Two-sided tilting complexes for Gorenstein orders
Derived equivalences have been used in representation theory of finite groups since \textit{J. Rickard} proved his fundamental theorem [J. Lond. Math. Soc., II. Ser. 39, No. 3, 436-456 (1989; Zbl 0642.16034)] characterizing derived equivalences between rings by the existence of certain tilting complexes. Later on [ibid. 43, No. 1, 37-48 (1991; Zbl 0683.16030)] \textit{J. Rickard} has shown that under assumptions satisfied for group algebras, if there are any derived equivalences, then there are as well derived equivalences which are just derived tensor functors, and the object to be tensored with is a so-called two-sided tilting complex. Broué's conjecture on derived equivalences between certain group algebras of finite groups motivated an intensive checking of examples, which often goes in three steps. First, one constructs a suitable one-sided tilting complex \(T\). Then one computes the endomorphism ring \(B\) of \(T\). Finally, one may look for a two-sided tilting complex which restricted to one side coincides with \(T\). A paper by \textit{A. Zimmermann} and the reviewer [Q. J. Math., Oxf. II. Ser. 48, No. 191, 351-361 (1997; Zbl 0882.16010)] provided a pullback formula for \(B\) in case \(T\) has a special form which is often used in examples. The paper under review deals with the same situation and gives an explicit construction of a two-sided tilting complex for the \(T\) mentioned before. Related information can be found in a recent book by \textit{A. Zimmermann} and the reviewer [Derived equivalences for group rings. Lect. Notes Math. 1685 (1998; Zbl 0898.16002)].
scientific article; zbMATH DE number 1247618