The global dimension of the algebras of polynomial integro-differential operators ${\mathbb{I}}_n$ and the Jacobian algebras ${\mathbb{A}}_n$

Abstract

The aim of the paper is to prove two conjectures that the (left and right) global dimension of the algebra of polynomial integro-differential operators ${\mathbb{I}}_n$ and the Jacobian algebra ${\mathbb{A}}_n$ is equal to $n$ (over a field of characteristic zero). An analogue of Hilbert's Syzygy Theorem is proven for them. The algebras ${\mathbb{I}}_n$ and ${\mathbb{A}}_n$ are neither left nor right Noetherian. Furthermore, they contain infinite direct sums of nonzero left/right ideals and are not domains. It is proven that the global dimension of all prime factor algebras of the algebras ${\mathbb{I}}_n$ and ${\mathbb{A}}_n$ is $n$ and the weak global dimension of all the factor algebras of ${\mathbb{I}}_n$ and ${\mathbb{I}}_n$ is $n$.

22 pages. arXiv admin note: text overlap with arXiv:0912.0723