Derived equivalences induced by big cotilting modules

We prove that given a Grothendieck category G with a tilting object of finite projective dimension, the induced triangle equivalence sends an injective cogenerator of G to a big cotilting module. Moreover, every big cotilting module can be constructed like that in an essentially unique way. We also prove that the triangle equivalence is at the base of an equivalence of derivators, which in turn is induced by a Quillen equivalence with respect to suitable abelian model structures on the corresponding categories of complexes.Comment: 33 pages; version 2: more details added in Remark 1.11 and in the proofs of Lemmas 1.17 and 4.4, discussion of well-known or inessential results has been reduced (e.g. basics on model structures or the relation to co-t-structures), references have been update