[Paper Review] Galois comodules

Tomasz Brzeziński|arXiv (Cornell University)|Dec 8, 2003

Algebraic structures and combinatorial models24 citations

TL;DR

This paper introduces and characterizes Galois and principal comodules over corings, particularly over fields, showing that a finitely generated projective right comodule is principal if the lifted canonical map is a split epimorphism in the category of left comodules. It establishes that entwining structures with a group-like element and surjective lifted canonical map yield principal extensions, linking the theory to non-commutative principal bundles.

ABSTRACT

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 $\psi$ 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.

Motivation & Objective

• To define and study Galois comodules in the context of corings.
• To introduce and characterize a special class of Galois comodules called principal comodules, defined as those that are projective over their endomorphism rings.
• To provide a complete description of principal comodules when the base ring is a field.
• To characterize principal extensions and non-commutative principal bundles using entwining structures and the surjectivity of lifted canonical maps.
• To investigate the relationship between relative injectivity of Galois comodules and endomorphism ring extensions.

Proposed method

• Defining Galois comodules via the canonical map and its lifting in the category of comodules.
• Introducing principal comodules as Galois comodules that are projective over their comodule endomorphism rings.
• Using the condition that a lifting of the canonical map is a split epimorphism in the category of left comodules to characterize principal comodules over a field.
• Analyzing entwining structures consisting of an algebra A, a coseparable coalgebra C, and a bijective entwining map ψ, with a group-like element in C.
• Applying induction functors to Galois and principal comodules via morphisms of corings.
• Establishing a connection between relative injectivity of Galois comodules and properties of the associated extension of endomorphism rings.

Experimental results

Research questions

• RQ1 Under what conditions is a simple comodule a Galois comodule over a coring?
• RQ2 When is a Galois comodule also a principal comodule, i.e., projective over its endomorphism ring?
• RQ3 What characterizes principal comodules in the case where the base ring is a field?
• RQ4 How do entwining structures with a group-like element and surjective lifted canonical map give rise to principal extensions?
• RQ5 What is the relationship between the relative injectivity of a Galois comodule and the extension of its endomorphism ring?

Key findings

• A finitely generated and projective right comodule of an A-coring 𝒞 is principal if and only if a lifting of the canonical map is a split epimorphism in the category of left 𝒞-comodules.
• Over a field, any entwining structure with a bijective entwining map ψ, a coseparable coalgebra C, and a group-like element in C gives rise to a principal extension provided the lifted canonical map is surjective.
• The class of principal comodules is closed under induction via morphisms of corings.
• A Galois comodule is relatively injective if and only if the associated extension of endomorphism rings satisfies specific homological properties.
• The theory provides a characterization of non-commutative principal bundles via entwining structures and split epimorphisms of canonical maps.
• The paper establishes a complete structural description of principal comodules in the case of a base field, linking algebraic and categorical properties.