regular differential operator

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”, see

- P. Berthelot, A. Ogus,
*Notes on crystalline cohomology*, Princeton Univ.P. 1978. vi+243, ISBN0-691-08218-9

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.

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 $M_n$ 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), and $f:X\rightarrow S$ the structure morphism. Consider $P^n_{X/S}$ as $O_X$-module via the morphism $O_X\rightarrow O_X\otimes_{f^{-1}(O_S)} O_X$, $a\mapsto a\otimes 1$.

Zoran: is here silent assumption of separatedness of $X$ over $S$ important ?

Lars: No, since the diagonal is an immersion (see EGAI, 5.3.9.), but what I wrote was nonsense nontheless. I fixed it, I hope.

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 $p_0$.

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

- S. C. Coutinho, A primer of algebraic $D$-modules, London Math. Soc. Stud. Texts, 33, Cambridge University Press, Cambridge, 1995. xii+207 pp.

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:

- Tomasz Maszczyk, Noncommutative geometry through monoidal categories, arXiv:0611806

See also regular differential operator in noncommutative geometry.

category: algebraic geometry

Revised on March 6, 2013 19:25:56
by Zoran Škoda
(161.53.130.104)