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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Global dimension of differential operator rings


Author: K. R. Goodearl
Journal: Proc. Amer. Math. Soc. 45 (1974), 315-322
MSC: Primary 16A72
DOI: https://doi.org/10.1090/S0002-9939-1974-0382358-X
MathSciNet review: 0382358
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Abstract: This paper is concerned with finding the global homological dimension of the ring of differential operators $R[\theta ]$ over a differential ring $R$ with a single derivation. Examples are constructed to show that $R[\theta ]$ may have finite dimension even when $R$ has infinite dimension. For a commutative noetherian differential algebra $R$ over the rationals, with finite global dimension $n$, it is shown that the global dimension of $R[\theta ]$ is the supremum of $n$ and one plus the projective dimensions of the modules $R/P$, where $P$ ranges over all prime differential ideals of $R$. One application derives the global dimension of the Weyl algebra over a commutative noetherian ring $S$ of finite global dimension, where $S$ either is an algebra over the rationals or else has positive characteristic.


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Keywords: Global dimension, rings of linear differential operators, differential algebra, Weyl algebras
Article copyright: © Copyright 1974 American Mathematical Society