Applications of balanced pairs

Let（X, Y） be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to（X, Y）, and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the X-resolution dimension of Y（resp. Y-coresolution dimension of X）is finite, then the bounded homotopy category of Y（resp. X） is contained in that of X（resp. Y）. As a consequence, we get that the right X-singularity category coincides with the left Y-singularity category if the X-resolution dimension of Y and the Y-coresolution dimension of X are finite.