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arXiv · 1705.05227

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

V. V. Bavula
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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 $n$.

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V. V. Bavula. 2017-05-11. The global dimension of the algebras of polynomial integro-differential operators $\mathbb{I}_n$ and the Jacobian algebras $\mathbb{A}_n$. https://arxiv.org/abs/1705.05227

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