# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## Global dimension of differential operator rings. IIHTML articles powered by AMS MathViewer

by K. R. Goodearl
Trans. Amer. Math. Soc. 209 (1975), 65-85 Request permission

## 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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