Two-sided tilting complexes for Gorenstein orders

Abstract

Work of J. Rickard proves that the derived module categories of two rings A and B are equivalent as triangulated categories if and only if there is a particular object T , a so-called tilting complex, in the derived category of A such that B is the endomorphism ring of T . The functor inducing the equivalence however is not explicit by the knowledge of T . Suppose the derived categories of A and B are equivalent. If A and B are R-algebras and projective of finite type over the commutative ring R, then Rickard proves the existence of a so-called two-sided tilting complex X, which is an object in the derived category of bimodules. The left derived tensor product by X is then an equivalence between the derived categories of A and B. There is no general explicit construction known to derive X from the knowledge of T . In an earlier paper S. König and the author gave for a class of algebras a tilting complex T by a general procedure with prescribed endomorphism ring. Under some mild additional hypothesis 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 paper. 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 equivalences is determined.

Cite this paper

@inproceedings{ZimmermannTwosidedTC, title={Two-sided tilting complexes for Gorenstein orders}, author={Alexander Zimmermann} }