Mathematics – Representation Theory
Scientific paper
2010-08-11
Mathematics
Representation Theory
Scientific paper
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.
Rickard Jeremy
Rouquier Raphael
No associations
LandOfFree
Stable categories and reconstruction does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Stable categories and reconstruction, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Stable categories and reconstruction will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-480699