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

Abstract

Let A A be the coordinate ring of a smooth affine algebraic variety defined over a field k k . Let D D be the module of k k -linear derivations on A A and form A [ D ] A[D] , the ring of differential operators on A A , as follows: consider A A and D D as subspaces of End k A {\operatorname {End}_k}A ( A A acting by left multiplication on itself), and define A [ D ] A[D] to be the subalgebra generated by A A and D D . Let rk ⁡ D \operatorname {rk} D denote the torsion-free rank of D D (that is, rk ⁡ D = dim F F ⊗ A D \operatorname {rk}D = {\dim _F}F{ \otimes _A}D where F F is the quotient field of A A ). The ring A [ D ] A[D] is a finitely generated k k -algebra so its Gelfand-Kirillov dimension GK ( A [ D ] ) {\text {GK}}(A[D]) may be defined. The following is proved. Theorem. GK ( A [ D ] ) = tr de g k A + rk ⁡ D = 2 tr de g k A {\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.