On some triangulated categories over group algebras

In this paper, we introduce the cofibrant derived category of a group algebra $kG$ and study its relation to the derived category of $kG$. We also define the cofibrant singularity category of $kG$, whose triviality characterizes the regularity of $kG$ with respect to the cofibrant dimension, and examine its significance as a measure of the obstruction to the equality between the classes of Gorenstein projective and cofibrant modules. We show that the same obstruction can be measured by certain localization sequences between stable categories.