Abstract. We study the bounded derived category D b (R-mod) of a left Noetherian ring R. We give a version of the Generalized Auslander-Reiten Conjecture for D b (R-mod) that is equivalent to the classical statement for the module category and is preserved under derived equivalence.

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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

Abstract. We study self-extensions of modules over symmetric artin algebras. We show that non-projective modules with eventually vanishing self-extensions must lie in AR components of stable type ZA∞. Moreover, the degree of the highest non-vanishing self-extension of these modules is determined by their quasilength. This has implications for the Auslander-Reiten Conjecture and the Extension Conjecture.

We study the complexity of a family of finite-dimensional self-injective fc-algebras where k is an algebraically closed field. More precisely, let T be the trivial extension of an iterated tilted algebra of type H. We prove that modules over the trivial extension T all have complexities either 0, 1, 2 or infinity, depending on the representation type of the hereditary algebra H.