# Generators and Defining Relations for the Ring of Differential Operators on a Smooth Affine Algebraic Variety

Research paper by V. V. Bavula

Indexed on: 07 Feb '09Published on: 07 Feb '09Published in: Algebras and Representation Theory

#### 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 $${\rm Der}_K({\cal O} (X))$$ of derivations on the algebra $${\cal O} (X)$$ of regular functions on the variety X. For the variety X which is not necessarily smooth, a set of natural derivations $${\rm der}_K({\cal O} (X))$$ of the algebra $${\cal O} (X)$$ and a ring $$\mathfrak{D} ({\cal O} (X))$$ of natural differential operators on $${\cal O} (X)$$ are introduced. The algebra $$\mathfrak{D} ({\cal O} (X))$$ is a Noetherian algebra of Gelfand–Kirillov dimension 2dim (X). When X is smooth then $${\rm der}_K({\cal O} (X))={\rm Der}_K({\cal O} (X))$$ and $$\mathfrak{D} ({\cal O} (X))={\cal D} ({\cal O} (X))$$. A criterion of smoothness of X is given when X is irreducible (X is smooth iff $$\mathfrak{D} ({\cal O} (X))$$ is a simple algebra iff $${\cal O} (X)$$ is a simple $$\mathfrak{D} ({\cal O} (X))$$-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 $${\cal D} ({\cal O} (X))$$ needs not be finitely generated nor (left or right) Noetherian, it is proved that each term $${\cal D} ({\cal O} (X))_i$$ of the order filtration $${\cal D} ({\cal O} (X))=\cup_{i\geq 0}{\cal D} ({\cal O} (X))_i$$ is a finitely generated left $${\cal O} (X)$$-module.