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

TL;DR

This paper proves that the global dimension of certain polynomial integro-differential and Jacobian algebras is equal to their dimension n, extending Hilbert's Syzygy Theorem to these non-Noetherian algebras.

Contribution

It establishes the exact global dimension of the algebras $ ext{I}_n$ and $ ext{A}_n$, and proves an analogue of Hilbert's Syzygy Theorem for them.

Findings

Global dimension of $ ext{I}_n$ and $ ext{A}_n$ is n.

All prime factor algebras of these algebras have global dimension n.

The weak global dimension of all factor algebras of $ ext{I}_n$ is 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 $n$.