Twisted multi-homogeneous coordinate rings
type: Item
instance of: scholarly article
title: Twisted multi-homogeneous coordinate rings
zbMATH Open document ID: 0957.14003
published in: Journal of Algebra
publication date: 2001-03-20T00:00:00Z
review text: Given two line bundles \(L\) and \(M\) on a projective scheme \(X\), one defines the bi-homogeneous coordinate ring \(B(X;L,M)=\bigoplus_{i,j\geq 0}H^{0}(X,L^{\otimes i}\otimes M^{\otimes j})\). The author gives an analogous construction in the setting of non-commutative algebraic geometry. To this end the line bundles are replaced by invertible bimodules and, to accommodate an analog of commutativity of the tensor product, a bimodule isomorphism \(M\otimes L\rightarrow L\otimes M\) is added as part of the structure. In fact this construction is carried out for a multi-homogeneous coordinate ring with an arbitrary number of invertible bimodules. This allows, under a suitable ampleness condition on the bimodules, to prove an analog of Serre's theorem giving an equivalence between the category of quasi-coherent sheaves and the category of graded modules modulo torsion submodules. In general, multi-homogeneous coordinate rings are not noetherian. A criterion for the ascending chain condition to hold is given. This gives, as applications, the noetherian property for the Rees algebra (which is the analog of blow up) and for the tensor product of homogeneous coordinate rings (which correspond to the fiber product).
reviewed by: Alex Martsinkovsky
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 16S38
Mathematics Subject Classification ID: 13A30
Mathematics Subject Classification ID: 13E05
zbMATH DE Number: 1423878
zbMATH Keywords: multi-homogeneous coordinate ring
zbMATH Keywords: invertible bimodule
zbMATH Keywords: Rees algebra
zbMATH Keywords: projective scheme
zbMATH Keywords: ascending chain condition
zbMATH Keywords: non-commutative algebraic geometry
zbMATH Keywords: quasi-coherent sheaves
zbMATH Keywords: graded modules modulo torsion submodules
zbMATH Keywords: homogeneous coordinate rings
author: Daniel Chan
MaRDI profile type: Publication
full work available at URL: jabr.1999.7938
OpenAlex ID: W2087768020
cites work: Q3972720
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DOI: 10.1006/JABR.1999.7938
description: scientific article; zbMATH DE number 1423878