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.
Cite
@article{arxiv.0707.1344,
title = {Piecewise principal comodule algebras},
author = {Piotr M. Hajac and Ulrich Kraehmer and Rainer Matthes and Bartosz Zielinski},
journal= {arXiv preprint arXiv:0707.1344},
year = {2007}
}