Weak global dimension and CM-freeness of rings

Abstract

Let $R$ be a ring with finite weak global dimension. We prove that $R$ is (strongly) CM-free. As applications,we obtain some triangulated equivalences between some compactly-generated homotopy categories and relative derived categories.We classify the definable categories of such equivalent triangulated categories when the weak global dimension of $R$ is at most 1, and show that such equivalent triangulated categoriessatisfy Telescope conjecture when $R$ is a von Neumann regular ring. We also improve the condition such that a triangulated equivalence about the extended Grothendieck duality holdsfrom the finiteness of the left and right global dimension of $R$ to only the finiteness of the right global dimension of $R$.