Formal Geometry for Noncommutative Manifolds

TL;DR

This paper extends formal algebraic geometry to noncommutative manifolds, introducing noncommutative coordinate systems, and explores their existence, uniqueness, and relation to classical geometry and D-modules.

Contribution

It develops the theory of noncommutative coordinate systems and shows they are reductions of classical coordinate bundles, linking noncommutative geometry with D-modules.

Findings

Noncommutative coordinate systems always arise as reductions of classical bundles.

Every noncommutative manifold has an underlying smooth variety via abelianization.

New connections between noncommutative structures and classical D-modules are established.

Abstract

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.