For a (commutative) scheme $X$, the sheaf $D(X)$ of regular differential operators is a sheaf of noncommutative rings, more precisely a sheaf of noncommutative algebras in the monoidal category of ${\mathrm{\u0111\u0165\u2019\u015e}}_{X}$-modules. Thus it may be considered as a case of noncommutative algebraic geometry, namely it is sort of a space $X$ with a noncommutative â€śstructure ringâ€ť $D(X)$. In the usual algebraic geometry, if $X$ is affine, i.e. of the form $X=\mathrm{Spec}A$, where $A$ is a commutative ring, the global sections $\mathrm{\xce\u201c}{O}_{X}\xe2\u2030\dots A$, and this extends for quasicoherent modules (this is sometimes called the affine Serreâ€™s global sections theorem). This phenomenon that global sections determine the sheaf is hence an affine phenomenon. An analogues phenomenon in the world of $D$-modules holds for $D$-modules on some nonaffine varieties, for example the flag varieties. Such schemes are called D-affine.

A. Beilinson, I. N. Bernstein, A proof of Jantzen conjecture, Adv. in Soviet Math. 16, Part 1 (1993), 1-50, MR95a:22022, pdf

The phenomenon has also its abstract counterpart in the language of differential monads of Lunts and Rosenberg, see here especially part I:

V. A. Lunts, A. L. Rosenberg, Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings, MPI 1996-53 pdf, II. D-Calculus in the braided case. The localization of quantized enveloping algebras, MPI 1996-76 pdf

Created on May 8, 2011 10:51:48
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