Width and Local Homology Dimension for Triangulated Categories

Let T be a compactly generated triangulated category. In this paper, the width and local homology dimension of an object X with respect to a homogeneous ideal a, widthR(a,X) and hd(a,X), respectively, are introduced. The local nature and some basic properties of widthR(a,X) and hd(a,X) are provided. In addition, we give an upper bound and lower bound of widthR(a,X). What is more, we give the relationship between the local homology dimension hd(a,X) and the arithmetic rank of a and dimR.