Morita equivalence revisited
Paul Baum (Pennsylvania State University)
Wednesday 6th May, 2009 16:00-17:00 Mathematics Building, Room 204
Let X be a complex affine variety. Denote the co-ordinate algebra of X by k. A k-algebra is an algebra A over the complex numbers such that A has been given the structure of a (left) k-module. An evident compatibility is required between the algebra operations of A and the given k-module structure. A is not assumed to be unital or commutative. A is of finite type if as a k-module A is finitely generated. For k-algebras of finite type, this talk introduces a new equivalence relation which is a weakening of Morita equivalence. Thus two algebras which are Morita equivalent are equivalent in the new equivalence relation, but there are many examples of non-Morita-equivalent algebras which are equivalent in the new equivalence relation. From a geometric point of view, the new equivalence relation permits a tearing apart of strata in the primitive ideal space which is not allowed by Morita equivalence. The new equivalence relation preserves periodic cyclic homology. An application to the representation theory of reductive p-adic groups will be briefly outlined. This talk is intended for non-specialists. All the basic definitions will be precisely stated.