Naïve noncommutative blowups at zero-dimensional schemes
type: Item
instance of: scholarly article
title: Naïve noncommutative blowups at zero-dimensional schemes
zbMATH Open document ID: 1141.14001
author: D. Rogalski
author: J. T. Stafford
published in: Journal of Algebra
publication date: 2008-02-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0612658
review text: \textit{D. S. Keeler} and the authors [Duke Math. J. 126, No. 3, 491--546 (2005; Zbl 1082.14003)] introduced the concept of naïve blowup of a scheme. The algebras studied in this work were obtained by blowing up in a single point, and they proved that these objects had properties quite unlike their commutative counterparts. In a companion paper [A class of noncommutative projective surfaces, in press], the authors use the concept of naïve blowups to classify a large class of noncommutative algebras. This classification requires one to naïvely blow up any suitably general zero-dimensional subscheme. The aim of the paper is therefore to study this more general case. The underlying data for a naïve blowup is as follows. Fix an integral scheme \(X\), with automorphism \(\sigma\) and a \(\sigma\)-ample sheaf \(\mathcal{L}\). Let \(Z=Z_{\mathcal{I}}\subset X\) be a zero-dimensional subscheme, with defining ideal \(\mathcal I\subset\mathcal{O}_X\). The \textit{bimodule algebra} is given by \(\mathcal{R}=\mathcal{R}(X,Z,\mathcal{L},\sigma)=\mathcal{O}_X \oplus\mathcal{R}_1 \oplus\mathcal{R}_2 \oplus\cdots,\) where \(\mathcal{R}_n=\mathcal{L}_n \otimes_{\mathcal{O}_X}\mathcal{I}_n,\) for \(\mathcal{L}_n=\mathcal{L} \otimes\sigma^\ast\mathcal{L} \otimes\cdots \otimes(\sigma^{n-1})^\ast\mathcal{L}\), and \(\mathcal{I}_n=\mathcal{I}\cdot\sigma^\ast\mathcal{I} \cdots(\sigma^{n-1})^\ast\mathcal{I}.\) \textit{The naïve blowup algebra of \(X\) at \(Z\)} is then the algebra of sections \[ R=R(X,Z,\mathcal{L},\sigma)= \text{H}^0(X,\mathcal{R})=k\oplus \text{H}^0(X,\mathcal{R}_1) \oplus\text{H}^0(X,\mathcal{R}_2) \oplus\cdots. \] If \(Z\) is the empty set, \(R\) is simply the twisted homogeneous coordinate ring \(B(X,Z,\mathcal{L},\sigma)\) which is essential in noncommutative projective geometry. If \(B=B(X,Z,\mathcal{L},\sigma)\) for a \(\sigma\)-ample invertibele sheaf \(\mathcal{L}\), then \(B\) has extremely pleasant properties, among others: (a) \(B\) is \textit{strongly noether…
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 14A22
zbMATH DE Number: 5235096
zbMATH Keywords: naive blowup
zbMATH Keywords: bounding function
MaRDI profile type: Publication
OpenAlex ID: W1985582232
arXiv ID: math/0612658
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DOI: 10.1016/J.JALGEBRA.2007.02.017
description: scientific article