[Paper Review] Adjoint and Frobenius Pairs of Functors, Equivalences, and the Picard Group for Corings

Mohssin Zarouali-Darkaoui|arXiv (Cornell University)|Mar 26, 2007

Advanced Algebra and Geometry3 citations

TL;DR

This paper establishes a unified framework for adjoint and Frobenius pairs of functors between comodule categories over corings, generalizes the comatrix coring construction to quasi-finite comodules, and extends the Picard group to corings. It provides a characterization of Frobenius induction functors in graded module categories and proves an exact sequence relating automorphisms, kernels, and the Picard group of graded corings.

ABSTRACT

We study adjoint and Frobenius pairs of functors, equivalences, and the Picard group for corings.

Motivation & Objective

• To study adjoint and Frobenius pairs of functors between comodule categories over corings.
• To generalize the comatrix coring construction to quasi-finite comodules and characterize equivalences of comodule categories.
• To extend the Picard group from algebras and coalgebras to corings and establish its structural properties.
• To apply the results to induction functors, entwined modules, and graded ring theory for new, concrete characterizations.
• To unify Frobenius extensions of rings and corestriction functors over coalgebras under a single coring-theoretic framework.

Proposed method

• Introduces the notion of a (right) Frobenius extension of corings, generalizing Morita’s theorem for rings and the dual result for coalgebras.
• Uses the cotensor product functor and cohom functors to analyze adjoint and Frobenius pairs in comodule categories.
• Applies the generalized comatrix coring construction to characterize equivalences between comodule categories via quasi-finite comodules.
• Constructs a faithful functor from the category of graded corings to a derived category of complexes to relate automorphisms and Picard groups.
• Derives an exact sequence involving the kernel of a composition of functors and the Picard group of a graded coring.
• Employs bimodule isomorphisms and cohomological conditions to characterize Frobenius functors in graded module categories.

Experimental results

Research questions

• RQ1 When is the induction functor between categories of graded modules over G-sets a Frobenius functor?
• RQ2 What conditions ensure that a morphism of corings induces a Frobenius pair of functors between their comodule categories?
• RQ3 How can the Picard group of a coring be defined and what properties does it inherit from algebras and coalgebras?
• RQ4 What is the structure of the Picard group for graded corings arising from G-sets and entwining structures?
• RQ5 How do automorphisms of a graded coring relate to the kernel of a composition of functors in the Picard group?

Key findings

• The paper establishes a generalization of Morita’s theorem for Frobenius extensions of rings and the dual result for coalgebras, unifying them under a single coring-theoretic framework.
• It characterizes Frobenius induction functors in categories of graded modules over G-sets, providing a necessary and sufficient condition involving bimodule isomorphisms and quasi-finiteness.
• An exact sequence is constructed: 1 → Ker(Ω∘F∘G∘H) → Aut_k((G,X,A)) → Pic_k(gr-(A,X,G)) with the kernel described by invertible elements satisfying specific equivariance and support conditions.
• The generalized comatrix coring construction yields concrete characterizations of equivalences between comodule categories, extending previous results on comatrix coalgebras.
• The Picard group of a graded coring gr-(A,X,G) is shown to be isomorphic to the image of a composition of functors, with the kernel described via a system of equations on invertible elements.
• The subgroup of Pic_k(gr-(A,X,G)) constructed here is simpler and more natural than the Beattie-Del Río subgroup, offering a cleaner algebraic description.