A fundamental question in classical stable homotopy theory is to understand the stable homotopy groups of the spheres. A relatively new method is via the motivic approach. Motivic stable homotopy theory has an algebro-geometric root and closely connects to questions in number theory. Besides, it relates to classical theory. One example of motivic applications in the classical theory is the Adams spectral sequence computations by Isaksen—Wang—Xu. They base their approach on a theoretical result by Gheorghe—Wang—Xu about a t-structure on the p-complete cellular C-motivic category. I will first talk about a generalization of this result, joint with Tom Bachmann, Guozhen Wang, and Zhouli Xu. The generalization leads to computational applications in the classical and motivic Adams spectral sequences.