Coring stabilizers for a Hopf algebra coaction

The coring stabilizer Stab(P) is introduced for any prime ideal P of a right H-comodule algebra A such that the factor ring A/P is either right or left Goldie. This notion is used to obtain Hopf algebraic analogs of two category equivalences associated with a homogeneous space. The category of linearized quasicoherent sheaves on a noncommutative homogeneous space is interpreted as a suitable quotient category of the category of Hopf modules. Birational invariance of such quotient categories is proved. It is shown that for a birational H-coequivariant extension B of A properly defined subsets of prime ideals of A and B correspond to each other bijectively.