nLab regular differential operator

Idea

Grothendieck has developed a deep version of differential calculus, based on a linearization of $O_X$-bimodules. It is also related to the (de Rham) descent data for the stack of $O_X$-modules over the simplicial scheme resolving the diagonal of $X$. As abstract descent data correspond to the flat connections for the corresponding monad, this was historically the first case in which this correspondence was noted; in positive characteristics Grothendieck called the corresponding descent data for the de Rham site “costratifications”, the corresponding connection is nowdays called Grothendieck connection, see Berthelot, Ogus 1978 for the fundamentals and the related notion of crystal (algebraic geometry).

This corresponds to looking at a sequence of infinitesimal neighborhoods of the diagonal. This geometrical principle can be applied to other categories; it is the basis of the study of jet schemes and close in spirit to some constructions in synthetic differential geometry.

Definitions

Given a commutative unital ring $R$, a filtration $M_n$ ($n\geq -1$) on a $R$-$R$-bimodule $M$ is a differential filtration if the commutator $[r,P]$ for any $P$ in $M_n$ and $r$ in $R$ is in $M_{n-1}$, and $M_{-1} = 0$. A bimodule is differential if it has an exhaustive ($\cup_N M_n = M$) differential filtration. Every $R$-$R$-bimodule has a differential part, i.e. the maximal differential submodule of $M$.

Regular differential operators, as defined by Grothendieck, are the elements of the differential part $Diff(R,R)$ of $Hom(R,R)$ i.e. a maximal differential subbimodule in $Hom(R,R)$. The operators in $Diff(R,R)_n$ are called the differential operators of degree $\leq n$. If $R\to B$ is a ring morphism, then the differential part of $B$ via its natural $R$-$R$-bimodule structure is also an object of $R\backslash \mathrm{Ring}$; in particular $Diff(R,R)$ is a ring and $R\hookrightarrow Diff(R,R)$ is an embedding of rings.

More generally (and in the affine case equivalently), for a $S$-scheme $X$, let $P^n_{X/S}$ denote the sheaf $(O_X\otimes_{f^{-1}(O_S)} O_X)/I^{n+1}$, where $I$ is the ideal of the diagonal (this makes sense since the diagonal morphism is an immersion, cf. EGAI, 5.3.9.), and $f:X\rightarrow S$ the structure morphism. Consider $P^n_{X/S}$ as $O_X$-module via the morphism $O_X\rightarrow O_X\otimes_{f^{-1}(O_S)} O_X$, $a\mapsto a\otimes 1$.

For $O_X$-modules $E,F$, $Diff_S(F,E)_n$ is defined to be $Hom_{O_X}(P_{X/S}^n\otimes_{O_X} F, E)$. Note that $P^n_{X/S}$ has two canonical structures as $O_X$-module given by the projections $p_i: X \times_S X\rightarrow X$. The tensor product $P_{X/S}^n \otimes_{O_X} F$ is understood to be constructed via $p_1$ and considered as an $O_X$-module via $p_0$.

Properties

In the affine case, and in characteristics zero, the sheaf of regular differential operators is locally isomorphic to the Weyl algebra. For that simple case, a good reference is Coutinho 1995.

Noncommutative analogues

Regular differential operators have been nontrivially generalized to noncommutative rings (and schemes) by V. Lunts and A. L. Rosenberg, as well as to the setting of braided monoidal categories. Their motivation is an analogue of a Beilinson-Bernstein localization theorem for quantum groups. The category of differential bimodules is categorically characterized in their work as the minimal coreflective topologizing monoidal subcategory of the abelian monoidal category of $R$-$R$-bimodules which is containing $R$. In the case of noncommutative rings, Lunts-Rosenberg definition of differential operators has been recovered from a different perspective in the setup of noncommutative algebraic geometry represented by monoidal categories; the emphasis is on the duality between infinitesimals and differential operators, see Maszczyk 2006.

See also regular differential operator in noncommutative geometry.

References

• P. Berthelot, A. Ogus, Notes on crystalline cohomology, Princeton Univ.P. 1978. vi+243, ISBN0-691-08218-9
• S. C. Coutinho, A primer of algebraic $D$-modules, London Math. Soc. Stud. Texts 33, Cambridge University Press, Cambridge, 1995. xii+207 pp.
• Tomasz Maszczyk, Noncommutative geometry through monoidal categories, arXiv:0611806

For regular differential operators in Lie theory see

• I. N. Bernstein, I. M. Gelfand, S. I. Gelfand, Differential operators on the base affine space and a study of $\mathfrak{g}$-modules, In: I. M. Gelfand, ed., Lie groups and their representations, 21–64. Adam Hilger, 1975

Lunts-Rosenberg work in noncommutative setting

• V. A. Lunts, A. L. Rosenberg, Differential operators on noncommutative rings, Selecta Math. (N.S.) 3:3 (1997) 335–359 (doi)

• V. A. Lunts, A. L. Rosenberg, Localization for quantum groups, Selecta Math. (N.S.) 5:1 (1999) 123–159 (doi).

• V. A. Lunts, A. L. Rosenberg, Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings, MPI 1996-53 pdf

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

MO questions

• MathOverflow: Equivalence of “Weyl Algebra” and “Crystalline” definitions of rings of differential operators between modules?, Ring of differential operators of a quotient ring

category: algebraic geometry