  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).
