Piecewise principal comodule algebras

Abstract

A comodule algebra P over a Hopf algebra H with bijective antipode is called principal if the coaction of H is Galois and P is H-equivariantly projective (faithfully flat) over the coaction-invariant subalgebra B. We prove that principality is a piecewise property: given N comodule-algebra surjections P->Pi whose kernels intersect to zero, P is principal if and only if all Pi's are principal. Furthermore, assuming the principality of P, we show that the lattice these kernels generate is distributive if and only if so is the lattice obtained by intersection with B. Finally, assuming the above distributivity property, we obtain a flabby sheaf of principal comodule algebras over a certain space that is universal for all such N-families of surjections P->Pi and such that the comodule algebra of global sections is P.

References in corpus (4)

• Symmetries of quantum spaces. Subgroups and quotient spaces of quantum $SU(2)$ and $SO(3)$ groups
• Strong Connections and Chern-Connes Pairing in the Hopf-Galois Theory
• Noncommutative Geometry Approach to Principal and Associated Bundles
• Quantum Principal Bundles and Corresponding Gauge Theories

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• Topics in Noncommutative Geometry
• On the Relationship between Classical and Deformed Hopf Fibrations
• Piecewise Principal Coactions of Co-Commutative Hopf Algebras
• Toeplitz algebras in quantum Hopf fibrations
• Nontrivial Deformation of a Trivial Bundle
• Non-commutative integral forms and twisted multi-derivations