Auslander–Reiten Triangles and Grothendieck Groups of Triangulated Categories

We prove that if the Auslander–Reiten triangles generate the relations for the Grothendieck group of a Hom-finite Krull–Schmidt triangulated category with a (co)generator, then the category has only finitely many isomorphism classes of indecomposable objects up to translation. This gives a triangulated converse to a theorem of Butler and Auslander–Reiten on the relations for Grothendieck groups. Our approach has applications in the context of Frobenius categories.


Introduction
The notion of almost split sequences was introduced by Auslander and Reiten in [4], and has played a fundamental role in the representation theory of finite dimensional algebras ever since [5]. The theory of almost split sequences, later called Auslander-Reiten sequences or just AR-sequences, has also greatly influenced other areas, such as algebraic geometry and algebraic topology [2,14].
Happel defined Auslander-Reiten triangles in triangulated categories [11]. These play a similar role in the triangulated setting as AR-sequences do for abelian or exact categories. While it is known that AR-sequences always exist in the category of finitely generated modules over a finite dimensional algebra, the situation in the triangulated case turns out to be more complicated, and the associated bounded derived category will not necessarily have AR-triangles. In fact, Happel proved that this category has AR-triangles if and only if the algebra is of finite global dimension [10,11]. Reiten and van den Bergh showed that a Hom-finite Krull-Schmidt triangulated category has AR-triangles if and only if it admits a Serre functor [18]. More recently, Diveris, Purin and Webb proved that if a category as above is connected and has a stable component of the Auslander-Reiten quiver of Dynkin tree class, then this implies existence of AR-triangles [8].
In the abelian setting, there is a well-studied relationship between AR-sequences, representation-finiteness and relations for the Grothendieck group. From Butler [7], Auslander-Reiten [3, Proposition 2.2] and Yoshino [21,Theorem 13.7], we know that if a complete Cohen-Macaulay local ring is of finite representation type, then the Auslander-Reiten sequences generate the relations for the Grothendieck group of the category of Cohen-Macaulay modules. Here we say that our ring is of finite representation type if the category of Cohen-Macaulay modules has only finitely many isomorphism classes of indecomposable objects. A converse to this theorem is given by Auslander for artin algebras [1] and by Hiramatsu in the case of a Gorenstein ring with an isolated singularity [13,Theorem 1.2], where the latter is extended by Kobayashi [15,Theorem 1.2]. Results of the type described above were recently generalized to the setup of exact categories by Enomoto [9] and to certain extriangulated categories by Padrol, Palu, Pilaud and Plamondon [16].
A natural question to ask is whether there is a similar connection between AR-triangles, representation-finiteness and the relations for the Grothendieck group in the triangulated case. Xiao and Zhu give a partial answer to this question. Namely, they show that if our triangulated category is locally finite, then the AR-triangles generate the relations for the Grothendieck group [20, Theorem 2.1]. Beligiannis generalizes and gives a converse to this result for compactly generated triangulated categories [6,Theorem 12.1].
In this paper we consider the reverse direction of Xiao and Zhu from a different point of view. We prove that if the Auslander-Reiten triangles generate the relations for the Grothendieck group of a Hom-finite Krull-Schmidt triangulated category with a (co)generator, then the category has only finitely many isomorphism classes of indecomposable objects up to translation. We conclude by an application in the context of Frobenius categories. As an example, we see that our approach recovers results of Hiramatsu and Kobayashi for Gorenstein rings.

Auslander-Reiten Triangles and Grothendieck Groups
Let R be a commutative ring. An R-linear category T is called Hom-finite provided that Hom T (X, Y ) is of finite R-length for every pair of objects X, Y in T . An additive category is called a Krull-Schmidt category if every object can be written as a finite direct sum of indecomposable objects having local endomorphism rings. In a Krull-Schmidt category, it is well known that every object decomposes essentially uniquely in this way.
Throughout the rest of this paper, we let T be an essentially small R-linear triangulated category. We also assume that T is a Krull-Schmidt category which is Hom-finite over R. We let ind(T ) consist of the indecomposable objects of T , while the translation functor of T is denoted by . For simplicity, we use the notation (A, B) = Hom T (A, B) and [A, B] = length R (Hom T (A, B)).
We say that T has finitely many isomorphism classes of indecomposable objects up to translation if there is a finite subset of ind(T ) such that for any U ∈ ind(T ), there is an integer n such that n U is isomorphic to an object in our finite subset.

Recall from [12] that a distinguished triangle
Auslander-Reiten triangle if the following conditions are satisfied: Let F(T ) denote the free abelian group generated by all isomorphism classes By abuse of notation, objects in K 0 (T , 0) are also denoted by [A]. As T is a Krull-Schmidt category, the quotient K 0 (T , 0) is isomorphic to the free abelian group generated by isomorphism classes of objects in ind(T ).
Let Ex(T ) be the subgroup of K 0 (T , 0) generated by the subset Similarly, we let AR(T ) denote the subgroup of K 0 (T , 0) generated by Recall from for instance [12] that the Grothendieck group of T is defined as In the proof of our main results, Theorem 2.4 and Theorem 2.5, we use the well-known fact that an equality in K 0 (T , 0) can yield an equality in Z. We need this in the case of [U, −] and [−, U] for an object U in T , but note that the following lemma could be phrased more generally in terms of additive functors.
Proof Let a 1 [X 1 ] + · · · + a r [X r ] = 0 in K 0 (T , 0). If a i ≥ 0 for every i = 1, 2, . . . , r, we use the defining relations for K 0 (T , 0) to obtain where a i X i denotes the coproduct of the object X i with itself a i times. Consequently, the object a 1 X 1 ⊕ · · · ⊕ a r X r is zero in T . Applying [U, −] or [−, U] and using additivity hence yields our desired equations.
If some of the coefficients a i are negative, we start by moving all negative terms to the right-hand side of our equality and proceed similarly.
The lemmas below, which yield a triangulated analogue of [15,Proposition 2.8], provide an important step in the proofs of Theorem 2.4 and Theorem 2.5. Note that parts of our proof of Lemma 2.2 is much the same as the proof of [8,Lemma 2.2]. Observe also that Lemma 2.3 follows from [19,Proposition 3.1] in the case where R is an algebraically closed field, and that the argument generalizes to our context. We include complete proofs for the convenience of the reader.

Lemma 2.2 Let
The following statements hold for an object U in T :

U ) is injective if and only if
A is not a direct summand in U .
Proof Note that C is a direct summand in U if and only if there exists a split epimorphism U → C. By the definition of an AR-triangle, this is equivalent to g * not being surjective, which proves (1).
Our triangle yields the long-exact sequence The morphism f * is hence injective if and only if ( −1 g) * is surjective. By applying part (1) to the object U , we see that ( −1 g) * is surjective if and only if C is not a direct summand in U , which is equivalent to −1 C not being a direct summand in U . This shows (2). The statements (3) and (4) are verified dually, using that AR-triangles equivalently can be defined in terms of a factorization property for the leftmost morphism, see for instance [12]. Proof From the long exact Hom-sequence arising from our triangle, we get the exact sequence

Lemma 2.3 Let
Splitting into short exact sequences and using our finiteness assumption, we see that the alternating sum of the lengths of the objects in the sequence vanishes. This gives the equation

Consequently, we have [U, A] − [U, B] + [U, C]
= 0 if and only if the right-hand side of the equation is also non-zero. This means that either K or L (or both) must be non-zero. The object K is non-zero if and only if f * is not injective. By part Lemma 2.2 part (2), this is the case if and only if −1 C is a direct summand in U . Similarly, the object L is non-zero if and only if g * is not surjective. Using part (1) of Lemma 2.2, this is equivalent to C being a direct summand in U . As U is indecomposable, a direct summand in U is necessarily isomorphic to U , which finishes our proof of part (1).
Our second statement is shown dually, using part (3) and (4) of Lemma 2.2.
We are now ready to prove our two main results, which show that we can study representation-finiteness of our category T by considering the relations for the associated Grothendieck group. (T , 0), then T has only finitely many isomorphism classes of indecomposable objects.

Theorem 2.4 Assume there is an object X in T such that Hom T (Y, X) = 0 or an object
Proof Let X be an object satisfying Hom T (Y, X) = 0 for every nonzero Y in T , and and integers a i for i = 1, 2, . . . , r such that in K 0 (T , 0). Given an object U in T , Lemma 2.1 now yields the equality  (1), this means that any indecomposable object in T is isomorphic to an object in the finite set {C i , −1 C i } r i=1 , which yields our desired conclusion. The proof in the dual case is similar, using Lemma 2.3 part (2).
In the theorem below, an object X in T is called a generator of T if for any non-zero object Y in T . Dually, an object X is called a cogenerator of T if Hom * T (Y, X) = 0 for any non-zero Y . K 0 (T , 0), then T has only finitely many isomorphism classes of indecomposable objects up to translation.

Theorem 2.5 Assume that T has a generator or a cogenerator. If Ex(T ) = AR(T ) in
Proof Let X be a cogenerator and consider an indecomposable object U in T . Notice that as X is a cogenerator, there exists an integer n such that Hom T ( n U, X) = 0. As in the proof of Theorem 2.4, our assumption Ex(T ) = AR(T ) implies existence of a finite family of AR-triangles  (1), this yields that either n U C i or n+1 U C i . Consequently, every indecomposable object in T can be obtained as a translation of an object in the finite set {C i } r i=1 , which yields our desired conclusion.
The proof in the case where our category T has a generator is dual, using Lemma 2.3 part (2).

Application to Frobenius Categories
We now move on to an application of Theorem 2.4. Throughout the rest of the paper, let C be an essentially small R-linear Frobenius category. Recall that a Frobenius category is an exact category with enough projectives and injectives, and in which these two classes of objects coincide. The stable category of C, i.e. the quotient category modulo projective objects, is denoted by C. We assume C to be a Krull-Schmidt category and that the stable category C is Hom-finite.
As C is a Frobenius category, the associated stable category is triangulated. Recall that distinguished triangles in C are isomorphic to triangles of form X → Y → Z → −1 X, where 0 → X → Y → Z → 0 is a short exact sequence in C and −1 X denotes the first cosyzygy of X. Note that −1 is a well-defined autoequivalence on the stable category. The morphism Z → −1 X in our distinguished triangle above is obtained from the diagram where I (X) is injective and both rows are short exact sequences. For a more thorough introduction to exact categories and the stable category of a Frobenius category, see for instance [12]. Based on the correspondence between short exact sequences in a Frobenius category and distinguished triangles in its stable category, we get results also for Frobenius categories. In order to see this, we need to rephrase some of our terminology in the context of exact categories. Let us first recall that a short exact sequence 0 → A → B g − → C → 0 in C is an Auslander-Reiten sequence if the following conditions are satisfied: (1) A, C ∈ ind(C); (2) the sequence does not split; (3) given any morphism t : W → C which is not a split-epimorphism, there is a morphism t : W → B such that g • t = t.
As before, let K 0 (C, 0) be the free abelian group generated by isomorphism classes of objects in C modulo the subgroup generated by {[A⊕B] − [A] − [B] | A, B ∈ C}. Again, we can define the subgroups Ex(C) and AR(C) of K 0 (C, 0), but now in terms of short exact sequences instead of distinguished triangles. Namely, we let Ex(C) be the subgroup generated by the subset there exists a short exact sequence 0 → X → Y → Z → 0 in C and AR(C) the subgroup generated by The next lemma describes a well-known correspondence between AR-sequences in C and AR-triangles in C, see [17,Lemma 3].
We are now ready to show the following lemma regarding the subgroups Ex(C) and AR(C) of K 0 (C, 0) and the analogous subgroups of K 0 (C, 0).
Proof Assume Ex(C) = AR(C) in K 0 (C, 0) and consider a distinguished triangle in C. As we work with isomorphism classes of objects, we can assume that our triangle is of the form X → Y → Z → −1 X, where 0 → X → Y → Z → 0 is a short exact sequence in C.
Since Ex(C) = AR(C), there exist AR-sequences 0 → A i → B i → C i → 0 and integers a i for i = 1, 2, . . . , r such that in K 0 (C, 0), and hence also in K 0 (C, 0). By Lemma 3.1, the right-hand side of this equation is contained in AR(C). Thus, we have shown that Ex(C) ⊆ AR(C). The reverse inclusion is clear.
We hence have the following corollary to Theorem 2.4.

Corollary 3.3
Assume there is an object X in C such that Hom C (Y, X) = 0 or an object X in C such that Hom C (X , Y ) = 0 for every non-zero Y in C. If Ex(C) = AR(C) in K 0 (C, 0), then the following statements hold: (1) The category C has only finitely many isomorphism classes of non-projective indecomposable objects. (2) If C has only finitely many indecomposable projective objects up to isomorphism, then C has only finitely many isomorphism classes of indecomposable objects.
Proof As C is an essentially small R-linear Krull-Schmidt category, the same is true for the stable category C. As Ex(C) = AR(C) in K 0 (C, 0), Lemma 3.2 yields that Ex(C) = AR(C) in K 0 (C, 0). The result now follows from Theorem 2.4.