Two sided tilting complexes for Green orders and Brauer tree algebras
Derived equivalences became popular in representation theory of finite groups because of results of \textit{J. Rickard}. First of all, his fundamental theorem [J. Lond. Math. Soc., II. Ser. 39, No. 3, 436-456 (1989; Zbl 0672.16034)] characterizes derived equivalences between rings by the existence of certain tilting complexes. Moreover, \textit{J. Rickard} provided examples of non-trivial derived equivalences by constructing some tilting complexes [J. Pure Appl. Algebra 61, No. 3, 303-317 (1989; Zbl 0685.16016)] inducing derived equivalences between Brauer tree algebras. This covers blocks of group algebras (over fields) having cyclic defect. Later on [J. Lond. Math. Soc., II. Ser. 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 twosided tilting complex. The article under review gives a rather explicit combinatorial construction of twosided tilting complexes (whose existence had been predicted by Rickard's result) for both Brauer tree algebras and Green orders, the latter covering blocks of cyclic defect of group algebras over discrete valuation rings. The article also contains a detailed exposition of the results and concepts mentioned above.
scientific article; zbMATH DE number 1014019