Projectivity and freeness over comodule algebras

2006/10/22 by Sergey Skryabin, Skryabin, S. · 6 citations
Mathematics · #Algebraic structures and combinatorial models #Rings, Modules, and Algebras #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.math/0610657

Abstract

Let H be a Hopf algebra and A an H-simple right H-comodule algebra. It is shown that under certain hypotheses every (H,A)-Hopf module is either projective or free as an A-module and A is either a quasi-Frobenius or a semisimple ring. As an application it is proved that every weakly finite (in particular, every finite dimensional) Hopf algebra is free both as a left and a right module over its finite dimensional right coideal subalgebras, and the latter are Frobenius algebras. Similar results are obtained for H-simple H-module algebras.

Cited by

• Freeness and divisibility for right H-simple left H-comodule algebras over a pointed Hopf algebra H
• H-Equivariant Morita equivalences of Loewy-graded comodule algebras
• The structure of iterated Hopf Ore extensions
• Nichols algebras, tensor categories and Kazhdan-Lusztig correspondences
• On two notions of torsion and metric compatibility of connections in noncommutative geometry
• Morita invariants of quasitriangular comodule algebras

Related

• Freeness and divisibility for right H-simple left H-comodule algebras over a pointed Hopf algebra H
• Projectivity of a relative Hopf module over the subring of coinvariants
• A structure theorem for quasi-Hopf comodule algebras
• Two-sided (two-cosided) Hopf modules and Doi-Hopf modules for quasi-Hopf algebras
• Projective modules over Frobenius algebras and Hopf comodule algebras