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 \rightarrow 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 \rightarrow P_i and such that the comodule algebra of global sections is P .