Curves on quasi-schemes
type: Item
instance of: scholarly article
title: Curves on quasi-schemes
zbMATH Open document ID: 0947.16029
DOI: 10.1023/A:1009984608942
author: J. J. Zhang
author: S. Paul Smith
published in: Algebras and Representation Theory
publication date: 2000-08-10T00:00:00Z
review text: Let \(X\) be a non-commutative scheme. A Grothendieck category of quasi-coherent sheaves on \(X\) is called a quasi-scheme and in a certain sense it can be identified with \(X\). Suppose that there is a regularly embedded hypersurface \(Y\subset X\). The authors investigate curves on \(X\) which are in general position with respect to \(Y\). In particular it is shown that the category of quasi-coherent sheaves on such a curve \(C\) is isomorphic to a certain quotient category of graded modules over the \(n\)-dimensional commutative polynomial ring graded by \(\mathbb{Z}^n\), where \(n\) is the number of points of the intersection \(C\cap Y\). The results obtained are applied to some interesting examples of non-commutative schemes such as the enveloping algebra of the two-dimensional non-Abelian Lie algebra, the quantum affine and projective planes, the quantum projective space of \textit{M. Vancliff} [J. Algebra 165, No. 1, 63-90 (1994; Zbl 0837.16023)], and others.
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14H81
Mathematics Subject Classification ID: 18E15
Mathematics Subject Classification ID: 19A49
Mathematics Subject Classification ID: 16S38
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 1287957
zbMATH Keywords: non-commutative schemes
zbMATH Keywords: quasi-schemes
zbMATH Keywords: quasi-coherent sheaves
zbMATH Keywords: quantum algebras
zbMATH Keywords: quantum planes
zbMATH Keywords: curves
zbMATH Keywords: Grothendieck categories
zbMATH Keywords: graded modules
zbMATH Keywords: enveloping algebras
reviewed by: Aleksandr G. Aleksandrov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1287957