# Morita equivalence for $k$-algebras

## TL;DR

This paper reviews Morita equivalence and introduces stratified equivalence for finite type $k$-algebras, highlighting their differences and implications for spectra and cyclic homology, with applications to affine Hecke algebras.

## Contribution

It introduces stratified equivalence as a weakening of Morita equivalence, allowing for more flexible relations while preserving key invariants.

## Key findings

- Stratified equivalence preserves spectra and cyclic homology.
- Stratified equivalence allows tearing apart of primitive ideal strata.
- Affine Hecke algebras exemplify the distinction between equivalences.

## Abstract

We review Morita equivalence for finite type $k$-algebras $A$ and also a weakening of Morita equivalence which we call stratified equivalence. The spectrum of $A$ is the set of equivalence classes of irreducible $A$-modules. For any finite type $k$-algebra $A$, the spectrum of $A$ is in bijection with the set of primitive ideals of $A$. The stratified equivalence relation preserves the spectrum of $A$ and also preserves the periodic cyclic homology of $A$. However, the stratified equivalence relation permits a tearing apart of strata in the primitive ideal space which is not allowed by Morita equivalence. A key example illustrating the distinction between Morita equivalence and stratified equivalence is provided by affine Hecke algebras associated to extended affine Weyl groups.