The global dimension of the algebras of polynomial integro-differential operators \mathbbI_n and the Jacobian algebras \mathbbA_n

2017/05/11 by V. V. Bavula, Bavula, V. V.
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Polynomial and algebraic computation #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1705.05227

openalex publication_date 2017/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \mathbbI_n and the Jacobian algebra \mathbbA_n is equal to n (over a field of characteristic zero). An analogue of Hilbert's Syzygy Theorem is proven for them. The algebras \mathbbI_n and \mathbbA_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 \mathbbI_n and \mathbbA_n is n and the weak global dimension of all the factor algebras of \mathbbI_n and \mathbbI_n is n.

Citations

• An analogue of the Conjecture of Dixmier is true for the algebra of polynomial integro-differential operators

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