Morita Equivalence but not Equivalence
Non-Commutative Algebra with an Example
This is technically a counterexample, it is not one in the traditional sense. Normally, a counterexample is an object for which some intuition says it should not exist. Nobody thinks all Morita equivalent algebras and equivalent. But, just as nobody thinks all exact sequences split, it is helpful to have examples of non-splitting exact sequences at one’s elbow.
Morita equivalence is a property of (associative) algebras and It means they have the same representation theory.
Definition 1. Let be associative algebras over a field Then, is said to be Morita equivalent to , written , if there is an equivalence of categories
To paraphrase: being acted on by is essentially the same as being acted on by
Notation: Throughout, let denote associative algebras over a fixed field
For commutative algebras, Morita equivalence coincides with isomorphism.
Lemma 2. where denotes the center of and
Proof. Let be central. Then, define a natural transformation whose component at a left -module is the action of on . By centrality of , is a module homomorphism. Conversely, given a natural transformation define This is central by naturality of Q.E.D.
Proposition 3. Let be commutative. Then, if and only if is isomorphic to
Proof. Clearly, if then Conversely, suppose Then, By Lemma 2, we are done. Q.E.D.
To show this result is “sharp” we construct non-isomorphic algebras that are Morita equivalent. By Proposition 3. (at least) one algebra must be non-commutative.
Theorem 4. For ,
Proof. Let denote the matrix with a in the spot and zeros everywhere else. Then, and So by Morita’s Theorem [1, Corollary 2.3.2] we’re done. Q.E.D.
For , Mat is non-commutative. So, for a commutative algebra we cannot have an isomorphism between and Mat
 Ginzburg, V. Lectures on Noncommutative Geometry. https://arxiv.org/pdf/math/0506603.pdf