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 be a commutative ring, let
be a subring of
and let
and
be two
-modules. A homomorphism of
-modules
is said to be a differential operator of order
, where
is a non-negative integer, if for any
the mapping
defined by the formula
![]() |
is a differential operator of order . A differential operator of order zero is a homomorphism of
-modules
. The set of all differential operators of order
forms a submodule
of the
-module of all homomorphisms of
-modules
. In particular,
![]() |
and the quotient module
![]() |
is isomorphic to the module of -derivations
of
with values in
. The union
of the increasing family of submodules
![]() |
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 -module
over the subring
, while the corresponding graded ring
![]() |
where
![]() |
is said to be the module of symbols. The image of a differential operator in the ring
is said to be the symbol of the differential operator.
If is an algebra over the field of rational numbers and the module of differentials
is projective, then there exists an isomorphism between the
-algebra
and the enveloping algebra of the Lie algebra of
-derivations
. In this case the ring
is isomorphic to the symmetric algebra of the
-module
.
For example, let be the ring of polynomials over a field
; the mappings
, defined by the formula
![]() |
are differential operators of over
of order
. The ring of differential operators
is a free module over
with basis
. Multiplication is given by the formula
![]() |
In particular,
![]() |
(Taylor's formula) which, if the characteristic of is equal to zero, yields
![]() |
If is an affine group
-scheme, invariant differential operators of
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 |
Differential operator on a module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential_operator_on_a_module&oldid=46695