# Differential operator on a module

A mapping of modules over a commutative ring which is an analogue of the concept of a differential operator. Let $R$ be a commutative ring, let $S$ be a subring of $R$ and let $N$ and $M$ be two $R$- modules. A homomorphism of $S$- modules $D : N \rightarrow M$ is said to be a differential operator of order $\leq m$, where $m$ is a non-negative integer, if for any $x \in R$ the mapping $D _ {x} : N \rightarrow M$ defined by the formula

$$D _ {x} ( n) = D ( xn) - x D ( n) ,$$

is a differential operator of order $\leq m - 1$. A differential operator of order zero is a homomorphism of $R$- modules $N \rightarrow M$. The set of all differential operators of order $\leq m$ forms a submodule $\mathop{\rm Diff} _ {S} ^ {m} ( N , M )$ of the $R$- module of all homomorphisms of $S$- modules $\mathop{\rm Hom} _ {S} ( N , M )$. In particular,

$$\mathop{\rm Diff} _ {S} ^ {0} ( N , M ) \simeq \mathop{\rm Hom} _ {R} ( N , M ) ,$$

and the quotient module

$$\mathop{\rm Diff} _ {S} ^ {1} ( R , M ) / \mathop{\rm Diff} _ {S} ^ {0} ( R , M )$$

is isomorphic to the module of $S$- derivations $\mathop{\rm Der} _ {S} ( R , M )$ of $R$ with values in $M$. The union $\mathop{\rm Diff} _ {S} ( M)$ of the increasing family of submodules

$$\mathop{\rm Diff} _ {S} ^ {0} ( M , M ) \subset \mathop{\rm Diff} _ {S} ^ {1} ( M ,\ M ) \subset \dots$$

is a filtered associative ring with respect to the operation of composition of mappings. This ring is known as the ring of differential operators of the $R$- module $M$ over the subring $S$, while the corresponding graded ring

$$\mathop{\rm Symb} _ {S} ( M) = \oplus _ {i \geq 0 } \mathop{\rm Symb} _ {S} ^ {i} ( M) ,$$

where

$$\mathop{\rm Symb} _ {S} ^ {i} ( M) = \mathop{\rm Diff} _ {S} ^ {i} ( M , M ) / \mathop{\rm Diff} _ {S} ^ {i-} 1 ( M , M ) ,$$

is said to be the module of symbols. The image of a differential operator $D \in \mathop{\rm Diff} _ {S} ^ {i} ( M , M )$ in the ring $\mathop{\rm Symb} _ {S} ^ {i} ( M)$ is said to be the symbol of the differential operator.

If $R$ is an algebra over the field of rational numbers and the module of differentials $\Omega _ {R/S} ^ {1}$ is projective, then there exists an isomorphism between the $S$- algebra $\mathop{\rm Diff} _ {S} ( R)$ and the enveloping algebra of the Lie algebra of $S$- derivations $\mathop{\rm Der} _ {S} ( R , R)$. In this case the ring $\mathop{\rm Symb} _ {S} ( R)$ is isomorphic to the symmetric algebra of the $R$- module $\mathop{\rm Der} _ {S} ( R , R )$.

For example, let $R = k [ T]$ be the ring of polynomials over a field $k$; the mappings $\partial / \partial T ^ {i} : R \rightarrow R$, defined by the formula

$$\frac \partial {\partial T ^ {i} } ( T ^ {r} ) = \left ( \begin{array}{c} r \\ i \end{array} \right ) T ^ {r-} i ,$$

are differential operators of $R$ over $k$ of order $i$. The ring of differential operators $\mathop{\rm Diff} _ {k} ( R)$ is a free module over $R$ with basis $\partial / \partial T ^ {0} \dots \partial / \partial T ^ {i} ,\dots$. Multiplication is given by the formula

$$\frac \partial {\partial T ^ {i} } \circ \frac \partial {\partial T ^ {j} } = \ \left ( \begin{array}{c} i+ j \\ i \end{array} \right ) \frac \partial {\partial T ^ {i+} j } .$$

In particular,

$$\left ( \frac \partial {\partial T ^ {1} } \right ) ^ {n} = n ! \frac \partial { \partial T ^ {n} }$$

(Taylor's formula) which, if the characteristic of $k$ is equal to zero, yields

$$\mathop{\rm Diff} _ {k} ( R) \cong R \left [ \frac \partial {\partial T ^ {1} } \right ] .$$

If $\mathop{\rm Spec} ( R)$ is an affine group $S$- scheme, invariant differential operators of $R$ may also be considered [2].

#### References

 [1] A.M. Vinogradov, I.S. Krasil'shchikov, "What is the Hamilton formalism?" Russian Math. Surveys , 30 : 1 (1975) pp. 177–202 Uspekhi Mat. Nauk. , 30 : 1 (1975) pp. 173–198 [2] A. Grothendieck, "Eléments de géométrie algébrique. IV Etude locale des schémas et des morphisms des schémas I" Publ. Math. IHES , 20 (1960) [3] M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , Masson (1970) [4] J.E. Björk, "The global homological dimension of some algebras of differential operators" Invent. Math. , 17 : 1 (1972) pp. 67–78
How to Cite This Entry:
Differential operator on a module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential_operator_on_a_module&oldid=46695
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article