Morita (uniqueness) theorem

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

Theorem 0.1   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

$\displaystyle A-mod \mapsto B-mod,$

$\displaystyle M \mapsto Q \bigotimes {}_A M.$

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

Theorem 0.2   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

$\displaystyle M_B \bigotimes {}B N_B \simeq A$

as $ A$ -bimodules and

$\displaystyle N_B \bigotimes{}_A M_b \simeq B$

as $ B$ -bimodules. With these assumptions, one obtains:

$\displaystyle End_{A-mod}(M_b) = B^{op},$

$\displaystyle End_{B-mod}(N_b) = A^op$

. Also $ M_b$ is projective as an $ A$ -module, whereas $ N_B$ is projective as a $ B$ -module.

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

Theorem 0.3   Corollary 2.

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

For proposition (i), one utilizes the functors $ (− \bigotimes{}_A M_b$ and $ (− \bigotimes{}_B N_b)$ to prove the equivalence of the two categories.

For the second proposition (ii), one needs to employ the functor

$\displaystyle N_b \bigotimes{}_A - \bigotimes{}_A M_b : {\bf A-bimod} \longrightarrow {\bf B-bimod}$

to prove the natural equivalence of the latter two categories.



Contributors to this entry (in most recent order):

As of this snapshot date, this entry was owned by bci1.