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\section*{regular differential operator}

Grothendieck has developed a deep version of \emph{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

\begin{itemize}%
\item P. Berthelot, A. Ogus, \emph{Notes on crystalline cohomology}, Princeton Univ.P. 1978. vi+243, ISBN0-691-08218-9

\end{itemize}
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 \textbf{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 \textbf{differential part}, i.e. the maximal differential submodule of $M$.

\textbf{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 $p_0$.

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

\begin{itemize}%
\item S. C. Coutinho, A primer of algebraic $D$-modules, London Math. Soc. Stud. Texts, 33, Cambridge University Press, Cambridge, 1995. xii+207 pp.

\end{itemize}
Regular differential operators have been nontrivially generalized to noncommutative rings (and schemes) by V. Lunts and [[Alexander Roseberg|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 subcategory|coreflective]] [[topologizing subcategory| topologizing]] [[monoidal category|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:

\begin{itemize}%
\item [[Tomasz Maszczyk]], Noncommutative geometry through monoidal categories, \href{http://arxiv.org/abs/math/0611806}{arXiv:0611806}

\end{itemize}
See also [[regular differential operator in noncommutative geometry]].

MathOverflow: \href{http://mathoverflow.net/questions/210891/equivalence-of-weyl-algebra-and-crystalline-definitions-of-rings-of-differen}{Equivalence of ``Weyl Algebra'' and ``Crystalline'' definitions of rings of differential operators between modules?}, \href{http://mathoverflow.net/questions/194218/ring-of-differential-operators-of-a-quotient-ring}{Ring of differential operators of a quotient ring}

category: algebraic geometry [[!redirects regular differential operators]]



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