapping cone construction for γ: iN ⟶ pN ⊗_{Λ} DΛ.

We know from (4.1) that

D Tr N ≅ Z^{0}(pN ⊗_{Λ} DΛ)[−1].

This module is indecomposable and has a local endomorphism ring. Here we use that N is indecomposable and that the resolution pN is minimal. The functor Z^{0} takes the sequence (6.2) to an exact sequence

(6.3)  0 → D Tr N ⟶^{Z^{0}α} Z^{0}Y ⟶^{Z^{0}β} N.

Now observe that the map Z^{0}β is right almost split. This is clear because β is right almost split and Z^{0} induces a bijection Hom_{K(Inj Λ)}(iM, iN) → Hom_{Λ}(M, N) for all M. In particular, Z^{0}β is an epimorphism since N is non-projective. We conclude from the following Lemma 6.4 that the sequence (6.3) is almost split. □

Lemma 6.4. An exact sequence 0 → X ⟶^{α} Y ⟶^{β} Z → 0 in an abelian category is almost split if and only if β is right almost split and the endomorphism ring of X is local.

Proof. See Proposition II.4.4 in [1]. □

Remark 6.5. There is an analogue of Theorem 6.3 for a projective module N. Then D Tr N = 0 and Z^{0}β is the right almost split map ending in N.

From the mapping cone construction for complexes, we get an explicit recipe for the construction of an almost split sequence. Note that the computation of almost split sequences is a classical problem in representation theory [6]. In particular, the middle term is considered to be mysterious.

Corollary 6.6. Let N be a finitely presented Λ-module which is indecomposable and non-projective. Denote by

P_{1} ⟶^{δ_{1}} P_{0} → N → 0 and 0 → N → I^{0} ⟶^{δ^{0}} I^{1}

a minimal projective presentation and an injective presentation of N respectively. Choose a non-zero k-linear map End_{Λ}(N) → E annihilating the radical of End_{Λ}(N), and extend it to a k-linear map ϕ: Hom_{Λ}(P_{0}, I^{0}) → E. Let ϕ̄ denote the image of ϕ under the isomorphism

D Hom_{Λ}(P_{0}, I^{0}) ≅ Hom_{Λ}(I^{0}, P_{0} ⊗_{Λ} DΛ).

Then we have a commutative diagram with exact rows and columns

[figure: 0; 0; 0; 0; L; α; M; β; N; 0; 0; P_{1} ⊗_{Λ} DΛ; _{[\begin{matrix}1 \\ 0 \end{matrix}]}; δ_{1}⊗1; (P_{1} ⊗_{Λ} DΛ) ⨿ I^{0}; _{[_{0 1} ]}; _{[ \begin{matrix}δ_{1}⊗1 & ϕ̄ \\ 0 & δ^{0} \end{matrix} ]}; I^{0}; δ^{0}; 0; 0; P_{0} ⊗_{Λ} DΛ; _{[\begin{matrix}1 \\ 0 \end{matrix}]}; (P_{0} ⊗_{Λ} DΛ) ⨿ I^{1}; _{[_{0 1} ]}; I^{1}; 0]

such that the upper row is an almost split sequence in the category of Λ-modules.

Example 6.7. Let k be a field and Λ = k[x]/(x^{2}). Let iS denote the injective resolution of the unique simple Λ-module S = k[x]/(x). The corresponding Auslander-Reiten triangle in K(Inj Λ) has the form

pS[−1] ⟶^{α} Y ⟶^{β} iS ⟶^{γ} pS,

where γ denotes an arbitrary non-zero map. Viewing Λ as a complex concentrated in degree zero, the corresponding Auslander-Reiten triangle has the form

Λ[−1] ⟶^{α} Y ⟶^{β} Λ ⟶^{γ} Λ,

where γ denotes the map induced by multiplication with x.

7. An adjoint of Happel’s functor

Let Λ be an artin k-algebra, that is, we assume that Λ is artinian as a module over k. We denote by Λ̂ its repetitive algebra. In this section, we extend Happel’s functor [8]

D^{b}(mod Λ) ⟶ \underline{mod} Λ̂

to a functor which is defined on unbounded complexes, and we give a right adjoint.

The repetitive algebra is by definition the doubly infinite matrix algebra without identity

Λ̂ = [ \begin{matrix}. & . & . \\ ... & Λ & & & 0 \\ & DΛ & Λ \\ & & DΛ & Λ \\ 0 & & & ... & ... \end{matrix} ]

in which matrices have only finitely many non-zero entries and the multiplication is induced from the canonical maps Λ ⊗_{Λ} DΛ → DΛ , DΛ ⊗_{Λ} Λ → DΛ , and the zero map DΛ⊗_{Λ} DΛ → 0. Note that projective and injective modules over Λ̂ coincide. We denote by K_{ac}(Inj Λ̂) the full subcategory of K(Inj Λ̂) which is formed by all acyclic complexes. The follwing description of the stable category \underline{Mod} Λ̂ is well-known; see for instance [13, Example 7.6].

Lemma 7.1. The functor Z^{0} : K_{ac}(Inj Λ̂) → \underline{Mod} Λ̂ is an equivalence of triangulated categories.

We consider the algebra homomorphism

ϕ: Λ̂ ⟶ Λ, (x_{ij}) ↦ x_{00},

and we view Λ as a bimodule _{Λ}ΛΛ̂ via ϕ. Let us explain the following diagram.

[figure: mod Λ; −⊗_{Λ}Λ; inc; mod Λ̂; inc; can; \underline{mod} Λ̂; D^{b}(mod Λ); −⊗_{Λ}Λ; D^{b}(mod Λ̂); _{\underline{mod} Λ̂}; K^{c}(Inj Λ); F^{c}; inc; ≀; K^{c}(Inj Λ̂); G^{c}; inc; ≀; K_{ac}^{c}(Inj Λ̂); inc; Z^{0}; ≀; K(Inj Λ); F; inc; K(Inj Λ̂); HomΛ̂(Λ,−); inc; G; K_{ac}(Inj Λ̂); inc; K(Mod Λ); j_{Λ}; −⊗_{Λ}Λ; K(Mod Λ̂); jΛ̂; HomΛ̂(Λ,−)]

The top squares show the construction of Happel’s functor D^{b}(mod Λ) → \underline{mod} Λ̂ for which we refer to [9, 2.5].

The bimodule_{Λ}ΛΛ̂ induces an adjoint pair of functors between K(Mod Λ) and K(Mod Λ̂). Note that HomΛ̂(Λ, −) takes injective Λ̂-modules to injective Λ-modules. Thus we get an induced functor K(Inj Λ̂) → K(Inj Λ). This functor preserves products and has therefore a left adjoint F, by Brown’s representability theorem [15, Theorem 8.6.1]. A left adjoint preserves compactness if the right adjoint preserves coproducts; see [14, Theorem 5.1]. Clearly, HomΛ̂(Λ, −) preserves coproducts since Λ is finitely generated over Λ̂. Thus F induces a functor F^{c}.

The inclusion K(Inj Λ) → K(Mod Λ) preserves products and has therefore a left adjoint j_{Λ}, by Brown’s representability theorem [15, Theorem 8.6.1]. Note that j_{Λ}M = iM is an injective resolution for every Λ-module M. We have the same for Λ̂, of course. Thus we have

F ^{∘} j_{Λ} = jΛ̂ ^{∘}(− ⊗_{Λ} Λ).

It follows that F takes the injective resolution of a Λ-module M to the injective resolution of the Λ̂-module M ⊗_{Λ} Λ. This shows that F^{c} coincides with − ⊗_{Λ} Λ when one passes to the derived category D^{b}(mod Λ) via the canonical equivalence K^{c}(Inj Λ) → D^{b}(mod Λ).

The inclusion K_{ac}(Inj Λ̂) → K(Inj Λ̂) has a left adjoint G; see [13, Theorem 4.2]. This left adjoint admits an explicit description. For instance, it takes the injective resolution iM of a Λ̂-module M to the mapping cone of the canonical map pM → iM, which is a complete resolution of M. The functor G preserves compactness and induces therefore a functor G^{c}, because its right adjoint preserves coproducts [14, Theorem 5.1].

The following result summarizes our construction.

Theorem 7.2. The composite

\underline{Mod} Λ̂ ⟶^{∼} K_{ac}(Inj Λ̂) ⟶^{HomΛ̂(Λ,−)} K(Inj Λ)

has a fully faithful left adjoint

K(Inj Λ) ⟶^{G ∘ F} K_{ac}(Inj Λ̂) ⟶^{∼} \underline{Mod} Λ̂

which extends Happel’s functor

D^{b}(mod Λ) ⟶^{−⊗_{Λ}Λ} D^{b}(mod Λ̂) → \underline{mod} Λ̂.

8. The computation of Auslander-Reiten triangles

In this section, we explain a method for computing Auslander-Reiten triangles in K(Inj Λ). It is shown that the adjoint of Happel’s functor reduces the computation to the problem of computing almost split sequences in mod Λ̂. This is based on the following result.

Proposition 8.1. Let F: S → T be a fully faithful exact functor between triangulated categories which admits a right adjoint G: T → S. Suppose

X_{S} ⟶^{α_{S}} Y_{S} ⟶^{β_{S}} Z_{S} ⟶^{γ_{S}} X_{S}[1] and X_{T} ⟶^{α_{T}} Y_{T} ⟶^{β_{T}} Z_{T} ⟶^{γ_{T}} X_{T} [1]

are Auslander-Reiten triangles in S and T respectively, where Z_{T} = F Z_{S}. Then

GX_{T} ⟶^{Gα_{T}} GY_{T} ⟶^{Gβ_{T}} GZ_{T} ⟶^{Gγ_{T}} GX_{T} [1]

is the coproduct of X_{S} ⟶^{α_{S}} Y_{S} ⟶^{β_{S}} Z_{S} ⟶^{γ_{S}} X_{S}[1] and a triangle W ⟶^{id} W → 0 → W [1].

Proof. We have a natural isomorphism Id_{S} ≅ G^{∘} F which we view as an identification. In particular, G induces a bijection

(8.1)  Hom_{T} (F X, Y) → Hom_{S}((G^{∘} F)X, GY)

for all X ∈ S and Y ∈ T . Next we observe that for any exact triangle X ⟶^{α} Y ⟶^{β} Z ⟶^{γ} X[1], the map β is a retraction if and only if γ = 0.

The map F β_{S} is not a retraction since F γ_{S} ≠ 0. Thus F β_{S} factors through β_{T}, and G(F β_{S}) = β_{S} factors through Gβ_{T}. We obtain the following commutative diagram.

[figure: X_{S}; α_{S}; ϕ; Y_{S}; β_{S}; ψ; Z_{S}; γ_{S}; X_{S}[1]; ϕ[1]; GX_{T}; Gα_{T}; GY_{T}; Gβ_{T}; GZ_{T}; Gγ_{T}; GX_{S}[1]]

On the other hand, Gβ_{T} is not a retraction since the bijection (8.1) implies Gγ_{T} ≠ 0. Thus Gβ_{T} factors through β_{S}, and we obtain the following commutative diagram.

[figure: GX_{T}; Gα_{T}; ϕ′; GY_{T}; Gβ_{T}; ψ′; GZ_{T}; Gγ_{T}; GX_{T} [1]; ϕ′[1]; X_{S}; α_{S}; Y_{S}; β_{S}; Z_{S}; γ_{S}; X_{S}[1]]

We have β_{S} ^{∘}(ψ′ ^{∘} ψ) = β_{S}, and this implies that ψ′ ^{∘} ψ is an isomorphism, since β_{S} is right minimal. In particular, GY_{T} = Y_{S} ⨿ W for some object W. It follows that

GX_{T} ⟶^{Gα_{T}} GY_{T} ⟶^{Gβ_{T}} GZ_{T} ⟶^{γ_{T}} GX_{T} [1]

is the coproduct of X_{S} ⟶^{α_{S}} Y_{S} ⟶^{β_{S}} Z_{S} ⟶^{γ_{S}} X_{S}[1] and the triangle W ⟶^{id} W → 0 → W [1]. □

Now suppose that Λ is an artin algebra. We fix an indecomposable compact object Z in K(Inj Λ), and we want to compute the Auslander-Reiten triangle X → Y → Z → X[1]. We apply Happel’s functor

H: K^{c}(Inj Λ) ⟶^{∼} D^{b}(mod Λ) → \underline{mod} Λ̂

and obtain an indecomposable non-projective Λ̂-module Z′ = HZ. For instance, if Z = iN is the injective resolution of an indecomposable Λ-module N, then HiN = N where N is viewed as a Λ̂-module via the canonical algebra homomorphism Λ̂ → Λ. Now take the almost split sequence 0 → D Tr Z′ → Y′ → Z′ → 0 in Mod Λ̂. This gives rise to an Auslander-Reiten triangle D Tr Z′ → Y′ → Z′ → D Tr Z′[1] in \underline{Mod} Λ̂. We apply the composite

\underline{Mod} Λ̂ ⟶^{∼} K_{ac}(Inj Λ̂) ⟶^{HomΛ̂(Λ,−)} K(Inj Λ).

It follows from Proposition 8.1 that the result is a coproduct of the Auslander-Reiten triangle X → Y → Z → X[1] and a split exact triangle.

Acknowledgement. The authors wish to thank Igor Burban for some helpful discussions on the topic of this paper. Moreover, we are grateful to Helmut Lenzing and Dieter Vossieck for pointing out some less well-known references.

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Henning Krause, Institut für Mathematik, Universität Paderborn, 33095 Paderborn, Germany

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

Jue Le, Department of Mathematics, Beijing Normal University, 100875 Beijing, P.R.China

E-mail address: lejue@vip.sina.com