On Galois comodules

2004/08/18 by Robert Wisbauer, Wisbauer, Robert
Mathematics · #16D80 #16D90 #16W30 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:16D80 #msc:16D90 #msc:16W30

paper · pdf · doi:10.48550/arxiv.math/0408251

24 pages

arxiv created 2004/10/20 · arxiv updated 2009/12/01

Abstract

Generalising the notion of Galois corings, Galois comodules were introduced as comodules P over an A-coring \cC for which P_A is finitely generated and projective and the evaluation map μ_\cC:\Hom^\cC(P,\cC)\ot_SP→ \cC is an isomorphism (of corings) where S=\End^\cC(P). It was observed that for such comodules the functors \Hom_A(P,-)\ot_SP and -\ot_A\cC from the category of right A-modules to the category of right \cC-comodules are isomorphic. In this note we call modules P with this property \em Galois comodules without requiring P_A to be finitely generated and projective. This generalises the old notion with this name but we show that essential properties and relationships are maintained. These comodules are close to being generators and have some common properties with tilting (co)modules. Some of our results also apply to generalised Hopf Galois (coalgebra Galois) extensions.

Related

• Galois comodules
• Equivalences between categories of modules and categories of comodules
• Galois functors and entwining structures
• The structure of corings: Induction functors, Maschke-type theorem, and\n Frobenius and Galois-type properties
• Group corings