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SIMPLICITY CRITERIA FOR RINGS OF DIFFERENTIAL OPERATORS

Part of: Local theory Surfaces and higher-dimensional varieties Differential algebra Rings and algebras arising under various constructions Modules, bimodules and ideals

Published online by Cambridge University Press:  11 May 2021

V. V. BAVULA
V. V. BAVULA*
Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK, e-mail: [email protected]
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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.

MSC classification

Primary: 16D25: Ideals 16S32: Rings of differential operators 16D30: Infinite-dimensional simple rings (except as in ) 13N15: Derivations
Secondary: 13N10: Rings of differential operators and their modules 14J17: Singularities 14B05: Singularities
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 2 , May 2022 , pp. 347 - 351
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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