Tilting Selfinjective Algebras and Gorenstein Orders

BY Rickard's fundamental theorem [8], the rings which are derived equivalent to a ring A are precisely the endomorphism rings of tilting complexes over A. A tilting complex T is a finitely generated complex of finitely generated projective modules, which does not admit selfextensions and which has the property that the smallest triangulated subcategory of D(A) which contains T also contains all the finitely generated projective modules. Rickard constructed tilting complexes between blocks of cyclic defect group over algebraically closed fields of characteristic p and their Brauer correspondent blocks in the group ring of the normalizer of their defect group [9]. Linckelmann [5] proved an analogous result for an extension of the p-adic integers having large enough residue field. Other known examples are derived equivalences between certain blocks with dihedral defect groups (Linckelmann [6]) including the 2-adic principal blocks of the alternating groups A4 and A5 over an algebraically closed field of characteristic 2 (Rickard [10], [8, Section 8]). The aim of this note is to show that tilting complexes giving all these, and many more, derived equivalences can be constructed and studied in a unified way. The innovation in our approach is that we describe endomorphism rings of tilting complexes via endomorphism rings of modules. To compute homomorphisms in the module category turns out to be easier than the usual computation in the homotopy category. In case A is an order, the endomorphism ring T := EndD>w(T) of the tilting complex T with homology concentrated in the two degrees 0 and 1 (to be defined in Section 2) is the pullback in the following diagram (see Proposition 2.1):