2003/12/08 by Tomasz Brzeziński, Tomasz Brzezinski, Brzezinski, Tomasz
Computer Science · Engineering · Mathematics · #13B02 #16W30 #Advanced Numerical Analysis Techniques #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Rings and Algebras (math.RA) #math.RA #msc:13B02 #msc:16W30
paper · pdf · doi:10.48550/arxiv.math/0312159
36 pages, LaTeX; a few corrections to version 2
openalex publication_date 2003/12/08 · arxiv created 2004/04/28 · arxiv updated 2009/12/01 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
Galois comodules of a coring are studied. The conditions for a simple comodule to be a Galois comodule are found. A special class of Galois comodules termed principal comodules is introduced. These are defined as Galois comodules that are projective over their comodule endomorphism rings. A complete description of principal comodules in the case a background ring is a field is found. In particular it is shown that a (finitely generated and projective) right comodule of an A-coring \mathcal C is principal provided a lifting of the canonical map is a split epimorphism in the category of left \mathcal C-comodules. This description is then used to characterise principal extensions or non-commutative principal bundles. Specifically, it is proven that, over a field, any entwining structure consisting of an algebra A, a coseparable coalgebra C and a bijective entwining map ψ together with a group-like element in C give rise to a principal extension provided the lifted canonical map is surjective. Induction of Galois and principal comodules via morphisms of corings is described. A connection between the relative injectivity of a Galois comodule and the properties of the extension of endomorphism rings associated to this comodule is revealed.