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SIMPLICITY CRITERIA FOR RINGS OF DIFFERENTIAL OPERATORS
Part of:
Local theory
Differential algebra
Surfaces and higher-dimensional varieties
Modules, bimodules and ideals
Rings and algebras arising under various constructions
Published online by Cambridge University Press: 11 May 2021
Abstract
Let K be a field of arbitrary characteristic, 
$${\cal A}$$
be a commutative K-algebra which is a domain of essentially finite type (e.g., the algebra of functions on an irreducible affine algebraic variety), 
$${a_r}$$
be its Jacobian ideal, and 
$${\cal D}\left( {\cal A} \right)$$
be the algebra of differential operators on the algebra
$${\cal A}$$
. The aim of the paper is to give a simplicity criterion for the algebra
$${\cal D}\left( {\cal A} \right)$$
: the algebra
$${\cal D}\left( {\cal A} \right)$$
is simple iff
$${\cal D}\left( {\cal A} \right)a_r^i{\cal D}\left( {\cal A} \right) = {\cal D}\left( {\cal A} \right)$$
for all i ≥ 1 provided the field K is a perfect field. Furthermore, a simplicity criterion is given for the algebra 
$${\cal D}\left( R \right)$$
of differential operators on an arbitrary commutative algebra R over an arbitrary field. This gives an answer to an old question to find a simplicity criterion for algebras of differential operators.
- Type
- Research Article
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- Copyright
- © The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust
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