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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0382358-X
Keywords: Global dimension, rings of linear differential operators, differential algebra, Weyl algebras
Article copyright: © Copyright 1974 American Mathematical Society

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