We will study the resolution dimension of functorially finite subcategories. The subcategories with the resolution dimension zero correspond to ring epimorphisms, and rejective subcategories correspond to surjective ring morphisms. We will study a chain of rejective subcategories to construct modules with endomorphisms rings of finite global dimension. We apply these result to study a function ...

We classify triangulated categories that are equivalent to finitely generated thick subcategories T⊂Db(cohC) for smooth projective curves C over an algebraically closed field.

Let D be a triangulated category with a cluster tilting subcategory U . The quotient category D/U is abelian; suppose that it has finite global dimension. We show that projection from D to D/U sends cluster tilting subcategories of D to support tilting subcategories of D/U , and that, in turn, support tilting subcategories of D/U can be lifted uniquely to weak cluster tilting subcategories of D.