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arXiv:1101.1123 [math.RA]AbstractReferencesReviewsResources

Rings of differential operators on curves

Jason P. Bell, Agata Smoktunowicz

Published 2011-01-06, updated 2011-01-16Version 2

Let $k$ be an algebraically closed field of characteristic 0 and let $A$ be a finitely generated $k$-algebra that is a domain whose Gelfand-Kirillov dimension is in $[2,3)$. We show that if $A$ has a nonzero locally nilpotent derivation then $A$ has quadratic growth. In addition to this, we show that $A$ either satisfies a polynomial identity or $A$ is isomorphic to a subalgebra of $\mathcal{D}(X)$, the ring of differential operators on an irreducible smooth affine curve $X$, and $A$ is birationally isomorphic to $\mathcal{D}(X)$.

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