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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Prime ideals in differential operator rings. Catenarity

Authors: K. A. Brown, K. R. Goodearl and T. H. Lenagan
Journal: Trans. Amer. Math. Soc. 317 (1990), 749-772
MSC: Primary 16A05; Secondary 16A66
MathSciNet review: 946215
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Abstract: Let $ R$ be a commutative algebra over the commutative ring $ k$, and let $ \Delta = \{ {\delta _1}, \ldots ,{\delta _n}\} $ be a finite set of commuting $ k$-linear derivations from $ R$ to $ R$. Let $ T = R[{\theta _1}, \ldots ,{\theta _n};{\delta _1}, \ldots ,{\delta _n}]$ be the corresponding ring of differential operators. We define and study an isomorphism of left $ R$-modules between $ T$ and its associated graded ring $ R[{x_1}, \ldots ,{x_n}]$, a polynomial ring over $ R$. This isomorphism is used to study the prime ideals of $ T$, with emphasis on the question of catenarity. We prove that $ T$ is catenary when $ R$ is a commutative noetherian universally catenary $ k$-algebra and one of the following cases occurs: (A) $ k$ is a field of characteristic zero and $ \Delta $ acts locally finitely; (B) $ k$ is a field of positive characteristic; (C) $ k$ is the ring of integers, $ R$ is affine over $ k$, and $ \Delta $ acts locally finitely.

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Keywords: Differential operator ring, prime ideal, catenarity, height, derivation
Article copyright: © Copyright 1990 American Mathematical Society