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 $$M_B \bigotimes {}B N_B \simeq A$$ as $$A$$-bimodules and $$N_B \bigotimes{}_A M_b \simeq B$$ as $$B$$-bimodules. With these assumptions, one obtains:

$$End_{A-mod}(M_b) = B^{op},$$ $$End_{B-mod}(N_b) = A^op$$. 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.

\end{theorem}
 * (i). If $$A$$ and $$B$$ are Morita equivalent rings, then the corresponding categories $${\mathbf mod-A}$$ and $${\mathbf mod-B}$$ 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.

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

For proposition (i), one utilizes the functors $$(âˆ’ \bigotimes{}_A M_b$$ and $$(âˆ’ \bigotimes{}_B N_b)$$ 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.