Closed Subbifunctors of the Extension Functor

Given a subbifunctor F of Ext1( , ), one can ask if one can generalize the construction of the derived category to obtain a relative derived category, where one localizes with respect to F-acyclic sequences. We show that this is possible if and only if F is closed. We also show that for artin algebras the closed subbifunctors correspond to Serre subcategories of a category of finitely presented functors that vanish on projectives, and we use this to find new examples of closed subbifunctors. Using relatively derived categories, we give a relative version of Happelʹs result on derived equivalences induced by tilting, and we show in an example how this can be used to find ordinary derived equivalences.