Two-Sided Tilting Complexes for Gorenstein Orders

✍ Scribed by Alexander Zimmerman

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 308 KB
• Volume: 209
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1998.7525

✦ Synopsis

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 desired 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. Konig and the author gave for a class of älgebras a tilting complex T by 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.