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A mapping of modules over a commutative ring which is an analogue of the concept of a differential operator. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d0322601.png" /> be a commutative ring, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d0322602.png" /> be a subring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d0322603.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d0322604.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d0322605.png" /> be two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d0322606.png" />-modules. A homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d0322607.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d0322608.png" /> is said to be a differential operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d0322609.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226010.png" /> is a non-negative integer, if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226011.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226012.png" /> defined by the formula
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226013.png" /></td> </tr></table>
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is a differential operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226014.png" />. A differential operator of order zero is a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226015.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226016.png" />. The set of all differential operators of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226017.png" /> forms a submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226018.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226019.png" />-module of all homomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226020.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226021.png" />. In particular,
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226022.png" /></td> </tr></table>
+
$$
 +
D _ {x} ( n)  = D ( xn) - x D ( n) ,
 +
$$
 +
 
 +
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
 
and the quotient module
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226023.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Diff} _ {S}  ^ {1} ( R , M ) /  \mathop{\rm Diff} _ {S}  ^ {0} ( R , M )
 +
$$
  
is isomorphic to the module of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226024.png" />-derivations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226025.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226026.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226027.png" />. The union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226028.png" /> of the increasing family of submodules
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226029.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Diff} _ {S}  ^ {0} ( M , M )  \subset    \mathop{\rm Diff} _ {S}  ^ {1} ( M ,\
 +
M )  \subset  \dots
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226030.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226031.png" /> over the subring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226032.png" />, while the corresponding graded ring
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226033.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Symb} _ {S} ( M)  = \oplus _ {i \geq  0 }  \mathop{\rm Symb} _ {S}  ^ {i}
 +
( M) ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226034.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Symb} _ {S}  ^ {i} ( M)  =   \mathop{\rm Diff} _ {S}  ^ {i} ( M , M ) /
 +
\mathop{\rm Diff} _ {S}  ^ {i-} 1 ( M , M ) ,
 +
$$
  
is said to be the module of symbols. The image of a differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226035.png" /> in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226036.png" /> is said to be the symbol of the differential operator.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226037.png" /> is an algebra over the field of rational numbers and the module of differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226038.png" /> is projective, then there exists an isomorphism between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226039.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226040.png" /> and the enveloping algebra of the Lie algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226041.png" />-derivations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226042.png" />. In this case the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226043.png" /> is isomorphic to the symmetric algebra of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226044.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226045.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226046.png" /> be the ring of polynomials over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226047.png" />; the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226048.png" />, defined by the formula
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226049.png" /></td> </tr></table>
+
$$
  
are differential operators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226050.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226051.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226052.png" />. The ring of differential operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226053.png" /> is a free module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226054.png" /> with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226055.png" />. Multiplication is given by the formula
+
\frac \partial {\partial  T  ^ {i} }
 +
( T  ^ {r} )  = \left ( \begin{array}{c}
 +
r \\
 +
i
 +
\end{array}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226056.png" /></td> </tr></table>
+
\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,
 
In particular,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226057.png" /></td> </tr></table>
+
$$
 +
\left (
 +
\frac \partial {\partial  T  ^ {1} }
 +
\right )  ^ {n}  = n !
 +
\frac \partial {
 +
\partial  T  ^ {n} }
 +
 
 +
$$
  
(Taylor's formula) which, if the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226058.png" /> is equal to zero, yields
+
(Taylor's formula) which, if the characteristic of $  k $
 +
is equal to zero, yields
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226059.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Diff} _ {k} ( R)  \cong  R \left [
 +
\frac \partial {\partial  T  ^ {1} }
 +
\right
 +
] .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226060.png" /> is an affine group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226061.png" />-scheme, invariant differential operators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032260/d03226062.png" /> may also be considered [[#References|[2]]].
+
If $  \mathop{\rm Spec} ( R) $
 +
is an affine group $  S $-
 +
scheme, invariant differential operators of $  R $
 +
may also be considered [[#References|[2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Grothendieck,  "Eléments de géométrie algébrique. IV Etude locale des schémas et des morphisms des schémas I"  ''Publ. Math. IHES'' , '''20'''  (1960)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Demazure,  P. Gabriel,  "Groupes algébriques" , '''1''' , Masson  (1970)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.E. Björk,  "The global homological dimension of some algebras of differential operators"  ''Invent. Math.'' , '''17''' :  1  (1972)  pp. 67–78</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Grothendieck,  "Eléments de géométrie algébrique. IV Etude locale des schémas et des morphisms des schémas I"  ''Publ. Math. IHES'' , '''20'''  (1960)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Demazure,  P. Gabriel,  "Groupes algébriques" , '''1''' , Masson  (1970)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.E. Björk,  "The global homological dimension of some algebras of differential operators"  ''Invent. Math.'' , '''17''' :  1  (1972)  pp. 67–78</TD></TR></table>

Latest revision as of 19:35, 5 June 2020


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)
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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