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

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