Coordinate systems and associative algebras

This dissertation applies and extends the techniques of formal algebraic geometry in the setting of certain "smooth" associative algebras and their globalizations, 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 noncommu- tative 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 D-modules on the commutative variety and sheaves of modules for the noncommutative structure sheaf. The lower central series invariants M[subscript k] of an associative algebra A are the two-sided ideals generated by k-fold nested commutators; the M[subscript k] give a decreasing filtration of A. We study the relationship between the geometry of X = Spec A[subscript ab] and the associated graded components N[subscript k] of this filtration. We show that the N[subscript k] form coherent sheaves on a certain nilpotent thickening of X, and that Zariski localization on X coincides with noncommutative localization of A. We then construct the N[subscript k] in terms of the bundle of coordinate systems on X and the N[subscript k] invariants for the free associative algebra; in particular, since this is independent of A, we exhibit the N[subscript k] as natural vector bundles on the category of smooth schemes.