PlanetPhysics/Morita Equivalence

Morita equivalence
This entry presents both the definition of Morita equivalent algebras and the Morita equivalence theorem, with a brief proof included. Let $$A$$ and $$B$$ be two associative, but not necessarily commutative, algebras. Such algebras $$A$$ and $$B$$ are called Morita equivalent, if there is an equivalence of categories between $$A$$-mod and $$B$$-mod.

\begin{theorem}{\mathbf Morita Equivalence Theorem} Commutative algebras $$A$$ and $$B$$ are Morita equivalent if and only if they are isomorphic.\end{theorem}

Proof. Following the above definition, isomorphic algebras are Morita equivalent. Let us assume that $$A$$ and $$B$$ are any two such Morita equivalent associative algebras. It follows then that $$A-mod \sim B-mod$$, and thus one also has that $$Z(A-mod) \simeq Z(B-mod).$$ If $$A$$ and $$B$$ are both commutative, then by the Associative Algebra Lemma one also has that $$A = Z_A$$ and $$B = Z_B.$$