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 $P^{\mathrm{co}H}$. We prove that principality is a piecewise property: given $N$ comodule-algebra surjections $P\to P_i$ whose kernels intersect to zero, $P$ is principal if and only if all $P_i$'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 $P^{\mathrm{co}H}$. 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 $P\to P_i$ and such that the comodule algebra of global sections is $P$.