PlanetPhysics/Morita Uniqueness Theorem

The main result for Morita equivalent algebras is provided by the following proposition.

\begin{theorem}Morita theorem.

Let and be two arbitrary rings, and also let be an additive, right exact functor. Then, there is a -bimodule , which is unique up to isomorphism, so that is isomorphic to the functor given by \end{theorem}

There are also two important and fairly straightforward corollaries of the Morita (uniqueness) theorem.

\begin{theorem} {\mathbf Corollary 1.}

Two rings, and , are Morita equivalent if and only if there is an -bimodule and a -bimodule so that as -bimodules and as -bimodules. With these assumptions, one obtains:

. Also is projective as an -module, whereas is projective as a -module. \end{theorem}

Proof . All equivalences of categories are exact functors, and therefore they preserve projective objects as required by Corollary 1.

\begin{theorem}Corollary 2.

  • (i). If and are Morita equivalent rings, then the corresponding categories and are also equivalent.
  • (ii). Furthermore, there exists a natural equivalence of categories which takes to , of course along with their natural bimodule structures.


Proof. Let and be the bimodules already defined in Corollary 1 .

For proposition (i), one utilizes the functors Failed to parse (syntax error): {\displaystyle (− \bigotimes{}_A M_b} and Failed to parse (syntax error): {\displaystyle (− \bigotimes{}_B N_b)} to prove the equivalence of the two categories.

For the second proposition (ii), one needs to employ the functor to prove the natural equivalence of the latter two categories.