  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.
