PlanetPhysics/Morita Uniqueness Theorem

The main result for [[../MoritaEquivalentAlgebras2/|Morita equivalent algebras]] is provided by the following [[../Predicate/|proposition]].

\begin{theorem}Morita [[../Formula/|theorem]].

Let $A$ and $B$ be two arbitrary rings, and also let $F:A-mod\to B-mod$ be an additive, right exact [[../TrivialGroupoid/|functor]]. Then, there is a $(B,A)$-bimodule $𝒬$, which is unique up to [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphism]], so that $F$ is isomorphic to the functor $G$ given by $A-mod\mapsto B-mod,$ $M\mapsto Q⨂{}_{A}M.$ \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 $M_B⨂B{N}_B\simeq A$ as $A$-bimodules and $N_B⨂{}_A{M}_b\simeq B$ as $B$-bimodules. With these assumptions, one obtains:

$En{d}_{A-mod}(M_b)=B^{op},$ $En{d}_{B-mod}(N_b)=A^{o}p$. Also $M_b$ is projective as an $A$-module, whereas $N_B$ is projective as a $B$-module. \end{theorem}

Proof . All equivalences of [[../Cod/|categories]] are exact functors, and therefore they preserve [[../ProjectiveObject/|projective objects]] as required by Corollary 1.

\begin{theorem}Corollary 2.

• (i). If $A$ and $B$ are Morita equivalent rings, then the corresponding categories $𝐦od-A$ and $𝐦od-B$ are also equivalent.
• (ii). Furthermore, there exists a natural equivalence of categories $𝐀-bimod\to 𝐁-bimod$ 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 $N_b⨂{}_A-⨂{}_A{M}_b:𝐀-bimod⟶𝐁-bimod$ to prove the natural equivalence of the latter two categories.