Morita context functors

Abstract

Given a Morita context (R, V, W, S), there are functors W ⊗ () and hom (V,) from R-mod to S-mod and a natural transformation A from the first to the second. This has an epi-mono factorization and the intermediate functor we denote by ()° with natural transformations v: W ⊗ () → ()° and μ: ()° → hom(V,). The tensor functor is exact if and only if WR is flat, whilst the hom functor is exact if and only if RV is projective. We begin by determining conditions under which ()° is exact; this is Theorem 1. Section 2 is concerned with a lattice isomorphism theorem and it is necessary to restrict attention to trace-closed submodules of RM. When ()° is exact there is an isomorphism between the lattice of trace-closed submodules of RM and the submodule lattice of M°. This is proved in Theorem 3. In the final section we involve the corresponding functor ()*: S-mod → R-mod determined by (S, W, V, R). In Theorem 5 we show that ()° is an equivalence, with inverse ()*, between the subcategories of trace-accessible, trace-torsion-free submodules. © 1988, Cambridge Philosophical Society. All rights reserved.