Stable categories and reconstruction

Abstract: This work is an attempt towards a Morita theory for stable equivalences between self-injective algebras. More precisely, given two self-injective algebras A and B and an equivalence between their stable categories, consider the set S of images of simple B-modules inside the stable category of A. That set satisfies some obvious properties of Hom-spaces and it generates the stable category of A. Keep now only S and A. Can B be reconstructed ? We show how to reconstruct the graded algebra associated to the radical filtration of (an algebra Morita equivalent to) B. We also study a similar problem in the more general setting of a triangulated category T. Given a finite set S of objects satisfying Hom-properties analogous to those satisfied by the set of simple modules in the derived category of a ring and assuming that the set generates T, we construct a t-structure on T. In the case T=D^b(A) and A is a symmetric algebra, the first author has shown that there is a symmetric algebra B with an equivalence from D^b(B) to D^b(A) sending the set of simple B-modules to S. The case of a self-injective algebra leads to a slightly more general situation: there is a finite dimensional differential graded algebra B with H^i(B)=0 for i>0 and for i<<0 with the same property as above.