Morita equivalence

From Wikipedia, the free encyclopedia

In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. They are named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958.

Contents

[edit] Motivation

Rings are commonly studied in terms of their modules, as modules can be viewed as representations of rings. Every ring has a natural R-module structure on itself where the module action is defined as the multiplication in the ring, so the approach via modules is more general and gives useful information. Because of this, one often studies a ring by studying the category of modules over that ring.

Morita equivalence takes this viewpoint to a natural conclusion by defining rings to be equivalent if their module categories are equivalent.

[edit] Formal definition

Two rings R and S are said to be Morita equivalent (or equivalent) if there is an additive equivalence of the category of (left) modules over R, RM, and the category of (left) modules over S, SM.

It can be shown that the left module categories are equivalent if and only if the right module categories are equivalent.

Equivalences can be characterized as follows: if F:RM \to SM and G:SM \to RM are additive (covariant) functors, then F and G are an equivalence if and only if there is a balanced (S,R)-bimodule P such that SP and PR are finitely generated projective generators and natural isomorphisms F \cong (P \otimes_R -) and G \cong \operatorname{Hom}_R(P,-).

[edit] Properties preserved by equivalence

Many properties are preserved by equivalence for the objects in the module category. Taking the ring as a special case, we have the following list of preserved properties between equivalent rings. If R and S are equivalent rings, then R is

if and only if S is. Furthermore, we have that Cen(R) is isomorphic to Cen(S), where the Cen denotes the center of the ring, and R/J(R) is equivalent to S/J(S), where J denotes the Jacobson radical.

However, Morita equivalence is not equivalent to isomorphism. It is possible, but extremely difficult, to distinguish between non-isomorphic but Morita equivalent rings. One important special case where Morita equivalence implies isomorphism is the case of commutative rings.

[edit] Examples

The ring of matrices with elements in R, Mn(R) is equivalent to R for any n > 0. Notice that this generalizes the classification of simple artinian rings given by Artin-Wedderburn theory. To see the equivalence, notice that if M is a left R-module then Mn is a Mn(R)-module where the module structure is given by applying matrices to vectors in the standard way. This allows the definition of a functor from the category of left R-modules to the category of left Mn(R)-modules. The inverse functor is defined by realizing that for any Mn(R)-module there is a left R-module V and a positive integer n such that the Mn(R)-module is obtained from V as described above.

[edit] Criterion for equivalence

For every right-exact functor F from the category of left-R modules to the category of left-S modules that commutes with direct sums, a theorem of homological algebra shows that there is a (S,R)-bimodule E such that F is naturally equivalent to E \otimes_R -. This means that R and S are Morita equivalent if and only if there are bimodules M and N such that M \otimes N \cong R and N \otimes M \cong S. Moreover, N \cong \operatorname{Hom}(M,S).

[edit] Further directions

Dual to the theory of equivalences is the theory of dualities between the module categories, where the functors used are contravariant rather than covariant. This theory, though similar in form, has significant differences because there is no duality between the categories of modules for any rings, although dualities may exist for subcategories. In other words, because infinite dimensional modules are not generally reflexive, the theory of dualities applies more easily to finitely generated algebras over noetherian rings. Perhaps not surprisingly, the criterion above has an analogue for dualities, where the natural isomorphism is given in terms of the hom functor rather than the tensor functor.

Morita Equivalence can also be defined in more structured situations, such as for symplectic groupoids and C*-algebras. In the case of C*-algebras, a stronger type equivalence, called strong Morita equivalence, is needed because of the additional structure to obtain results useful in applications.

[edit] Significance in K-theory

If two rings are Morita equivalent, there is an induced equivalence of the respective categories of projective modules since the Morita equivalences will preserve exact sequences (and hence projective modules). Since the algebraic K-theory of a ring is defined (in Quillen's approach) in terms of the homotopy groups of the classifying space of the nerve of the category of projective modules over the ring, Morita equivalent rings must have isomorphic K-groups.

[edit] References

  • F.W. Anderson and K.R. Fuller: Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, 2 nd Ed., Springer-Verlag, New York, 1992, ISBN 0-387-97845-3, ISBN 3-540-97845-3