# Publishing house / Banach Center Publications / All volumes

## Morita equivalence for $k$-algebras

### Volume 120 / 2020

Banach Center Publications 120 (2020), 245-265 MSC: 14A22, 16G30, 20C08, 33D80. DOI: 10.4064/bc120-16

#### Abstract

We review Morita equivalence for finite type $k$-algebras $A$ and also a weakening of Morita equivalence which we call \lt em \gt stratified equivalence \lt /em \gt . 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 affine Weyl groups.

#### Authors

• Anne-Marie AubertCNRS
Sorbonne Université
Université Paris Diderot
Institut de Mathématiques de Jussieu
Paris Rive Gauche
IMJ-PRG
F-75005 Paris, France
ORCID: 0000-0002-9613-9140
e-mail
• Paul BaumMathematics Department
Pennsylvania State University
University Park, PA 16802, USA
e-mail
• Roger PlymenSchool of Mathematics
Alan Turing Building
Manchester University
Manchester M13 9PL, UK
ORCID: 0000-0002-2071-6925
e-mail
• Maarten SolleveldIMAPP