Finiteness Conditions and Relative Singularity Categories

We introduce the n-pure projective(resp.,injective)dimension of complexes in n-pure derived categories,and give some criteria for computing these dimensions in terms of the n-pure projective(resp.,injective)resolutions(resp.,coresolutions)and n-pure derived functors.As a consequence,we get some equivalent characterizations for the finiteness of n-pure global dimension of rings.Finally,we study Verdier quotient of bounded n-pure derived category modulo the bounded homotopy category of n-pure projective modules,which is called an n-pure singularity category since it can reflect the finiteness of n-pure global dimension of rings.