Piecewise principal comodule algebras. (Q645834)

Comodule algebras provide a natural noncommutative geometry generalization of spaces equipped with group actions. Principal extensions appear to be a proper analogue of principal bundles in this context. They can be considered as functors from the category of finite-dimensional corepresentations of the Hopf algebra to the category of finitely generated projective modules over the coaction-invariant subalgebra. 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~\(P^{\text{co\,}H}\). The aim of the paper under review is to establish a viable concept of locality of comodule algebras, and to analyse its relationship with principality. The following main results are shown. 1) Let \(\pi_i\colon P\to P_i\), \(i\in\{1,\dots,N\}\), be surjections of right \(H\)-comodule algebras such that \(\bigcap_{i=1}^N\ker\pi_i=0\). Then \(P\) is principal if and only if all the \(P_i\)'s are principal. 2) The lattice these kernels generate is distributive if and only if so is the lattice obtained by intersection with \(P^{\text{co\,}H}\) if \(P\) has the principality. Under the assumption of the above distributivity property, the authors also obtain a flabby sheaf of principal comodule algebras over a certain space that is universal for all such \(N\)-families of surjections \(P\to P_i\) and such that the comodule algebra of global sections is \(P\).