A comodule algebra over a Hopf algebra with bijective antipode is called principal if the coaction of is Galois and is -equivariantly projective (faithfully flat) over the coaction-invariant subalgebra . We prove that principality is a piecewise property: given comodule-algebra surjections whose kernels intersect to zero, is principal if and only if all 's are principal. Furthermore, assuming the principality of , we show that the lattice these kernels generate is distributive if and only if so is the lattice obtained by intersection with . 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 -families of surjections and such that the comodule algebra of global sections is .
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Piotr M. Hajac, Ulrich Krähmer, Rainer Matthes, Bartosz Zieliński, Piecewise principal comodule algebras. J. Noncommut. Geom. 5 (2011), no. 4, pp. 591–614DOI 10.4171/JNCG/88