Dimensions of triangulated categories

Abstract

We define a dimension for a triangulated category. We prove a representabilityTheorem for a certain class of functors on finite dimensional triangulatedcategories. We study the dimension of the boundedderived category of an algebra or a scheme and we show in particularthat the bounded derived category of coherent sheaves over avariety has a finite dimension. For a self-injective algebra, a lowerbound for Auslander's representation dimension is given by the dimensionof the stable category. We use this to compute the representationdimension of exterior algebras. This provides the first known examplesof representation dimension >3. We deduce that theLoewy length of the group algebra over F_2 of a finite group is strictly bounded below by2-rank of the group (a conjecture of Benson).