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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.