Combinatorial restrictions on the tree class of the Auslander–Reiten quiver of a triangulated category

Abstract

Abstract. We show that if a connected, Hom-finite, Krull-Schmidt triangulated category has an Auslander-Reiten quiver component with Dynkin tree class then the category has Auslander-Reiten triangles and that component is the entire quiver. This is an analogue for triangulated categories of a theorem of Auslander, and extends a previous result of Scherotzke. We also show that if there is a quiver component with extended Dynkin tree class, then other components must also have extended Dynkin class or one of a small set of infinite trees, provided there is a non-zero homomorphism between the components. The proofs use the theory of additive functions. Main resultsLet k be a field and let C be a k-linear triangulated category which is Hom-finite, Krull-Schmidt and connected. The Auslander-Reiten quiver of C is the graph whose vertices are the indecomposable objects of C (up to isomorphism) and where we draw an arrow from X to Y , labelled with certain multiplicity information, if there is an irreducible morphism from X to Y . We will be concerned with parts of the Auslander-Reiten quiver where Auslander-Reiten triangles ex-is an Auslander-Reiten triangle we write U = τ W and W = τ −1 U to define the Auslander-Reiten translate τ . By a stable component Γ of the Auslander-Reiten quiver we mean a subgraph with the properties:(1) for every indecomposable object M ∈ Γ, for every n ∈ Z, τ n M also lies in Γ and, (2) every irreducible morphism beginning or ending at M lies in Γ. A stable component Γ has the form ZT /G where T is a labelled tree (the tree class of Γ) and G is a group, by [5] and [11]. Note that we do not suppose that Γ is closed under the shift operation of C, and also that

Auslander-Reiten triangles and Grothendieck groups of triangulated categories

“…It is a triangulated analogue of [13,Proposition 2.8]. Note that our proof of (1) part b) is much the same as the proof of [7,Lemma 2.2]. Observe also that (2) follows from [16, Proposition 3.1] in the case where R is an algebraically closed field, and that the argument generalizes to our context.…”

“…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 [15]. 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 [7].…”

Auslander–Reiten Triangles and Grothendieck Groups of Triangulated Categories

“…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.…”

“…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].…”

Bilinear Forms on Grothendieck Groups of Triangulated Categories

“…Evidently the long exact sequence becomes the three-term sequence shown. This lemma has many consequences: one application of it is described in [6]. The rest of this paper is devoted to studying its implications for bilinear forms on Grothendieck groups.…”