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