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Morita (uniqueness) theorem (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.
  • (i). If $A$ and $B$ are Morita equivalent rings, then the corresponding categories ${\bf mod-A}$ and ${\bf mod-B}$ are also equivalent.
  • (ii). Furthermore, there exists a natural equivalence of categories

    $\displaystyle {\bf A-bimod} \to {\bf B-bimod}$
    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 $(\^aˆ’ \bigotimes{}_A M_b$ and $(\^aˆ’ \bigotimes{}_B N_b)$ 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.



"Morita (uniqueness) theorem" is owned by bci1.

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Also defines:  $(B,A)$-bimodule
Keywords:  Morita theorem

Cross-references: projective objects, categories, isomorphism, functor, theorem, proposition, Morita equivalent algebras

This is version 9 of Morita (uniqueness) theorem, born on 2009-06-15, modified 2009-06-15.
Object id is 808, canonical name is MoritaUniquenessTheorem.
Accessed 698 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
Pending Errata and Addenda
None.
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