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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Gelfand-Kirillov dimension of rings of formal differential operators on affine varieties


Author: S. P. Smith
Journal: Proc. Amer. Math. Soc. 90 (1984), 1-8
MSC: Primary 16A55; Secondary 14L99, 16A56, 16A72, 17B35, 17B40
MathSciNet review: 722404
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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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0722404-0
PII: S 0002-9939(1984)0722404-0
Keywords: Gelfand-Kirillov dimension, differential operators, derivations
Article copyright: © Copyright 1984 American Mathematical Society