# Proceedings of the American Mathematical Society

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

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Gel′fand-Kirillov dimension of rings of formal differential operators on affine varietiesHTML articles powered by AMS MathViewer

by S. P. Smith
Proc. Amer. Math. Soc. 90 (1984), 1-8 Request permission

## Abstract:

Let $A$ be the coordinate ring of a smooth affine algebraic variety defined over a field $k$. Let $D$ be the module of $k$-linear derivations on $A$ and form $A[D]$, the ring of differential operators on $A$, as follows: consider $A$ and $D$ as subspaces of ${\operatorname {End}_k}A$ ($A$ acting by left multiplication on itself), and define $A[D]$ to be the subalgebra generated by $A$ and $D$. Let $\operatorname {rk} D$ denote the torsion-free rank of $D$ (that is, $\operatorname {rk}D = {\dim _F}F{ \otimes _A}D$ where $F$ is the quotient field of $A$). The ring $A[D]$ is a finitely generated $k$-algebra so its Gelfand-Kirillov dimension ${\text {GK}}(A[D])$ may be defined. The following is proved. Theorem. ${\text {GK}}(A[D]) = {\text {tr de}}{{\text {g}}_k}A + \operatorname {rk} D = 2{\text { tr de}}{{\text {g}}_k}A$. Actually we work in a more general setting than that just described, and although a more general result is obtained, this is the most natural and important application of the main theorem.
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