Semiprime graded algebras of dimension two
type: Item
instance of: scholarly article
title: Semiprime graded algebras of dimension two
zbMATH Open document ID: 0967.16021
author: J. T. Stafford
author: Michael Artin
published in: Journal of Algebra
publication date: 2001-06-06T00:00:00Z
review text: In a previous article [Invent. Math. 122, No. 2, 231-276 (1995; Zbl 0849.16022)], the authors have classified Noetherian graded domains of quadratic growth over algebraically closed fields; the paper under review extends this classification to semiprime algebras over an arbitrary base field \(k\). In some sense, the description of noncommutative projective curves that was begun in the earlier paper can thus be considered complete. Let \(R\) be a semiprime, Noetherian, connected graded \(k\)-algebra of quadratic growth. The description of \(R\) in terms of geometric data proceeds, roughly speaking, along the following lines: Let \(Y\) be a projective variety of dimension \(\leq 1\) over the base field \(k\) and let \(\mathcal E\) be an \({\mathcal O}_Y\)-order in a finite dimensional semisimple algebra \(A\) over the field \(K=k(Y)\). Also, let \(\tau\) be an automorphism of \(A\) that restricts to an automorphism \(\sigma\) of \(Y\), and let \(\mathcal B\) be an ample, invertible \(\mathcal E\)-module. Following \textit{M.~Van den Bergh} [J. Algebra 184, No. 2, 435-490 (1996; Zbl 0876.17011)], a Noetherian ``twisted homogeneous coordinate ring'' can be constructed as \(B=\bigoplus_n H^0(Y,{\mathcal B}\otimes\cdots\otimes{\mathcal B}^{\tau^{n-1}})\). The ring \(R\) is then Noetherian if and only if some Veronese subring \(R^{(m)}\) of \(R\) has the form \(k+I\), where \(I\) is a left ideal of such a ring \(B\) and where \(I=B\) at each point \(p\in Y\) at which \(\sigma\) has finite order. A lot of detailed information concerning semiprime graded algebras of Gelfand-Kirillov dimension 2 is obtained from this characterization. For example, a finitely generated, semiprime graded Goldie ring \(S\) with \(\text{GKdim}(S)=2\) is non-Noetherian if and only if there exists a second layer link from a prime ideal \(P\) with \(\text{GKdim}(S/P)=1\) to a maximal ideal \(M\) of \(S\).
reviewed by: Günter Krause
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 16S38
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 16P10
zbMATH DE Number: 1448638
zbMATH Keywords: semiprime graded algebras
zbMATH Keywords: Noetherian rings
zbMATH Keywords: Gelfand-Kirillov dimension
zbMATH Keywords: twisted homogeneous coordinate rings
zbMATH Keywords: Veronese rings
zbMATH Keywords: noncommutative projective geometry
zbMATH Keywords: noncommutative projective curves
zbMATH Keywords: algebras of quadratic growth
MaRDI profile type: Publication
full work available at URL: jabr.1999.8226
OpenAlex ID: W2001524566
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DOI: 10.1006/JABR.1999.8226
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description: scientific article; zbMATH DE number 1448638