  This paper develops the tools of formal algebraic geometry in the setting of
noncommutative manifolds, roughly ringed spaces locally modeled on the free
associative algebra. We define a notion of noncommutative coordinate system,
which is a principal bundle for an appropriate group of local coordinate
changes. These bundles are shown to carry a natural flat connection with
properties analogous to the classical Gelfand-Kazhdan structure.
  Every noncommutative manifold has an underlying smooth variety given by
abelianization. A basic question is existence and uniqueness of noncommutative
thickenings of a smooth variety, i.e., finding noncommutative manifolds
abelianizing to a given smooth variety. We obtain new results in this direction
by showing that noncommutative coordinate systems always arise as reductions of
structure group of the commutative bundle of coordinate systems on the
underlying smooth variety; this also explains a relationship between
$\mathcal{D}$-modules on the commutative variety and sheaves of modules for the
noncommutative structure sheaf.
