Abstract :
Work of J. Rickard proves that the derived module categories of two ringsAandBare equivalent as triangulated categories if and only if there is a particular objectT, a so-called tilting complex, in the derived category ofAsuch thatBis the endomorphism ring ofT. The functor inducing the equivalence, however, is not explicit by the knowledge ofT. Suppose the derived categories ofAandBare equivalent. IfAandBareR-algebras and projective of finite type over the commutative ringR, then Rickard proves the existence of a so-called two-sided tilting complexX, which is an object in the derived category of bimodules. The left derived tensor product byXis then an equivalence between the desired categories ofAandB. There is no general explicit construction known to deriveXfrom the knowledge ofT. In an earlier paper S. König and the author gave for a class of algebras a tilting complexTby a general procedure with prescribed endomorphism ring. Under some mild additional hypotheses, we construct in the present paper an explicit two-sided tilting complex whose restriction to one side is any given one-sided tilting complex of the type described in the above-cited work. This provides two-sided tilting complexes for various cases of derived equivalences, making the functor inducing this equivalence explicit. In particular, the perfect isometry induced by such a derived equivalence is determined.
