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

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

2005-04-22

arxiv.org/abs/math/0504475v1

Mathematics

Rings and Algebras

29 pages

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Generators and defining relations for the ring of differential operators on a smooth affine algebraic variety does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Generators and defining relations for the ring of differential operators on a smooth affine algebraic variety, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Generators and defining relations for the ring of differential operators on a smooth affine algebraic variety will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-716513