For a (commutative) scheme , the sheaf of regular differential operators is a sheaf of noncommutative rings, more precisely a sheaf of noncommutative algebras in the monoidal category of -modules. Thus it may be considered as a case of noncommutative algebraic geometry, namely it is sort of a space with a noncommutative “structure ring” . In the usual algebraic geometry, if is affine, i.e. of the form , where is a commutative ring, the global sections , and this extends for quasicoherent modules (this is sometimes called the affine Serre’s global sections theorem). This phenomenon that global sections determine the sheaf is hence an affine phenomenon. An analogues phenomenon in the world of -modules holds for -modules on some nonaffine varieties, for example the flag varieties. Such schemes are called D-affine.
Beilinson-Bernstein localization theorem and generalizations…
D-affinity is studied in
The phenomenon has also its abstract counterpart in the language of differential monads of Lunts and Rosenberg, see here especially part I: