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
