Tilting objects in the stable category of vector bundles on a weighted projective line of type \((2,2,2,2;\lambda )\)
type: Item
instance of: scholarly article
title: Tilting objects in the stable category of vector bundles on a weighted projective line of type \((2,2,2,2;\lambda )\)
zbMATH Open document ID: 1307.14019
published in: Journal of Algebra
publication date: 2014-09-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1302.0504
reviewed by: Alfonso Zamora
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 16G50
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 18G05
zbMATH DE Number: 6340902
zbMATH Keywords: tilting object
zbMATH Keywords: weighted projective line
zbMATH Keywords: vector bundle
zbMATH Keywords: stable category
author: Jianmin Chen
author: Yanan Lin
author: Shiquan Ruan
MaRDI profile type: Publication
OpenAlex ID: W1993835077
arXiv ID: 1302.0504
cites work: Q3795770
cites work: Exceptional vector bundles on projective spaces
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cites work: Q5496074
cites work: Q4288697
cites work: Higher algebraic K-theory: I
cites work: Tame algebras and integral quadratic forms
cites work: Faisceaux algébriques cohérents
DOI: 10.1016/J.JALGEBRA.2013.07.038
review text: The paper under review explores the category of vector bundles on a weighted projective line \(\mathbb{X}\) of type \((2,2,2,2;\lambda)\), constructing a tilting object for the category.NEWLINENEWLINEA weighted projective line can be seen as a projective line endowed with finitely many weights. Its category of coherent sheaves is proven to be derived equivalent to the category of finite-dimensional modules over certain algebra and its category of vector bundles is equivalent, as an additive category, to the category of graded Cohen-Macaulay modules over its corresponding graded ring.NEWLINENEWLINEThe stable category of vector bundles (obtained by factoring out all line bundles) turns out to be a triangulated category equivalent to \(D^{b}(coh \mathbb{X})\). However, this equivalence is not explicit hence, although the stable category of vector bundles has tilting objects, it is dificult to find one and this is the main purpose of this article.NEWLINENEWLINEThe paper shows the basic definitions and concepts in section \(2\) and explores projective covers in section \(3\). In section \(4\) a formula to compute slopes of vector bundles under shifting is given, and all exceptional objects of the category are described in section \(5\). Theorem 6.2 in section \(6\) presents the main result by constructing an explicit tilting object consisting of a direct sum of five rank two bundles and one rank three bundle, whose endomorphism algebra is a canonical algebra of type \((2,2,2,2)\), remarking that there are no simpler tilting objects for this category.
description: scientific article; zbMATH DE number 6340902