Noncommutative graded domains with quadratic growth
type: Item
instance of: scholarly article
title: Noncommutative graded domains with quadratic growth
zbMATH Open document ID: 0849.16022
DOI: 10.1007/BF01231444
author: J. T. Stafford
author: Michael Artin
published in: Inventiones Mathematicae
publication date: 1996-11-03T00:00:00Z
full work available at URL: https://eudml.org/doc/144320
review text: The Gelfand-Kirillov dimension GK(--) is an extremely useful tool for the study of algebras over a field \(k\), since information about the growth of a \(k\)-algebra \(A\) often has far-reaching consequences for its structure. Thus, for example, if \(A\) is a domain (i.e. a ring without zero divisors) with \(\text{GK}(A)<\infty\), then \(A\) is left and right Ore. Or, by an important result of \textit{L. Small} and \textit{R. B. Warfield} jun. [J. Algebra 91, 386-389 (1984; Zbl 0545.16011)], if \(A\) is affine and prime with \(\text{GK}(A)=1\), then \(A\) is finite as a module over its center which turns out to be a finitely generated domain of dimension 1. Furthermore, if in this case \(A\) is a domain and \(k\) is algebraically closed, then \(A\) is commutative. Since no \(k\)-algebra can have Gelfand-Kirillov dimension strictly between 1 and 2, it is natural to investigate domains \(A\) with \(\text{GK}(A)=2\). In full generality this still seems to be intractable, so the authors restrict themselves to graded domains of Gelfand-Kirillov dimension two and provide a fairly complete classification of these. Let \(A=\bigoplus_{n\geq 0}A_n\) be a finitely graded domain (i.e. \(A\) is finitely generated and \(A_0\) is a finite dimensional vector space over \(k\)) with \(2\leq\text{GK}(A)\leq{11\over 5}\). Then the nonzero homogeneous elements of \(A\) form an Ore set, yielding the so-called graded quotient ring of \(A\), which is known to be a skew Laurent polynomial ring \(D[z,z^{-1};\sigma]\), where \(D\) is a division algebra, \(z\) is a homogeneous element of positive degree and \(\sigma\) is an automorphism of \(D\). The first major result of the paper under review is that \(D\) is finitely generated as a module over it center \(K\), and that \(K\) is a finitely generated field extension of \(k\) of transcendence degree one. It turns out that \(\text{GK}(A)=2\), which is proved by establishing \(\text{GK}(D)=1\) and then invoking the result of Small and Warfield m…
reviewed by: Günter Krause
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16U20
Mathematics Subject Classification ID: 16U10
zbMATH DE Number: 861963
zbMATH Keywords: Noetherian rings
zbMATH Keywords: Gelfand-Kirillov dimension
zbMATH Keywords: growth
zbMATH Keywords: finitely generated domains
zbMATH Keywords: graded domains of Gelfand-Kirillov dimension two
zbMATH Keywords: Ore sets
zbMATH Keywords: graded quotient rings
zbMATH Keywords: skew Laurent polynomial rings
zbMATH Keywords: division algebras
zbMATH Keywords: finitely generated field extensions
zbMATH Keywords: Hilbert series
zbMATH Keywords: Ore condition
zbMATH Keywords: function fields of smooth projective curves
zbMATH Keywords: homogeneous coordinate rings
zbMATH Keywords: PI-rings
MaRDI profile type: Publication
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OpenAlex ID: W2154008761
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description: scientific article; zbMATH DE number 861963