Morita Theory for Comodules Over Corings

Abstract

By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule Σ of an A-coring 𝒞. This allows to extend (weak and strong) structure theorems in the literature, in particular beyond the cases when any of the coring 𝒞 or the comodule Σ is finitely generated and projective as an A-module. That is, we obtain relations between the category of 𝒞-comodules and the category of firm modules for a firm ring R, which is an ideal of the endomorphism algebra End ^𝒞(Σ). For a firmly projective comodule of a coseparable coring we prove a strong structure theorem assuming only surjectivity of the canonical map.

Key Words:

• Comodules over corings
• Firm rings
• Morita contexts

2000 Mathematics Subject Classification:

• 16D90
• 16W30

ACKNOWLEDGMENTS

The first author acknowledges a Bolyai János Research Scholarship and financial support of the Hungarian Scientific Research Fund OTKA F67910.

The second author thanks the Fund for Scientific Research–Flanders (Belgium) (F.W.O.–Vlaanderen) for a Postdoctoral Fellowship.

Notes

Communicated by J. Alev.