Finite generation of rings of differential operators of semigroup algebras.
type: Item
instance of: scholarly article
title: Finite generation of rings of differential operators of semigroup algebras.
zbMATH Open document ID: 1054.16021
author: Mutsumi Saito
author: William N. Traves
published in: Journal of Algebra
publication date: 2004-09-23T00:00:00Z
review text: Let \(A\) be an integer matrix of size \(d\times n\). Assume that the columns of \(A\) generate \(\mathbb{Z}^d\) as an additive Abelian group. For a column \(a_i\), \(1\leqslant i\leqslant n\), from \(A\) denote by \(t^{a_i}\) the monomial with the multi-index \(a_i\) in the complex Laurent polynomial algebra \(B\) with variables \(t_1,\dots,t_d\). Consider the subalgebra \(R_A\) in \(B\) generated by \(t^{a_i}\), \(1\leqslant i\leqslant n\). Then the algebra of differential operators \(D(R_A)\) is defined as the subalgebra of the Weyl algebra \(B\langle\partial_1,\dots,\partial_d\rangle\) mapping \(R_a\) into itself. The main result of the paper states that \(D(R_A)\) is a finitely generated algebra. There is a natural filtration on \(D(R_A)\). There is given a criterion of finite generation of the associated graded algebra.
Mathematics Subject Classification ID: 16S32
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16S15
Mathematics Subject Classification ID: 16W70
zbMATH DE Number: 2102489
zbMATH Keywords: differential operators
zbMATH Keywords: semigroup algebras
zbMATH Keywords: graded rings
zbMATH Keywords: order filtrations
zbMATH Keywords: toric varieties
zbMATH Keywords: finitely generated rings
reviewed by: Vyacheslav A. Artamonov
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.01.009
OpenAlex ID: W2166815903
cites work: Q4317713
cites work: Q3819327
cites work: Hypergeometric functions and toric varieties
cites work: Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas (Quatrième partie). Rédigé avec la colloboration de J. Dieudonné
cites work: Q4205532
cites work: Rings of differential operators on toric varieties
cites work: Q4485772
cites work: Differential operators on toric varieties
cites work: Invariants under tori of rings of differential operators and related topics
cites work: Isomorphism classes of \(A\)-hypergeometric systems
cites work: Gröbner deformations of hypergeometric differential equations
cites work: Q4329405
cites work: Q3818127
cites work: Lifting differential operators from orbit spaces
cites work: Groups of simple algebras
cites work: Affine Semigroups and Cohen-Macaulay Rings Generated by Monomials
DOI: 10.1016/J.JALGEBRA.2004.01.009
description: scientific article; zbMATH DE number 2102489