nLab
Morita context

Contents

Idea

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 k=k = \mathbb{Z}, rings) AA and BB, it consists of bimodules AP B{}_A P_B, BQ A{}_B Q_A, and bimodule homomorphisms f:PQAf: P\otimes Q\to A, g:QPBg: Q\otimes P\to B satisfying mixed associativity conditions.

Properties

Theorem. (Bass II.3.4) If ff is surjective, then:

(i) ff is an isomorphism

(ii) PP and QQ are generators in the categories of AA-modules

(iii) PP and QQ are finitely generated and projective

(iv) gg induces isomorphisms of bimodules PHom B(Q,B)P\cong Hom_B(Q,B) and QHom B(P,B)Q\cong Hom_B(P,B)

(v) homomorphisms of AA-algebras End B(P)AEnd B(Q)End_B(P)\leftarrow A\rightarrow End_B(Q) are isomorphisms

(Bass II.4.1) A Morita context can be constructed from an AA-algebra BB and a right BB-module PP. Then set A=End B(P)A = End_B(P) and Q=Hom B(P,B)Q=Hom_B(P,B). Then f=f Pf = f_P and g=g Pg = g_P are defined by (bq)p=b(qp)(b q) p = b (q p) and (qa)p=q(ap)(q a) p = q(a p).

(Bass II.4.4) (i) f Pf_P s surjective iff PP is finitely generated projective BB-module. Then f Pf_P s iso.

(ii) g Pg_P is surjective iff PP s a generator of mod Bmod_B, then g Pg_P is iso

(iii) The Morita context (A,B,P,Q,f,g)(A,B,P,Q,f,g) is a Morita equivalence iff PP is both projective and a generator. Then P:mod Amod B\otimes_P : mod_A\to mod_B and its right adjoint Hom B(P,)Hom_B(P,-) form the equivalence.

Literature

There are generalizations in more general bicategories:

category: algebra