Relative derived categories with respect to subcategories

The notion of relative derived category with respect to a subcategory is introduced. A triangle-equivalence, which extends a theorem of Gao and Zhang [Gorenstein derived categories, J. Algebra 323 (2010) 2041–2057] to the bounded below case, is obtained. Moreover, we interpret the relative derived functor [Formula: see text] as the morphisms in such derived category and give two applications.

Relative Quotient Triangulated Categories

Let A be a finite dimensional algebra over a field k. We consider a subfunctor F of [Formula: see text], which has enough projectives and injectives such that [Formula: see text] is of finite type, where [Formula: see text] denotes the set of F-projectives. One can get the relative derived category [Formula: see text] of A-mod. For an F-self-orthogonal module TF, we discuss the relation between the relative quotient triangulated category [Formula: see text] and the relative stable category of the Frobenius category of TF-Cohen-Macaulay modules. In particular, for an F-Gorenstein algebra A and an F-tilting A-module TF, we get a triangle equivalence between [Formula: see text] and the relative stable category of TF-Cohen-Macaulay modules. This gives the relative version of a result of Chen and Zhang.