Noncommutative ampleness from finite endomorphisms
type: Item
instance of: scholarly article
title: Noncommutative ampleness from finite endomorphisms
zbMATH Open document ID: 1345.14005
author: Dennis S. Keeler
author: K. Retert
published in: Journal of Algebra
publication date: 2015-04-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1502.01668
review text: A homogeneous coordinate ring \(R\) on a projective integral scheme \(X\) can be made from global sections of \(\mathcal L^{\otimes n}\) with \(\mathcal L\) an ample invertible sheaf. The sheaf \(\mathcal L\) has been used to prove the Serre correspondence Theorem which shows an equivalence of categories between the tails of finitely generated graded \(R\)-modules and coherent sheaves on \(X\). A twisted coordinate ring \(B\) was constructed by \textit{M. Artin} and \textit{M. Van den Bergh} [J. Algebra 133, No. 2, 249--271 (1990; Zbl 0717.14001)] by using an automorphism \(\sigma\) of \(X\) to twist the usual multiplication. This leads to a noncommutative definition of ampleness, and they proved an equivalence of categories between the category of tails of graded (right) \(B\)- modules and the category of coherent sheaves on \(X\). It was not proved by Van den Bergh or Artin that the definition of ampleness is equivalent on the left and the right, but it was proved by the first author [\textit{D. S. Keeler}, J. Am. Math. Soc. 13, No. 3, 517--532 (2000; Zbl 0952.14002)] by studying the behaviour of the automorphism \(\sigma\) on the numerical equivalence classes of divisors. \textit{M. Van den Bergh} [J. Algebra 184, No. 2, 435--490 (1996; Zbl 0876.17011)] then generalized the definitions to the case where \(\sigma\) is a general finite endomorphism. In this article, the authors examine the noncommutative ampleness in the finite endomorphism case. In this case, one never has ampleness on the left, while ampleness on the right is possible: Let \(X\) be a regular projective integral scheme with a finite endomorphism \(\sigma\) and an invertible sheaf \(\mathcal L\). If \(X\) is not an automorphism, the sequence of \(\mathcal O_X\)-bimodules defined by \(\sigma\) and \(\mathcal L\) is not left ample. If \(\mathcal L\) is ample in the commutative sense and \(\sigma^\ast\mathcal L\simeq\mathcal L^{\otimes r}\) for some \(r\in\mathbb Z\), then the sequence is right ample.…
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 6429886
zbMATH Keywords: noncommutative ampleness
zbMATH Keywords: left ample
zbMATH Keywords: right ample
zbMATH Keywords: twisted homogeneous coordinate ring
zbMATH Keywords: bimodule algebra
MaRDI profile type: Publication
OpenAlex ID: W1990400744
arXiv ID: 1502.01668
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DOI: 10.1016/J.JALGEBRA.2015.01.025
description: scientific article