PlanetPhysics/Morita Uniqueness Theorem

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

\begin{theorem}Morita theorem.

Let $A$ and $B$ be two arbitrary rings, and also let $F:A-mod\to B-mod$ be an additive, right exact functor. Then, there is a $(B,A)$-bimodule ${\mathcal {Q}}$, which is unique up to isomorphism, so that $F$ is isomorphic to the functor $G$ given by $A-mod\mapsto B-mod,$ $M\mapsto Q\bigotimes {}_{A}M.$ \end{theorem}

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

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

Two rings, $A$ and $B$, are Morita equivalent if and only if there is an $(A,B)$-bimodule $M_{b}$ and a $(B,A)$-bimodule $N_{b}$ so that $M_{B}\bigotimes {}BN_{B}\simeq A$ as $A$-bimodules and $N_{B}\bigotimes {}_{A}M_{b}\simeq B$ as $B$-bimodules. With these assumptions, one obtains:

$End_{A-mod}(M_{b})=B^{op},$ $End_{B-mod}(N_{b})=A^{o}p$. Also $M_{b}$ is projective as an $A$-module, whereas $N_{B}$ is projective as a $B$-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 $A$ and $B$ are Morita equivalent rings, then the corresponding categories ${\mathbf {m} od-A}$ and ${\mathbf {m} od-B}$ are also equivalent.
• (ii). Furthermore, there exists a natural equivalence of categories ${\mathbf {A} -bimod}\to {\mathbf {B} -bimod}$ which takes $A$ to $B$, of course along with their natural bimodule structures.

\end{theorem}

Proof. Let $M_{b}$ and $N_{b}$ 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 $N_{b}\bigotimes {}_{A}-\bigotimes {}_{A}M_{b}:{\mathbf {A} -bimod}\longrightarrow {\mathbf {B} -bimod}$ to prove the natural equivalence of the latter two categories.