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Global dimension of differential operator rings. II


Author: K. R. Goodearl
Journal: Trans. Amer. Math. Soc. 209 (1975), 65-85
MSC: Primary 16A72
DOI: https://doi.org/10.1090/S0002-9947-1975-0382359-7
MathSciNet review: 0382359
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Abstract: The aim of this paper is to find the global homological dimension of the ring of linear differential operators $ R[{\theta _1}, \ldots ,{\theta _u}]$ over a differential ring $ R$ with $ u$ commuting derivations. When $ R$ is a commutative noetherian ring with finite global dimension, the main theorem of this paper (Theorem 21) shows that the global dimension of $ R[{\theta _1}, \ldots ,{\theta _u}]$ is the maximum of $ k$ and $ q + u$, where $ q$ is the supremum of the ranks of all maximal ideals $ M$ of $ R$ for which $ R/M$ has positive characteristic, and $ k$ is the supremum of the sums $ rank(P) + diff\;dim(P)$ for all prime ideals $ P$ of $ R$ such that $ R/P$ has characteristic zero. [The value $ diff\;dim(P)$ is an invariant measuring the differentiability of $ P$ in a manner defined in §3.] In case we are considering only a single derivation on $ R$, this theorem leads to the result that the global dimension of $ R[\theta ]$ is the supremum of gl $ dim(R)$ together with one plus the projective dimensions of the modules $ R/J$, where $ J$ is any primary differential ideal of $ R$. One application of these results derives the global dimension of the Weyl algebra in any degree over any commutative noetherian ring with finite global dimension.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0382359-7
Keywords: Global dimension, rings of linear differential operators, differential algebra, Weyl algebras
Article copyright: © Copyright 1975 American Mathematical Society

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