Morita (uniqueness) theorem

The main representation result underlying Morita theory is the Eilenberg–Watts theorem.

Theorem 1 (Eilenberg–Watts). Let A and B be rings, and let

F : A -Mod −→ B -Mod

be an additive, right-exact functor that preserves arbitrary direct sums. Then there exists a (B,A)-bimodule Q, unique up to isomorphism, such that F is naturally isomorphic to the tensor-product functor

Q ⊗ (− ) : A-Mod −→ B -Mod. A

One may take

Q = F (A),

with its right A-module structure induced by the endomorphisms of the left regular A-module.

This theorem yields the standard bimodule characterization of Morita equivalence.

Corollary 1. Two rings A and B are Morita equivalent if and only if there exist an (A,B)-bimodule P and a (B,A)-bimodule Q such that

P ⊗B Q ∼= A

as (A,A)-bimodules and

Q ⊗ P ∼= B A

as (B,B)-bimodules.

In this situation the functors

Q ⊗A (− ) : A-Mod − → B -Mod

and

P ⊗B (− ) : B -Mod −→ A -Mod

are quasi-inverse equivalences.

Moreover,

End (P ) ∼= Bop, End (Q ) ∼= Aop, A B

and P and Q are finitely generated projective generators on the appropriate sides.

Proof sketch. If the module categories are equivalent, choose quasi-inverse equivalences F and G. By the Eilenberg–Watts theorem,

F ∼= Q ⊗A (− ), G ∼= P ⊗B (− )

for suitable bimodules Q and P. Since

G ∘ F ∼= Id and F ∘ G ∼= Id , A-Mod B -Mod

evaluating at the regular modules gives

P ⊗B Q ∼= A, Q ⊗A P ∼= B.

Conversely, these bimodule isomorphisms immediately show that the two tensor functors above are quasi-inverse equivalences.

Corollary 2. If A and B are Morita equivalent, then:

the categories of right modules $Mod -A and Mod -B$
are equivalent;
the categories of bimodules $A -Mod -A and B-Mod -B$
are equivalent.

With bimodules P and Q as above, an equivalence on bimodules is given by

M ↦−→ Q ⊗A M ⊗A P.

It sends the regular (A,A)-bimodule A to the regular (B,B)-bimodule B, since

Q ⊗ A ⊗ P ∼= Q ⊗ P ∼= B. A A A

A quasi-inverse is

N ↦−→ P ⊗B N ⊗B Q.

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