We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin-Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin-Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of selfinjective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.

Let A be a hereditary algebra. We construct a fundamental domain for the cluster category C A inside the category of modules over the duplicated algebraĀ of A. We then prove that there exists a bijection between the tilting objects in C A and the tiltingĀ-modules all of whose nonprojectiveinjective indecomposable summands lie in the left part of the module category ofĀ.

Abstract. We find a simple condition which implies finiteness of (little) finitistic global dimension for artin algebras. As a consequence we obtain a short proof of the finitistic global dimension conjecture for radical cubed zero algebras. The same condition also holds for algebras of representation dimension less then or equal to three. Hence the conjecture holds in that case as well.Let Λ be an Artin algebra (an algebra of finite length over a commutative Artinian ring). We will consider only finitely generated (f.g.) modules over Λ, and hence we will only address the little finitistic global dimension conjecture which states that there exists a uniform bound called f indimΛ for the finite projective dimensions of all f.g. (left) Λ-modules of finite projective dimension. This conjecture implies the Nakayama conjecture. Some of the known cases in which the finitistic global dimension conjecture holds are the radical cubed zero case [GZ] and the monomial relation case [GKK] (see also [IZ], [BFGZ]). The conjecture is also true in the case the category of modules of finite projective dimension is contravariantly finite in the category of all f.g. modules [AR]. However, the converse is not true [IST]. In this paper we give a short proof that sup{pdM | M f.g. with rad 2 M = 0 and pdM < ∞} is finite. This is a generalization of the radical cubed zero case since all syzygies have radical square zero in that case. A thorough overview of the state of both little and big finitistic global dimension conjectures is given by B. Huisgen-Zimmerman [H].As another consequence of the main theorem we prove the finitistic dimension conjecture for algebras with representation dimension repdimΛ ≤ 3. The notion of representation dimension was introduced by M. Auslander [A1]. O. Iyama showed

Abstract.Let A be an artin algebra 21 and a two-sided ideal of A. Then 21 is the trace of a projective A-module P in A . We study how the homological properties of the categories of finitely generated modules over the three rings A/21, A and the endomorphism ring of P are related. We give some applications of the ideas developed in the paper to the study of quasi-hereditary algebras.

We define and study virtual representation spaces for vectors having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the first fundamental theorem, the saturation theorem and the canonical decomposition theorem. In the special case of Dynkin quivers with n vertices, this gives the fundamental interrelationship between supports of the semi-invariants and the tilting triangulation of the (n − 1)-sphere. Contents