# Morita Equivalence Revisited

MORITA EQUIVALENCE REVISITED

Let $X$ be a complex affine variety and $k$ its coordinate algebra. Equivalently, $k$ is a unital algebra over the complex numbers which

is commutative, finitely generated, and nilpotent-free. A $k$-algebra is an algebra $A$ over the complex numbers $\mathbb{C}$ which is a $k$-module

(with an evident compatibility between the algebra structure of $A$ and the $k$-module structure of $A$). $A$ is not required to have a unit.

A $k$-algebra $A$ is of finite type if as a $k$-module $A$ is finitely generated. This talk will review Morita equivalence for $k$-algebras and will then introduce --- for finite type $k$-algebras --- a weakening of Morita equivalence called geometric equivalence.

The new equivalence relation preserves the primitive ideal space (i.e. the set of equivalence classes of irreducible $A$-modules) and the periodic cyclic homology of $A$. However, the new equivalence relation permits a tearing apart of strata in the primitive ideal space which is not allowed by Morita equivalence.

The ABPS (Aubert-Baum-Plymen-Solleveld) conjecture asserts that if $G$ is a connected split reductive $p$-adic group, then the finite type algebra which Bernstein assigns to any given Bernstein component is geometrically equivalent to the coordinate algebra of the associated extended quotient --- and that the geometric equivalence can be chosen so that the resulting bijection between the Bernstein component and the extended quotient has properties as in the statement of ABPS.

The above is joint work with Anne-Marie Aubert, Roger Plymen, and Maarten Solleveld.