Finiteness conditions and relative derived categories

In this paper, we introduce a class of exact structures in terms of finiteness conditions of modules, which are called n-pure exact structures. We investigate the properties of n-pure derived categories of module categories using n-pure exact structures, and show that n-pure derived categories share many nice properties of classical derived categories. In particular, we show that bounded n-pure derived categories can be realized as certain homotopy categories. We also compare the classical bounded derived categories and the bounded n-pure derived categories, and show that the former can be described by n-pure projective modules.