A Morita context or, in some authors (e.g. Bass) the pre-equivalence data is a generalization of Morita equivalence between categories of modules. In the case of right modules, for two associative k-algebras (or, in the case of , rings) and , it consists of bimodules , , and bimodule homomorphisms , satisfying mixed associativity conditions.
Theorem. (Bass II.3.4) If is surjective, then:
(i) is an isomorphism
(ii) and are generators in the categories of -modules
(iii) and are finitely generated and projective
(iv) induces isomorphisms of bimodules and
(v) homomorphisms of -algebras are isomorphisms
(Bass II.4.1) A Morita context can be constructed from an /algebra and a right -module . Then set and . Then and are defined by and .
(Bass II.4.4) (i) s surjective iff is finitely generated projective -module. Then s iso.
(ii) is surjective iff s a generator of , then is iso
(iii) The Morita context is a Morita equivalence iff is both projective and a generator. Then and its right adjoint form the equivalence.
There are generalizations in more general bicategories:
Last revised on February 24, 2014 at 12:36:34. See the history of this page for a list of all contributions to it.