Structure sheaves on almost commutative algebras
type: Item
instance of: scholarly article
title: Structure sheaves on almost commutative algebras
zbMATH Open document ID: 0963.16024
author: Huishi Li
published in: Journal of Algebra
publication date: 2001-06-25T00:00:00Z
review text: An algebra \(R\) over a field \(k\) of characteristic \(0\) is said to be almost commutative if there exists a positive filtration \(0\subset F_0R\subset F_1R\subset\cdots\) of \(R\) such that \(F_0R=k\) and the associated graded ring \(G(R)\) is a commutative affine \(k\)-algebra generated by homogeneous elements of degree \(1\). In previous work, the author constructed a sheaf \({\mathcal O}^F_Y\) on the scheme \(Y=\text{Proj}(G(R))\), in such a way that for every \(p\in Y\), the stalk \({\mathcal O}^F_{Y,p}\) coincides with the localization \(S^{-1}_pR\) of \(R\) at the saturated subset \(S_p\) of \(R\) corresponding to the Ore set consisting of homogeneous elements in \(G(R)-p\). In this paper, the author first constructs the sheaf \({\mathcal O}^F_Y\) in an alternative way, modulo the extra assumption that \(G(R)\) be a domain and provides a detailed description of the rings of sections of this sheaf over basic open sets. It is then shown that the sheaf \({\mathcal O}^F_Y\) can really be very helpful to represent the base ring \(R\) geometrically: the author gives general conditions (satisfied by large classes of concrete rings) under which \(R\) reduces to the ring of global sections of the sheaf \({\mathcal O}^F_Y\). The remainder of the paper is essentially devoted to a thorough study of the lifting properties of the stalks \({\mathcal O}^F_{Y,p}\), the local study of \(R\) and the application of local-global results, using the fact that the induced map \(R\to\bigoplus S^{-1}_pR\) is faithfully flat. As a final result, the author generalizes to the present situation the continuity of the local-rank function.
Mathematics Subject Classification ID: 16S38
Mathematics Subject Classification ID: 16S60
Mathematics Subject Classification ID: 16W70
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 1448609
zbMATH Keywords: almost commutative algebras
zbMATH Keywords: filtrations
zbMATH Keywords: graded rings
zbMATH Keywords: affine algebras
zbMATH Keywords: sheaves
zbMATH Keywords: schemes
zbMATH Keywords: stalks
zbMATH Keywords: localizations
zbMATH Keywords: Ore sets
zbMATH Keywords: homogeneous elements
zbMATH Keywords: rings of sections
zbMATH Keywords: local-rank functions
MaRDI profile type: Publication
full work available at URL: jabr.1999.8137
OpenAlex ID: W2030112067
cites work: A New Algebraic Approach to Microlocalization of Filtered Rings
cites work: Q3360348
cites work: Differential operators on homogeneous spaces. I: Irreducibility of the associated variety for annihilators of induced modules
cites work: Certaines algèbres de type fini sont des algèbres de Jacobson
cites work: Characteristic varieties and vanishing cycles
cites work: Noncommutative deformations of type-\(A\) Kleinian singularities
cites work: On the Global Dimension of Certain Primitive Factors of the Enveloping Algebra of a Semi-Simple Lie Algebra
cites work: Q3677877
cites work: Quantum sections and gauge algebras
cites work: On the stability of graded rings
cites work: Lifting Ore sets of Noetherian filtered rings and applications
cites work: zariskian filtrations
cites work: Dehomogenization of Gradings to Zariskian Filtrations and Applications to Invertible Ideals
cites work: Zariski rings and flatness of completion
cites work: Q3786586
cites work: Q3035452
cites work: Gelfand-Kirillov dimension and associated graded modules
cites work: Q3957087
cites work: Q3882615
cites work: Absolutely torsion-free rings
cites work: A microstructure sheaf and quantum sections over a projective scheme
cites work: A Class of Algebras Similar to the Enveloping Algebra of sl(2)
cites work: Lifting maximal orders
DOI: 10.1006/JABR.1999.8137
description: scientific article; zbMATH DE number 1448609