  Viewing comodule algebras as the noncommutative analogues of affine varieties
with affine group actions, we propose rudiments of a localization approach to
nonaffine Hopf algebraic quotients of noncommutative affine varieties
corresponding to comodule algebras. After reviewing basic background on
noncommutative localizations, we introduce localizations compatible with
coactions. Coinvariants of these localized coactions give local information
about quotients. We define Zariski locally trivial quantum group algebraic
principal and associated bundles. Compatible localizations induce localizations
on the categories of Hopf modules. Their interplay with the functor of taking
coinvariants and its left adjoint is stressed out.
  Using localization approach, we constructed a natural class of examples of
quantum coset spaces, related to the quantum flag varieties of type A of other
authors. Noncommutative Gauss decomposition via quasideterminants reveals a new
structure in noncommutative matrix bialgebras. Particularily, in the "quantum"
case, calculations with quantum minors yield the structure theorems.
