Generators and defining relations for the ring of differential operators on a smooth affine algebraic variety

Abstract

For the ring of differential operators on a smooth affine algebraic variety $X$ over a field of characteristic zero a finite set of algebra generators and a finite set of defining relations are found explicitly. As a consequence, a finite set of generators and a finite set of defining relations are given for the module $\Der_K(\OO (X))$ of derivations on the algebra $\OO (X)$ of regular functions on the variety $X$. For the variety $X$ which is not necessarily smooth, a set of natural derivations ${\rm der}_K(\OO (X))$ of the algebra $\OO (X)$ and a ring $\gD (\OO (X))$ of natural differential operators on $\OO (X)$ are introduced. The algebra $\gD (\OO (X))$ is a Noetherian algebra of Gelfand-Kirillov dimension $2\dim (X)$. When $X$ is smooth then ${\rm der}_K(\OO (X))=\Der_K(\OO (X))$ and $\gD (\OO (X))=\CD (\OO (X))$. A criterion of smoothness of $X$ is given when $X$ is irreducible ($X$ is smooth iff $\gD (\OO (X))$ is a simple algebra iff $\OO (X)$ is a simple $\gD (\OO (X))$-module). The same results are true for regular algebras of essentially finite type. For a singular irreducible affine algebraic variety $X$, in general, the algebra of differential operators $\CD (\OO (X))$ needs not be finitely generated nor (left or right) Noetherian, it is proved that each term $\CD (\OO (X))_i$ of the order filtration $\CD (\OO (X))=\cup_{i\geq 0}\CD (\OO (X))_i$ is a finitely generated left $\OO (X)$-module.