Generalized Tilting Theory

Abstract

Given small dg categories A and B and a B-A-bimodule T, we give necessary and sufficient conditions for the associated derived functors of Hom and the tensor product to be fully faithful. Special emphasis is put on the case when RHom $$_\mathrm{A}$$ (T,?) is fully faithful and preserves compact objects, in which case nice recollements situations appear. It is also shown that, given an algebraic compactly generated triangulated category D, all compactly generated co-smashing triangulated subcategories which contain the compact objects appear as the image of such a RHom $$_\mathrm{A}$$ (T,?). The results are then applied to the case when A and B are ordinary algebras, comparing the situation with the well-stablished tilting theory of modules. In this way we recover and extend recent results by Bazzoni–Mantese–Tonolo, Chen-Xi and D. Yang.