PlanetPhysics/Morita Equivalence

Morita equivalence

This entry presents both the definition of Morita equivalent algebras and the Morita equivalence [[../Formula/|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 [[../Cod/|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 [[../CenterOfAbelianCategory/|Associative Algebra Lemma]] one also has that $A=Z_A$ and $B=Z_B.$