Gel′fand-Kirillov dimension of rings of formal differential operators on affine varieties

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.